Finite Groups 2003
John Thompson in his Little Hall office Photo: Jane Dominguez, Courtesy CLAS News & Publications.
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Finite Groups 2003

John Thompson in his Little Hall office Photo: Jane Dominguez, Courtesy CLAS News & Publications.

Finite Groups 2003 Proceedings of the Gainesville Conference on Finite Groups March 612, 2003

Editors C. Y. Ho P. Sin P. H. Tiep A. Turull

≥ Walter de Gruyter · Berlin · New York

Editors Chat Yin Ho Department of Mathematics University of Florida PO Box 118105 358 Little Hall Gainesville, FL 32611-8105 USA e-mail: [email protected]

Peter Sin Department of Mathematics University of Florida PO Box 118105 358 Little Hall Gainesville, FL 32611-8105 USA e-mail: [email protected]

Pham Huu Tiep Department of Mathematics University of Florida PO Box 118105 358 Little Hall Gainesville, FL 32611-8105 USA e-mail: [email protected]

Alexandre Turull Department of Mathematics University of Florida PO Box 118105 358 Little Hall Gainesville, FL 32611-8105 USA e-mail: [email protected]

Mathematics Subject Classification 2000: 20-06; 05Bxx, 17Bxx, 20Cxx, 20Dxx, 20Exx, 20Fxx, 20Gxx, 20Jxx Keywords: buildings, classification of finite simple groups, cohomology of groups, finite geometries, finite p-groups, Lie algebras and superalgebras, representation theory of finite and algebraic groups

P Printed on acid-free paper which falls within the guidelines of the E ANSI to ensure permanence and durability.

Library of Congress Cataloging-in-Publication Data Finite Groups 2003 (2003 : Gainesville, Fla.) Finite Groups 2003 : proceedings of the Gainesville conference on finite groups, March 612, 2003 / edited by Chat Yin Ho … [et al.]. p. cm. ISBN 3-11-017447-2 (cloth : alk. paper) 1. Finite groups Congresses. I. Ho, Chat-Yin, 1946 II. Title. QA174.F56 2004 5121.23dc22 2004021279

ISBN 3-11-017447-2 Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at http://dnb.ddb.de. ” Copyright 2004 by Walter de Gruyter GmbH & Co. KG, 10785 Berlin, Germany. All rights reserved, including those of translation into foreign languages. No part of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording or any information storage and retrieval system, without permission in writing from the publisher. Printed in Germany. Cover design: Thomas Bonnie, Hamburg. Typeset using the authors’ TEX files: I. Zimmermann, Freiburg. Printing and binding: Hubert & Co. GmbH & Co. KG, Göttingen.

To John Thompson on the occasion of his 70th birthday

Preface

During the Academic Year 2002–2003, a number of events took place on both sides of the Atlantic to celebrate the 70th Birthday of our illustrious colleague and friend Professor John G. Thompson. A conference was held in Cambridge, England, in Thompson’s honor. A special issue of the Journal of Algebra was edited to mark the occasion. Here in Gainesville, the occasion was marked by the University of Florida mainly with two events. The Thompson Assistant Professorship was launched, and the whole Academic Year 2002–2003 was a Special Year in Algebra. One of the highlights of the Special Year in Algebra was the conference Finite Groups 2003 of which the present volume is the Proceedings. We are grateful to the Mathematics Department, the College of Liberal Arts and Sciences, and the Office of Research and Graduate Programs at the University of Florida, as well as the National Science Foundation, and the National Security Agency for their support of the Special Year in Algebra in general, and the conference Finite Groups 2003 in particular. Finite Groups 2003 took place in Gainesville from March 6, 2003 to March 12, 2003. As befits a conference in honor of John Thompson, talks on many different topics were given. A list of speakers and their topics is included below. Many of the speakers emphasized the relationship of their work to the work of John Thompson. At the conference banquet, Professor Broué presented John Thompson with a copy of the special issue of the Journal of Algebra edited in honor of his 70th Birthday. The present Proceedings is a collection of articles in honor of John Thompson. They do not represent a record of the talks as given, but, instead, they are articles prepared specifically for these Proceedings. Each article has undergone a strict refereeing process, and we are sorry that, because of the scope and the size of the proceedings, not all interesting articles submitted could be accepted. Most of the articles here present original research results with their proofs. A few are survey articles written by leading researchers. It was the intention of all participants at the conference to honor John Thompson. We hope these Proceedings do honor him, for we, as indeed all group theorists do, owe him a great debt of gratitude for helping make our subject as interesting as it is today. Gainesville, August 2004

C. Y. Ho, P. Sin, P. H. Tiep and A. Turull

List of talks M. Aschbacher (California Institute of Technology): A question of Farjoun C. Bachoc (Université Bordeaux): Codes and designs in Grassmannian spaces A. Bereczky (Sultan Qaboos University): Standard bases for unipotent elements in classical groups R. Boltje (University of California, Santa Cruz): Characterizations of Adams operations on certain representation rings M. Broué (Institut Henri Poincaré): Representations of reductive groups in transverse characteristic W. K. Chan (Wesleyan University): On almost strong approximation for algebraic groups defined over number fields U. Dempwolff (Universität Kaiserslautern): On automorphisms of symmetric designs R. Dipper (Universität Stuttgart): The rational Schur algebra G. Ebert (University of Delaware): Odd order flag-transitive affine planes D. Frohardt (Wayne State University): Genus zero actions M. Geck (Université Claude Bernard-Lyon I): On the computation of some Schur indices for cuspidal unipotent characters S. Glasby (Central Washington University): Modules over non-algebraically closed fields induced from a normal subgroup of prime index G. Glauberman (University of Chicago): An extension of Thompson’s replacement theorem by algebraic group methods D. Gluck (Wayne State University): The final stage of the proof of the k(GV )conjecture J. González (Universidad del País Vasco): The power structure of potent p-groups R. Gow (National University of Ireland): The arithmetic of Steinberg lattices of a Chevalley group R. Gramlich (Technische Universität Darmstadt): Phan-type theorems R. Guralnick (University of Southern California): Conjugacy classes and the noncoprime k(GV )-problem D. Hemmer (University of Georgia): Specht module filtrations for representations of the symmetric group M. Herzog (Tel-Aviv University): The existence of normal subgroups in finite groups G. Hiss (RWTH Aachen): Some observations on products of characters of finite classical groups A. Jaikin-Zapirain (Universidad Autónoma de Madrid): The number of finite pgroups with bounded number of generators

x

List of talks

M. Kallaher (Washington State University): On semiovals T. Keller (Southwest Texas State University): On the k(GV )-conjecture R. Kessar (Ohio State University): Shintani descent and perfect isometries for blocks of finite general linear groups A. Kleshchev (University of Oregon): W -algebras B. Klopsch (Universität Düsseldorf): Enumerating highly non-soluble groups I. Korchagina (Rutgers University): Groups of simultaneously even- and p-type P. Lescot (Université de Picardie): Variations on Thompson’s J -subgroup M. Lewis (Kent State University): Character degree graphs with 5 vertices K. Lux (University of Arizona): A database for basic algebras of simple groups R. Lyons (Rutgers University): Second generation proof G. Malle (Universität Kassel): Families of characters for complex reflection groups D. McNeilly (University of Alberta): Realizability of Weil representations A. Moretó (Universidad del País Vasco): Heights of characters and defect groups G. Navarro (Universitat de València): Problems on characters and Sylow subgroups S. Onofrei (Kansas State University): A point-line characterization of the building of type E7 C. Pillen (University of South Alabama): Extensions for finite Chevalley groups, Frobenius kernels, and algebraic groups A. Prince (Heriot-Watt University): Oval configurations of involutions in the symmetric group D. Riley (Western Ontario University): Beyond the restricted Burnside problem G. Robinson (University of Birmingham): Upper bounds for numbers of characters M. Ronan (University of Illinois at Chicago): Classification of buildings J. Sangroniz (Universidad del País Vasco): Characters of algebra groups N. Sastry (Indian Statistical Institute): On the conjugacy classes of finite Suzuki and Ree groups M. Sawabe (Naruto University of Education): On a p-local geometry associated with a p-subgroup complex C. Scoppola (Università degli Studi di L’Aquila): Space and loop Lie algebras L. Scott Jr. (University of Virginia): Some new examples in 1-cohomology G. Seitz (University of Oregon): Unipotent centralizers and saturation J. Shareshian (Washington Universtity): Topology of intervals in subgroup lattices S. Shpectorov (Bowling Green State University): Classification of amalgams

List of talks

xi

S. Smith (University of Illinois at Chicago) : Quasithin groups – in retrospect R. Solomon (Ohio State University): The signalizer method B. Srinivasan (University of Illinois at Chicago): Remarks on Dade’s conjecture for GL(n, q) K. Uno (Osaka University): Some variations of conjectures on character degrees L. Wilson (University of Florida): Large powerful subgroups of p-groups T. Wolf (Ohio University): Regular and large orbits of induced modules A. E. Zalesski (University of East Anglia): Minimal polynomials of group elements in linear representations of finite groups J. Zhang (Peking University): On p-local rank of finite groups

Table of contents

Preface

vii

List of talks

ix

Michael Aschbacher On a question of Farjoun

1

Christopher P. Bendel, Daniel K. Nakano and Cornelius Pillen Extensions for finite groups of Lie type: twisted groups

29

David J. Benson On the classifying space and cohomology of Thompson’s sporadic simple group 47 Stephen Doty New versions of Schur–Weyl duality

59

Norberto Gavioli, Valerio Monti and Carlo M. Scoppola Just infinite periodic Lie algebras

73

Meinolf Geck On the Schur indices of cuspidal unipotent characters

87

George Glauberman An extension of Thompson’s Replacement Theorem by algebraic group methods

105

Ralf Gramlich Simple connectedness of the geometry of nondegenerate subspaces of a symplectic space over arbitrary fields

111

Robert M. Guralnick and Gunter Malle Classification of 2F -modules, II

117

Allen Herman and Barry Monson On the real Schur indices associated with infinite Coxeter groups

185

Gerhard Hiss and Frank Lübeck Some observations on products of characters of finite classical groups

195

Andrei Jaikin-Zapirain The number of finite p-groups with bounded number of generators

209

xiv

Table of contents

Benjamin Klopsch Enumerating highly non-soluble groups

219

Inna Korchagina 3-Signalizers in almost simple groups

229

Mark L. Lewis Classifying character degree graphs with 5 vertices

247

Alexander Moretó Heights of characters and defect groups

267

Gabriel Navarro Problems on characters and Sylow subgroups

275

Alan R. Prince Ovals in finite projective planes via the representation theory of the symmetric group

283

Urmie Ray Rank invariance and automorphisms of generalized Kac–Moody superalgebras 291 Geoffrey R. Robinson Bounding numbers and heights of characters in p-constrained groups

307

Mark A. Ronan Classification of buildings

319

Josu Sangroniz Characters of algebra groups and unitriangular groups

335

Ronald Solomon The signalizer method

351

M. Chiara Tamburini and Alexandre E. Zalesski Classical groups in dimension 5 which are Hurwitz

363

Franz G. Timmesfeld A classification of Weyl-groups as finite {3, 4}-transposition groups

373

Lawrence E. Wilson Powerful subgroups of 2-groups

381

Thomas R. Wolf Regular orbits of induced modules of finite groups

389

Table of contents

xv

Jiping Zhang Radical subgroups and p-local ranks

401

List of contributors

407

List of participants

411

On a question of Farjoun Michael Aschbacher∗

Let G be a group, H a subgroup of G, Hom(H, G) the set of all group homomorphisms from H into G, and End(G) = Hom(G, G). Following E. Farjoun, we say the embedding of H in G is closed if each member of Hom(H, G) extends uniquely to a member of End(G). Example 1. G is closed in itself for each group G. Example 2. Let H and G be simple. Then H is closed in G iff Aut(G) is transitive on subgroups of G isomorphic to H , Aut Aut(G) (H ) = Aut(H ), and CAut(G) (H ) = 1. In particular if G is the alternating group of degree n, and H is the stabilizer of a point, then H is closed in G for each n > 5 with n = 7. As part of his investigation of idempotent augmented functors, Farjoun posed the following question: Question 1. If H is a finite closed nilpotent subgroup of a group G, is G = H ? For technical reasons it is sometimes easier to work with a slightly weaker condition. Define H to be nearly closed in G if each member of End(H ) extends uniquely to a member of End(G). Question 2. If H is a finite nearly closed nilpotent subgroup of a group G, is G = H ? In this paper we introduce some machinery useful in investigating Question 2 and present some partial results which show the Question has a positive answer in various special cases. For example we reduce Question 2 to the case where H is a p-group for some prime p, we show Question 2 has an affirmative answer when G is finite and the nilpotence class of H is at most 3, and we show Question 2 has an affirmative answer when G is finite for other interesting classes of p-groups. Here are some specifics: Let C be a class of groups such that if G ∈ C and H ≤ G then H ∈ C. Define a group P to be C-rigid or rigid with respect to C if whenever P is nearly closed in G ∈ C then P = G. Our first result says: Theorem 1. Let C be a class of groups and P1 and P2 groups which are C-rigid. Then P1 × P2 is C-rigid. ∗ This work was partially supported by NSF-0203417.

2

Michael Aschbacher

As finite nilpotent groups are the direct product of their Sylow groups, Theorem 1 reduces Question 2 to the case where H is a p-group, so in the remainder of this introduction assume G is a group, p is a prime, and P is a finite p-subgroup of G. Moreover in the rest of the introduction we work in the class C of finite groups, so G is assumed to be finite. The next result shows Question 2 has a positive answer when P is large: Theorem 2. If P is nearly closed in a finite group G and P ∈ Sylp (G) then G = P . Theorem 3. If P is metabelian or of nilpotence class at most 3, then P is rigid with respect to the class of finite groups. A. Mann has a slick proof that groups of class 2 are rigid in the class of all groups; indeed he shows that if P is closed in G then Z2 (P ) ≤ Z2 (G). The method used to prove Theorem 3 can be used to treat other classes of p-groups, but there are obstructions to a simple minded extension of the approach. See Section 9 for more discussion and a sketch of a proof that Question 2 has a positive answer for groups P of class at most 5 generated by 3 elements of order p. One tool for investigating Question 2 involves the relationship between “splittings” of P and G, where a splitting of P is a decomposition of P as a split extension. In Section 2 we consider a “splitting category” and define a class of “splitting solvable” groups, which includes many interesting p-groups. Then we prove: Theorem 4. If P is splitting solvable then P is rigid with respect to the class of finite groups. Finally the near closure property does not inherit to subgroups or homomorphic images, so while the condition is very strong, it is not easy to exploit using traditional techniques from finite group theory. On the other hand various properties following from near closure do inherit. Thus in Section 10 we investigate classes of finite groups satisfying such properties. We would like to show the subnormal closure of P in each group G in the class is P -nilpotent (i.e. G = Op (G)P ); such a result would give an affirmative answer to Question 2 when G is solvable, and give very strong information in the nonsolvable case. Unfortunately examples show this result is not true in general for the classes we’ve considered, even when G is solvable. Nevertheless we include an abbreviated discussion of this approach in one case, since that discussion does at least lead to our final theorem, and since in addition the result may be true if other properties are adjoined to obtain smaller classes. Theorem 5. If p is an odd prime, G is a finite solvable group, and P is a p-subgroup of G such that CG (P ) = Z(G), then Op (G) is the largest p -signalizer for P . Recall a p -signalizer for P is a P -invariant p -subgroup of G. The reader is directed to [2] for basic notation, terminology, and results involving finite groups.

On a question of Farjoun

3

The author would like to thank Yoav Segev for drawing his attention to Farjoun’s question, for many helpful conversations about the problem, and for suggesting improvements to the proofs of a number of lemmas in this paper.

1. Monoids and semigroups We will be concerned with the monoid End(G) of a finite group G, so we begin with some discussion of monoids. In this section M is a finite semigroup; that is M is a set with an associative binary operation · . Recall M is a monoid if M has an identity 1. If an identity exists, it is unique. A zero for M is an element 0 ∈ M such that for each x ∈ M, 0 · x = 0 = x · 0. Again if M has a 0 then it is unique. For X ⊆ M, write [X] for the subsemigroup of M generated by X. Thus [X] consists of all products x1 . . . xn with xi ∈ X for all i. In particular for x ∈ X, [x] = {x n : 0 < n ∈ Z}. An idempotent of M is an element x ∈ M such that x 2 = x. An element y ∈ M is said to be split if y n = y for some n > 1. 1.1. Let a ∈ M, k the least positive integer such that a k = a j for some j > k, and q the least positive integer such that a k+q = a k . Set B = {a j : j ≥ k}. Then: (1) B is a cyclic subgroup of M of order q. (2) B is the set of split elements in [a]. (3) Let sq be the first positive multiple of q such that k ≤ sq. Then a sq is the identity of B and B = a t , for each t ≥ k with (q, t) = 1. Proof. As a k = a k+q , by induction on i, a k+i = a k+i+q for each i ≥ 0. Then by induction on r: (i) a m+rq = a m for each m ≥ k and r ≥ 0. Hence: (ii) B = {a j : j ∈ J }, where J = {k + i : 0 ≤ i < q}. From (i) we see that e := a rq is the identity element of B, for each r such that rq ≥ k, and hence, for each b ∈ B, bq = e. It follows that for each b ∈ B, bq−1 is the inverse of b and B is a group. If a j1 = a j2 for some j1 ≤ j2 in J , then multiplying by a (s+1)q−j1 , we get that e = a (s+1)q = a (s+1)q+j2 −j1 . Multiplying by a k we get a k = a k+(s+1)q+j2 −j1 = a k+j2 −j1 , because a (s+1)q is the identity element of B. But the equality a k = a k+j2 −j1 contradicts the minimality of q, unless j1 = j2 . It follows that |B| = q. Let k ≤ t ∈ Z with (t, q) = 1, and set c := a t . As (t, q) = 1: (iii) tJ = J mod q, so B = [c] by (i)–(iii). This establishes (1) and (3). Finally if a r is split then a r = a t for some t > r, so r ≥ k by minimality of k. Thus (2) holds.

4

Michael Aschbacher

For a ∈ M, let sp(a) = {b ∈ [a] : b is split}. By 1.1, sp(a) is a group, so it has a unique identity id(a). 1.2. Let a ∈ M. (1) sp(a) is a cyclic subgroup of M. (2) id(a) is the unique idempotent in [a]. Proof. Part (1) follows from 1.1. Let e = id(a). Then e2 = e so e is an idempotent. Conversely if f ∈ [a] is an idempotent then f is split, so f ∈ sp(a), and hence e = f as groups have a unique idempotent. Define an equivalence relation ∼ on M by a ∼ b if id(a) = id(b). Write a˜ for the equivalence class of a under ∼, and set sp(a) ˜ = {b : a ∼ b and b is split} 1.3. sp(a) ˜ is a subgroup of M with identity id(a). Proof. Let e = id(a) and H the set of x ∈ M such that xe = ex = x and yx = xy = e for some y ∈ M. It is easy to check that H is a subgroup of M. Then since M is finite, e ∈ [h] and h is split for each h ∈ H , so e = id(h) by 1.2.2. Thus H ⊆ sp(a), ˜ and of course sp(a) ˜ ⊆ H , so the lemma holds. Define a relation ≤ on M by b ≤ a if ab = ba = b. For a ∈ M let M(≤ a) = {b ∈ M : b ≤ a}. Write for the set of idempotents in M. 1.4. .

(1) ≤ is antisymmetric.

(2) ≤ is transitive. (3) a ≤ a iff a ∈ . (4) ≤ is a partial order on . (5) If M has 0 or 1 then 0 ≤ a ≤ 1 for all a ∈ M. (6) For each a ∈ , M(≤ a) is a subsemigroup of M with identity a. (7) If a ∈ then M(≤ a) = aMa and sp(a) ˜ ⊆ M(≤ a). ˜ ⊆ M(≤ a). (8) If a, b ∈ and b ≤ a then sp(b) Proof. If b ≤ a and a ≤ b then a = ab = b, so (1) holds. Let c ≤ b ≤ a. Then ac = a(bc) = (ab)c = bc = c,

On a question of Farjoun

5

and similarly ca = c. Thus (2) holds. Next a ≤ a iff a 2 = a, so (3) holds. Of course (1)–(3) imply (4). Part (5) is easy. If b, c ≤ a then (bc)a = b(ca) = bc and similarly a(bc) = bc, so (6) holds. Let a ∈ . Then aMa ⊆ M(≤ a) and for x ∈ M(≤ a), x = axa ∈ aMa. Then as a is an identity for the group sp(a), ˜ (7) holds. By (2) and ˜ ⊆ M(≤ b) ⊆ M(≤ a), so (8) holds. (7), if a, b ∈ with b ≤ a, then sp(b)

2. Splittings of groups In this section we establish a category to investigate certain questions involving split extensions of groups which arise in the study of Farjoun’s Question. There is some overlap between our treatment and that of Bechtell in [1]. For example our notion of “splitting series” corresponds to Bechtell’s “complementary splitting systems” and our “splitting simple groups” are his “inseperable groups”. We thank Yoav Segev for calling our attention to Bechtell’s work. In this section G is a group. A splitting of G is a pair (K, C) where K is a normal subgroup of G and C is a complement to K in G. Let S be the set of splittings of G and partially order S by (K, C) ≤ (H, B) if C ≤ B and H ≤ K. Call K the kernel of the splitting and C the complement. Write 0 for the splitting (G, 1) and 1 for the splitting (1, G). Thus 0 and 1 are the least and greatest elements of the poset S, respectively. These splitting are the trivial splittings and the remaining splittings are said to be nontrivial. Define G to be splitting simple if G has no nontrivial splittings. That is G is splitting simple if G cannot be realized as a nontrivial split extension; equivalently there are no nontrivial semidirect product decompositions of G. Of course simple groups are splitting simple but the converse is not true: Example 2.1. A finite abelian group G is splitting simple iff G is a cyclic p-group for some prime p. Example 2.2. Quaternion groups are splitting simple. A splitting series for G is a series 0 = S0 ≤ S1 ≤ · · · ≤ Sn = 1 of splittings Si = (Ki , Ci ). Define a splitting series S = (Si : 0 ≤ i ≤ n) to be maximal if for each 0 ≤ i < n, Si < Si+1 and there exists no splitting S with Si < S < Si+1 . For ≤ Aut(G), define a splitting (K, C) of G to be a -splitting if K and C are -invariant. Define G to be -splitting simple (or -simple for short) if there are no nontrivial -splittings of G. Define a splitting (K, C) of G to be simple if K is C-splitting simple.

6

Michael Aschbacher

2.3. Let S = (Si : 0 ≤ i ≤ n) be a splitting series for G. Then the following are equivalent: (1) S is maximal. (2) For each 0 ≤ i < n, Ki ∩ Ci+1 is Ci -simple. (3) For each 0 ≤ i < n, the splitting (Ki ∩ Ci+1 , Ci ) of Ci+1 is simple. Proof. By definition, (2) and (3) are equivalent. If Si < (K, C) < Si+1 , then 1 < Ki ∩ C < Ki ∩ Ci+1 . Now by the modular property of groups (K∩Ci+1 )C∩Ki = (K∩Ci+1 )(C∩Ki ), and Ci+1 = CK∩Ci+1 = C(K ∩ Ci+1 ). Thus (K ∩ Ci+1 , C ∩ Ki ) is a nontrivial Ci -splitting of Ci+1 ∩ Ki , so (2) implies (1). Conversely suppose (J0 , A0 ) is a nontrivial Ci -splitting of Ki ∩ Ci+1 . Set J = J0 Ki+1 and A = A0 Ci . As (J0 , A0 ) is a Ci -splitting, Ci acts on J0 and A0 , so A is a subgroup of G and Ci+1 = Ci Ki ∩ Ci+1 = Ci (Ki ∩ Ci+1 ) acts on J0 . Thus J is normal in Ki+1 Ci+1 = G. By construction, Ki+1 ≤ J ≤ Ki , and Ci ≤ A ≤ Ci+1 . As Ci is a complement to Ki ∩Ci+1 in Ci+1 and A0 is a complement to J0 in Ki ∩Ci+1 , A is a complement to J0 in Ci+1 . Next J ∩ Ci+1 = J0 Ki+1 ∩ Ci+1 = J0 (Ki+1 ∩ Ci+1 ) = J0 , so as A is a complement to J0 in Ci+1 , J ∩ A = 1 and G = Ki+1 Ci+1 = Ki+1 J0 A = J A. Thus (J, A) is a splitting of G with (Ki , Ci ) < (J, A) < (Ki+1 , Ci+1 ), completing the proof that (1) implies (2). We next define a category we call the splitting category. We first need a lemma and some notation. 2.4. Let α : G → G be a surjective group homomorphism, J = ker(α), and (K, C) a splitting of G. Then (Kα, Cα) is a splitting of G iff (J ∩ K, J ∩ C) is a splitting of J . Proof. As G = KC, G = Gα = KαCα, and as K G, Kα Gα = G . Thus (Kα, Cα) is a splitting of G iff Kα ∩ Cα = 1. Let x ∈ G ; then x ∈ Kα ∩ Cα iff x = kα = cα for some k ∈ K and c ∈ C iff kc−1 ∈ J . Thus (Kα, Cα) is a splitting iff (a) If k ∈ K and c ∈ C with kc−1 ∈ J , then k, c ∈ J . Now observe that (a) is equivalent to (b): (b) J = (K ∩ J )(C ∩ J ). Suppose (a) holds and let j ∈ J . As G = KC, j = ya for some y ∈ K and a ∈ C. Then y, a ∈ J by (a), so (b) holds. Conversely assume (b) holds and kc−1 ∈ J . By (b), kc−1 = zb for some z ∈ J ∩ K and b ∈ J ∩ C, so as K ∩ C = 1, k = z and c−1 = b; that is (a) holds. Finally as K ∩ C = 1, (b) holds iff (K ∩ J, C ∩ J ) is a splitting for J . Thus the lemma is established.

On a question of Farjoun

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Write sub(G) for the set of subgroups of G and trip(G) for the set product {G} × sub(G) × sub(G). Given a group homomorphism α : G → G , define αˇ : trip(G) → trip(Gα) by αˇ : (G; K, C) → (Gα; Kα, Cα). The objects in the splitting category are the triples S = (G; K, C) such that (K, C) is a splitting of G. The objects in the splitting category are called splittings. A morphism from S to S = (G ; K , C ) is a map of the form αˇ for some surjective group homomorphism α : G → G such that S αˇ = S . Composition of morphisms is composition of functions. ˇ = S . The The image or cokernel of a morphism αˇ : S → S is the splitting im(α) kernel of αˇ is the triple ker(α) ˇ = (ker(α); K ∩ ker(α), C ∩ ker(α)); by 2.4, ker(α) ˇ is a splitting. Define the set of normal splittings N (S) of S to be the set of splittings (H ; K ∩ H, C ∩ H ) such that H G. Given T = (H, K ∩ H, C ∩ H ) ∈ N(S), define S/T to be the triple (Gπ ; Kπ, Cπ ), where π : G → G/H is the natural map. Call S/T the factor splitting of S modulo T . By 2.4: 2.5 (Homomorphism Theorem for Splittings). Let S = (G; K, C) be a splitting, T = (H ; K ∩ H, C ∩ H ) ∈ N (S), and π : G → G/H the natural map. Then: (1) S/T is a splitting. (2) πˇ : S → S/T is a morphism of splittings. ˇ ∈ N(S). (3) If αˇ : S → S is a morphism of splittings then ker(α) (4) If ker(α) ˇ = T then there exists a unique isomorphism σ : G/H → G such that α = π σ ; moreover σˇ : S/T → S is an isomorphism of splittings. (5) The map R → R πˇ is an injection from the set of splittings R = (G; J, B) with H ≤ J into the set of splittings of S/T . 2.6. Let S = (G; K, C) and T = (G; J, B) be splittings with (K, C) ≤ (J, B), and define ι : G → B by ι : j b → b for j ∈ J and b ∈ B (cf. 3.11.2). Then R = (J ; J, 1) ∈ N (S) and ιˇ : S → (B; K ∩ B, C) is a morphism with kernel R. Let S = (Si : 0 ≤ i ≤ n) be a splitting series for G. For 1 ≤ i ≤ n, define the ith factor of S to be the splitting Fi = (Ci ; Ki−1 ∩ Ci , Ci−1 ). By 2.6 and 2.5.4, Fi ∼ = Si /Ri , where Ri = (Ki ; Ki , 1). Thus for example F1 = (C1 ; C1 , 1) and Fn = Sn−1 . Remark 2.7. Question: Is there a Jordan–Hölder Theorem for splittings? The strongest result definitely does not hold: Different maximal splitting series can have different families of factors. For example we just saw that Sn−1 is a factor in the series S, and in a finite group any maximal proper splitting is the penultimate term in some maximal series. One could ask for something weaker. Let K(S) = (Ki−1 /Ki : 1 ≤ i ≤ n) be the family of factor groups in the series of kernels of S; equivalently K(S) is the family (Ki−1 ∩ Ci : 1 ≤ i ≤ n) of kernels in the factors Si /Ri of S. Is the family K(S)

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independent of the maximal splitting series S? If G is an abelian p-group the answer is yes. Namely G = G1 × · · · × Gn is the direct product of cyclic groups and this decomposition is unique up to isomorphism. Moreover the family of factors in this decomposition is the family K(S) in any maximal splitting series for G. However this property also fails in general. If (K, C) is a simple splitting for G such that C is splitting simple, then, by 2.3 S = 0 < (K, C) < 1 is a maximal splitting series for G with K(S) = {K, C}. If G is the dihedral group of order 8 then up to isomorphism there are two such splittings: G is the split extension of E4 by Z2 and of Z4 by Z2 , and the corresponding series have different kernel families. In Example 2.9 we will see more instances of maximal series with different kernel families. However in all these examples, the length of a maximal splitting series is at least constant. Define G to be splitting solvable if there exists a splitting series S for G with each member of K(S) abelian. Of course splitting solvable groups are solvable, but the converse is not true; cf. Example 2.2. From Example 2.9 below, the Sylow p-groups of general linear groups over fields of characteristic p are splitting solvable. A similar argument shows the Sylow groups of symmetric groups are splitting solvable, as those Sylow groups are direct products of iterated wreath products. 2.8. Let p be a prime, G a p-group, and assume |Z(G)| = p. Then each nontrivial splitting of G with an abelian kernel is simple. Proof. Suppose (K, C) is a nontrivial splitting of G with K abelian; we must show K is C-simple. Assume (J, B) is a nontrivial C-splitting of K. As K is abelian, K = J × B, and as the splitting is C-invariant, CJ (C) = 1 = CB (C). But as K = J × B and G = KC, CJ (C)CB (C) ≤ Z(G), contradicting |Z(G)| = p. Example 2.9. Let L = L1 × · · · × Lm be the direct product of general linear groups Li ∼ = GLni (pei ) for some prime p, and let P ∈ Sylp (L); then P = P1 × · · · × Pm , where Pi = P ∩ Li ∈ Sylp (Li ). Let M be a maximal parabolic of L over P . Then M = KCM , where K is the unipotent radical of M and CM is a Levi factor. Moreover (K, CM ) is a splitting of M; (K, C) is a splitting of P , where C = P ∩ CM ; K is abelian; and CM is the product of general linear groups. Thus proceeding by induction on the Lie rank of L, P is splitting solvable. Indeed the Lie rank of L is m l= li , where li = ni − 1 is the Lie rank of Li . i=1

Further the series S of maximal length we obtain via this process are of length l, and K(S) consists of the radicals of maximal parabolics in Levi factors. Now specialize to the case ei = 1 for all i. In this case we claim the series S of length l are maximal splitting series. As series obtained by pasting together maximal series of factors of a direct product is a maximal series for the product, proceeding by

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induction on the Lie rank of L, we may assume m = 1. Then |Z(P )| = p, so by 2.8, the splitting (K, C) is simple. Then as C is a direct product of general linear groups, the claim follows by induction on the Lie rank. Notice however that in general the families of kernels obtained from different series are different. For example in L4 (p) the series are of length 3 and up to isomorphism, the families are {Ep3 , Ep2 , Ep } and {Ep4 , Ep , Ep }.

3. The homomorphism monoid In this section G is a group and M = End(G) regarded as a monoid under the operation of composition of functions. Let α, β, γ ∈ M. 3.1. .

(1) ker(α) ≤ ker(αβ).

(2) Gαβ ≤ Gβ. (3) For m ≤ n positive integers, ker(α m ) ≤ ker(α n ) and Gα n ≤ Gα m . Proof. As ker(α)αβ = 1β = 1, (1) holds. As Gα ≤ G, (2) holds. Then (3) follows from (1) and (2) by induction on n − m. 3.2. If αβ = β then ker(α) ≤ ker(β) and G = ker(β)Gα. Proof. Suppose αβ = β. Then by 3.1.1, ker(α) ≤ ker(αβ) = ker(β). Also for g ∈ G, g(g −1 )α ∈ ker(β), so G = ker(β)Gα. 3.3. If βα = β then α|Gβ = 1Gβ so Gβ ≤ Gα. Proof. Suppose βα = β. Then for a ∈ Gβ, a = gβ for some g ∈ G and aα = gβα = gβ = a. Recall from Section 1 that β is split in M if β m = β for some m > 1. 3.4. If β is split in M then (ker(β), Gβ) is a splitting of G. Proof. By hypothesis β = ββ m−1 = β m−1 β, so by 3.2, G = ker(β)Gβ m−1 and Gβ m−1 ≤ Gβ by 3.1.3, so G = ker(β)Gβ. By 3.3, ker(β m−1 ) ∩ Gβ = 1, while by 3.1.3, ker(β) ≤ ker(β m−1 ), so ker(β) ∩ Gβ = 1. 3.5. Assume α is an idempotent in M. Then: (1) α|Gα = 1Gα , and (2) (ker(α), Gα) is a spltting of G. Proof. Part (1) follows from 3.3 and (2) from 3.4. 3.6. If Gα is finite then α is split in M iff (ker(α), Gα) is a splitting of G.

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Proof. The first statement implies the second by 3.4. Conversely assume the second statement. By 3.1.2, ϕ = α|Gα ∈ Hom(Gα, Gα), and as (ker(α), Gα) is a splitting, 1 = ker(α) ∩ Gα, so ϕ ∈ Aut(Gα). Thus as Gα is finite, n = |ϕ| is an integer and ϕ n+1 = ϕ. Now for g ∈ G, g = xa with x ∈ ker(α) and a ∈ Gα, so gα = xαaα = aα = aϕ, and similarly gα n+1 = aϕ n+1 = aϕ, so α = α n+1 . 3.7. The following are equivalent: (1) α is an idempotent in M. (2) (ker(α), Gα) is a splitting of G and α|Gα = 1Gα . Proof. By 3.5, (1) implies (2). Assume (2). Adopting the notation of the proof of 3.6, that proof shows that gα = aϕ = a, so gα 2 = aα = a = gα; that is (1) holds. 3.8. .

(1) If ker(α) ≤ ker(β) and α is idempotent then αβ = β.

(2) If α is idempotent then ker(α) ≤ ker(β) iff αβ = β. Proof. Assume the hypotheses of (1). Then by 3.7, G = ker(α)Gα, so as ker(α) ≤ ker(β), G = ker(β)Gα and ker(β) = ker(α)(ker(β) ∩ Gα). By 3.7, α|Gα = 1Gα , so the second equality says ker(β)α ≤ ker(β), and hence ker(β)αβ = 0. Let g ∈ G. By the first equality, g = ka for some k ∈ ker(β) and a ∈ Gα. Then gβ = (ka)β = kβaβ = aβ and gαβ = (kαβ)(aαβ) = aαβ = aβ, so β = αβ. This establishes (1). Then (1) and 3.2 imply (2). 3.9. .

(1) If Gβ ≤ Gα and α is idempotent then βα = β.

(2) If α is idempotent then Gβ ≤ Gα iff βα = β. Proof. Assume the hypotheses of (1); then for g ∈ G, gβα = gβ as α|Gα = 1Gα . Thus (1) holds, while (1) and 3.3 imply (2). Recall from Section 1 that β ≤ α iff αβ = βα = β. 3.10. Assume α is idempotent; then the following are equivalent: (1) β ≤ α. (2) ker(α) ≤ ker(β) and Gβ ≤ Gα. Proof. The lemma follows from 3.8 and 3.9. 3.11. Let be the set of idempotents in M and S the set of splittings of G. (1) The map ξ : α → (ker(α), Gα) is a bijection of with S. (2) ξ is an isomorphism of posets.

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Proof. Let α ∈ . By 3.7, ξ(α) ∈ S. Given S = (K, C) ∈ S, define λ(S) : G → G by (ka)λ = a for k ∈ K and a ∈ C. Then λ(S) ∈ and λ is an inverse for ξ , so (1) is established. By 3.10, the bijection ξ is an isomorphism of posets. 3.12. If G is finite and Gα is splitting simple then either α is nilpotent or α is split in M. Proof. Recall the definition of id(α) from Section 1. Let H = Gα. Since id(α)|H is an idempotent of H , and since H is splitting simple, H id(α) = 1 or H . If H id(α) = 1, then α id(α) = 0, which that id(α) = (id(α))2 = 0, so α is nilpotent. In the second case id(α) ∈ Aut(H ), which implies that α|H ∈ Aut(H ), so ker(α) ∩ Gα = 1, and α is split.

4. Extending and restricting homomorphisms In this section G is a group, A is a subgroup of G, H G, and G = AH . 4.1. Assume α ∈ Hom(A, G) and β ∈ Hom(H, G) such that (1) α|A∩H = β|A∩H , and (2) (ha )β = (hβ)aα for all a ∈ A and h ∈ H . Define ψ : G → G by ψ : ah → aαhβ for a ∈ A and h ∈ H . Then ψ ∈ End(G) extends α and β. Proof. As G = AH , (1) insures that ψ is well defined. Next by (2): (ahbk)ψ = (abhb k)ψ = aαbα(hb )βkβ = aαbα(hβ)bα kβ = aαhβbαkβ = (ah)ψ(bk)ψ, so ψ ∈ End(G). By construction, ψ extends α and β. For g ∈ G, let cg ∈ Aut(H ) be conjugation by g, and AutA (H ) = {ca : a ∈ A}. 4.2. Assume α ∈ Hom(A, G) and β ∈ Aut(H ) such that α|A∩H = β|A∩H and for each a ∈ A, caβ = caα .

(∗)

Define ψ : G → G by ψ : ah → aαhβ for a ∈ A and h ∈ H . Then ψ ∈ End(G) extends α and β. Proof. Condition (∗) says for each h ∈ H : (hβ)aα = (hβ)caα = (hβ)caβ = hca β = (ha )β; that is (*) implies condition (2) of 4.1, so the lemma follows from 4.1.

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4.3. Let β ∈ CAut(H ) (A ∩ H ) ∩ CAut(H ) (AutA (H )). Define ψ : G → G by ψ : ah → ahβ for a ∈ A and h ∈ H . Then ψ ∈ Aut(G). Proof. We apply 4.2 to β and the identity map α on A. As β centralizes AutA (H ), caβ = ca = caα , for each a ∈ A. Further as β centralizes A ∩ H , β|A∩H = α|A∩H . Thus the lemma follows from 4.2. 4.4. Consider the following two conditions: (a) 1A extends to a unique member of End(G). (b) CAut(G) (A) = 1 and there exists no nontrivial splitting (K, C) of G with A ≤ C. Then: (1) Condition (a) implies condition (b). (2) If G is finite then (a) and (b) are equivalent. Proof. Assume (a). Each member of CAut(G) (A) is a member of End(G) extending 1A , so CAut(G) (A) = 1. If (K, C) is a splitting of G with A ≤ C then by 4.1 there is ψ ∈ End(G) with kernel K and ψ|C = 1C , so ψ = 1G and hence K = 1. Conversely assume G is finite and (b) holds. Let θ ∈ End(G) extend 1A , and let ζ = id(θ) in the language of Section 1; thus ζ is an idempotent in End(G). As k = 1 for each positive integer k, ζ θ|A A |A = 1A . Thus by 3.7, (ker(ζ ), Gζ ) is a splitting of G and A ≤ Gζ , so by (b), ζ is an injection and hence an automorphism of G centralizing A. Hence θ = ζ = 1G by (b). 4.5. The following are equivalent: (1) 0A extends to a unique member of End(G). (2) Hom(G/AG , G) = 0. Proof. Let K = AG , π : G → G/K the natural map, and ψ ∈ Hom(G/K, G). Then πψ extends 0A , so if (1) holds then π ψ = 0G , and hence ψ = 0; that is (1) implies (2). Conversely if θ ∈ End(G) extends 0A then A ≤ ker(θ ), so K ≤ ker(θ ). Thus there exists a unique α ∈ Hom(G/K, G) with π α = θ. Hence if (2) holds then α = 0, so θ = 0.

5. (c, d)-embeddings In this section G is a group, p is a prime, and P is a finite subgroup of G. Write 1P for the identity map on P and 0P for the 0-homomorphism mapping each element of P to 1. If P is nilpotent write cl(P ) for the nilpotence class of P .

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Let c, d be nonnegative integers, and define Endc (P ) = {ϕ ∈ End(P ) : P ϕ is nilpotent and cl(P ϕ) ≤ c}, and EndP (G) = {θ ∈ End(G) : P θ ≤ P }. We say P is (c, d)-embedded in G if (1) 1P extends to a unique member of End(G), (2) Each ϕ ∈ Endc (P ) extends to at most one member of End(G), (3) Each ϕ ∈ Endd (P ) extends to at least one member of End(G). A c-embedding is a (c, c)-embedding. Remark 5.1. P is 0-embedded in G iff 1P and 0P extend to unique members of End(G). Remark 5.2. P is nearly closed in G iff P is cl(P )-embedded in G. 5.3. Assume P is (c, 0)-embedded in G and θ ∈ EndP (G) with cl(P θ ) ≤ c. Then: (1) The restriction map ρ : EndP θ (Gθ) → End(P θ ) is injective. (2) P θ is (c, 0)-embedded in Gθ . Proof. Assume ηi ∈ End(Gθ), i = 1, 2, extend φ ∈ End(P θ ). Then ϕ = θ|P φ ∈ End(P ) and θ ηi ∈ End(G) extend ϕ with cl(P θ ηi ) ≤ cl(P θ ) ≤ c. Therefore as P is (c, 0)-embedded in G, θη1 = θη2 . Hence as the image of θ is the domain of ηi , η1 = η2 . Thus (1) holds, and of course (1) implies (2). In the remainder of the section assume P is a finite p-group. 5.4. Assume 0P extends to a unique member of End(G), and G = Z(G)P . Then G = P. Proof. Let Z = Z(G) and e the exponent of P ∩ Z. Then the map β : x → x e is in End(Z) with (P ∩ Z)β = 1. Let α = 0P . Then α and β satisfy the hypotheses of 4.1 with Z, P in the roles of H, A, so by that lemma ψ ∈ End(G), where ψ : az → ze for a ∈ P , z ∈ Z. Thus ψ extends α, so by hypothesis, ψ = 0G . That is e is the exponent of Z. By 4.5, Hom(G/P , G) = 0. But G/P ∼ = Z/(Z ∩ P ) has exponent dividing the power e of p, so if Z/Z ∩P = 1, then Zp is isomorphic both to an image of Z/(Z ∩P ) and to a subgroup of G, so Hom(G/P , G) = 0, a contradiction. 5.5. Assume P is 0-embedded in G. Then: (1) CG (P ) = Z(G). (2) If P is abelian then G = P . (3) If K G with G/K a finite p-group then G = P K.

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Proof. For g ∈ CG (P ) the inner automorphism cg induced on G by g is in CAut(G) (P ), so (1) follows from 4.4. Assume P is abelian. Then P ≤ Z(G) by (1), so G is abelian by a second application of (1). Thus P = G by 5.4, establishing (2). Assume the hypotheses of (3) and suppose G = P K. Then as G/K is a finite p-group, P K ≤ J with |G : J | = p. Then an argument in the final paragraph of the proof of 5.4 supplies a contradiction. Remark 5.6. Notice that 5.5.2 says that Question 2 has a positive answer for finite abelian p-groups. 5.7. Assume P is (1, 0)-embedded in G and θ ∈ EndP (G) with P θ abelian. Then Gθ = P θ . Proof. By 5.3.2, P θ is (1, 0)-embedded in Gθ , so Gθ = P θ by 5.5.2 applied to the embedding of P θ in Gθ.

6. Monoids and (c, d)-embeddings In this section P is a subgroup of a group G. We use the ideals Endc (P ) and Endd (P ) of End(P ) to study (c, d)-embeddings of P . Define EndPc (G) = {ϕ ∈ EndP (G) : P ϕ is nilpotent and cl(P ϕ) ≤ c}. Given a monoid M, an ideal of M is a subset I of M with MI M ⊆ I . Let ρ : ψ → ψ|P be the restriction map from EndP (G) to End(P ). 6.1. .

(1) EndP (G) is a submonoid of End(G).

(2) Endc (P ) and EndPc (G) are ideals of End(P ) and EndP (G), respectively. (3) ρ(EndPc (G)) ≤ Endc (P ). (4) P is nearly closed in G iff ρ : EndP (G) → End(P ) is an isomorphism of monoids; the inverse of ρ is the extension map ex : ϕ → ϕ. ˆ (5) P is (c, d)-embedded in G iff (a) 1G is the unique extension of 1P , (b) ρ : EndPc (G) → Endc (P ) is injective, (c) ρ : EndPd (G) → Endd (P ) is surjective. (6) P is c-embedded in G iff 1G is the unique extension of 1P and ρc = ρ| EndP (G) : EndPc (G) → Endc (P ) is an isomorphism. In that event c exc = ex| Endc (P ) is the inverse of ρc .

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Proof. Straightforward; to illustrate we give a proof of (4). Visibly ρ : EndP (G) → End(P ) is a homomorphism of monoids. Further by definition, P is nearly closed in G iff ρ is a bijection, and of course in that event ex is the inverse of ρ. Using the fact that various group theoretic properties of P , G can be retrieved from the properties of the monoids Endc (P ) and EndPc (G), we can use the fact that ρc is an isomorphism to see that properties of P lift to analogous properties of G. Define a c-embedding (G, P ) (i.e. P is c-embedded in G) to be c-minimal if for each pair (H, Q) with H < G, Q ≤ P , and Q c-embedded in H , we have H = Q. Write ξ for the bijection of 3.11 in the case of both P and G. 6.2. Assume P is c-embedded in G and (PK , PC ) is a splitting of P with cl(PC ) ≤ c. Set α = ξ −1 (PK , PC ). Then: ˆ is the unique splitting of (1) αˆ is an idempotent in EndPc (G) and (K, C) = ξ(α) G with PK ≤ K and PC ≤ C. (2) PC is c-embedded in C. (3) If (G, P ) is c-minimal and PC = P then C = PC so if G is finite and P is a p-group for some prime p, then O p (G) ∩ P ≤ PK . Proof. By definition, PK = ker(α) and PC = P α. As P is c-embedded in G, αˆ ˆ and PC ≤ C = Gα. ˆ As α 2 = α and ex exists by 6.1.6. Then PK ≤ K = ker(α) 2 ˆ so (K, C) is a splitting of G. If (K , C ) were a second is an isomorphism, αˆ = α, −1 such splitting then βˆ = ξ (K , C ) is an idempotent with PK ≤ ker(β) ∩ ker(α) and β|PC = 1PC = α|PC , so α = β and hence (K, C) = (K , C ). Thus (1) holds. Let ϕ ∈ End(PC ) and set ψ = αϕ. Then ψ ∈ Endc (P ), so ψˆ exists. As ψ = αϕ and α is idempotent, αψ = ψ. As P ψ = P αϕ ≤ P α, ψα = ψ. Thus ψ ≤ α, so ψˆ ≤ αˆ by 6.1.6, and hence Gψˆ ≤ Gαˆ by 3.10, so ψˆ |C ∈ End(C). Next for c ∈ PC , c = aα for some a ∈ P and cψˆ = cψ = aαψ = aα 2 ϕ = aαϕ = cϕ, so ψˆ |C extends ϕ on C. Thus ρc : EndPc C (C) → Endc (PC ) is surjective, while ρc is injective by 5.3. This completes the proof of (2). If (G, P ) is c-minimal and PC = P , then C = PC by (2). Thus if G is finite and P is a p-group then O p (G) ≤ K and hence O p (G) ∩ P ≤ K ∩ P ≤ PK . 6.3. Assume P is nearly closed in G. Then: (1) Let (PK , PC ) be a splitting of P and set α = ξ −1 (PK , PC ). Then αˆ is an ˆ is the unique splitting of G with idempotent in EndP (G) and (K, C) = ξ(α) PK ≤ K and PC ≤ C. (2) PC is nearly closed in C. (3) If PC is rigid in some class C containing G then C = PC .

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Proof. Parts (1) and (2) follow from 6.2 and Remark 5.2. Under the hypotheses of (3), as C ≤ G ∈ C, C ∈ C, while by (2), PC is nearly closed in C, so PC = C by the rigidity of PC . 6.4. Assume P = P1 × P2 with P1 = 1 = P2 . Then either of the following imply that G = P: (1) P is c-embedded in G, cl(P ) ≤ c, and (G, P ) is c-minimal. (2) P is nearly closed in G and P1 and P2 are C-rigid for some class C containing G. Proof. Assume hypothesis (1) or (2) holds. Observe Si = (P3−i , Pi ) is a splitting of P for i = 1, 2, so αi = ξ −1 (Si ) is an idempotent in End(P ). Thus by 6.2.1 and 6.3.1, (Ki , Ci ) = ξ(αˆ i ) is a splitting of G with P3−i ≤ Ki , while Pi = Ci by 6.2.3 and 6.3.3. Let K = K1 ∩ K2 . Then G = Ki Ci = Ki Pi and P2 ≤ K1 so K1 = G ∩ K1 = K2 P2 ∩ K1 = KP2 , and hence G = K1 P1 = KP1 P2 = KP . Further K ∩ P = K ∩ P1 P2 ≤ K1 ∩ P1 P2 = P2 (K1 ∩ P1 ) = P2 , and similarly K ∩ P ≤ P1 , so K ∩ P ≤ P1 ∩ P2 = 1. Therefore (K, P ) is a splitting of G, so K = 1 by 4.4.1. Remark 6.5. Notice that 6.4 implies Theorem 1.

7. P -perfect groups In this section G is a finite group, p is a prime, and P is a subgroup of G. Write (G, P ) for the subnormal closure of P in G; that is is the smallest subnormal subgroup of G containing P . We say G is P-nilpotent if G = Op (G)P and G is P-perfect if G = P G . 7.1. Let = (G, P ). Then: (1) is P -perfect. (2) If P ≤ K ≤ G then (K, P ) ≤ . (3) If ≤ K ≤ G then = (K, P ). (4) If α : G → K is a surjective homomorphism then α = (K, P α). (5) If G is nilpotent and P -perfect then G = P .

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(6) If G is nilpotent then = P . Proof. Because P ≤ P , (1) follows from minimality of . Since P ≤ ∩ K K, it follows that (K, P ) ≤ ∩ K, giving (2). Then if ≤ K, (K, P ) is subnormal in , so = (K, P ) by minimality of , establishing (3). Assume the hypotheses of (4). Then α is subnormal in Gα = K, so (K, P α) ≤ α. Then the preimage I in of (K, P α) is a subnormal subgroup of containing P , so I = , and hence α ≤ I α = (K, P α), thus (4) holds. Assume G is nilpotent. Then P is subnormal in G, so (5) and (6) hold. In the remainder of the section assume P is a p-group and set L = O p (G). 7.2. Assume G is P -perfect. Then: (1) G = P L. (2) L = [L, P ]. ¯ = G/L. Then G ¯ is a P¯ -perfect p-group, so P¯ = G ¯ by 7.1.5, and hence Proof. Let G (1) holds. Let K = [L, P ] and G∗ = G/K. Then P ∗ centralizes L∗ so as G∗ = P ∗ L∗ by ∗ (1), P ∗ G∗ . Thus G∗ = P ∗G = P ∗ , so L∗ = O p (G∗ ) = 1. That is (2) holds. 7.3. Assume G is P -perfect and ϕ ∈ Hom(P , G). Let be the set of ϕˆ ∈ End(G) extending ϕ such that Gϕˆ is nilpotent. Then: (1) = ∅ iff P ∩ L ≤ ker(ϕ). (2) If = ∅ then: (a) consists of a unique element ϕ, ˆ (b) Gϕˆ = P ϕ, (c) ker(ϕ) ˆ = L ker(ϕ). Proof. Suppose = ∅ and let ϕˆ ∈ . As G is P -perfect, Gϕˆ is P ϕ-perfect, ˆ so as Gϕˆ is nilpotent, (b) holds by 7.1.5. By (b), L ≤ ker(ϕ), ˆ so P ∩ L ≤ P ∩ ker(ϕ) ˆ = ker(ϕ). This proves half of (1). Also by 7.2.1, G = P L, so ker(ϕ) ˆ = L(P ∩ ker(ϕ)) ˆ = L ker(ϕ), establishing (c). Then as G = LP , (b) and (c) imply ϕˆ is unique, completing the proof of (2). Conversely suppose P ∩L ≤ ker(ϕ). Define ϕˆ : G → G by (al)ϕˆ = aϕ for a ∈ P and l ∈ L. By 4.1 (taking A = P , H = L, α = ϕ and β the trivial homomorphism), ϕˆ ∈ End(G) extends ϕ.

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8. c-embeddings in finite groups In this section G is a finite group, p is a prime, and P is a p-subgroup of G. 8.1. Assume P is 0-embedded in G. Then: (1) G = P G G∞ . (2) If Y G and G/Y is nilpotent then G = Y P . (3) If G is solvable then G is P -perfect. (4) If G is nilpotent then G = P . (5) G = LP . (6) If G is p-nilpotent then G = P . (7) Let P ≤ T ∈ Sylp (G). Then Z(P ) ≤ Z(T ) ≤ Z(G). Proof. Let K = P G and H = G∞ . If KH = G then as G/H is solvable there is a normal subgroup X of G containing KH with |G : X| = q prime. Then as G has a subgroup of order q, Hom(G/K, G) = 0, contrary to 4.5 and the hypothesis that P is 0-embedded in G. This establishes (1). Further (2) follows from (1), since if G/Y is nilpotent and Y P = G then also Y K = G by 7.1.5. Also (3) and (4) follow from (1) and (2). Similarly (5) follows from (2) applied to Y = O p (G). Assume G is p-nilpotent and let Y = Op (G). Then by (2), G = Y P so P is a complement to Y in G. Then (6) follows from 4.4. Part (7) follows from 5.5.1. 8.2. Assume P is 0-embedded in G. Then CAut(L) (P ∩ L) ∩ CAut(L) (AutP (L)) = 1. Proof. Let β ∈ CAut(L) (P ∩ L) ∩ CAut(L) (AutP (L)). Then by 4.3 applied to P and L in the roles of A and H , we get ψ ∈ Aut(G) with ψ|P = 1P and hψ = hβ for h ∈ L. As P is 0-embedded in G, ψ = 1G , so β = 1. 8.3. Assume P is 0-embedded in G and let P ≤ T ∈ Sylp (G). Then: (1) CT (P ∩ L) = CT (L). (2) If P ∩ L is abelian then G = P . Proof. If CT (L) < CT (P ∩ L) then there is g ∈ CT (P ∩ L) − CT (L) with [g, P ] ≤ CT (L). But then 1 = cg ∈ Aut(L) centralizes P ∩ L and Aut P (L), so 8.2 supplies a contradiction. This establishes (1). Assume X = P ∩ L is abelian. Then X ≤ Z(L) by (1). Thus Y = T ∩ L centralizes X, so Y ≤ Z(L) by (1). Hence as L = O p (L) and Y ∈ Sylp (L), L is a p -group. Thus G = P by 8.1.6.

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8.4. Assume P is d-embedded in G, let be a set of subgroups of P such that for each X ∈ , cl(X) ≤ d and X ∩ L is abelian. Let be the set of normal subgroups K of P such that P /K is isomorphic to a member of , and set K. K() = K∈

Then P ∩ L ≤ K(). Proof. Claim it suffices to show for each K ∈ , that there exists ϕˆK ∈ End(G) with K = P ∩ ker(ϕˆK ) and GϕˆK ∼ = P /K. For if so then L ≤ ker(ϕˆK ), so ker(ϕˆK ), L≤ K∈

and hence P ∩L≤

P ∩ ker(ϕˆK ) =

K∈

K = K().

K∈

Thus the claim is established. Next by definition of , for K ∈ there exists X ∈ and ϕ ∈ End(P ) with K = ker(ϕ) and X = P ϕ. As X ∈ , cl(X) ≤ d, so as P is d-embedded in G, ϕ extends to ϕˆ ∈ End(G). Let H = Gϕˆ and LH = O p (H ); by the claim it remains to show H = X. But X ∩ LH ≤ X ∩ L, so X ∩ LH is abelian as X ∈ . Hence H = X by 5.3 and 8.3.2, completing the proof. 8.5. Assume P is 1-embedded in G and let ϕ ∈ End1 (P ). Then: (1) There exists a unique ϕˆ ∈ Hom(G, G) extending ϕ, and Gϕˆ = P ϕ. (2) P ∩ L ≤ [P , P ]. Proof. The first part of (1) is immediate from the definition of 1-embedding, and the second follows from 5.7. P ∗ = R1∗ × · · · × Rn∗ where Ri∗ is cyclic. Let Ki be the Let P ∗ = P /[P , P ]; then preimage in P of j =i Rj∗ and ri ∈ P with Ri∗ = ri∗ . Then |Ri∗ | divides |ri |, so P contains a subgroup Xi ∼ = Ri∗ . Set = {X1 , . . . , Xn }. As Xi is abelian, Xi ∩ L is abelian and cl(Xi ) = 1. Thus P ∩ L ≤ K() by 8.4. Finally P /Ki ∼ = Ri∗ ∼ = Xi , so K() ≤

n

Ki = [P , P ],

i=1

completing the proof. 8.6. Assume P ∈ Sylp (G) and P is 1-embedded in G. Then G = P . Proof. By 8.5.2, P ∩ L ≤ [P , P ] ≤ (P ). Therefore G is p-nilpotent by 9.5, so G = P by 8.1.6.

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8.7. Assume P is 1-embedded in G and [P , P ] is abelian. Then G = P . Proof. By 8.5.2, P ∩ L ≤ [P , P ], so P ∩ L is abelian by hypothesis. Then the lemma follows from 8.3.2. 8.8. Assume P is 1-embedded in G and P is of class at most 3. Then G = P . Proof. As P is of class at most 3, [P , P ] is abelian by 9.4, so the lemma follows from 8.7. Remark 8.9. Notice that 8.6 implies Theorem 2 and 8.7 and 8.8 imply Theorem 3. 8.10. If P is 0-embedded in G of class c and P ∩ L ≤ Lk (P ) for some k ≥ (c − 1)/2, then G = P . Proof. As k ≥ (c − 1)/2, Lk (P ) is abelian by 9.4. Then as P ∩ L ≤ Lk (P ), P ∩ L is abelian, so the lemma follows from 8.3.2. We close this section with a proof of Theorem 4. Assume the hypotheses of Theorem 4 and let S = (Si : 0 ≤ i ≤ n) be a splitting series for P such that each member of K(S) is abelian. Let Sn−1 = (PK , PC ). Then PK is abelian and PC is splitting solvable, so by induction on the order of P , PC is rigid in the class of finite groups. Hence by 6.3, there is a splitting (K, C) of G with PK = P ∩ K and C = PC . In particular P ∩ L ≤ PK , and hence P ∩ L is abelian, so G = P by 8.3.

9. Finite p-groups In this section P is a finite p-group of class c. Let Zi (P ) be the ith term in the ascending central series; that is Z0 (P ) = 1 and for i > 1, Zi (P ) is the preimage in P of Z(P /Zi−1 (P )). Let Li (P ) be the ith term in the descending central series; that is L0 (P ) = P and for i > 0, Li (P ) = [Li−1 (P ), P ]. We record some well known facts: 9.1. Li (P ) ≤ Zc−i (P ). Proof. See for example the proof of 9.6 in [2]. 9.2. [Li (P ), Zj (P )] ≤ Zj −i−1 (P ), where Zr (P ) = 1 for r ≤ 0. Proof. Let P ∗ = P /Zj −i−1 (P ). If i + 1 ≤ j then Zj (P )∗ = Zi+1 (P ∗ ) and Zj −i−1 (P )∗ = 1, while if i +1 ≥ j then P = P ∗ and Zj −i−1 (P ) = 1. Thus replacing P by P ∗ , we may assume j ≤ i + 1 and it remains to show [Li (P ), Zj (P )] = 1. We induct on i; if i = 0 then j ≤ 1 and [Li (P ), Zj (P )] ≤ [P , Zj (P )] ≤ Z0 (P ) = 1, so the induction is anchored and we may assume i is positive. Let L = Li−1 (P ) and U = Zj (P ). We show [L, U, P ] = 1 = [U, P , L] = 1; then by the 3-subgroup lemma, [Li (P ), Zj (P )] = [P , L, U ] = 1.

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Now by induction on i, [L, U ] ≤ Zj −i (P ) ≤ Z(P ) as j ≤ i +1, so [L, U, P ] = 1. Also [U, P ] ≤ Zj −1 (P ) ≤ CP (L) by induction on i as j ≤ i + 1; thus [U, P , L] = 1. 9.3. Li (P ) centralizes Zi+1 (P ). Proof. This is a special case of 9.2. 9.4. If i ≥ (c − 1)/2 then Li (P ) is abelian. Proof. By 9.1, Li (P ) ≤ Zc−i (P ) and if i ≥ (c − 1)/2 then Zc−i (P ) ≤ Zi+1 (P ). Thus the lemma follows from 9.3. 9.5. Assume G is a finite group, P ∈ Sylp (G), and P ∩ O p (G) ≤ (P ). Then G is p-nilpotent. Proof. This is a theorem of Tate in [4]; there is a proof in Theorem 4.7 in [3]. We close this section with a sketch of a proof of the following result: 9.6. Let P be a p-group 2-embedded in a finite group G and assume P is generated by at most three elements of order p. Let L = O p (G). Then: (1) Either G = P ∼ = Q8 ∗ Z4 or P ∩ L ≤ L2 (P ). (2) If cl(P ) ≤ 5 then G = P . We include this discussion to give an indication of how to use 8.4 to extend Theorem 3 to suitable families of p-groups of class greater than 3. The discussion also indicates the limitations of the method and the nature of the obstructions to its application. As 9.6 is so weak, it does not seem worthwhile to give all the details of the proof. We need the following facts which we use without proof; p1+2 denotes the extraspecial p-group of order p 3 generated by elements of order p. Facts. Assume P is of class at least 2 and generated by at most 3 elements of order p. Let be the set of normal subgroups K of P with P /K ∼ = p 1+2 . Then: (1) Either p = 2 and P ∼ K. = Q8 ∗ Z4 or L2 (P ) = (2) There exist

p 1+2

∼ = D ≤ P with D [P , P ].

K∈

Fact 1 can be proved by considering the universal p-group P of class 2 generated by 3 elements of order p. The corresponding group generated by 4 or more elements fails to have this property, so to extend the method, one would have to work with a larger class H of groups then {p1+2 }. Fact 2 is easy, but it would not be easy to establish the analogous statement for the larger class H, if indeed that statement is true. Now here is a proof of 9.6: By 8.7 we may assume [P , P ] is nonabelian and by 8.5, P ∩ L ≤ [P , P ].

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We first prove (1) by applying 8.4 to the set of subgroups X of P such that X∼ = p 1+2 and X L. Define and K() as in 8.4. As P ∩ L ≤ [P , P ], Fact 2 says = ∅. Thus by 8.4 it suffices to show that K() = L2 (P ). This follows from Fact 1 unless P ∼ = Q8 ∗ Z4 . But in that case P = G by 8.7. This completes the proof of (1). By (1) we may assume P ∩ L ≤ L2 (P ). Thus (2) follows from 8.10.

10. Hall–Higman reductions In this section G is a finite group, p is a prime, P is a p-subgroup of G, and L = O p (G). Write E = E (P ) for the set of all (isomorphism types) of embeddings (H, P ) of P in finite groups H . For (Gi , P ) ∈ E , write (G1 , P ) ≤ (G2 , P ) if there exists P ≤ H ≤ G2 such that (G1 , P ) ∼ = (H /K, P K/K) for some normal p -subgroup K of H . Evidently ≤ is a partial order on E . Define D ⊆ E to be a Hall–Higman class if whenever (G, P ) ∈ D and (H, P ) ≤ (G, P ), then also (H, P ) ∈ D. Observe that the intersection of Hall–Higman classes is a Hall–Higman class. An embedding (G, P ) is P-centered if CG (P ) = Z(G). 10.1. The following subclasses of E are Hall–Higman: (1) The class of embeddings (G, P ) such that P is subnormal in G. (2) The class of embeddings (G, P ) such that the subnormal closure in G of P is P -nilpotent. (3) The class of P -centered embeddings. Proof. This is straightforward; for example (1) and (2) follow from 7.1.2 and 7.1.4. However we do (3). Let P ≤ H ≤ G, K a normal p -subgroup of H , and set H ∗ = H /K. Assume G is P -centered. Then CH (P ) ≤ CG (P ) ∩ H ≤ Z(G) ∩ H ≤ Z(H ). As K is a p -group, CH (P )∗ = CH ∗ (P ∗ ) and Z(H )∗ ≤ Z(H ∗ ). Given subclasses D and P of E , define D to be a P -class if D ⊆ P . Example 10.2. Write (G, P ) for the set of all overgroups I of P in G such that I is P -perfect and P -nilpotent, but Op (I ) Op (G). Consider the class PI of extensions (G, P ) such that (G, P ) = ∅. A subclass D of E is P-signalizer closed if D ⊆ PI . Given a subclass P of E , define an embedding (G, P ) to be minimal non-P if (a) (G, P ) ∈ / P , but (b) whenever (H, P ) < (G, P ), (H, P ) ∈ P .

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10.3. Let D and P be subclasses of E , and assume D is a Hall–Higman class. Then either: (a) D is a P -class, or (b) D contains a minimal non-P embedding. Proof. Assume D P . Then there exists (G, P ) ∈ D minimal subject to (G, P ) ∈ / P under the partial order ≤. As D is a Hall–Higman class, (H, P ) ∈ D for each (H, P ) < (G, P ). Thus (H, P ) ∈ P by minimality of (G, P ). That is (G, P ) is a minimal non-P embedding. Define IG (P , p ) to be the set of all P -invariant p -subgroups of G. Define to consist of those X ∈ IG (P , p ) such that X = [X, P ]. Recall the definition of (G, P ) from Section 7.

I− G (P , p )

10.4. Set = (G, P ). Then:

(1) G is P -perfect and P -nilpotent iff G = LP and L ∈ I− G (P , p ).

(2) (G, P ) = ∅ iff [Op (G), P ] is the unique maximal member of I− G (P , p ).

(3) If O p (CG (P )) ≤ Op (G) then (G, P ) = ∅ iff Op (G) is the unique maximal member of IG (P , p ).

(4) contains each P -perfect P -nilpotent subgroup of G and I− G (P , p ) ⊆ .

(5) (, P ) = (G, P ). (6) If is P -nilpotent then (G, P ) = ∅. Proof. Parts (1)–(3) are straightforward. The first statement in (4) follows from 7.1.2; then the second statement in (4) follows from the first and (1). As is subnormal in G, Op () ≤ Op (G), so (4) implies (5). Then (5) implies (6). 10.5. Assume G is minimal non-P -signalizer closed. Then: (1) G is P -perfect. (2) Op (G) = 1.

∗ (3) For each 1 = Y ∈ I− G (P , p ), Y is faithful on F (G) and Y P ∈ (G, P ).

(4) G = I F ∗ (G) for each I ∈ (G, P ), and Op (I ) is faithful on F ∗ (G).

(5) There exists I ∈ (G, P ) such that X = Op (I ) is a q-group of class at most 2 for some prime q, X is of exponent q if q is odd, P centralizes (X), and P is irreducible on X/(X). (6) Either (i) G is solvable and F ∗ (G) = Op (G), or (ii) G is not solvable, F ∗ (G) = Op (G))E(G), Op (I ) centralizes Op (G), and E(G) is the product of components permutated transitively by I .

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Proof. Part (1) follows from 10.4.5 and minimality of G. Let K = Op (G) and G∗ = G/K. If K = 1 then by minimality of G, ∗

(G∗ , P ∗ ) = ∅. Thus [Op (G∗ ), P ∗ ] is the unique maximal member of I− G∗ (P , p ) − − by 10.4.2. But as K is a p -group, IG∗ (P ∗ , p ) = IG (P , p )∗ , so as Op (G∗ ) = 1, (G, P ) = ∅ by another application of 10.4.2, contrary to the choice of G. Thus (2) is established. Let I ∈ (G, P ) and F = F ∗ (G). Then by 10.4.1, I = P X where X = [P , X] is a p -group. Conversely if Y = [Y, P ] is a nontrivial p -group then CY (F ) ≤ Z(F ), so Y is faithful on F by (2). Further [Y ∩ Op (F Y ), F ] ≤ Op (F ) = 1, so Y ∩ Op (F X) = 1. Therefore Y P ∈ (I F, P ), while if G = I F then (I F, P ) = ∅ by minimality of G. Thus (3) and (4) hold. As I is P -nilpotent and X is a p -group, P acts on a Sylow q-group U of X for each q ∈ π(X). Further as X = [X, P ], [U, P ] = 1 for some choice of q. Let V be a supercritical subgroup of U ; as [U, P ] = 1, [V , P ] = 1. Pick W ≤ V minimal subject to 1 = W = [W, P ]. Then P W ∈ (G, P ) by (3), establishing (5). Pick I as in (5) and let R = Op (G). If F ∗ (G) = R then as I and R are solvable and G = I R by (4), G is solvable, so (6i) holds. Thus we may assume L1 is a component of G and let J = LI1 . In particular G is not solvable, so G = RI and hence X ≤ Op (RI ), so X centralizes R. Similarly if G = I J then X centralizes L1 , so as X is faithful on F , it follows that G = I J = P XJ . Thus (6ii) hold, completing the proof of the lemma. 10.6. Assume G is minimal non-P -signalizer closed and G is solvable. Let R = Op (G) and Q = Op (L). Then: (1) G = LP is P -perfect. (2) R = F ∗ (G). (3) L = QX where X and the action of P on X is described in 10.5.5. (4) P Q ∈ Sylp (G). (5) R = QCP (X). (6) [Q, CP (X)] = 1. (7) P X is irreducible on Q/(Q). (8) Either (a) Q is elementary abelian, or (b) Q is special, Q is of exponent p if p is odd, and X centralizes (Q). (9) If CG (P ) = Z(G) then Q is special, Z(Q) = Z(L), and X is elementary abelian. Proof. Part (1) and (2) follow from 10.5. Further by 10.5, G = RXP , where X = [P , X] is described in 10.5.5. Thus L = QX, so (3) holds and (4) follows from (1) and (3).

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By (4), R = R ∩ P Q = Q(R ∩ P ). As P X is p-nilpotent and G = P QX, R ∩ P = CP (X). That is (5) holds. Let M be minimal subject to M ≤ Q, P X ≤ NG (M), and [M, X] = 1. Then as P acts on X, M = [M, X] by minimality of M. If G = P XM then X ≤ Op (P XM) by minimality of G, so X centralizes M, contrary to the choice of M. Thus G = P XM, so M = Q. Then (7) and (8) follow from the minimality of M = Q. Finally assume CG (P ) = Z(G). Let Z = Z(Q). By coprime action, Z = [Z, X] ⊕ CZ (X). As P acts on X it acts on [Z, X], so if [Z, X] = 1 then 1 = C[Z,X] (P ) ≤ Z(G), a contradiction. Thus [Z, X] = 1, so Z = Z(L). As Z(Q) = Z(L) and X is nontrivial on Q, case (b) of (8) holds. By (3) and 10.5.5, (X) ≤ CG (P ) = Z(G), so as X is faithful on Q by (4) and 10.5.4, (X) = 1. This completes the proof of (9). 10.7. If p is odd then the class of solvable P -centered groups is P -signalizer closed. Proof. Assume otherwise. By 10.1.3 and 10.3, there is an embedding (G, P ) which is P -centered and minimal non-P -signalizer closed. Thus G is described in 10.6. Adopt the notation of 10.6 and let Z = Z(Q), G∗ = G/Z, and V = Q∗ . As Q is special, we may regard Z and V as Fp G-modules, and as R = QCP (X) centralizes these modules, they are also modules for G+ = G/R. Further G+ = X+ P + ∼ = XP /CP (X), X centralizes Z, and X+ P + is faithful and irreducible on V . Define f : V × V → Z by f (a ∗ , b∗ ) = [a, b]. Then f is a G-equivariant alternating bilinear map. Let pn be the smallest power of p such that p n ≡ 1 mod q and let F = Fpn . Let AF = A ⊗Fp F for A ∈ {V , Z}. Then V F and Z F are F G-modules and we get a G-equivariant alternating F -bilinear map f F : V F × V F → Z F with f F (1 ⊗ v, 1 ⊗ u) = f (v, u) for u, v ∈ V . Let σ ∈ Aut(F ) be defined by σ : a → a p ; that is = Gal(F /Fp ) = σ ). Then (cf. 25.7 in [2]) AF is an Fp -module via (a × v)σ = aσ ⊗ v for a ∈ F , v ∈ A, and in this representation, commutes with G. Moreover if W is a G-submodule of AF then W = B F for some G-submodule of A. Therefore as G+ is irreducible on V , G is irreducible on V F , so as X+ G+ , V F is a semisimple F X-module. As X is elementary abelian, CV (Y ), V = Y ∈Y

where Y is the set of hyperplanes of Y of X with CV (Y ) = 0. As P X is irreducible on V , P is transitive on Y and for Y ∈ Y, XNP (Y ) is irreducible on CV (Y ). Then CV F (Y ), VF = Y ∈Y

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and writing Y for the set of weights of X on CV (Y ), CV (Y )λ . CV F (Y ) = λ∈Y

Let vi ∈ CV (Yi ), λi ∈ Yi , and x ∈ X. Then x fixes f F (v1 , v2 ) ∈ Z F , so f F (v1 , v2 ) = f F (v1 , v2 )x = f F (v1 x, v2 x) = f F (λ1 (x)v1 , λ2 (x)v2 ) = λ1 (x)λ2 (x)f F (v1 , v2 ), so f F (v1 , v2 ) = 0 unless λ2 = λ−1 1 . But in the latter case Y1 = ker(λ1 ) = ker(λ2 ) = Y2 , so [CQ (Y1 ), CQ (Y2 )] = 1 if Y1 = Y2 . Thus Q= QY , Y ∈Y

is a central product of the subgroups QY = [CQ (Y ), X]. Therefore QY is special as Q is special and XNP (X) is irreducible on QY /Z(QY ) as XP is irreducible on Q/Z. Suppose NP (Y ) centralizes a ∈ QY − Z. Pick a set C of coset representatives for NP (Y ) in P and let ac. b= c∈C

Then b ∈ CG (P ) − Z(G), a contradiction. Therefore CQY (NP (Y )) ≤ Z. Next X = XY × Y for some NP (Y ) complement XY to Y in X, and XY NP (Y ) is irreducible on QY /Z(QY ), so by the Thompson A × B-Lemma, PY = CP (X/Y ) centralizes QY . Therefore as CQY (NP (Y )) ≤ Z, NP (Y ) = PY . Thus q ≡ 1 mod p and P Y = NP (Y )/PY is cyclic. Then P Y is semiregular on Y , Y = λP Y is an orbit under P Y , and CV F (Y )λ is 1-dimensional. ρ : µ → As λ−1 ∈ Y = λP Y , ρ ∈ P Y , where ρ is the involution such that −1 µ for µ ∈ Y . Let v be a generator of CV (Y )Fλ , = Op (), and u = δ∈ vδ. Then W0 = uOp () is a free module for Op () invariant under , and W1 = W0PY is a sum of free modules for P Y . Then by a remark above W1 = W F for some P Y -submodule of CV (Y ). Further as p is odd, ρ ∈ , so f F (u, uγ ) = 0 for γ ∈ Op ()P Y , and hence f F is 0 on W1 . Thus the preimage QW in QY of W is abelian and indeed as p is odd, Q is of exponent p, so QW is elementary abelian. Thus as P Y is free on QW /Z(QY ), QW splits over Z(QY ) as a P -module, so CQY (NP (Y )) Z, a contradiction. 10.8. Assume p is odd and G is solvable and P -centered. Then Op (G) is the unique maximal p -signalizer for P . Proof. This follows from 10.7 and 10.4.3. Remark 10.9. Notice that 10.8 is just Theorem 5.

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Remark 10.10. When p = 2 there is a counter example to 10.8 which is minimal non-P -signalizer closed with X elementary abelian. Take G = P L where L = L1 ×L2 with Li ∼ = SL2 (3), and P is a 2-group acting on L and X ∈ Syl3 (L) with Aut P (L) ∼ = D8 and P ∩ O2 (G) = CP (L). Choose P so that Z2 (P ) ≤ CP (L). For example let P = a, b be metacyclic with |a| = 16, |b| = 4, and a b = a 3 . Then P is of class 4 with Z2 (P ) = a 4 , b2 . Take CP (L) = Z2 (P ) (so that P /CP (L) ∼ = D8 ) and P ∩ L ≤ b2 , a 8 .

References [1] H. Bechtell, Splitting systems for finite solvable groups, Arch. Math. 36 (1981), 295–301. [2] M. Aschbacher, Finite group theory, Cambridge Stud. Adv. Math. 10, Cambridge Univ. Press, Cambridge 1986. [3] B. Huppert, Endliche Gruppen I, Springer-Verlag, Berlin 1967. [4] J. Tate, Nilpotent quotient groups, Topology 3 (1964), 109–111. M. Aschbacher, California Institute of Technology, Pasadena, CA 91125, U.S.A. E-mail: [email protected]

Extensions for finite groups of Lie type: twisted groups Christopher P. Bendel∗, Daniel K. Nakano† and Cornelius Pillen

Abstract. In [BNP1], [BNP2], [BNP3] the authors relate the extensions for modules for finite Chevalley groups to certain extensions for the corresponding reductive groups and Frobenius kernels. In this paper we will show that these results can be extended to the twisted finite groups of Lie type. The applications are most pertinent to the twisted groups of types A, D, and E6 . 2000 Mathematics Subject Classification: Primary 20C, 20G; Secondary 20J06, 20G10

1. Introduction 1.1. Let G be a connected simply connected almost simple algebraic group defined and split over the field Fp with p elements, and let k be the algebraic closure of Fp . Let G(Fq ) be the finite Chevalley group consisting of Fq -rational points of G where q = p r for a positive integer r and let Gr denote the rth Frobenius kernel of G. In a series of papers [BNP1], [BNP2], [BNP3], [BNP4] the authors investigated the connections between the extension theory of modules for the semisimple algebraic group G, the finite Chevalley group G(Fq ) and the Frobenius kernel Gr via the use of spectral sequences involving truncated (bounded) subcategories of G-modules. In particular these methods enabled us to provide explicit formulas for the extensions between simple modules of finite Chevalley groups via algebraic groups and Frobenius kernels. As a direct application, several open problems involving the existence of selfextensions for finite Chevalley groups posed by Humphreys [Hum] were resolved by proving suitable generalizations of Andersen’s work on line bundle cohomology for the flag variety [And1], [And2]. Recent work by Tiep and Zalesskiˇı [TZ, Prop. 1.4] has shown that questions involving extensions between simple modules for finite groups arise naturally when one studies the modular representation theory of finite groups. They prove that the ∗ Research of the first author was supported in part by NSA grant MDA904-02-1-0078. † Research of the second author was supported in part by NSF grant DMS-0102225.

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Christopher P. Bendel, Daniel K. Nakano and Cornelius Pillen

ability to lift certain simple modular representations to characteristic zero can happen only when the simple module admits a self-extension. In the mid 1980s, Smith [Smi] showed that the existence of self-extensions is also relevant in the study of sheaf homology. 1.2. The purpose of this paper is to demonstrate that the ideas from the authors’ earlier work can be used to study the extension theory for the other finite groups of Lie type. Following the description of the classification of the finite Lie groups given in [Car, 1.19], we refer to the untwisted groups as Chevalley groups or split forms of G. If G is of type A, D, or E6 there exist non-trivial automorphisms of the Dynkin diagram of G. Such an automorphism, σ , induces a group automorphism of G that commutes with the Frobenius morphism, which we will also denote by σ . Let Gσ (Fq ) be the finite group consisting of the fixed points of σ composed with the rth Frobenius morphism on G. We will refer to the groups Gσ (Fq ) as twisted groups or quasi-split forms of G [Car, 1.19]. Since our results hold mostly for large primes, we will limit our discussion of the Suzuki and Ree groups to Section 2.4. In the remaining sections the extension theory of modules for the split and quasi-split forms Gσ (Fq ) are compared to those for G and Gr . In our discussion, we will leave out several of the details of earlier proofs which have straight forward generalizations to the twisted groups. Instead we will focus on places where there is care needed in accounting for the twist σ . The new formulas that we obtain involving extensions of simple modules for Gσ (Fq ) do indeed involve using the automorphism σ in the calculations. We also indicate several instances in the text where improvements have been made to our earlier work. 1.3. Notation. The conventions in the paper will follow the ones used in [Jan1]. Throughout this paper G is a connected simply connected almost simple algebraic group defined and split over the finite field Fp with p elements. The field k is the algebraic closure of Fp and G will be considered as an algebraic group scheme over k. Let be a root system associated to G with respect to a maximal split torus T . Moreover, let + (resp. − ) be positive (resp. negative) roots and be a base consisting of simple roots. Let B be a Borel subgroup containing T corresponding to the negative roots. Let X(T ) be the integral weight lattice obtained from contained in the Euclidean space E with the inner product denoted by , . The set X(T ) has a partial ordering given by λ ≥ µ if and only if λ−µ ∈ α∈ Nα for λ, µ ∈ X(T ). Let α ∨ = 2α/α, α be the coroot corresponding to α ∈ . The set of dominant integral weights is defined by X(T )+ = {λ ∈ X(T ) : 0 ≤ λ, α ∨ for all α ∈ }. Furthermore, the set of pr -restricted weights is Xr (T ) = {λ ∈ X(T ) : 0 ≤ λ, α ∨ < pr for all α ∈ }.

Extensions for finite groups of Lie type: twisted groups

31

The Weyl group W is the group generated by the reflections sα : E → E given by sα (λ) = λ − λ, α ∨ α. The group W acts on X(T ) via the “dot action” given by w · λ = w(λ + ρ) − ρ where ρ is the half sum of the positive roots. The longest element in W will be denoted by w0 and the Coxeter number for is h = ρ, α0∨ + 1, where α0 is the maximal short root. 1.4. We adopt the set-up described in [Jan3, 1.3]. Let σ denote an automorphism of the Dynkin diagram of G. The automorphism σ can be extended to a linear bijection on X(T ) that permutes the fundamental weights and preserves the inner product , as well as the partial order on X(T ). Moreover, σ (α0 ) = α0 . The graph automorphism σ induces an automorphism on G which will also be denoted by σ . The group automorphism σ commutes with the standard Frobenius morphism F and is compatible with the action of σ on X(T ). Let Gσ (Fq ) be the group of fixed points of F r σ = σ F r , where q = p r . Throughout the paper Gσ (Fq ) will always either denote a split form, if σ acts trivially on the Dynkin diagram of G, or a quasi-split form of G, if G is of type A, D, or E6 and σ is non-trivial [Car, 1.19]. The Suzuki and Ree groups will be dealt with separately in Section 2.4. For each λ ∈ X(T )+ , let H 0 (λ) = indG B λ. The simple modules for G are labeled by the set X(T )+ and are given by the correspondence λ → L(λ) = socG (H 0 (λ)). Every G-module V becomes a Gσ (Fq )-module by restriction. A complete set of non-isomorphic simple Gσ (Fq )-modules and simple Gr -modules are obtained by restricting {L(λ) : λ ∈ Xr (T )} [Cur], [Ste1]. For any G-module V and simple Gσ (Fq )-module L(λ) we denote by [V : L(λ)]Gσ (Fq ) the multiplicity of L(λ) as a Gσ (Fq )-composition factor of V . One obtains [V : L(λ)]Gσ (Fq ) = [V : L(γ )]G [L(γ ) : L(λ)]Gσ (Fq ) . γ ∈X(T )+

Moreover, twisting a G-module V with the rth Frobenius morphism yields the same result as twisting by σ −1 upon restriction to Gσ (Fq ). According to [Jan3, 1.3] one obtains for all µ ∈ X(T )+ that L(µ)(r) ∼ = L(σ (µ))

(1.4.1)

as a Gσ (Fq )-module.

2. General Ext-formulas 2.1. We briefly review the techniques developed in [BNP1]. For a fixed positive integer r let Cs be the full subcategory of Mod(G) with objects having composition factors whose highest weights lie in πs where πs = {λ ∈ X(T )+ : λ + ρ, α0∨ < 2pr s(h − 1)}.

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Christopher P. Bendel, Daniel K. Nakano and Cornelius Pillen

Notice that the category C1 differs slightly from Jantzen’s pr -bounded category where it is assumed that λ, α0∨ < 2pr (h − 1) [Jan1, p. 360]. Let FCs be the truncation functor from Mod(G) to Cs which takes M ∈ Mod(G) to the largest submodule of j M in Cs (see [Don, §1.1]). Let Gs = FCs indG Gσ (Fq ) and R Gs be the higher right derived functors of Gs . We remark that if M is a finite-dimensional Gσ (Fq )-module, then Gs (M) is also finite-dimensional. In order to see this fact, for any weight λ in πs , let P (λ) be the projective cover of L(λ) in Cs . Using right adjointness of Gs to the restriction functor we have [Gs (M) : L(λ)] = dim HomG (P (λ), Gs (M)) = dim HomGσ (Fq ) (P (λ), M). The last term is finite because πs is a finite set and Cs is equivalent to the module category for some finite-dimensional quasi-hereditary algebra [BNP1, §4.5]. Since σ preserves the inner product and the order on X(T )+ the arguments in [BNP1, 4.1- 4.4] yield an analog of [BNP1, Thm. 4.4a] as follows. Theorem. For M ∈ Cs and N ∈ Mod(Gσ (Fq )) there exists a first quadrant spectral sequence i,j

i+j

E2 = ExtiG (M, R j Gs (N)) ⇒ ExtGσ (Fq ) (M, N ). 2.2. The rth Steinberg module St r is projective and injective in Mod(Gσ (Fq )) and in Mod(Gr ). For p ≥ 2(h − 1) the projectivity and injectivity extends to the category C1 [Jan2]. The injective hulls of all p r -restricted simple modules in all three categories appear as respective direct summands of tensor products of Str with appropriate p r -restricted simple modules. Moreover, the injective hull Qr (λ) of a simple Gr -module L(λ) lifts to a G-structure and is also injective in C1 [Jan2]. A theorem due to Chastkofsky [Cha] and Jantzen [Jan3] says that the injective hull of L(λ) in Mod(Gσ (Fq )) appears exactly once as a summand of Qr (λ), both for the split forms and for the quasi-split forms. Therefore, the arguments in [BNP1, 4.6–5.2], which were addressing only the split forms extend to the quasi-split forms. One obtains a generalization of [BNP1, 5.2a]. Theorem. Let s ≥ 1, p ≥ 2s(h − 1), M ∈ Mod(Gσ (Fq )), and let L ∈ Cs such that L has only p r -restricted composition factors in its head (as a G-module), then Ext iGσ (Fq ) (L, M) ∼ = ExtiG (L, Gs (M)) for all 0 ≤ i ≤ s. 2.3. For the remainder of the paper we will look at the Ext1 groups. We will simply denote G1 by G, π1 by π , and C1 by C. It turns out that the module G(k) is of particular interest. The theorem below justifies this statement. It is an analog to [BNP2, Thm. 2.2]. The proof in [BNP2] can easily be adapted to the quasi-split case.

Extensions for finite groups of Lie type: twisted groups

33

The Suzuki and Ree groups will be discussed separately in the next section. Notice that the result holds for any prime. Theorem. Let λ, µ ∈ Xr (T ). Then Ext 1Gσ (Fq ) (L(λ), L(µ)) ∼ = Ext1G (L(λ), L(µ) ⊗ G(k)). 2.4. Suzuki and Ree groups. The purpose of this section is to obtain a version of Theorem 2.3 for the Suzuki and Ree groups. Let G be of type C2 or F4 and p = 2 or G of type G2 and p = 3. Then there exists a special isogeny σ on G with σ 2 = F , the Frobenius morphism, which leads to a stronger Steinberg Tensor Product Theorem [Ste1, Section 11], [Sin, 1.4 (2)]. Define Xσ (T ) ⊂ X1 (T ) to be the subset of those weights that are orthogonal to iall long simple roots. Then every λ ∈ X(T )+ has a σ -adic expansion λ = m i=0 σ λi , λi ∈ Xσ (T ). For an odd positive integer m we denote by G(m) the subgroup of G fixed by ) and Ree groups (for types G2 , F4 ). The set σ m . These are the Suzuki (for type C2 i of σ m -restricted weights Xσ m (T ) = { m−1 i=0 σ λi | λi ∈ Xσ (T )} parametrizes the simple modules of G(m). The Steinberg module St(m) with highest weight (σ m −1)ρ is injective and projective. For fixed m we denote by C the subcategory of Mod(G) whose highest weights γ satisfy γ , α0∨ ≤ 2(σ m − 1)ρ, α0∨ . As in Section 2.1 we let FC be the truncation functor from Mod(G) to C which takes M ∈ Mod(G) to the largest submodule of M in C and set G = FC indG G(m) . As before we obtain for any M ∈ C and any N ∈ Mod(G(m)) a spectral sequence i,j

i+j

E2 = ExtiG (M, R j G(N)) ⇒ ExtG(m) (M, N ).

(2.4.1)

In the proof of the following theorem we make use of the fact that the E21,0 -term of the spectral sequence embeds in the E 1 -term. Theorem. Let λ, µ ∈ Xσ m (T ). Then Ext1G(m) (L(λ), L(µ)) ∼ = Ext1G (L(λ), L(µ) ⊗ G(k)). Proof. All simple modules are self-dual and Ext 1G(m) (L(λ), L(µ)) ∼ = Ext1G(m) (L(µ), L(λ)). Therefore, we may assume that µ, α0∨ ≤ λ, α0∨ and furthermore that µ ≯ λ. Set M = St(m)⊗L((σ m −1)ρ −λ). Then M is injective and projective as a G(m)-module and M ⊗ L(µ) is in C. In order to proceed as in the proof of [BNP2, Thm. 2.2] we need the following HomG(m) (M ⊗ L(µ), k) ∼ = HomG(m) (L(λ) ⊗ L(µ), k)

(2.4.2)

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Christopher P. Bendel, Daniel K. Nakano and Cornelius Pillen

and HomG (M, L(λ) ∼ = k.

(2.4.3)

Clearly,

k HomG(m) (L(λ) ⊗ L(µ), k) ∼ = HomG (L(λ) ⊗ L(µ), k) ∼ = 0

for λ = µ, otherwise.

Both (2.4.2) and (2.4.3) follow immediately from the Lemma below. Equation (2.4.2) now replaces [BNP2, (2.2.1)] and (2.4.3) allows us to define the G-module R via 0 → R → M ⊗ L(µ) → L(λ) ⊗ L(µ) → 0. This last equation replaces [BNP2, (2.2.2)]. For the remainder of the proof one can proceed as in the proof of [BNP2, Thm. 2.2] by replacing the spectral sequence [BNP2, (2.1.1)] with (2.4.1). Lemma. Let λ, µ ∈ Xσ m (T ) with µ ≯ λ. Then (a) HomG (St(m) ⊗ L((σ m − 1)ρ − λ), L(λ)) ∼ = k. (b) HomG(m)

(St(m) ⊗ L((σ m

k for λ = µ, ∼ − 1)ρ − λ), L(µ)) = 0 otherwise.

Proof. (a) Set r = (m−1)/2 and let ρ˜ be the sum of all fundamental weights in Xσ (T ). Then St(m) ∼ ˜ (r) . There are unique λ0 ∈ Xr (T ) and λ1 ∈ Xσ (T ) = St r ⊗L((p − 1)ρ) 0 r 1 with λ = λ + p λ . Let Gσ = Ker σ . Then Gσ is an infinitesimal group scheme [Sin, 1.2]. One can argue as follows: HomG (St(m) ⊗ L((σ m − 1)ρ − λ), L(λ)) ∼ HomG (St(m), L((σ m − 1)ρ − λ) ⊗ L(λ)) = ∼ = (HomGr (Str , L((pr − 1)ρ − λ0 ) ⊗ L(λ0 ))G ⊗ HomG (L((p − 1)ρ), ˜ L((p − 1)ρ˜ − λ1 ) ⊗ L(λ1 )) ∼ ˜ L((p − 1)ρ˜ − λ1 ) ⊗ L(λ1 )) (by [Jan1, II.11.9 (2)]) = HomG (L((p − 1)ρ), ∼ ˜ L((p − 1)ρ˜ − λ1 ) ⊗ L(λ1 )))G (by [Sin, 1.8 (1)]). = (HomG (L((p − 1)ρ), σ

The assertion now follows from the fact that L((p −1)ρ) ˜ is projective as a Gσ -module [Sin, 1.7 (1)] and that the multiplicity of L((p − 1)ρ) ˜ in L((p − 1)ρ˜ − λ1 ) ⊗ L(λ1 ) is one. (b) St(m) is projective as a G(m)-module. Therefore dim HomG(m) (St(m) ⊗ L((σ m − 1)ρ − λ) ⊗ L(µ), k)

= dim HomG(m) (St(m), L((σ m − 1)ρ − λ) ⊗ L(µ)) = [L((σ m − 1)ρ − λ) ⊗ L(µ) : St(m)]G(m)

Extensions for finite groups of Lie type: twisted groups

=

35

[L((σ m − 1)ρ − λ) ⊗ L(µ) : L(ν)]G · [L(ν0 ) ⊗ L(σ m ν1 ) : St(m)]G(m)

ν∈X(T )+

=

[L((σ m − 1)ρ − λ) ⊗ L(µ) : L(ν)]G · [L(ν0 ) ⊗ L(ν1 ) : St(m)]G(m) .

ν∈X(T )+

Notice that ν1 = 0 forces ν0 = (σ m − 1)ρ and [L((σ m − 1)ρ − λ) ⊗ L(µ) : St(m)]G(m) = [L((σ m − 1)ρ − λ) ⊗ L(µ) : St(m)]G . The latter is zero unless µ ≥ λ, which implies µ = λ. In this case the multiplicity is one, as desired. Assume that there exists a ν with ν1 = 0 that contributes to the above sum. Observe that σ m ν1 > ν1 . It follows that (σ m − 1)ρ − λ + µ ≥ ν = ν0 + σ m ν1 > ν0 + ν1 . By using [L(ν0 ) ⊗ L(ν1 ) : St(m)]G(m) = [L(ν0 ) ⊗ L(ν1 ) : L(γ )]G · [L(γ0 ) ⊗ L(σ m γ1 ) : St(m)]G(m) γ ∈X(T )+

=

[L(ν0 ) ⊗ L(ν1 ) : L(ν)]G · [L(γ0 ) ⊗ L(γ1 ) : St(m)]G(m) ,

γ ∈X(T )+

we can argue inductively on the size of the weight ν that [L(ν) : St(m)]G(m) = 0 unless ν ≥ (σ m − 1)ρ. But in that case (σ m − 1)ρ − λ + µ ≥ ν = ν0 + σ m ν1 > ν0 + ν1 ≥ (σ m − 1)ρ gives the desired contradiction to µ ≯ λ. 2.5. From now on all finite groups considered are either split or quasi-split forms of G. The remainder of Section 2 is devoted to collecting information on G(k). Observe that if ν ∈ π is expressed as ν = ν0 + pr ν1 for weights ν0 ∈ Xr (T ) and ν1 ∈ X(T )+ , then the weight ν1 satisfies ν1 , α0∨ < 2(h − 1). We will show that the highest weight of any composition factor of G(k) is contained in a prescribed finite set which is independent of p and r. For a positive integer m, we define the set

m = {ν ∈ X(T )+ : ν, α0∨ < m}. Proposition. If L(ν0 + p r ν1 ) is a composition factor of G(k) then ν0 , ν1 ∈ 2(h−1) . Proof. We know already that ν1 ∈ 2(h−1) . Assume that [G(k) : L(ν0 + p r ν1 )]G = 0.

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Christopher P. Bendel, Daniel K. Nakano and Cornelius Pillen

Then [G(k) : L(ν0 )]Gr = 0. This implies that HomGr (Qr (ν0 ), G(k)) = 0. Set ν) [Jan1, II.11.9]. ν = (pr − 1)ρ + w0 ν0 . Then Qr (ν0 ) is a Gr -summand of Str ⊗L( One obtains ν), G(k)) = HomGr (Str , G(k) ⊗ L( ν)∗ ) = 0. HomGr (Str ⊗L( ν)∗ is not zero. Therefore The St r -isotypical component of the Gr -socle of G(k) ⊗ L( there exists a weight γ such that ν)∗ ) = HomG (Str ⊗L(γ )(r) ⊗ L( ν), G(k)) = 0. HomG (Str ⊗L(γ )(r) , G(k) ⊗ L( The G-module Str ⊗L(γ )(r) is simple with highest weight (p r − 1)ρ + p r γ . This weight has to be less than or equal to any highest weight µ of G(k) ⊗ L( ν)∗ . But any ∨ ∨ r such µ satisfies µ + ρ, α0 < (3p − 1)(h − 1). Hence, γ , α0 < 2(h − 1). Since G(k) is the truncation of indG Gσ (Fq ) (k) it follows that HomG (Str ⊗L(γ )(r) ⊗ L( ν), IndG Gσ (Fq ) (k)) = 0. and by adjointness ν), k) = 0. HomGσ (Fq ) (Str ⊗L(γ )(r) ⊗ L( As a Gσ (Fq )-module L(γ )(r) ∼ = L(σ (γ )) and one obtains ν), Str ) = 0. HomG(Fq ) (L(σ (γ )) ⊗ L( It follows from [Jan2, Satz 1.5] that ν), Str ) dim HomGσ (Fq ) (L(σ (γ )) ⊗ L( = [L(σ (γ )) ⊗ L( ν) : Str ]Gσ (Fq ) = [L(σ (γ )) ⊗ L( ν) ⊗ L(σ (µ)) : Str ⊗L(µ)(r) ]G . µ∈X(T )+

ν = −w0 ν0 + pr µ − σ (µ) for some Therefore σ (γ ) ≥ (p r − 1)ρ + pr µ − σ (µ) − µ. Since σ preserves the inner product one obtains 2(h − 1) > γ , α0∨ ≥ ν0 , α0∨ .

2.6. An immediate consequence of Theorem 2.3 and Proposition 2.5 is that any extension between simple modules for the finite group Gσ (Fq ) has to come from some extension between a finite set of modules for the algebraic group. For large primes this will be made more explicit later. Corollary. Let λ, µ ∈ Xr (T ). If Ext1Gσ (Fq ) (L(λ), L(µ)) = 0 then there exist weights

ν0 , ν1 ∈ 2(h−1) such that Ext1G (L(λ) ⊗ L(ν0 ), L(µ) ⊗ L(ν1 )(r) ) = 0.

Proof. By Theorem 2.3 we have Ext1Gσ (Fq ) (L(λ), L(µ)) ∼ = Ext 1G (L(λ), L(µ)⊗G(k)). Our discussion in 2.1 and Proposition 2.5 show that G(k) is finite-dimensional with

37

Extensions for finite groups of Lie type: twisted groups

simple composition factors of the form L(γ + p r ν1 ) where γ , ν1 ∈ 2(h−1) . If Ext 1Gσ (Fq ) (L(λ), L(µ)) = 0 then there exist γ , ν1 ∈ 2(h−1) such that Ext 1G (L(λ), L(µ)⊗L(γ )⊗L(ν1 )(r) ) ∼ = Ext 1G (L(λ)⊗L(−w0 γ ), L(µ)⊗L(ν1 )(r) ) = 0. Set ν0 = −w0 γ and the assertion follows. 2.7. Proposition 2.5 gives useful information only for q > 2(h − 1). In that case we are also able to describe the G-socle of G(k). Lemma. Let p r ≥ 2(h − 1) then socG G(k) =

L(σ (ν) + p r (−w0 ν)).

ν∈ 2(h−1)

Proof. The condition p r ≥ 2(h − 1) forces any weight in 2(h−1) to be p r -restricted. By Proposition 2.5 and right adjointness of G to the restriction functor one obtains HomG (L(ν0 + p r ν1 ), G(k)) ⊗ L(ν0 + p r ν1 ) socG G(k) = ν0 ,ν1 ∈ 2(h−1)

=

HomGσ (Fq ) (L(ν0 ) ⊗ L(ν1 )(r) , k) ⊗ L(ν0 + p r ν1 )

ν0 ,ν1 ∈ 2(h−1)

=

HomGσ (Fq ) (L(ν0 ) ⊗ L(σ (ν1 )), k) ⊗ L(ν0 + p r ν1 )

ν0 ,ν1 ∈ 2(h−1)

=

(by (1.4.1))

HomGσ (Fq ) (L(ν0 ), L(−w0 σ (ν1 ))) ⊗ L(ν0 + p r ν1 )

ν0 ,ν1 ∈ 2(h−1)

=

L(−w0 σ (ν) + p r ν)

ν∈ 2(h−1)

=

L(σ (ν) + p r (−w0 ν)).

ν∈ 2(h−1)

2.8. The nicest and most explicit extension results can be obtained when G(k) is semisimple. The semisimplicity of G(k) was shown for the split forms in [BNP1, 7.4]. We present a slightly different proof that involves only the algebraic group and works for split and quasi-split forms of G. Proposition. Let p ≥ 3(h − 1), then G(k) is semisimple. Moreover, G(k) = L(σ (ν) + p r (−w0 ν)). ν∈ 2(h−1)

Proof. For type A1 we refer to [BNP1, Thm. 7.4], so we may therefore assume that G is not of type A1 . Assume further that L(λ0 + pr λ1 ) and L(ν0 + pr ν1 ) are composition

38

Christopher P. Bendel, Daniel K. Nakano and Cornelius Pillen

factors of G(k). We look at the first three terms of the five-term exact sequence corresponding to the Lyndon–Hochschild–Serre spectral sequence [Jan1, I.6.6]: 0 → Ext1G/Gr (L(λ1 )(r) , HomGr (L(λ0 ), L(ν0 )) ⊗ L(ν1 )(r) ) → Ext 1G (L(λ0 + p r λ1 ), L(ν0 + p r ν1 )) → HomG/Gr (L(λ1 )(r) , Ext1Gr (L(λ0 ), L(ν0 )) ⊗ L(ν1 )(r) ) → ··· . By Proposition 2.5 we have λ0 , λ1 , ν0 , ν1 ∈ 2(h−1) . The size of p forces all these weights to be inside the lowest alcove. Clearly we have Hom G1 (L(λ0 ), L(ν0 )) ∼ = HomG (L(λ0 ), L(ν0 )) and by the Linkage Principle Ext 1G (L(λ1 ), L(ν1 )) = 0. Therefore, the first term of the sequence vanishes. Since λ0 and ν0 are inside the lowest alcove it follows from [BNP3, 5.4 Cor B] that Ext 1G1 (L(λ0 ), L(ν0 )) = 0. Thus the third term in the sequence also vanishes. We conclude that there are no extensions between simple composition factors of G(k). The module is semisimple and the assertion follows from Lemma 2.7. 2.9. The semisimplicity of G(k) for p ≥ 3(h − 1) gives us an extremely useful Ext1 formula for simple Gσ (Fq )-modules. For the split forms this was shown in [BNP2, 2.5]. Theorem. For p ≥ 3(h − 1) and λ, µ ∈ Xr (T ), Ext 1Gσ (Fq ) (L(λ), L(µ)) ∼ Ext1G (L(λ) ⊗ L(ν)(r) , L(µ) ⊗ L(σ (ν))). = ν∈ h

Proof. For λ, µ ∈ Xr (T ), we have Ext1Gσ (Fq ) (L(λ), L(µ)) ∼ = Ext1G (L(λ), L(µ) ⊗ G(k)) (by Thm. 2.3) ∼ Ext1G (L(λ), L(µ) ⊗ L(σ (ν) + p r (−w0 ν))) = ν∈ 2(h−1)

∼ =

ν∈ 2(h−1)

∼ =

ν∈ h

(by Prop. 2.8) Ext1G (L(λ) ⊗ L(ν)(r) , L(µ) ⊗ L(σ (ν)))

Ext1G (L(λ) ⊗ L(ν)(r) , L(µ) ⊗ L(σ (ν))) (by [BNP2, Prop. 2.4 c]).

Extensions for finite groups of Lie type: twisted groups

39

3. Extensions for r ≥ 2 The formula given in the previous theorem involves extensions between a simple pr bounded G-module and the tensor product of two simple p r -restricted G-modules, one of them having a small highest weight. It turns out that for r ≥ 2 the extension and the tensor product can be calculated separately. This allows for even more explicit formulas. Also, for r ≥ 3, if 3.1. For γ , δ ∈ X(T )+ , let S(γ , δ) = HomG (L(γ ), L(δ)). r−2 i−1 i λ where λ ∈ X (T ), set λ ¨ p = λi and for λ ∈ Xr (T ) and λ = r−1 i i 1 i=0 i=1 p r = 2 set λ¨ = 0. Theorem 3.1 is a generalization of [BNP3, Thm. 2.4b]. It follows from Theorem 2.9 by reidentifying the G-extensions as done in [BNP3] using [BNP2, Thm. 3.2] (equivalently [BNP3, Thm. 2.1]) and [BNP3, Prop. 2.3]. Theorem. Let p ≥ 3(h − 1), r ≥ 2, and λ, µ ∈ Xr (T ). Then Ext 1Gσ (Fq ) (L(λ), L(µ)) ∼ = Ext1G (L(λ), L(µ)) ⊕ R where R=

ν∈ h −{0}

Ext1G (L(λr−1 ) ⊗ L(ν)(1) , L(µr−1 )) ¨ ⊗ HomG (L(λ0 ), L(µ0 ) ⊗ L(σ (ν))) ⊗ S(λ¨ , µ).

or equivalently HomG (L(ν)(1) , Ext 1Gr (L(λr−1 ), L(µr−1 )) R= ν∈ h −{0}

¨ ⊗ HomG (L(λ0 ), L(µ0 ) ⊗ L(σ (ν))) ⊗ S(λ¨ , µ). 3.2. The next theorem, which is an analog of [BNP3, Thm. 3.1], identifies the extensions between a pair of simple Gσ (Fq )-modules with the G-extensions of a single λ pair of p r -restricted simple G-modules. Note that if n = r − 1, then the weights and µ defined below simply equal λ and µ respectively. Theorem. Let p ≥ 3(h − 1) and λ, µ ∈ Xr (T ). Set n = min({i | λi = µi } ∪ {r − 1}) and define p r -restricted weights λ=

r−n−2 i=0

p i σ (λn+1+i ) +

r−1 i=r−n−1

pi λi−r+n+1 ,

40

Christopher P. Bendel, Daniel K. Nakano and Cornelius Pillen

and

µ=

r−n−2 i=0

r−1

p i σ (µn+1+i ) +

p i µi−r+n+1 .

i=r−n−1

(a) If r ≥ 3, then λ), L( µ)). Ext1Gσ (Fq ) (L(λ), L(µ)) ∼ = Ext1G (L( (b) If r = 2 and λ0 = µ0 or λ1 = µ1 , then λ), L( µ)). Ext 1Gσ (Fq )) (L(λ), L(µ)) ∼ = Ext1G (L( (c) If r = 2 and λi = µi for i = 0, 1, set λˆ = σ (λ1 )+pλ0 and µˆ = σ (µ1 )+pµ0 . Then ˆ Ext 1Gσ (Fq )) (L(λ), L(µ)) ∼ = Ext1G (L(λ), L(µ)) ⊕ Ext1G (L(λˆ ), L(µ)). Proof. The Frobenius morphism is an automorphism on Gσ (Fq ). Hence, Ext 1Gσ (Fq ) (L(λ), L(µ)) ∼ = Ext1Gσ (Fq ) (L(λ)(r−1−n) , L(µ)(r−1−n) ), where n = min({i | λi = µi } ∪ {r − 1}). Steinberg’s tensor product theorem together with (1.4.1) implies that L( λ) ∼ µ) ∼ = L(λ)(r−1−n) and L( = L(µ)(r−1−n) as Gσ (Fq )modules. Therefore, we may identify λ), L( µ)). Ext 1Gσ (Fq ) (L(λ), L(µ)) ∼ = Ext1Gσ (Fq ) (L( (a) For r ≥ 3, Theorem 3.1 and [BNP3, Thm. 2.4a] imply that Ext1Gσ (Fq ) (L(λ), L(µ)) = Ext1G (L( λ), L( µ)) = 0 unless there exists an i with 0 ≤ i ≤ r − 1 and λi = µi . We assume such an i exists and apply Theorem 3.1 to the pair of weights µr−1 which forces Ext1G1 (L( λr−1 ), L( µr−1 )) = 0 by λ and µ. Observe that λr−1 = [And1]. The remainder term R in Theorem 3.1 vanishes and λ), L( µ)) ∼ λ), L( µ)). Ext 1Gσ (Fq ) (L(λ), L(µ)) ∼ = Ext1Gσ (Fq ) (L( = Ext1G (L( This proves part (a). λr−1 = µr−1 and one can (b) Assume r = 2. If either λ0 = µ0 or λ1 = µ1 , then argue as above. (c) Assume r = 2 and λi = µi for i = 0, 1. By Theorem 3.1, we have Ext1Gσ (Fq ) (L(λ), L(µ)) ∼ = Ext1G (L(λ), L(µ)) ⊕ R

Extensions for finite groups of Lie type: twisted groups

where R=

ν∈ h −{0}

41

Ext 1G (L(λ1 ) ⊗ L(ν)(1) , L(µ1 )) ⊗ HomG (L(λ0 ), L(µ0 ) ⊗ L(σ (ν))).

Since λ0 = µ0 , we have R= Ext1G (L(λ1 ) ⊗ L(ν)(1) , L(µ1 )) ⊗ HomG (L(λ0 ), L(µ0 ) ⊗ L(σ (ν))). ν∈ h

Since σ is an automorphism on G, we have Ext 1G (L(λ1 ) ⊗ L(ν)(1) , L(µ1 )) ∼ = Ext1G (L(σ (λ1 )) ⊗ L(σ (ν))(1) , L(σ (µ1 ))). We can re-write R as R= Ext1G (L(σ (λ1 )) ⊗ L(ν)(1) , L(σ (µ1 ))) ⊗ HomG (L(λ0 ), L(µ0 ) ⊗ L(ν)). ν∈ h

By [BNP3, Thm. 2.4a], the remainder term R is exactly Ext 1G (L(λˆ ), L(µ)). ˆ 3.3. Various conjectures have been made about the dimensions of Ext1 -groups. Here it is shown that in most cases the dimensions of the Ext 1 groups between simple modules for the finite groups are bounded by the dimensions of Ext1 groups for the reductive algebraic groups. The corollary below is a generalization of [BNP3, Thm. 3.3]. The proof follows along the same lines and details are left to the reader. Corollary. Let p ≥ 3(h − 1), r ≥ 2, and λ, µ ∈ Xr (T ). Then max{dimk Ext1Gσ (Fq ) (L(λ), L(µ)) | λ, µ ∈ Xr (T )} = max{dimk Ext1G (L(λ), L(µ)) | λ, µ ∈ Xr (T )}, unless, r = 2, σ = 1, and the underlying root system is of type Cn , n ≥ 1. In that case max{dimk Ext1Gσ (Fq ) (L(λ), L(µ)) | λ, µ ∈ Xr (T )} ≤ 2 max{dimk Ext1G (L(λ), L(µ)) | λ, µ ∈ Xr (T )}.

4. Self-extensions Let p ≥ 3(h − 1). In [BNP2] it was shown that self-extensions, i.e. extensions of a simple G(Fq )-module by itself, do not exist unless q = p and the root system of the underlying algebraic group is of type Cn . Since there are no quasi-split forms of type Cn it is not surprising that the result generalizes to the twisted case.

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4.1. Self-extensions, r ≥ 2. Theorem 3.2 gives a nice one-to-one correspondence between extensions of simple modules for the finite group Gσ (Fq ) and extensions of p r -restricted simple modules for the algebraic group. Since it is well-known that self-extensions do not exist for algebraic groups it is an immediate consequence of Theorem 3.2 that self-extensions also vanish for simple Gσ (Fq )-modules as long as q > p and p ≥ 3(h − 1). Theorem. For p ≥ 3(h − 1), r ≥ 2, and λ ∈ Xr (T ), Ext1Gσ (Fq ) (L(λ), L(λ)) = 0. 4.2. Self-extensions, r = 1. In the case p = q the analysis in [BNP2] was considerably harder since self-extensions were known to exist for groups of type Cn [Hum], [TZ, Rem. 3.18], [Pil]. It turns out, however, that the quasi-split forms do not add to this list. One obtains the following result. Theorem. Let p ≥ 3(h − 1) and λ ∈ X1 (T ). If either (a) G does not have underlying root system of type Cn (n ≥ 1) or

(b) λ, αn∨ = p−2−c 2 , where αn is the unique long simple root and c is odd with 0 < |c| ≤ h − 1, then Ext 1Gσ (Fp ) (L(λ), L(λ)) = 0. Proof. By Theorem 2.9 one has Ext1Gσ (Fp ) (L(λ), L(µ)) ∼ =

ν∈ h

Ext 1G (L(λ) ⊗ L(ν)(r) , L(µ) ⊗ L(σ (ν))),

where h = {ν ∈ X(T )+ |ν, α0∨ < h}. The only complication in the quasi-split case versus the split case is that the weights ν and σ (ν) may be distinct. However, Corollary 4.9 in [BNP2] says that Ext1G (L(λ) ⊗ L(ν1 )(r) , L(µ) ⊗ L(σ (ν2 ))) vanishes for any ν1 , ν2 ∈ h as long (a) or (b) holds.

5. First cohomology for simple modules In this section we consolidate our results on the first cohomology for simple groups that can be found in [BNP1] [BNP3] and extend them to the quasi split forms of G. Notice that Theorem 5.1 offers some improvements on the condition of the prime (compare to [BNP1, Thm. 7.4]) and the size of the weights ( h−1 versus h ). For technical reasons we exclude the well-known results for groups of type A1 [AJL]. The most striking result is a one-to-one correspondence between p r -bounded simple Gmodules with non-vanishing cohomology and simple Gσ (Fq )-modules with non-zero cohomology. The correspondence is simply the identity for pr -restricted G-modules, while non-restricted G-modules correspond to simple Gσ (Fq )-modules whose pr restricted highest weights are not linked to 0. Naturally, this gives also a one-to-one

Extensions for finite groups of Lie type: twisted groups

43

correspondence between the cohomology of simple modules for the quasi-split and the split forms over a given G. p we denote the extended affine Weyl group generated by the ordinary Weyl 5.1. By W group and the translation by elements in pX(T ). Two weights λ and ν are said to be G1 -linked if λ = w · ν + pγ for some w ∈ W and γ ∈ X(T ) or, equivalently, if there p such that λ = u · ν. In this section we make use of the exists an element u ∈ W strong linkage principle [Jan1, II.6.13] and replace the order relation “≤” on X(T ) by the stronger relation “↑” as defined in [Jan1, II.6.4]. Theorem. Let p ≥ 3(h−1), λ ∈ Xr (T ), and h−1 = {ν ∈ X(T )+ | ν, α0∨ < h−1}. Assume that G is not of type A1 . (a) If λ is not G1 -linked to any weight in h−1 , then H1 (Gσ (Fq ), L(λ)) = 0. (b) If λ is G1 -linked to the zero weight, then H1 (Gσ (Fq ), L(λ)) ∼ = H1 (G, L(λ)). (c) If r = 1 and λ is G1 -linked to some element in h−1 , then there exist unique p such that λ = u · ν and ν ∈ h−1 and u ∈ W H1 (Gσ (Fq ), L(λ)) ∼ = H1 (G, L(u · 0 + pσ −1 (ν))). In addition, if pσ −1 (ν) ↑ u · 0 then k if λ = pωα + sα · (−w0 σ (ωα ))where α ∈ 1 H (Gσ (Fq ), L(λ)) = and ωα is the corresponding fundamental weight, 0 else. (d) If r > 1, then H1 (Gσ (Fq ), L(λ)) = 0 unless there exist unique ν ∈ h−1 and p such that λ = pr−1 u · 0 + ν. In that case u∈W H1 (Gσ (Fq ), L(λ)) ∼ = H1 (G, L(u · 0 + pσ −1 (ν))). In addition, if pσ −1 (ν) ↑ u · 0 then r r−1 k if λ = p ωα − p α − w0 σ (ωα ) where α ∈ H1 (Gσ (Fq ), L(λ)) = and ωα is the corresponding fundamental weight, 0 else. Proof. Since p ≥ 3(h − 1) all weights ν ∈ h are in the interior of the lowest alcove. Consequently, from Jantzen’s translation principle [Jan1, II. 7.9], L(ν) ∼ = T0ν (k). One

44

Christopher P. Bendel, Daniel K. Nakano and Cornelius Pillen

can now argue as follows: H1 (G(Fq ), L(λ)) ∼ = Ext1G(Fq ) (k, L(λ)) ∼ Ext1G (L(ν)(r) , L(λ) ⊗ L(σ (ν))) (by Theorem 2.9) = ν∈ h

∼ =

ν∈ h

∼ =

ν∈ h

∼ =

ν∈ h

∼ =

ν∈ h

Ext1G (L(−w0 σ (ν)), L(λ) ⊗ L(−w0 ν)(r) ) Ext1G (L(ν), L(λ) ⊗ L(σ −1 (ν))(r) ) Ext1G (T0ν (k), L(λ + pr σ −1 (ν))) Ext1G (k, Tν0 L(λ + p r σ −1 (ν))) ([Jan1, II.7.6])

Assume that Tν0 L(λ+p r σ −1 (ν)) is not zero and denote by λν the unique pr -restricted weight such that L(λν + p r σ −1 (ν)) ∼ = Tν0 L(λ + p r σ −1 (ν)). We have Ext 1G (k, Tν0 L(λ + p r σ −1 (ν))) ∼ = Ext1G (k, L(λν + p r σ −1 (ν))) ∼ = Ext 1 (L(−w0 σ −1 (ν))(r) , L(λν )). G

Define Q via the exact sequence of G-modules 0 → L(λν ) → H 0 (λν ) → Q → 0 and obtain the exact sequence HomG (L(−w0 σ −1 (ν))(r) , Q) → Ext1G (L(−w0 σ −1 (ν))(r) , L(λν )) → Ext 1G (L(−w0 σ −1 (ν))(r) , H 0 (λν )). We conclude that Ext1G (k, Tν0 L(λ+p r σ −1 (ν))) = 0 unless either −w0 p r σ −1 (ν) ↑ λν by [Jan1, II.6.6] or λν = pr ωα − p i α and −w0 σ −1 (ν) = ωα , where α is a simple root and ωα the corresponding fundamental weight. The last assertion follows from [BNP3, Prop. 4.3 b]. In the first case we conclude that pr −w0 σ −1 (ν), α0∨ = pr ν, α0∨ ≤ (p r − 1)(h − 1). In either case ν, α0∨ < h − 1, unless G is of type A1 . Obviously the cohomology vanishes unless λ is G1 -linked to some ν ∈ h−1 . This implies part (a) of the Theorem. Next assume that λ is G1 -linked to two elements ν1 and ν2 in h−1 . Then there exist w ∈ W and γ ∈ X(T ) such that w · ν1 + pγ = ν2 . This implies for any simple root α that ν2 + ρ, α ∨ − w(ν1 + ρ), α ∨ = pγ , α ∨ . Since νi , α0∨ ≤ h − 2, the absolute value of the left hand side is at most h − 2 + 1 + 2h − 3 = 3(h − 1) − 1 < p. Thus γ = 0, which forces w = 1 and ν1 = ν2 . Consequently, if λ is G1 -linked to some p and ν ∈ h−1 such that λ = u · ν. element in h−1 , then there exist unique u ∈ W

Extensions for finite groups of Lie type: twisted groups

45

This implies that λν = u·0 and H1 (Gσ (Fq ), L(λ)) ∼ = Ext1G (k, L(u·0+p r σ −1 (ν))) ∼ = 1 r −1 H (G, L(u · 0 + p σ (ν))). Now part (b) and the first part of (c) follow immediately. Assume further that r = 1 and pσ −1 (ν) ↑ u · 0. From our earlier observations we know that the cohomology vanishes unless u · 0 = pωα − α = pωα + sα · 0 for some simple root α and ν = −w0 σ (ωα ). This forces λ = pωα + sα · (−w0 σ (ωα )) and H1 (Gσ (Fq ), L(λ)) ∼ = Ext1G (k, L(pωα −α+p(−w0 ωα )) ∼ = Ext1G (L(ωα )(1) , L(pωα − α)). The remainder of part (c) now is a consequence of [BNP3, Prop. 4.5]. Finally, we assume that r > 1 and that λ is G1 -linked to some non-zero ν ∈ p , and H1 (Gσ (Fq ), L(λ)) ∼

h−1 . Then ν is unique, λ = w · ν with unique w ∈ W = 1 r −1 H (G, L(w · 0 + p σ (ν))). From [BNP3, Thm. 2.4a] one obtains that H1 (G, L(w · p . In this case [BNP3, 0+p r σ −1 (ν))) = 0, unless w·0 = pr−1 u·0 for a unique u ∈ W 1 1 ∼ Thm. 2.4a] implies further that H (Gσ (Fq ), L(λ)) = H (G, L(w · 0 + pr σ −1 (ν))) ∼ = H1 (G, L(u · 0 + pσ −1 (ν))). Moreover, λ = ν + pr−1 u · 0. Now assume that pσ −1 (ν) ↑ u · 0. Again it follows from [BNP3, Prop. 4.5] that 1 H (G, L(u · 0 + pσ −1 (ν))) = k, if u · 0 = pωα − α for some simple root α and ν = −w0 σ (ωα ). This implies that λ = pr ωα − p r−1 α − w0 σ (ωα ). In all other cases the cohomology vanishes. The modules L(pωα + sα · (−w0 σ (ωα ))) and L(pr ωα − p r−1 α − w0 σ (ωα )) are of interest because for appropriate choices of α they are the smallest simple modules with H1 (Gσ (Fp ), L(λ)) = 0 and H1 (Gσ (Fq ), L(λ)) ∼ = H1 (G, L(λ)), respectively (see [Jan4, Prop. 2.2], [BNP3, 4.8]).

References [And1]

H. H. Andersen, Extensions of modules for algebraic groups, Amer. J. Math. 106 (1984), 498–504.

[And2]

H. H. Andersen, Extensions of simple modules for finite Chevalley groups, J. Algebra 111 (1987), 388–403.

[AJL]

H. H. Andersen, J. Jørgensen, P. Landrock, The projective indecomposable modules of SL(2, pn ), Proc. London Math. Soc. (3) 46 (1983), 38–52.

[BNP1] C. P. Bendel, D. K. Nakano, C. Pillen, On comparing the cohomology of algebraic groups, finite Chevalley groups, and Frobenius kernels, J. Pure and Appl. Algebra 163 (2001), 119–146. [BNP2] C. P. Bendel, D. K. Nakano, C. Pillen, Extensions for finite Chevalley groups I, Adv. Math. 183 (2004), 380–408. [BNP3] C. P. Bendel, D. K. Nakano, C. Pillen, Extensions for finite Chevalley groups II, Trans. Amer. Math. Soc. 354 (2002), 4421–4454. [BNP4] C. P. Bendel, D. K. Nakano, C. Pillen, Extensions for Frobenius kernels, J. Algebra 272 (2004), 476–511.

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[Car]

R. W. Carter, Finite Groups of Lie Type. Conjugacy Classes and Complex Characters, Wiley & Sons, Chichester 1993.

[Cha]

L. Chastkofsky, Characters of projective indecomposable modules for finite Chevalley groups, Proc. Symp. Pure Math. 37 (1980), 359–362.

[Cur]

C. W. Curtis, Representations of Lie algebras of classical type with applications to linear groups, J. Math. Mech. 9 (1960), 307–326.

[Don]

S. Donkin, On Schur algebras and related algebras I, J. Algebra 104 (1986), 310–328.

[Hum]

J. E. Humphreys, Non-zero Ext1 for Chevalley groups (via algebraic groups), J. London Math. Soc. 31 (1985) 463–467.

[Jan1]

J. C. Jantzen, Representations of Algebraic Groups, Academic Press, Orlando 1987.

[Jan2]

J. C. Jantzen, Darstellungen halbeinfacher Gruppen und ihrer Frobenius-Kerne, J. reine angew. Math 317 (1980), 157–199.

[Jan3]

J. C. Jantzen, Zur Reduktion modulo p der Charaktere von Deligne und Lusztig, J. Algebra 70 (1981), 452–474.

[Jan4]

J. C. Jantzen, First cohomology groups for classical Lie algebras, Progr. Math. 95, Birkhäuser Verlag 1991, 289–315.

[Pil]

C. Pillen, Self-extensions for the finite symplectic groups, preprint.

[Sin]

P. Sin, Extensions of simple modules for special algebraic groups, J. Algebra 170 (1994), 1011–1034.

[Smi]

S. D. Smith, Sheaf homology and complete reducibility, J. Algebra 95 (1985), 72–80.

[Ste1]

R. Steinberg, Endomorphisms of linear algebraic groups, Mem. Amer. Math. Soc. 80, Amer. Math. Soc., Providence 1968.

[Ste2]

R. Steinberg, Representations of algebraic groups, Nagoya Math. J. 22 (1963), 33–56.

[TZ]

P. H. Tiep, A. E. Zalesskiˇı, Mod p reducibility of unramified representations of finite groups of Lie type, Proc. London Math. Soc. 84 (2002), 439–472.

Christopher P. Bendel, Department of Mathematics, Statistics and Computer Science, University of Wisconsin-Stout, Menomonie, WI 54751, U.S.A. E-mail: [email protected] Daniel K. Nakano, Department of Mathematics, University of Georgia, Athens, GA 30602, U.S.A. E-mail: [email protected] Cornelius Pillen, Department of Mathematics and Statistics, University of South Alabama, Mobile, AL 36688, U.S.A. E-mail: [email protected]

On the classifying space and cohomology of Thompson’s sporadic simple group David J. Benson∗

Abstract. This paper examines the classifying space of Thompson’s sporadic simple group. Two theorems are proved. The first gives a homotopy colimit decomposition for the 2-completion of the classifying space. The second shows that the mod two cohomology contains a copy of the Dickson invariants of rank five, over which it is finitely generated as a module.

1. Introduction In this paper, we consider the classifying space and the cohomology of John Thompson’s sporadic simple group Th at the prime two. We prove the following theorem. Theorem 1.1. Let C and N be maximal 2-local subgroups of Th of shape 21+8 A9 and 25 L5 (2) respectively, intersecting in a group K of shape 21+8 A8 = 25 24 L4 (2). Then the obvious map of groups from the amalgamated free product C ∗K N to Th is a mod two cohomology isomorphism. The corresponding map on classifying spaces from the homotopy colimit (double mapping cylinder) of the diagram BK BN

/ BC

to BTh is a homotopy equivalence after 2-completion. The proof of this theorem is straightforward, and really just involves combining the appropriate results from the literature. One consequence of this theorem is that the Mayer–Vietoris sequence of the amalgamated free product reduces to a short exact sequence 0 → H ∗ (Th, F2 ) → H ∗ (C, F2 ) ⊕ H ∗ (N, F2 ) → H ∗ (K, F2 ) → 0. In other words, H ∗ (C, F2 ) and H ∗ (N, F2 ) are subrings of H ∗ (K, F2 ) whose linear span is the whole ring, and whose intersection is H ∗ (Th, F2 ). Note that K contains a Sylow 2-subgroup of Th. ∗ The author is partly supported by a grant from the NSF.

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David J. Benson

We need to work harder to prove the following theorem, which is really the main theorem of the paper. Theorem 1.2. There is a polynomial subring of H ∗ (Th, F2 ) of the form R = F2 [x16 , x24 , x28 , x30 , x31 ] with the following properties. (i) If E is any rank five elementary abelian 2-subgroup of Th (note that all maximal elementary abelian 2-subgroups in Th have rank five), then the restriction of R to H ∗ (E, F2 ) is injective, and the image is equal to the rank five Dickson invariants. (ii) H ∗ (Th, F2 ) is finitely generated as an R-module. This theorem continues a thread relating the Dickson invariants to sporadic groups, described in the papers [2], [5], [6], [8]. This began when Adem, Maginnis and Milgram computed the cohomology of the Mathieu group M12 and noticed that there was a copy of the rank three Dickson invariants, over which H ∗ (M12 , F2 ) was a finitely generated free module. There is a maximal 2-local subgroup N of M12 which is a nonsplit extension 23 L3 (2), and these Dickson invariants restrict to the invariants of N/O2 (N ) in the cohomology of O2 (N). Benson and Wilkerson [8] proved that this inclusion was induced by a map of classifying spaces BM12 → BG2 , and speculated on higher rank cases. In the paper [5], a homotopy colimit decomposition of the classifying space B Co3 was found. Using this, a map of classifying spaces B Co3 → BDI(4) was constructed, realizing a copy of the rank four Dickson invariants as a subring of H ∗ (Co3 , F2 ) over which it is finitely generated as a module. Here, DI(4) is the finite loop space at the prime two constructed by Dwyer and Wilkerson [15]. This map corresponds to the existence of a maximal 2-local subgroup which is a nonsplit extension 24 L4 (2). In the article [6], some observations were made concerning the relationship between BDI(4) and some nonexistent sporadic simple groups considered by Solomon [33]. These observations and some unpublished work of Puig were among the motivations leading to the notion of a p-local finite group; just enough local information is given to be able to build a p-completed classifying space, and the p-local finite group can be recovered from its p-completed classifying space. For more information, see Broto, Levi and Oliver [11]. In terms of this language, Levi and Oliver [19] made more precise the connections between BDI(4) and Solomon’s groups. We complete the picture by displaying a copy of the rank five Dickson invariants as a subring of H ∗ (Th, F2 ), corresponding to the maximal 2-local subgroup 25 L5 (2). It is a theorem of Lin and Williams [20] that there can be no space whose cohomology is the rank five Dickson invariants, so there is no obvious way to realize the inclusion of the Dickson invariants topologically. Furthermore, there is no nonsplit extension of the form 26 L6 (2), so there is no next case to consider, and the game stops here. It is also worth observing that if we allow the generators to be raised to the power of a high enough p-power, there is always a copy of the Dickson invariants over which the cohomology of a finite group with coefficients in Fp is finitely generated

On the classifying space and cohomology of Thompson’s sporadic simple group

49

as a module. See for example Theorem 2.1 of [8], Theorem 8.1.8 of Neusel [25], or Theorem 8.2 of [7]. The Landweber–Stong conjecture, proved by Bourguiba and Zarati [9] (see also Henn [17]), says in this situation that the depth of the cohomology ring can be computed by seeing how many of the Dickson invariants form a regular sequence, taken in ascending order of degree. A simpler proof of the Landweber– Stong conjecture, but restricted to the context of cohomology of finite groups, can be found in [7]. Since the triality group 3D4 (2) : 3 is a maximal subgroup of Th, and has the same 2-rank, the following is an easy corollary of Theorem 1.2. Corollary 1.3. Both H ∗ (3D4 (2), F2 ) and H ∗ (3D4 (2) : 3, F2 ) contain a copy of the rank five Dickson invariants R = F2 [x16 , x24 , x28 , x30 , x31 ] as a subring over which they are finitely generated as a module. We remark that this can also be proved in a different way. Namely, 3D4 (2) : 3 is a finite subgroup of the compact Lie group F4 . Since H ∗ (BF4 ; F2 ) is known from the work of Borel to be a polynomial ring F2 [y4 , y6 , y7 , y16 , y24 ] (see for example Vavpetiˇc and Viruel [35]), it is not hard to find a suitable copy of the Dickson invariants as a subring over which this ring is a finitely generated module, and then restrict. One can show that the restriction maps from H ∗ (BF4 ; F2 ) to H ∗ (3D4 (2), F2 ) and H ∗ (3D4 (2) : 3, F2 ) are injective, by restricting further to a rank five elementary abelian 2-subgroup. Information on Thompson’s simple group of order 215 .310 .53 .72 .13.19.31 can be found in Thompson [34], Markot [24], Reifart [29], Smith [31], Parrott [26], Lyons [22], Holt [18], Yoshiara [38] and Havas, Soicher and Wilson [16]. The character table can be found in the Atlas [12], and the maximal subgroups were found by Wilson [37] and Linton [21]. Some authors write E or F3 or F3|3 for this group. Throughout this paper, the word “space” means “simplicial set.” When we refer to the p-completion of a space X, we mean Bousfield–Kan Fp -completion, denoted R∞ X (where R = Fp ) in Bousfield and Kan [10]. We note that the classifying spaces we are considering are Fp -good, so that their Fp -completions are Fp -complete. Specifically, it is shown in Ch. VII §§3–5 of [10] that spaces with finite fundamental group and spaces with perfect fundamental group are Fp -good; most of our groups are finite, and C ∗K N is perfect. Background material on the Dickson invariants can be found in Wilkerson’s primer [36], or in §8.1 of [4].

2. Local structure and the classifying space In this section, we give two closely related proofs of Theorem 1.1, and prepare the ground for the cohomological computation in the next section. We begin by recalling from Wilson [37] some details of the 2-local structure of Thompson’s sporadic simple group Th. All the involutions (elements of order two) in Th are conjugate. If t is

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David J. Benson

an involution then the centralizer C = CTh (t) is a nonsplit extension 21+8 A9 where the normal subgroup P = O2 (C) is an extraspecial group 21+8 of “plus” type (i.e., of 2-rank five) and the quotient is an alternating group A9 . The 8-dimensional 2modular representation of C/P ∼ = A9 on P /Z(P ) is not the deleted permutation representation, but is one of the two representations which may be obtained from this one by applying a triality automorphism of + 8 (2). So in the permutation action of the quotient C/P ∼ = A9 on the maximal elementary abelian subgroups of P , there is one orbit of size nine, which may be put in one-one correspondence in a natural way with the nine points permuted by A9 . A maximal elementary abelian subgroup of P is said to be nice if it is in this orbit. Of the 135 maximal elementary abelian subgroups of P , nine are nice and 126 are not. The 126 which are not nice also form a single orbit under the action of A9 , with stabilizer (A5 × A4 ) : 2. The centralizer of a maximal elementary abelian 2-subgroup which is not nice has shape (24 × Q8 ). The normalizer modulo the centralizer is GL2 (4) extended by a field automorphism, namely L2 (4) = 24 (A5 × 3) : 2. Each of the 270 noncentral involutions in P in is in a unique nice subgroup of P . These involutions are all conjugate in C, and if we choose one then the four group V4 it generates with t has centralizer in Th of shape 22 [29 ]L3 (2). An involution in C/P ∼ = A9 of cycle type (22 15 ) does not lift to an involution in C, so the only other conjugacy class of four group in Th has a representative V4 generated by t and an involution in C whose image in A9 has cycle type (24 1). The centralizer of V4 has order 210 .3. If we choose a four group V4 in P which does not contain t and is not contained in a nice elementary abelian subgroup of P (all such four groups are conjugate in C), then it is conjugate in Th to V4 . The involutions in the centralizer in C of V4 all lie in P . From this, Wilson deduces the following. Lemma 2.1. Every elementary abelian 2-subgroup of Th is conjugate to a subgroup of P . Every elementary abelian subgroup of P is either contained in a nice elementary abelian subgroup of P , or it is contained in a unique conjugate in Th of P , and its normalizer is contained in C. Remark 2.2. Note that an elementary abelian 2-subgroup which is not nice can be contained in more than one conjugate of C. Let Q be a nice elementary abelian subgroup of P . Then N = NTh (Q) is a nonsplit extension 25 L5 (2) where the normal subgroup is the elementary abelian subgroup Q = O2 (N ) ∼ = (Z/2)5 and the quotient is the group L5 (2) of five by five invertible matrices over the field of two elements. The group N is called the Dempwolff group [13]. There are exactly two conjugacy classes of maximal 2-local subgroups of Th, and they are represented by C and N . Proof of Theorem 1.1. We give two proofs. The first proof goes as follows. Let A2 (Th) be the poset of nontrivial elementary abelian 2-subgroups of Th. Denote by P the subposet of A2 (Th) consisting of the conjugates of Z(C) = Z(P ) ∼ = Z/2 and

On the classifying space and cohomology of Thompson’s sporadic simple group

51

of O2 (N ) ∼ = (Z/2)5 . Ryba, Smith and Yoshiara showed that the reduced Lefschetz module of P is projective. Denote by F the kernel of the map C ∗K N → Th. By the Kurosh subgroup theorem, this is a free group. The exact sequence coming from the simplicial cochains on |P | is 0 → F2 → F2 [Th /C] ⊕ F2 [Th /N ] → F2 [Th /K] → H 1 (F, F2 ) → 0. This sequence is split exact, because C, N and K all contain a Sylow 2-subgroup of Th. This shows that the reduced Lefschetz module of P is isomorphic to H 1 (F, F2 ). It follows that H 1 (F, F2 ) is a projective F2 Th-module. Lemma 3.1 of [8] then proves the theorem. For the second proof, it is proved in Smith andYoshiara [32] that the inclusion of P into A2 (Th) is a homotopy equivalence. This also follows directly from Lemma 2.1. Since the inclusion is also a Th-equivariant map, the theorem then follows from Theorem 1.10 of Dwyer [14]. The above description of the 2-local structure also allows us to describe the cohomology variety VTh , namely the maximal ideal spectrum of H ∗ (Th, k), where k is an algebraically closed field of characteristic two. Namely, there are just two conjugacy classes of maximal elementary abelian 2-subgroups in Th. They both have rank five. One is the nice subgroups, represented by O2 (N ), and the other is represented by a rank five elementary abelian subgroup of P which is not nice. Using the work of Quillen [27], [28] we see that VTh has two irreducible components, both of dimension five. Their intersection has dimension four.

3. The cohomology Next, we describe in a little more detail the structure of the subgroup C of Th of shape 21+8 A9 , and produce an element w of degree 16 in H ∗ (BC, F2 ) with nontrivial restriction to Z(C) = t . We shall show that for a suitable choice of such an element w, x16 = Tr C,Th (w) ∈ H 16 (Th, F2 ) plays the role of the lowest Dickson invariant. The remaining Dickson invariants are obtained from this one by applying Steenrod operations. We start by describing a different but closely related group manufactured from the extraspecial group P . Since P is an extraspecial 2-group of order 21+8 and 2-rank five, it may be generated by involutions a1 , . . . , a4 , b1 , . . . , b4 which commute with the exception of the following nontrivial commutators: [a1 , b1 ] = [a2 , b2 ] = [a3 , b3 ] = [a4 , b4 ] = t, where t is the central involution. The ordinary irreducible representations of P can all be written over the reals. They consist of 28 one dimensional representations and

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one irreducible representation on a real vector space V of dimension 16 (note that |P | = 28 + 162 ). The representation V may be described as an induced representation RP ⊗RQ V where V is the one dimensional representation of the subgroup Q = t, a1 , a2 , a3 , a4 in which t → (−1) and ai → (1) for 1 ≤ i ≤ 4. In terms of matrices, V is a representation in which t is a diagonal matrix of −1’s, a1 , a2 , a3 and a4 act as diagonal matrices with eight 1’s and eight −1’s, and b1 , b2 , b3 and b4 act as permutation matrices. + The automorphism group of P has shape 28 SO+ 8 (2), where SO8 (2) is the corresponding special orthogonal group (which has a simple subgroup of index 2 denoted in the Atlas [12] by O8+ (2) and elsewhere by + 8 (2)). Given any automorphism α of P , the conjugate of V by α is isomorphic to V . So there is a 16 by 16 matrix Aα , conjugation by which has the same effect as the action of α. In other words, if ρ : P → Aut(V ) is the representation of P on V , then for all g ∈ P we have ρ(α(g)) = Aα ρ(g)A−1 α . The matrix Aα is uniquely determined up to multiplication by scalars (by Schur’s lemma), and if we demand that Aα have determinant one then it is unique up to sign. It follows that the group formed by taking both possibilities of Aα for each automorphism α is a double cover of Aut(P ) of shape 21+8 SO+ 8 (2). There are three conjugacy classes of subgroups of + (2) isomorphic to A9 , per8 muted transitively by the triality automorphism. For one of them the restriction of the natural eight dimensional representation is the deleted permutation representation. The other two are conjugate in SO+ 8 (2). Choosing a representative of one of the latter two conjugacy classes, we obtain a group C0 of shape 21+8 A9 . This is still not isomorphic to the subgroup C of Th, but they are isomorphic modulo centers. To obtain C from C0 , we form the Baer sum of this central extension with the central extension pulled back from the nontrivial double cover of A9 . In other words, take C0 × C1 , where C1 is the double cover of A9 , pass to the subgroup consisting of pairs with the same image in A9 , and then quotient out the diagonal central element of order two. The smallest faithful complex representation of C is 128 dimensional, and is obtained by tensoring V with an irreducible faithful eight dimensional real representation of C1 . Let w16 ∈ H 16 (C0 , F2 ) be the degree 16 Stiefel–Whitney class of the real representation V . Since the restriction of V to Z(C0 ) is equal to a direct sum of 16 copies of the sign representation, the formula for the Stiefel–Whitney classes of a direct sum shows that w16 restricts to z016 ∈ H ∗ (BZ(C0 ), F2 ) = F2 [z0 ]. It follows that in the spectral sequence H ∗ (C0 /Z(C0 ), H ∗ (Z(C0 ), F2 )) ⇒ H ∗ (C0 , F2 ) the element z016 on the fiber is a universal cycle. The element x = d2 (z0 ) ∈ H 2 (C0 /Z(C0 ), F2 ) classifies the central extension by Z(C0 ), and Serre’s transgression theorem (§II.9c of [30]) shows that Sq8 Sq4 Sq2 Sq1 x = 0.

On the classifying space and cohomology of Thompson’s sporadic simple group

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Next, we examine C1 ∼ = 2A9 . The additive structure and a large part of the multiplicative structure of the cohomology of this group was calculated by Adem, Karagueuzian, Milgram and Umland [1]. We do not need to make use of this description, because all we need is a single Stiefel–Whitney class. The deleted permutation representation of A9 of dimension eight displays it as a subgroup of SO(8). Let 1 → γ → Spin(8) → SO(8) → 1 be the double cover map, where γ is a central element of order two. The inverse image of A9 in Spin(8) is isomorphic to C1 , so we regard C1 as a subgroup of Spin(8) in this way. Now if we let be one of the two spin representations of Spin(8) of dimension 8, then the Stiefel–Whitney class w8 () is an element of H 8 (B Spin(8); F2 ) whose restriction to H ∗ ( γ , F2 ) is the eighth power of the degree one generator. It follows that the restriction u = resSpin(8),C1 (w8 ()) ∈ H 8 (C1 , F2 ) is an element whose restriction to Z(C1 ) is nontrivial. So in the spectral sequence H ∗ (C1 /Z(C1 ), H ∗ (Z(C1 ), F2 )) ⇒ H ∗ (C1 , F2 ), the element z18 ∈ H ∗ (Z(C1 ), F2 ) = F2 [z1 ] on the fiber is a universal cycle. The element y = d2 (z1 ) ∈ H 2 (C1 /Z(C1 ), F2 ) classifies the central extension by Z(C1 ). Again using Serre’s transgression theorem [30], we have Sq4 Sq2 Sq1 y = 0, Now C/Z(C) ∼ = C1 /Z(C1 ) ∼ = A9 . = C0 /Z(C0 ), of shape 28 A9 , and C/O2 (C) ∼ So there is an inflation map inf : H ∗ (C1 /Z(C1 ), F2 ) → H ∗ (C/Z(C), F2 ), and the element x + inf(y) ∈ H 2 (C/Z(C), F2 ) classifies the central extension of C/Z(C) by Z(C). This is because the Baer sum of central extensions of a group corresponds to the sum of the degree two cohomology classes. See the exercises to Section IV.4 of Mac Lane [23], for example. Since the Steenrod operations are F2 -linear, and commute with the inflation map, we have Sq8 Sq4 Sq2 Sq1 (x + inf(y)) = Sq8 Sq4 Sq2 Sq1 (x) + inf(Sq8 Sq4 Sq2 Sq1 (y)) = 0 in H ∗ (C/Z(C), F2 ). Again using Serre’s transgression theorem [30], it follows that z16 is a universal cycle in the spectral sequence H ∗ (C/Z(C), H ∗ (Z(C), F2 )) ⇒ H ∗ (C, F2 ). From this, we deduce that there is an element w ∈ H 16 (C, F2 ) with the property that the restriction to Z(C) is z16 .

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Lemma 3.1. With this definition of w, the transfer from C to Th of w is an element x16 ∈ H ∗ (Th, F2 ) with the property that the restriction to Z(C) of x16 is equal to z16 . Proof. To compute the restriction of the transfer, we use the Mackey double coset formula. Double cosets where the intersection of C with the conjugate of Z(C) is trivial do not contribute to the sum. The fact that Z(C) is equal to the center of a Sylow 2-subgroup of C implies that the remaining double cosets apart from the one corresponding to the identity element correspond to conjugacy classes of involutions in C of even size, which therefore contribute zero in characteristic two. Lemma 3.2. The restriction resTh,O2 (N) (x16 ) is equal to the degree 16 Dickson invariant in H ∗ (O2 (N), F2 )N/O2 (N) . Proof. The restriction must be nonzero by Lemma 3.1, since Z(C) ≤ O2 (N ). It must also be invariant under the action of N/O2 (N). There is only one nonzero invariant in this degree, namely the degree 16 Dickson invariant. We can now create all the Dickson invariants, using the action of the Steenrod algebra. Namely, we set x24 = Sq8 (x16 ), x28 = Sq4 (x24 ), x30 = Sq2 (x28 ) and x31 = Sq1 (x30 ). Since the Steenrod operations commute with the restriction map, the formulae given in Wilkerson [36] show that the restrictions of x16 , x24 , x28 , x30 and x31 are the generators for the rank five Dickson invariants in H ∗ (O2 (N ), F2 ). In particular, these elements are algebraically independent. Lemma 3.3. Let E be an elementary abelian subgroup of P of rank five which is not nice. Then H 16 (E, F2 )NTh (E) is two dimensional, and is spanned by the degree 16 Dickson invariant and the element u formed by taking the product of the 16 elements of H ∗ (E, F2 ) whose restrictions to Z(C) are nonzero. Proof. Set H = NTh (E)/CTh (E) = 24 (A5 × 3)2. We work our way up a normal series for H . The invariants of O2 (H ) is a polynomial ring on four generators of degree one and a generator of degree 16. The invariants of A5 on the polynomial ring generated by the four generators of degree one with the given action of A5 (namely, a two dimensional representation over F4 with the scalars restricted) were computed by Adem and Milgram [3]. From Theorem 3.2 of that paper, it is not hard to read off the degree 16 invariants, together with the action of the extra elements of order three and two. There are five dimensions of A5 -invariants, of which only one dimension is (A5 × 3)2-invariant, namely the square of the degree eight generator. The remaining four dimensions are given by multiplying elements of degrees 5, 8 and 3. The fixed space of this four dimensional space is zero, even under the extra element of order three. This shows that the degree 16 invariants of 24 (A5 × 3)2 on H ∗ (O2 (N ), F2 ) are two dimensional. Since the degree 16 Dickson invariant and u have different restrictions to subgroups of order two in N other than Z(C), it follows that they span this two dimensional space.

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Lemma 3.4. Let E be any elementary abelian 2-subgroup of Th of rank five. Then the restrictions of x16 , x24 , x28 , x30 and x31 to E are equal to the Dickson invariants. Proof. If E is nice, this follows from Lemma 3.2. So suppose that E is not nice. By Lemma 2.1, we can assume that E is contained in P . The restriction of x16 is certainly invariant under NTh (E), and so by Lemma 3.3, it is in the two dimensional space spanned by the degree 16 Dickson invariant and u. The element x16 has nonzero restriction to every involution in Th. The element u restricts to zero on a cyclic subgroup of E other than Z(C), and u plus the Dickson invariant restricts to zero on Z(C). So the only possibility is that the restriction of x16 is equal to the degree 16 Dickson invariant. Applying the Steenrod operations shows that x24 , x28 , x30 and x31 restrict to the remaining Dickson invariants. Proposition 3.5. The cohomology ring H ∗ (Th, F2 ) is finitely generated as a module over the subring F2 [x16 , x24 , x28 , x30 , x31 ]. Proof. It follows from Quillen’s F-isomorphism theorem [27, 28] that it suffices to check that the cohomology of every elementary abelian 2-subgroup of Th is finitely generated over the restriction of this polynomial subring. There are two conjugacy classes of maximal elementary abelian 2-subgroups, both of rank five, and these are dealt with in Lemma 3.4. Remark 3.6. It would be very interesting to know whether H ∗ (Th, F2 ) is a free module over F2 [x16 , x24 , x28 , x30 , x31 ]; in other words, whether H ∗ (Th, F2 ) is a Cohen– Macaulay ring. We have no information on this question.

References [1]

A. Adem, D. Karagueuzian, R. J. Milgram, and K. Umland, The cohomology of the Lyons group and the double covers of the alternating groups, J. Algebra 208 (1998), 452–479.

[2]

A. Adem, J. Maginnis, and R. J. Milgram, The geometry and cohomology of the Mathieu group M12 , J. Algebra 139 (1991), 90–133.

[3]

A. Adem and R. J. Milgram, A5 -invariants, the cohomology of L3 (4) and related extensions, Proc. London Math. Soc. (3) 66 (1993), 187–224.

[4]

D. J. Benson, Polynomial invariants of finite groups, London Math. Soc. Lecture Note Series 190, Cambridge University Press, Cambridge 1993.

[5]

, Conway’s group Co3 and the Dickson invariants, Manuscripta Math. 85 (1994), 177–193.

[6]

, Cohomology of sporadic groups, finite loop spaces, and the Dickson invariants, inGeometry and Cohomology in Group Theory (Durham, 1994), ed. by P. H. Kropholler, G. A. Niblo, and R. Stöhr, London Math. Soc. Lecture Note Series 252, Cambridge Univ. Press, Cambridge 1998, 10–23.

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[7]

, Dickson invariants, regularity and computation in group cohomology, Illinois J. Math. 48 (2004), 171–197.

[8]

D. J. Benson and C. W. Wilkerson, Finite simple groups and Dickson invariants, in Homotopy Theory and Its Applications (Cocoyoc, Mexico, 1993), Contemp. Math. 188, Amer. Math. Soc., Providence, RI 1995, 39–50.

[9]

D. Bourguiba and S. Zarati, Depth and the Steenrod algebra, with an appendix by J. Lannes, Invent. Math. 128 (1997), 589–602.

[10] A. K. Bousfield and D. M. Kan, Homotopy limits, completions and localizations, Lecture Notes in Math. 304, Springer-Verlag, Berlin, New York 1972. [11] C. Broto, R. Levi and R. Oliver, The homotopy theory of fusion systems, J. Amer. Math. Soc. 16 (2003), 779–856. [12] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson, Atlas of Finite Simple Groups, Oxford University Press, Oxford 1985. [13] U. Dempwolff, On extensions of an elementary abelian group of order 25 by GL(5, 2), Rend. Sem. Mat. Univ. Padova 48 (1973), 359–364. [14] W. G. Dwyer, Homology approximations for classifying spaces of finite groups, Topology 36 (1997), 783–804. [15] W. G. Dwyer and C. W. Wilkerson, A new finite loop space at the prime two, J. Amer. Math. Soc. 6 (1993), 37–64. [16] G. Havas, L. H. Soicher, and R. A. Wilson, A presentation for the Thompson sporadic simple group, in Groups and computation III (Columbus, OH, 1999), Ohio State Univ. Math. Res. Inst. Publ. 8, de Gruyter, Berlin 2001, 193–200. [17] H.-W. Henn, A variant of the proof of the Landweber–Stong conjecture, in Group Representations: Cohomology, Group Actions and Topology (Seattle, WA, 1996), Proc. Symp. Pure Math. 63, Amer. Math. Soc., Providence, RI 1998, 271–275. [18] D. F. Holt, The triviality of the multiplier of Thompson’s group F3 , J. Algebra 94 (1985), 317–323. [19] R. Levi and R. Oliver, Construction of 2-local finite groups of a type studied by Solomon and Benson, Geom. Topol. 6 (2002), 917–990. [20] J. P. Lin and F. Williams, On 14-connected finite H -spaces, Israel J. Math. 66 (1989), 274–288. [21] S. A. Linton, The maximal subgroups of the Thompson group, J. London Math. Soc. 39 (1989), 79–88; Correction, J. London Math. Soc. 43 (1991), 253–254. [22] R. Lyons, The Schur multiplier of F3 is trivial, Comm. Algebra 12 (1984), 1887–1898. [23] S. Mac Lane, Homology, Grundlehren Math. Wiss. 114, Springer-Verlag, Berlin, New York 1963. [24] R. Markot, A 2-local characterization of the simple group E, J. Algebra 40 (1976), 585– 595. [25] M. Neusel, Inverse invariant theory and Steenrod operations, Mem. Amer. Math. Soc. 692, Amer. Math. Soc., Providence, RI 2000.

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[26] D. Parrott, On Thompson’s simple group, J. Algebra 46 (1977), 389–404. [27] D. G. Quillen, The spectrum of an equivariant cohomology ring, I, Ann. of Math. 94 (1971), 549–572. [28]

, The spectrum of an equivariant cohomology ring, II, Ann. of Math. 94 (1971), 573–602.

[29] A. Reifart, A characterization of Thompson’s sporadic simple group, J. Algebra 38 (1976), 192–200. [30] J.-P. Serre, Homologie singulière des espaces fibrés, Ann. of Math. 54 (1951), 425–505. [31] P. E. Smith, A simple subgroup of M? and E8 (3), Bull. London Math. Soc. 8 (1976), 161–165. [32] S. D. Smith and S. Yoshiara, Some homotopy equivalences for sporadic geometries, J. Algebra 192 (1997), 326–379. [33] R. Solomon, Finite groups with Sylow 2-subgroups of type .3, J. Algebra 28 (1974), 182–198. [34] J. G. Thompson, A simple subgroup of E8 (3), Finite Groups, ed. by N. Iwahori, Japan Soc. for the Promotion of Science, 1976, 113–116. [35] A. Vavpetiˇc andA. Viruel, On the homotopy type of the classifying space of the exceptional Lie group F4 , Manuscripta Math. 107 (2002), 521–540. [36] C. W. Wilkerson, A primer on Dickson invariants, in Proceedings of the Northwestern Homotopy Theory Conference, Contemp. Math. 19, American Math. Society, Providence,RI 1983, 421–434. [37] R. A. Wilson, Some subgroups of the Thompson group, J. Aust. Math. Soc. Ser. A 44 (1988), 17–32. [38] S. Yoshiara, The radical 2-subgroups of the sporadic simple groups J4 , Co2 and Th, J. Algebra 233 (2000), 309–341. Dave Benson, Department of Mathematics, University of Georgia, Athens GA 30602, U.S.A. E-mail: [email protected]

New versions of Schur–Weyl duality Stephen Doty

Abstract. After reviewing classical Schur–Weyl duality, we present some other contexts which enjoy similar features, relating to Brauer algebras and classical groups. 2000 Mathematics Subject Classification: 20

1. Classical Schur–Weyl duality 1.1. Schur’s double-centralizer result. Consider the vector space V = Cn . The symmetric group Sr acts naturally on its r-fold tensor power V ⊗r , by permuting the tensor positions. This action obviously commutes with the natural action of GLn = GLn (C), acting by matrix multiplication in each tensor position. So we have a C GLn CSr bimodule structure on V ⊗r . (Here CG denotes the group algebra of a group G.) In 1927, Schur [34] proved that the image of each group algebra under its representation equals the full centralizer algebra for the other action. More precisely, if we name the representations as follows ρ

σ

C GLn −−−−→ End(V ⊗r ) ←−−−− CSr

(1)

then we have equalities ρ(C GLn ) = EndSr (V ⊗r )

(2)

σ (CSr ) = EndGLn (V

(3)

⊗r

).

(Here, for a given set S operating on a vector space T through linear endomorphisms, EndS (T ) denotes the set of linear endomorphisms of T commuting with each endomorphism coming from S.) Results of Carter–Lusztig [7], de Concini and Procesi [8], and J. A. Green [19] (and others) show that all the above statements remain true if one replaces C by an arbitrary infinite field K. 1.2. Schur algebras. The finite-dimensional algebra in (2) above, for any K, is known as the Schur algebra, and often denoted by SK (n, r) or simply S(n, r). The Schur algebra “sees” the part of the rational representation theory of the algebraic

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group GLn (K) occurring (in some appropriate sense) in V ⊗r . More precisely, there is an equivalence between r-homogeneous polynomial representations of GLn (K) and SK (n, r)-modules. In characteristic 0, those representations (as r varies) determine all finite-dimensional rational representations, while in positive characteristic they still provide a tremendous amount of information. The representation σ in (1) is faithful if n r, so σ induces an isomorphism KSr EndGLn (V ⊗r ) = EndSK (n,r) (V ⊗r )

(n r).

(4)

This leads to intimate connections between polynomial representations of GLn (K) and representations of KSr , a theme that has been exploited by many authors in recent years. Perhaps the most dramatic example of this is the result of Erdmann [18] (building on previous work of Donkin [13] and Ringel [33]) which shows that knowing decomposition numbers for all symmetric groups in positive characteristic will determine the decomposition numbers for general linear groups in the same characteristic. Conversely, James [24] had already shown that the decomposition matrix for a symmetric group is a submatrix of the decomposition matrix for an appropriate Schur algebra. Thus the (still open) general problem of determining the modular characters of symmetric groups is equivalent to the similar problem for general linear groups (over infinite fields). 1.3. The enveloping algebra approach. Return to the basic setup, over C. One may differentiate the action of the Lie group GLn (C) to obtain an action of its Lie algebra gln . Replacing the representation ρ in (1) by its derivative representation dρ : U(gln ) → End(V ⊗r ) leads to the following alternative statement of Schur’s result: dρ(U(gln )) = EndSr (V ⊗r )

σ (CSr ) = Endgln (V ⊗r ).

(5) (6)

In particular, the Schur algebra (over C) is a homomorphic image of U(gln ). All of this works over an arbitrary integral domain K if we replace U(gln ) by its “hyperalgebra” UK := K ⊗Z UZ obtained by change of ring from a suitable Z-form of U(gln ); see [11]. (One can adapt the Kostant Z-form, originally defined for the enveloping algebra of a semisimple Lie algebra, to the reductive gln .) 1.4. The quantum case. Jimbo [25] extended the results of 1.3 to the quantum case (where the quantum parameter is not a root of unity). One needs to replace Sr by the Iwahori–Hecke algebra H(Sr ) and replace U(gln ) by the quantized enveloping algebra U(gln ). The analogue of the Schur algebra in this context is known as the q-Schur algebra, often denoted by S(n, r) or Sq (n, r). Dipper and James [9] have shown that q-Schur algebras are fundamental for the modular representation theory of finite general linear groups.

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61

As many authors have observed, the picture in 1.1 can also be quantized. For that one needs a suitable quantization of the coordinate algebra of the algebraic group GLn . There is a completely different (geometric) construction of q-Schur algebras given in [1]. 1.5. Integral forms. The Schur algebras SC (n, r) admit an integral form SZ (n, r) such that SK (n, r) K ⊗Z SZ (n, r) for any field K. In fact SZ (n, r) is simply the image of UZ (see 1.3) under the surjective homomorphism U(gln ) → SC (n, r). Similarly, the quantum Schur algebra SQ(v) (n, r) admits an integral form defining all specializations via base change. One needs to replace Z by A = Z[v, v −1 ]; then the integral form SA (n, r) is the image of the Lusztig A-form UA under the surjection U(gln ) → SQ(v) (n, r). (To match this up with various specializations in the literature, one often has to take q = v 2 .) 1.6. Generators and relations. Recently, in joint work with Giaquinto (see [16]), a very simple set of elements generating the kernel of the surjection U(gln ) → SC (n, r) was found. A very similar set of elements generates the kernel of the surjection U(gln ) → SQ(v) (n, r). These elements are expressible entirely in terms of the Chevalley generators for the zero part of U(gln ) or U(gln ). Thus we obtain a presentation of SC (n, r) and SQ(v) (n, r) by generators and relations, compatible with the usual Serre (Drinfeld–Jimbo) presentation of U(gln ) (resp., U(gln )). As a result, we find a certain subset of the integral PBW-basis for U(gln ) or U(gln ) the image of which gives an integral basis for SZ (n, r) or SA (n, r). This basis yields a similar basis in any specialization. Moreover, a subset of it provides a new integral basis of H(Sn ).

2. The Brauer algebra From now on I will assume, unless stated otherwise, that the underlying field is C (it could just as well be any field of characteristic zero). One expects that many statements will be valid over an arbitrary infinite field, via some appropriate integral form, similar to what happens in type A. 2.1. The algebra Br(x) . Let R be a commutative ring, and consider the free R[x](x) module Br with basis consisting of all r-diagrams. An r-diagram is an (undirected) graph on 2r vertices and r edges such that each vertex is incident to precisely one edge. One usually thinks of the vertices as arranged in two rows of r each, the top and bottom rows. (See Figure 1.) Edges connecting two vertices in the same row (different rows) are called horizontal (resp., vertical). We can compose two such diagrams D1 , D2 by identifying the bottom row of vertices in the first diagram with the top row of vertices in the second diagram. The result is a graph with a certain number, δ(D1 , D2 ), of

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Stephen Doty

interior loops. After removing the interior loops and the identified vertices, retaining the edges and remaining vertices, we obtain a new r-diagram D1 ◦ D2 , the composite diagram. Multiplication of r-diagrams is defined by the rule D1 · D2 = x δ(D1 ,D2 ) (D1 ◦ D2 ). (x)

One can check that this multiplication makes Br into an associative algebra; this is the Brauer algebra. (See Figures 1–3 for an illustration of the multiplication in the Brauer algebra.) Note that if we take x = 1 then the set of r-diagrams is a monoid under diagram (1) composition, and Br is simply the semigroup algebra of that monoid.

Figure 1. Two Brauer diagrams D1 , D2 for r = 5.

Figure 2. Computing the composite of D1 and D2 .

Figure 3. The composite diagram D1 ◦ D2 .

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New versions of Schur–Weyl duality

For any x the group algebra R[x]Sr may be identified with the subalgebra of (x) Br spanned by the diagrams containing only vertical edges. Such Brauer diagrams provide a graphical depiction of permutations. The group algebra R[x]Sr of Sr (x) also appears as a quotient of Br , the quotient by the two-sided ideal spanned by all diagrams containing at least one horizontal edge. Label the vertices in each row of an r-diagram by the indices 1, . . . , r. For any 1 i = j r let ci,j be the r-diagram with horizontal edges connecting vertices i, j on the top and bottom rows. All other vertices in the diagram ci,j are vertical, connecting vertex k on the top and bottom rows, for all k = i, j . Brauer observed that (x) Br is generated by the permutation diagrams together with just one of the ci,j . (x)

2.2. Schur–Weyl duality. Brauer [5] introduced the algebra Br in 1936 to describe the invariants of symplectic and orthogonal groups acting on V ⊗r . (Brauer’s conventions were slightly different; we are here following the approach of Hanlon and Wales (−n) is isomorphic with the algebra defined by Brauer [23], who pointed out that Br to deal with the symplectic case.) Let G be Spn or On , where n is even in the first instance. By restricting the action ρ considered in 1.1 we have an action of G on V ⊗r . (n) One can extend the action of Sr to an action of Br (over C) on V ⊗r , where = −1 if G = Spn and = 1 if G = On . To do this, it is enough to specify the action of the diagram ci,j . This acts on V ⊗r as one of Weyl’s contraction maps contracting in tensor positions i and j . So we have (commuting) representations ρ

σ

(n)

CG −−−−→ End(V ⊗r ) ←−−−− Br

(7)

which satisfy Schur–Weyl duality; i.e., the image of each representation equals the full centralizer algebra of the other action: ρ(CG) = EndB (n) (V ⊗r ) r

σ (Br(n) ) = EndG (V ⊗r ).

(8) (9)

The algebras in equality (8) are the symplectic and orthogonal Schur algebras (see [12], [14], [15]). If n r−1 the representation σ in (7) is faithful [6]; thus it induces an isomorphism (n) Br EndG (V ⊗r ). 2.3. Schur–Weyl duality in type D. In type Dn/2 (n even) the orthogonal group On is not connected, and contains the connected semisimple group SOn (special orthogonal group) as subgroup of index 2. In order to handle this situation, Brauer (see also [20]) (n) defined a larger algebra Dr , spanned by the usual r-diagrams previously defined, together with certain partial r-diagrams on 2r vertices and r − n edges, in which n vertices in each of the top and bottom rows are not incident to any edge, and showed (n) (n) that the action of Br can be extended to an action of this larger algebra Dr on V ⊗r .

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Thus we have representations ρ

σ

(n)

C SOn −−−−→ End(V ⊗r ) ←−−−− Dr (n)

Brauer showed that the actions of SOn and Dr

.

(10)

on V ⊗r satisfy Schur–Weyl duality:

ρ(C SOn ) = EndD (n) (V ⊗r )

(11)

σ (Dr(n) ) = EndSOn (V ⊗r ).

(12)

r

The algebra in (11) is a second Schur algebra in type D, a proper subalgebra of the algebra EndB (n) (V ⊗r ) appearing in (8) above. r

2.4. Generators and relations. One can formulate the above statements of Schur– Weyl duality using enveloping algebras, analogous to 1.3. This leads to a presentation (see [17]) of the symplectic and orthogonal Schur algebras which is compatible with (a slight modification of) the usual Serre presentation of the enveloping algebra U(g), where g = spn (n even) or son . 2.5. The quantum case. There is a q-version of the Schur–Weyl duality considered in this section, although not as developed as in type A. One needs to replace the Brauer algebra by its q-analogue, the Birman–Murakami–Wenzl (BMW) algebra (see [4], [31]), and replace the enveloping algebra by a suitable quantized enveloping algebra. One can think of the BMW algebra in terms of Kauffman’s tangle monoid; see [26], [22], [30]. (Roughly speaking, tangles are replacements for Brauer diagrams, in which one keeps track of under and over crossings, subject to certain natural relations.) There are applications of the BMW algebra to knot theory, as one might imagine. This leads to a q-analogue of the symplectic Schur algebras, in particular, which have been studied by Oehms [32]. (n) To the best of my knowledge, a q-analogue of the larger algebra Dr (n even) considered in 2.3 remains to be formulated.

3. The walled Brauer algebra (x) 3.1. The algebra Br,s . This algebra was defined in 1994 in [2]. It is the subalgebra (x) of Br+s spanned by the set of (r, s)-diagrams. By definition, an (r, s)-diagram is an (r + s)-diagram in which we imagine a wall separating the first r from the last s columns of vertices, such that: (a) all horizontal edges cross the wall; (b) no vertical edges cross the wall. An edge crosses the wall if its two vertices lie on opposite sides of the wall. The (x) (x) multiplication in Br,s is that of Br+s .

New versions of Schur–Weyl duality

65

Label the vertices on the top and bottom rows of an (r, s)-diagram by the numbers 1, . . . , r to the left of the wall and −1, . . . , −s to the right of the wall. Let ci,−j (1 i r; 1 j s) be the diagram with a horizontal edge connecting vertices i and −j on the top row and the same on the bottom row, and with all other edges connecting vertex k (k = i, −j ) in the top and bottom rows. It is easy to see that the walled Brauer algebra is generated by the permutations it contains along with just one of the ci,−j . (Note that ci,−j is the (r + s)-diagram denoted by ci,r+j in 2.1.)

(x)

3.2. Dimension. What is the dimension of Br,s ? One way to answer that question is to consider the map, flip, from (r + s)-diagrams to (r + s)-diagrams, defined by interchanging the top and bottom vertices to the right of the imaginary wall. For example, Figure 4 shows a (4, 2)-diagram (to the left) and its corresponding 6-diagram, obtained from the left diagram by applying flip. Note that flip is involutary: applying it twice gives the original diagram back again.

Figure 4. A (4, 2)-diagram and its corresponding permutation, after applying flip.

One easily checks that the map flip carries (r, s)-diagrams bijectively onto the set of (r +s)-diagrams with all edges vertical. Such diagrams correspond with permutations (x) of r + s objects, so the dimension of Br,s is (r + s)!.

(x) 3.3. Another view of Br,s . The above correspondence between (r, s)-diagrams and (x) permutations gives another way to think of the multiplication in Br,s . Given two (r, s)-diagrams D1 , D2 let D1 , D2 be their corresponding permutations obtained by applying flip. Define a new (rather bizarre) composition on permutations as follows. Given any two permutation diagrams D1 , D2 (with r + s columns of vertices) identify the first r vertices of the bottom row of D1 with the first r vertices of the top row of D2 , and identify the last s vertices of the top row of D1 with the last s vertices of the bottom row of D2 . After removing loops and identified vertices this gives a new permutation diagram D3 in which the vertices in the top (resp., bottom) row are the remaining top (bottom) row vertices from the original diagrams.

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Let δ(D1 , D2 ) be the number of loops removed in computing the composite permutation diagram D3 . Define multiplication of permutation diagrams by the rule

D1 · D2 = x δ(D1 ,D2 ) D3 In other words, we are multiplying permutations by composing maps “on the right” on one side of the wall, and “on the left” on the other side (roughly speaking). For example, Figure 5 below shows the computation of the composite diagram in the walled Brauer algebra (left column) and the computation in terms of the corresponding permutations (right column). Figure 6 shows the resulting diagrams after the single loop and identified vertices have been removed.

Figure 5. Composition of diagrams and permutations. The diagrams on the left correspond under flip with the permutations on the right.

One can check that D3 coincides with (D1 ◦ D2 ) and δ(D1 , D2 ) = δ(D1 , D2 ). (x) In other words, flip defines an algebra isomorphism between the algebra Br,s and the (x) r,s spanned by permutation diagrams with the multiplication defined above. algebra B (x) (R[x]Ss )opp . (x) R[x]Sr and B Note that in particular B r,0 0,s

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New versions of Schur–Weyl duality

Figure 6. The corresponding diagrams resulting from Figure 5. The two diagrams correspond under flip.

3.4. Schur–Weyl duality. Consider mixed tensor space V r,s := V ⊗r ⊗V ∗⊗s , where V ∗ is the usual linear dual space of V . Mixed tensor space is naturally a module for (n) GLn , and one obtains an action of Br,s on V r,s simply by restricting the action of (n) Br+s , which acts the same on V r,s as it does on V ⊗(r+s) , since on restriction to On we have V V ∗ . Thus we have the following commutative diagram ρ

C GLn −−−−→ ι

End(V r,s )

ρ

σ

(n)

←−−−− Br,s ι σ

(13)

(n)

C On −−−−→ End(V ⊗(r+s) ) ←−−−− Br+s (n)

in which the vertical maps ι, ι are inclusion. By [2], the actions of GLn and Br,s on V r,s in the first row of the diagram satisfy Schur–Weyl duality: ρ(C GLn ) = EndB (n) (V r,s )

(14)

(n) ) = EndGLn (V r,s ). σ (Br,s

(15)

r,s

The algebra in (14) is another Schur algebra S(n; r, s) in type A, studied in [10]. These Schur algebras provide us with a new family of quasihereditary algebras, generalizing the classical Schur algebras, since S(n; r, 0) S(n, r). In fact, the S(n; r, s) provide a new class of generalized Schur algebras in the sense of Donkin [11]. For fixed n, the family of S(n; r, s)-modules as r, s vary constitutes the family of all rational representations of GLn . Whence the name rational Schur algebras for the S(n; r, s). When n r + s, the representation σ in the top row of (13) above is faithful, so (n) induces an isomorphism Br,s EndGLn (V r,s ).

3.5. The quantum case. Quantizations of the walled Brauer algebra have been defined and studied in work of Halverson [21], Leduc [29], Kosuda–Murakami [28], and Kosuda [27].

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4. The deranged algebra 4.1. The problem. One might wonder if there are versions of Schur–Weyl duality in which the natural module V is replaced by some other representation. Perhaps the first choice would be to replace V with the adjoint module, i.e., an algebraic group acting on its Lie algebra via the adjoint representation. The simplest instance of this would be type A, where we consider the module sl⊗r n as an SLn -module, and ask for its ⊗r centralizer algebra EndSLn (sl⊗r ) = End (sl GL n n n ). (It makes no difference whether ⊗r we regard sln as module for SLn or for GLn . We look at sl⊗r n rather than gln since gln is not simple as a GLn -module or SLn -module.) 4.2. Relation with Brauer algebras. Even though this is a question about type A, its solution is intimately connected with the walled Brauer algebra. Here is a brief outline of the solution to this problem, recently obtained in [3]. The main idea is to utilize the decomposition gln = sln ⊕C (as SLn or GLn module) to write ⊗r ⊗t (16) gl⊗r n = sln ⊕ 0t

(n) e C eBr,r

(n 2r)

(17)

where e = i (1 − n−1 ci,−i ) (notation of 3.1). The algebra in (17) has a basis consisting of all elements of the form eDe as D ranges over the set of (r, r)-diagrams with no horizontal edges connecting i to −i. Such diagrams correspond under flip, after inverting the signs labeling the bottom row, with derangements of 2r objects. (A derangement is a permutation having no fixed points.) Let N(k) = the number of derangements of k objects, then we have by the Inclusion–Exclusion Principle (see [35, 2.2.1 Example]) k k−j k (−1) j! (18) N (k) = j j =0

which is the nearest integer to k!/e (e = 2.7182818 . . . ). Thus the dimension of (n) the centralizer algebra eBr,r e is given by N (2r). Because of this connection with (n) derangements, the algebra eBr,r e is known as the deranged algebra. In [3] explicit formulas are obtained for the number of times a given simple module L appears as a summand of sl⊗r n . In particular, it is shown that N(r) = the multiplicity of the trivial module in sl⊗r n

New versions of Schur–Weyl duality

69

and N (r − 1) = the multiplicity of sln in sl⊗r n demonstrating that the combinatorics of derangement numbers is inherent in this theory. 4.3. Schur–Weyl duality. At least when n 2r, the actions of GLn and C on sl⊗r n satisfy Schur–Weyl duality: ρ(C GLn ) = EndeB (n) e (sl⊗r n )

(19)

(n) e) = EndGLn (sl⊗r σ (eBr,r n ).

(20)

r,r

This will almost certainly hold for all n, r. Acknowledgements. These notes are based on a lecture, various versions of which I have given in the past year, in a number of locations, including Stuttgart, Birmingham, Queen Mary (London), Lancaster, Manchester, Oxford, and Cambridge. I’m grateful to the organizers of those events for the opportunity to present these ideas.

References [1]

Be˘ılinson, A. A., Lusztig, G., MacPherson, R. A geometric setting for the quantum deformation of GLn , Duke Math. J. 61 (1990), 655–677.

[2]

Benkart, G., Chakrabarti, M., Halverson, T., Leduc, R., Lee, C., Stroomer, J., Tensor product representations of general linear groups and their connections with Brauer algebras, J. Algebra 166 (1994), 529–567.

[3]

Benkart, G., Doty, S., Derangements and tensor powers of adjoint modules for sln , J. Algebraic Combin. 16 (2002), 31–42.

[4]

Birman, J. S., Wenzl, H., Braids, link polynomials and a new algebra, Trans. Amer. Math. Soc. 313 (1989), 249–273.

[5]

Brauer, R., On algebras which are connected with the semisimple continuous groups, Ann. of Math. 38 (1937), 857–872.

[6]

Brown, William P., An algebra related to the orthogonal group, Michigan Math. J. 3 (1955), 1–22; The semisimplicity of ωfn , Ann. of Math. 63 (1956), 324–335.

[7]

Carter, R. W., Lusztig, G., On the modular representations of the general linear and symmetric groups, Math. Z. 136 (1974), 193–242.

[8]

de Concini, C., Procesi, C., A characteristic free approach to invariant theory, Adv. in Math. 21 (1976), 330–354.

[9]

Dipper, R., James, G., The q-Schur algebra, Proc. London Math. Soc. (3) 59 (1989), 23–50; q-tensor space and q-Weyl modules, Trans. Amer. Math. Soc. 327 (1991), 251–282.

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[10] Dipper, R., Doty, S., Rational Schur algebras, in preparation. [11] Donkin, S., On Schur algebras and related algebras I, J. Algebra 104 (1986), 310–328. [12] Donkin, S., Good filtrations of rational modules for reductive groups, in: The Arcata Conference on Representations of Finite Groups (Arcata, Calif., 1986), Proc. Sympos. Pure Math. 47, Part 1, Amer. Math. Soc., Providence, RI 1987, 69–80. [13] Donkin, S., On tilting modules for algebraic groups, Math. Z. 212 (1993), 39–60. [14] Doty, S., Polynomial representations, algebraic monoids, and Schur algebras of classical type, J. Pure Appl. Algebra 123 (1998), 165–199. [15] Doty, S., Representation theory of reductive normal algebraic monoids, Trans. Amer. Math. Soc. 351 (1999), 2539–2551. [16] Doty, S., Giaquinto, A., Presenting Schur algebras, Internat. Math. Res. Notices 2002, 1907–1944. [17] Doty, S., Giaquinto, A., Sullivan, J., Presenting Schur algebras in types B, C, D, in preparation. [18] Erdmann, K., Decomposition numbers for symmetric groups and composition factors of Weyl modules, J. Algebra 180 (1996), 316–320. [19] Green, J. A., Polynomial representations of GLn , Lecture Notes in Math. 830, SpringerVerlag, Berlin, New York 1980. [20] Grood, C., Brauer algebras and centralizer algebras for SO(2n, C), J. Algebra 222 (1999), 678–707. [21] Halverson, T., Characters of the centralizer algebras of mixed tensor representations of GL(r, C) and the quantum group Uq (gl(r, C)), Pacific J. Math. 174 (1996), 359–410. [22] Halverson, T., Ram, A., Characters of algebras containing a Jones basic construction: the Temperley–Lieb, Okasa, Brauer, and Birman–Wenzl algebras, Adv. Math. 116 (1995), 263–321. [23] Hanlon, P., Wales, D., On the decomposition of Brauer’s centralizer algebras, J. Algebra 121 (1989), 409–445; Eigenvalues connected with Brauer’s centralizer algebras, J. Algebra 121 (1989), 446–476; Computing the discriminants of Brauer’s centralizer algebras, Math. Comp. 54 (1990), 771–796. [24] James, G. D., The decomposition of tensors over fields of prime characteristic, Math. Z. 172 (1980), 161–178. [25] Jimbo, M., A q-analogue of U (gl(N + 1)), Hecke algebra, and the Yang–Baxter equation, Lett. Math. Phys. 11 (1986), 247–252. [26] Kauffman, L. H., An invariant of regular isotopy, Trans. Amer. Math. Soc. 318 (1990), 417–471. [27] Kosuda, M., Representation of q-analogue of rational Brauer algebras, Tsukuba J. Math. 21 (1997), 707–728. [28] Kosuda, M., Murakami, J., The centralizer algebras of mixed tensor representations of Uq (gln ) and the HOMFLY polynomial of links, Proc. Japan Acad. Ser. A Math. Sci. 68 (1992), 148–151; Centralizer algebras of the mixed tensor representations of quantum

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group Uq (gl(n, C)), Osaka J. Math. 30 (1993), 475–507. [29] Leduc, R., A two-parameter version of the centralizer algebra of the mixed tensor representations of the general linear group and quantum general linear group, Ph.D. Dissertation, University of Wisconsin-Madison 1994. [30] Morton, H. R.; Wassermann, A. J., A basis for the Birman–Wenzl algebra, Preprint 1989, with foreword and references updated Jan. 2000. [http://www.liv.ac.uk/~su14/ knotprints.html] [31] Murakami, J., The Kauffman polynomial of links and representation theory, Osaka J. Math. 24 (1987), 745–758; The representations of the q-analogue of Brauer’s centralizer algebras and the Kauffman polynomial of links, Publ. Res. Inst. Math. Sci. 26 (1990), 935–945. [32] Oehms, S., Symplektische q-Schur-Algebren, Dissertation, Universität Stuttgart, Stuttgart, 1997. Berichte aus der Mathematik, Verlag Shaker, Aachen 1997. [33] Ringel, C., The category of modules with good filtrations over a quasi-hereditary algebra has almost split sequences, Math. Z. 208 (1991), 209–223. [34] Schur, I., Über die rationalen Darstellungen der allgemeinen linearen Gruppe (1927). Reprinted in Schur, I., Gesammelte Abhandlungen, Vol. III, pp. 68–85, Springer-Verlag, Berlin, 1973. [35] Stanley, R. P., Enumerative combinatorics, Vol. 1, Cambridge Stud. Adv. Math. 49, Cambridge Univ. Press, Cambridge 1997. Stephen Doty, Department of Mathematics and Statistics, Loyola University Chicago, Chicago, IL 60626, U.S.A. E-mail: [email protected]

Just infinite periodic Lie algebras Norberto Gavioli, Valerio Monti and Carlo M. Scoppola

Abstract. In this paper we give a characterization of just infinite Lie subalgebras of a loop algebra in terms of the existence of certain ideals and in terms of the existence of certain nontrivial L-endomorphisms of positive degree. Such an algebra has a faithful representation of finite degree over the ring of polynomials with coefficients in the ground field, and is shown to be ideally r-constrained for a suitable r. An upper bound for r is given in the solvable case. 2000 Mathematics Subject Classification: 17B70

1. Introduction The study of graded Lie algebras has received considerable attention in recent years (see [3], [10], [5], and [9] for complete references), beyond its obvious importance as a tool in the study of pro-p-groups. Indeed, it is possible to attach a Lie ring to any given pro-p-group G (see e.g. [8]), defining a suitable Lie bracket on the sum of the factors of a strongly central series of G. One can choose among different central series for this construction, and the properties of the corresponding Lie ring reflect this choice. In many interesting cases, the Lie ring turns out to be a Lie algebra over a finite field. An effective study of these classes of (groups and) algebras is based, of course, on a variety of technical features. Among these, an increasingly prominent role seems to be taken by the concept of periodicity: various kinds of repeating patterns in the structure of both the groups and the corresponding Lie rings or algebras have indeed been observed (see [6, III.17.9], [7], [1]). In this paper we focus on the study of just infinite(-dimensional) graded Lie algebras [9], and we propose a notion of periodicity for them. In Section 2 we begin the study of the ideal structure in just infinite Lie algebras, with a particular attention to abelian ideals, and with some remarks on the solvable case. Section 3 is then devoted to just infinite subalgebras of some Kac–Moody algebras. Our main result is their characterization in terms of the existence of proper ideals containing no non-trivial homogeneous elements, and also in terms of existence of certain homomorphisms that induce a periodicity on the algebra structure. We define here the notion of depth, that proves to be very handy in the computations. Then we

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Norberto Gavioli, Valerio Monti and Carlo M. Scoppola

show that these algebras are r-constrained (see [4]). Furthermore this section contains the construction of a faithful representation of a periodic just infinite Lie algebra over the ring of polynomials with coefficients in the ground field. The paper ends with some remarks on the solvable case.

2. Just infinite Lie algebras Lie algebras denoted by Latin capital letters appearing in this paper will be finitely generated N-graded Lie algebras generated by their homogeneous component of degree 1. We shall write g-algebra to mean graded algebra. We shall use curly capital letters to denote Lie algebras without any extra structure. In case of graded algebras we shall write g-ideal and g-module respectively to mean a homogeneous ideal or a finitely generated graded module.

If M is a g-module and x ∈ M we say that x = ni=0 xji is the homogeneous decomposition of x if {ji }ni=0 is a strictly increasing sequence and, for i = 0, . . . , n, the element xji is non-trivial and homogeneous of degree ji . The elements xj0 and xjn are called respectively the head and the

tail of x and are denoted by hd x and tl x. We shall use the notation M i to denote j ≥i Mj . Note that if L is a Lie g-algebra then Li is the usual i-th Lie power of L. We shall make repeated use of the following easy lemma without any further mention. Lemma 2.1. Let M be an L-g-module and N a g-submodule of finite codimension. Then there exists an index j such that M j ≤ N . Definition 2.2. Let L be a g-algebra and let M be an L-g-module. The g-module M is just infinite if it is infinite-dimensional and, for every non-trivial g-submodule N, dim(M/N ) is finite. A (necessarily infinite-dimensional) g-algebra L is just infinite if L is just infinite as an L-g-module. Lemma 2.3. Let M be a just infinite L-g-module. Then every non-trivial ( possibly non-graded ) submodule of M has finite codimension in M. Proof. Let N be a non-trivial submodule

of M. If x is a non-trivial element of N whose tail has degree i, then x − tl x ∈ j*
*

of (N + j*
*

above that some Ki For i ≥ 0, set Ki := (N + j*
*

i Mi for i ≥ k, that

Mj ≥ Mi . By an easy induction we get N

+ j

so that N + j

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As a consequence we have the following easy remarks which we shall often use without explicit reference: Remark 2.4. The intersection of two non-trivial submodules of a just infinite g-module is non-trivial. Remark 2.5. A just infinite g-module has no non-trivial finite-dimensional submodule. Lemma 2.6 (Schur). Let M be a just infinite L-g-module. If N is a submodule of M and f : N → M is a non-trivial L-homomorphism then f is injective. Proof. By Remark 2.5, f (N) has infinite dimension. As f (N) ∼ = N/ ker f , ker f has infinite codimension in N and, a fortiori, in M, so that ker f = 0 by Lemma 2.3. Remark 2.7. The centre of a just infinite Lie g-algebra L is trivial. Proof. If Z (L) = 0, then the vector space generated by a non-trivial element of Z (L) would be a non-trivial ideal with infinite codimension. The following application of Schur’s Lemma will be useful in the following: Lemma 2.8. Let I be a non-trivial ideal of a just infinite Lie g-algebra L. If CL (I ) = 0 then every ideal J which does not centralize I contains an element y (which we may choose to be homogeneous if J is a g-ideal ) such that ad y induces an injective L-endomorphism of I (in particular y does not centralize any non-trivial element of I ). Proof. We first assume that CL (I ) contains a non-trivial g-ideal: by Lemma 2.1, / CL (I ) but L/CL (I ) is nilpotent. We may then choose y ∈ J such that y ∈ [y, L] ≤ CL (I ). Moreover, we may choose y homogeneous if J is a g-ideal. It can be easily checked that ad y induces a non-trivial L-endomorphism of I , which is injective by Schur Lemma. We now come to the general case and claim that CL (I ) contains a non-trivial gideal, so that the previous proof applies. We consider the subspace gr I generated by the heads of the non-trivial elements of I : an easy check shows that gr I is a g-ideal centralized by the heads of the non-trivial elements of CL (I ). This means that gr I is a non-trivial ideal whose centralizer contains a non-trivial g-ideal, so that by the first part of the proof every ideal which does not centralize gr I contains an element y which does not centralize any non-trivial element of gr I . In particular assume that I does not centralize gr I , so that we may choose y in I : this is impossible since y centralizes at least the elements of CL (I ) ∩ gr I , which is non-trivial, since it is the intersection of two non-trivial ideals. We have then proved that I centralizes gr I , i.e., CL (I ) contains a non-trivial g-ideal, as claimed.

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We recall that the nil radical of a (not necessarily graded) Lie algebra is the sum of all nilpotent ideals. The nil radical is not in general nilpotent: this is true, however, for a just infinite Lie g-algebra L, since the non-trivial ideals of L have finite codimension so that the nil radical can be expressed as a finite sum of nilpotent ideals and is, therefore, nilpotent. We may actually say much more about the nil radical of a just infinite Lie g-algebra: Proposition 2.9. Let L be a just infinite Lie g-algebra and let M be the nil radical of L. Let I be a non-trivial ideal of L. The following conditions are equivalent: (i) I is abelian; (ii) I is nilpotent; (iii) I ≤ M; (iv) CL (I ) = 0. If one (and hence all) of the previous conditions is verified then CL (I ) = M. Proof. Clearly (i) implies (ii) and (ii) implies (iii). Since we already noted that M is nilpotent, its centralizer is non-trivial, so that the same is true for the ideals contained in M: (iii) implies (iv). To prove that (iv) implies (i) note that if I were not abelian then we could apply Lemma 2.8 and find an element y in I which does not centralize any non-trivial element of I : this is impossible, since y centralizes at least one non-trivial element of I , that is y itself. Now suppose that I satisfies the conditions above: CL (I ) is then a non-trivial ideal with non-trivial centralizer, i.e. it satisfies condition (iv), so that, by (iii), CL (I ) ≤ M. On the other hand, M satisfies trivially condition (iii): therefore it is abelian and then centralizes every ideal contained in it, so that CL (I ) ≥ M. Corollary 2.10. The nil radical M of a just infinite Lie g-algebra is an abelian g-ideal. Proof. If M = 0 there is nothing to prove. Otherwise M satisfies condition (iii) of Proposition 2.9 so that it is abelian and it is the centralizer of the non-trivial ideals contained in it. In the proof of Lemma 2.8 we showed that a non-trivial centralizer contains a non-trivial g-ideal: in particular M contains a non-trivial g-ideal, say I . Clearly CL (I ) is a g-ideal, and it is exactly M by Proposition 2.9. Corollary 2.11. For a just infinite Lie g-algebra L the following conditions are equivalent: (i) L is solvable; (ii) L contains a non-trivial abelian ideal; (iii) the nil radical of L is non-trivial. Proof. Clearly (i) implies (ii) and (ii) implies (iii). If (iii) holds, let M be the nil radical of L: then M is abelian (hence solvable) by Corollary 2.10, and L/M is nilpotent (hence solvable) by Lemma 2.1. Therefore L is solvable.

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3. Periodic algebras If L is a (non graded) finite-dimensional Lie algebra over the field F , the loop algebra1 Loop(L) is the tensor product L ⊗F (sF [s]) where F [s] is the ring of polynomials in the indeterminate s. The algebra Loop(L) is graded over N and it is not necessarily generated by its homogeneous component of degree 1. Definition 3.1. A non-trivial ideal I of a Lie g-algebra L is completely inhomogeneous if no non-trivial homogeneous element of L belongs to I . We have the following easy characterization of completely inhomogeneous ideals in just infinite Lie g-algebras: Remark 3.2. A non-trivial ideal I of a just infinite Lie g-algebra L is completely inhomogeneous if and only if L/I is not nilpotent. Proof. If L/I is nilpotent then I contains a term of the lower central series of L, that is a non-trivial homogeneous ideal. Conversely, if I contains a non-trivial homogeneous element, say x, then I contains the g-ideal generated by x, so that Lemma 2.1 implies that I contains a term of the lower central series of L and L/I is nilpotent. In the following theorem, the hypothesis on the codimension of I is needed only to construct the loop algebra Loop(L/I ). Proposition 3.3. Let L be a Lie g-algebra and let I be a completely inhomogeneous ideal of finite codimension in L. Then L is isomorphic to the g-subalgebra of Loop(L/I ) generated by L1I+I · s. Proof. We define a linear map ϕ : L → Loop(L/I ) by setting ϕ(x) := (x + I )s i for x ∈ Li . If x ∈ Li and y ∈ Lj then [ϕ(x), ϕ(y)] = [(x + I )s i , (y + I )s j ] = [(x + I ), (y + I )]s i+j = ([x, y] + I )s i+j = ϕ([x, y]). Thus ϕ is a graded Lie algebra homomorphism. By definition of ϕ, a homogeneous element k of ker ϕ must belong to I . Since I is completely inhomogeneous this means that k = 0 and ϕ is injective. As L is generated by L1 we have that ϕ(L) is generated by ϕ(L1 ) = L1I+I · s. As a partial converse we have: Proposition 3.4. An infinite-dimensional g-subalgebra L of a loop algebra contains a completely inhomogeneous ideal. 1 Strictly speaking, we are defining the positive part of a loop algebra, since the loop algebra is generally defined as L ⊗F (F [s, s −1 ]) where F [s, s −1 ] is the ring of the Laurent polynomials in the indeterminate s.

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Proof. If L ⊆ Loop(L), the intersection of L with the kernel of the homomorphism s=1

L ⊗ sF [s] −−→ L is a non-trivial ideal of L whose only homogeneous element is 0. Proposition 3.4 does not hold in general if we drop the hypothesis that L is infinite-dimensional. As a trivial example consider a finite-dimensional abelian algebra L. Then L · s is a subalgebra of Loop(L), all of whose non-trivial elements have degree 1, and then it does not contain any completely inhomogeneous ideal. We define the depth of a non-trivial element x of a Lie g-algebra as the difference between the degree of the tail of x and the degree of the head of x. Clearly x is homogeneous if and only if it has depth 0. We shall need the following lemma. Lemma 3.5. Let I be a non-trivial ideal of a Lie g-algebra L. Let d be the minimal depth of non-trivial elements of I and let J be the subspace of L generated by the heads of the elements of I of depth d. Then: (i) every homogeneous element of J is the head of a unique element of I of depth d; (ii) J is a g-ideal; (iii) there exists a non-trivial L-homomorphism t : J → L which sends homogeneous elements of degree i into homogeneous elements of degree i + d. Proof. Let H be the subset of L formed by the heads of the elements of I of depth d. If h ∈ H and x and x¯ are elements of I of depth d whose head is h, then x − x¯ is either 0 or it has depth less than d: by minimality of d the latter case cannot hold, so that x = x. ¯ This proves the uniqueness part of (i). Set b(h) := x. Let Hˆ := H ∪ 0. We define a map t : Hˆ → L by setting ht := tl b(h) if h = 0 and 0t := 0. If h ∈ Hˆ and z is a homogeneous element of L then either [h, z] = 0 or [h, z] is the head of [b(h), z], which is an element of I of depth at most d and hence exactly d. In both cases [h, z] belongs to J , thus proving (ii). Moreover, an easy inspection of the possible cases shows that [ht, z] = [h, z]t: then to prove (iii) we only need to show that t can be extended to a linear map to the whole of J . For every positive integer i set Hi := H ∩ Li . If h and k belong to Hi and α and β are scalars, then αh + βk is either 0 or it is the head of αb(h) + βb(k) which is an element of I of depth at most d, and then, once again, exactly d. We have then shown that Hˆ i := Hi ∪ 0 is a subspace of L and therefore it is equal to J ∩ Li , thus proving the existence part of (i). Moreover, it can be easily checked that t is linear on Hi , so that it can be extended to a linear map on J thus completing the proof of (iii). Definition 3.6. A just infinite Lie g-algebra L is periodic if there exists a g-ideal M and a non-trivial L-homomorphism t : M → L which sends homogeneous elements of degree i into homogeneous elements of degree i + d, with d positive integer. The homomorphism t is called an M-period of degree d.

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By Schur Lemma an M-period t of degree d is necessarily injective. As M is a non-trivial g-ideal it contains a term of the lower central series of L so that t is defined on Lj for all sufficiently large j , thus inducing an L-homomorphism between Lj and Lj +d . Since also Mt is a non-trivial g-ideal, it contains a term of the lower central series of L so that the homomorphisms above are surjective for all sufficiently large j . Remark 3.7. An M-period t of degree d induces an L-isomorphism between Lj and Lj +d for all sufficiently large j . By definition an M-period t does not necessarily send M into itself. We may, however, restrict t to a suitable g-ideal N (e.g. a term of the lower central series of L contained in M) and get: Remark 3.8. Given an M-period t there exists a g-ideal N contained in M such that the restriction of t to N is an N -period sending N into itself. Remark 3.9. A solvable just infinite Lie g-algebra L is periodic. Proof. By Corollaries 2.10 and 2.11 the nil radical M of L is an abelian non-trivial g-ideal. By Lemma 2.8 there exists a homogeneous element of L whose adjoint induces an M-period. We can give some characterizations of periodic algebras: Theorem 3.10. For a just infinite Lie g-algebra L the following are equivalent: (i) L is isomorphic to a g-subalgebra of a loop algebra; (ii) L has a completely inhomogeneous ideal; (iii) L is periodic. Proof. We have already seen the equivalence of (i) and (ii) in Propositions 3.3 and 3.4. Assume now (ii), and choose a completely inhomogeneous ideal I of L. The minimal depth of non-trivial elements of I is then a positive number d, so that Lemma 3.5 provides us with a g-ideal J and a J -period of degree d. Conversely, if t is an M-period of a periodic just infinite Lie g-algebra L then M(t − 1) is a non-trivial completely inhomogeneous ideal of L. The previous theorem shows that a periodic just infinite Lie g-algebra L is isomorphic to a g-subalgebra of a loop algebra Loop(L): note that, while L is not necessarily unique, it is necessarily not nilpotent for, otherwise, L would be finite-dimensional. However, it is possible to choose L to have: Proposition 3.11. A periodic just infinite Lie g-algebra L is isomorphic to a g-subalgebra of a loop algebra Loop(L) where L has the following properties: (i) every proper quotient of L is nilpotent;

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(ii) L has a unique minimal ideal ; (iii) is a differentiably simple algebra. Proof. Let I be an ideal of L which is maximal among the completely inhomogeneous ideals of L: by Proposition 3.3, L is isomorphic to a g-subalgebra of Loop(L), where L := L/I . By Remark 3.2, our choice of I implies (i). Since L is not nilpotent and finite-dimensional, there exists a non-trivial ideal such that = Li for all sufficiently large i. By (i), a non-trivial ideal of L contains a term of the lower central series of L, so that it contains : this proves (ii). If J is an ideal of invariant under Der( ) then J is clearly an ideal of L: minimality of implies (iii). In the previous proposition we chose an ideal maximal among completely inhomogeneous ideals of a periodic just infinite Lie g-algebra L: such an ideal is not unique. In fact, let t be an M-period: by Remark 3.8, we may assume that t sends M to itself, so that powers of t are defined. It can be easily checked that M(1 + t + t 2 ) and M(t + t 2 ) are completely inhomogeneous ideals of L. Since their sum contains the g-ideal M, there is no completely inhomogeneous ideal containing both of them, and therefore they are contained in different maximal completely inhomogeneous ideals. Definition 3.12. Let L be a Lie g-algebra and r a positive integer. As in [4] we shall say that L is ideally r-constrained if for every g-ideal I of L and every positive integer j one of the following holds: (1) I ≤ Lj , (2) I ≥ Lj +r−1 . Remark 3.13. In [2] it is proved that a similar condition holds in a pro-p-group G for some r if and only if G has finite obliquity (see [7] for the definition of obliquity). The definition of obliquity has been adapted to Lie g-algebras in [4], where it is proved that a Lie g-algebra is ideally r-constrained for some r if and only if it has finite obliquity. If L is an ideally r-constrained Lie g-algebra then every non-trivial g-ideal contains a term of the lower central series of L. As the terms of the lower central series have finite codimension in L (we are assuming that L is finitely generated) this implies that L is either finite-dimensional or just infinite. In the periodic case we have a converse: Proposition 3.14. A periodic just infinite Lie g-algebra L is ideally r-constrained for some integer r. Proof. For every positive integers i and j define the subset of Lj : Vi,j := {x ∈ Lj | [x, L1 , . . . , L1 ] = Li+j }. i times

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By choosing a system of affine coordinates in Lj and Li+j we see that Vi,j is described by the vanishing of a finite number of determinants in the coordinates of Lj , so that Vi,j is an affine subvariety of Lj . a g-ideal conSince L is just infinite, every non-trivial element x in Lj generates taining Li+j for a suitable i: this means that x ∈ / Vi,j , so that i Vi,j = 0. Clearly Vi+1,j ≤ Vi,j : as a decreasing sequence of affine varieties stabilizes after a finite number of steps there exists a minimum integer r(j ) such that Vr(j ),j = 0. A g-ideal I which is not contained in Lj +1 contains a non-trivial homogeneous element in Lj and then it contains Lj +r(j ) . We only need to show that the sequence of integers {r(j )}j is bounded above. By Remark 3.7 the period t induces an L-isomorphism between Lj and Lj +deg(t) for j ≥ k. This implies that r(j + deg(t)) = r(j ) for j ≥ k, so that sup{r(j )} = max{r(j ) | j ≤ k + deg(t) − 1}. Definition 3.15. Let M be an L-g-module. An L-endomorphism of M is graded of degree d if, for every i, it sends homogeneous elements of degree i into homogeneous element of degree i + d. Proposition 3.16. An L-endomorphism t of a just infinite L-g-module M can be expressed as the sum of a finite number of graded L-endomorphisms of nonnegative degree (in particular there is no graded L-endomorphism of negative degree). If t is not graded of degree 0 then t acts transcendentally on M, M is a finitely generated free module over the polynomial ring F [t] and L acts over M as an algebra of F [t]-endomorphisms. Proof. If m is a homogeneous element of M of degree i, say, we define mtd as the homogeneous component of degree i + d of mt. It can be easily checked that td is linear on Mi for every i, so that it can be extended to a linear map on the whole of M, and a further simple inspection shows that td is an L-endomorphism of M. For a given m in M we have clearly mtd = 0 for all but a finite number of d’s. If we moreover choose m to be a non-trivial homogeneous element of minimal degree we have mtd = 0 for d < 0. Schur Lemma then implies that td = 0 for all but a finite number of d’s and that td = 0 for d < 0. This proves that t is the sum of a finite number of graded L-endomorphisms of nonnegative degree. We now assume that t is not graded of degree 0: the maximum integer d such that td = 0 is then positive. Since Mtd is a non-trivial g-submodule of M, there exists a subspace V of M, generated by a finite number of homogeneous elements, such that M = V + Mtd . We claim that V generates M as an F [t]-module: to this aim we show that every homogeneous element m of M belongs to V F [t]. By our choice of V , there exist v ∈ V and l ∈ Mtd both homogeneous of the same degree of m, say i, and such that m = v + l. As l ∈ Mtd , there exists u in M such that l = utd : since td is injective, by Schur Lemma, and it is graded, u is a homogeneous element of degree less than i. By definition of td , utd − ut is a sum of homogeneous elements of degree less than i: arguing by induction on i, we may then assume that both u and utd − ut belong to V F [t]. Therefore m = v + (utd − ut) + ut belongs to V F [t].

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We have then shown that M is finitely generated as an F [t]-module. This implies that t is transcendental over F for, otherwise, F [t] would be finite-dimensional over F , so that M would be finite-dimensional too. A non-trivial polynomial p(t) in t is injective by Schur Lemma: therefore M is a torsion-free F [t]-module. As F [t] is a principal ideal domain, M is a finitely generated free F [t]-module. Clearly L acts over M as a Lie algebra of F [t]-endomorphisms since t centralizes the action of L. We cannot directly apply the previous proposition to an M-period t of a just infinite Lie g-algebra L, since t needs not to be an endomorphism of M. However, thanks to Remark 3.8, this problem can be overridden by taking the restriction of t to a suitable g-ideal: Corollary 3.17. If t is an M-period of a just infinite Lie g-algebra L such that Mt ≤ M then M is a non-trivial finitely generated free F [t]-module. Corollary 3.18. Let L be a just infinite solvable g-algebra with nil radical M. If x is a (not necessarily homogeneous) element of L \ M such that [x, L] ≤ M then M is a finitely generated free F [t]-module, where t is induced by ad x, and L acts on M as a Lie algebra of F [t]-endomorphisms. Proposition 3.19. A periodic just infinite Lie g-algebra L has a faithful representation τ of finite degree n over F [t] such that the kernel K of the ordinary representation τ

t=1

→ gl(n, F [t]) −−→ gl(n, F ) L− is a completely inhomogeneous ideal. Proof. We fix an M-period t such that Mt ≤ M and a homogeneous element x of L such that [x, L] ≤ M: moreover, if L is solvable, we ask that M is the nil radical of L (as in the proof of Remark 3.9) and that x ∈ / M (this is not necessarily the element whose adjoint induces the period t). By Corollary 3.17, N := F [t] ⊕ M is a finitely generated free F [t]-module. We now define τ : L → gl(N, F [t]) by setting: τ (l)

(f (t), m) −−→ (0, [x, l]f (t) + [m, l]). It can be easily verified that this actually defines a representation. We show that τ is faithful. If l ∈ / CL (M) then (0, m)τ (l) = 0 for some m ∈ M. If l is a non-trivial element in CL (M) then, by Corollary 2.11 and Proposition 2.9, L is solvable and M = CL (M): therefore l belongs to M and ad x induces a non-trivial L-endomorphism of M. Schur Lemma then implies that x does not centralize any element of M: in particular [x, l] = 0, so that (1, 0)τ (l) = (0, [x, l]) = 0. Now we show that K is a completely inhomogeneous ideal. Let k be a homogeneous element in K. Therefore N τ (k) ≤ N (1 − t). In particular, for every homogeneous m ∈ M, we have that (1, 0)τ (k) and (0, m)τ (k) belong to N(1 − t). By

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definition of τ this means that [x, k] and [m, k] belong to M(1 − t). As M(1 − t) is a completely inhomogeneous ideal, while [x, k] and [m, k] are homogeneous elements, this yields [x, k] = [m, k] = 0. For the arbitrariness of m we then get that k centralizes M + F x. Since M + F x is an ideal which is not abelian, Proposition 2.9 implies that k = 0, that is K does not contain any non-trivial homogeneous element. We now apply the above results to the solvable case. Corollary 3.20. Let L be a just infinite solvable g-algebra L with nil radical M. If L is not metabelian then char(F ) is non-zero and it divides the rank of the free F [t]-module M. Proof. Just note that if L is not metabelian, then, in the construction given in Proposition 3.19, we may choose x in the derived subalgebra L2 of L and we may assume that x is the element whose adjoint induces the period t so that the matrix 0 0 ... 0 0 t 0 τ (x) = . . . . . . t 0 0 must have zero trace. On the other hand, if n is the rank of M, Tr(τ (t)) = nt so that char(F ) divides n. Definition 3.21. A periodic just infinite Lie g-algebra is linear if it is isomorphic to a g-subalgebra of Loop(gl(n, F )) for some n. Corollary 3.22. A periodic just infinite Lie g-algebra L has a graded faithful representation and therefore it is a linear algebra. Proof. This is an easy consequence of the Ado–Iwasawa theorem, however we provide here a different self-contained proof. Proposition 3.19 ensures us that L has a completely inhomogeneous ideal K such that L/K is a subalgebra of gl(n, f ). Therefore Loop(L/K) is a g-subalgebra of Loop(gl(n, f )). By Proposition 3.3 L is isomorphic to the g-subalgebra of Loop(gl(n, f )) generated by L1K+K . In Proposition 3.14 we proved that a periodic just infinite Lie g-algebra is ideally r-constrained for a suitable r. In the solvable case we may give an estimate for such an r. To do this we need the following: Definition 3.23. Let L be a g-algebra and M an L-g-module. The width of M is defined as wd(M) := sup∞ i=0 dim(Mi ) Note that wd(M) is not necessarily finite. Clearly a periodic algebra (in particular a just infinite solvable g-algebra) has finite width. We are now ready to state and prove the:

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Theorem 3.24. Let L be a just infinite solvable Lie g-algebra with nil radical M. Let c be the nilpotency class of L/M and w be the width of M. Set r := wc + 1. Then L is ideally r-constrained. Proof. If I is a g-ideal of L not contained in Lj , we need to show that I contains Lj +wc . Clearly Ii = 0 for every i ≥ j − 1. In particular Ij +c = 0. Take a homogeneous element x of degree c which is not in M: like in Remark 3.9 ad x induces a M-period t of degree c. The period t sends Ii into Ii+c for i ≥ c + 1, so that ∞ dim(Ij +kc ) k=1 is a non-decreasing sequence of positive integers bounded above by ). This means w. Therefore there exists k¯ ≤ w such that dim(Ij +kc ¯ ) = dim(Ij +(k+1)c ¯ j +(k+1)c−1 ¯ . If we set I˜ := j +kc Ii we have that [I˜, L1 ] ≤ I˜ + I˜t that Ij +kc ¯ t = Ij +(k+1)c ¯ ¯ ∞ i so that Iˆ := 0 I˜t is a non-trivial g-ideal. As L is just infinite, Iˆ contains a term of the lower central series of L so that there exists h ≥ 0 such that Iˆ ¯ =L ¯ . j +(k+h)c

j +(k+h)c

h h Since Iˆj +(k+h)c = Ij +kc ¯ ¯ t and t induces an injective linear map from Lj +kc ¯ into ¯

j +kc ≥ Lj +wc , as required. , we have that Ij +kc Lj +(k+h)c ¯ ¯ = Lj +kc ¯ . Therefore I ≥ L

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R. Camina, The Nottingham group, in New Horizons in pro-p Groups (M. du Sautoy, D. Segal, and A. Shalev, eds.), Birkäuser, Basel 2000, 205–221.

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A. R. Camina and R. D. Camina, Pro-p groups of finite width, Comm. Algebra 29 (2001), 1583–1593.

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N. Gavioli and V. Monti, Ideally constrained Lie algebras, J. Algebra 253 (2002), 31–49.

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N. Gavioli, V. Monti, and D. Young, Metabelian thin Lie algebras, J. Algebra 77 (2001), 102–117.

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G. Klaas, C. R. Leedham-Green, and W. Plesken, Linear Pro-p-Groups of Finite Width, Lecture Notes in Math. 1674, Springer-Verlag, Berlin, Heidelberg, New York 1997.

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Norberto Gavioli, Dipartimento di Matematica Pura ed Applicata, Università degli Studi di L’Aquila, Via Vetoio, 67010 Coppito (L’Aquila) AQ, Italy E-mail: [email protected] Valerio Monti, Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate, Università degli Studi di Roma ‘La Sapienza’, via Scarpa, 16, 00161 Roma, Italy E-mail: [email protected] Carlo M. Scoppola, Dipartimento di Matematica Pura ed Applicata, Università degli Studi di L’Aquila, Via Vetoio, 67010 Coppito (L’Aquila) AQ, Italy E-mail: [email protected]

On the Schur indices of cuspidal unipotent characters Meinolf Geck

Abstract. In previous work of Gow, Ohmori, Lusztig and the author, the Schur indices of all unipotent characters of finite groups of Lie type have been explicitly determined except for six cases in groups of type F4 , E7 and E8 . In this paper, we show that the Schur indices of all cuspidal unipotent characters for type F4 and E8 are 1, assuming that the group is defined over a field of “good” characteristic. This settles four out of the six open cases. For type E7 , we show that the Schur indices are at most 2. 2000 Mathematics Subject Classification: Primary 20C15; Secondary 20G40

1. Introduction By the work of Feit, Gow, Ohmori, Turull, . . . , one expects that the Schur index of an irreducible character of a finite simple group is 1 or 2. This paper is a contribution to a solution of this problem, as far as the unipotent characters of a finite group of Lie type are concerned. By Lusztig [16], the Schur indices of all rational-valued unipotent characters are generally 1. However, there are exceptions in the finite unitary groups (Ohmori [20]) and the Ree groups 2F4 (q) ([6]) where the Schur indices are 2. In [6], we were also able to compute the Schur indices of some non-rational valued unipotent characters, but not all. This paper deals with some of the remaining cases, which arise in groups of type F4 , E7 and E8 . Our principal tool are Kawanaka’s generalized Gelfand–Graev characters [11]. These are characters induced from certain unipotent subgroups. It is known that every unipotent character occurs with “small” multiplicity in such an induced character. We shall show here that certain generalized Gelfand–Graev representations can actually be realized over Q (in type F4 or E8 ) or over a quadratic extension of Q (in type E7 ). The method for doing this is inspired by Ohmori [19], who considered the case of ordinary Gelfand–Graev representations. Furthermore, using the explicit multiplicity formulas obtained by Kawanaka [11] and Lusztig [14], we check that the unipotent characters that we wish to consider all occur with multiplicity 1 in those generalized Gelfand–Graev representations.

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Combining the previous work by Ohmori, Lusztig and the author with the results obtained in this paper, the known results on Schur indices of unipotent characters can be summarized as follows. Let G be a simple algebraic group and F : G → G be an endomomorphism such that GF is a finite (twisted or untwisted) group of Lie type. Let ρ be a unipotent irreducible character of GF . Then the Schur index of ρ is given as follows. (1) Assume that ρ is cuspidal and rational-valued. (a) If GF is of type 2An−1 where n = s(s + 1)/2 for some s 1, then the Schur index of the unique cuspidal unipotent character of GF is 1 if n is congruent to 0 or 1 modulo 4, and 2 otherwise; see Ohmori [20]. (b) If GF is a Ree group of type 2F4 , then the Schur index of the unique cuspidal unipotent character which occurs with even multiplicity in all Deligne–Lusztig generalized characters RT ,1 is 2; see [6], Theorem 1.6. (c) In all other cases, the Schur index of ρ is 1; see Lusztig [16], Theorem 0.2. (2) Assume that ρ is cuspidal and not rational-valued. Then the Schur index of ρ is 1, except possibly for the two cuspidal unipotent characters in type E7 (where the Schur index√ is at most 2) or the two cuspidal unipotent characters with character field Q( −1) in type E8 over a field of characteristic 5; see Table 1 and Corollary 1.5 in [6], and Sections 2, 3 in the present paper. See Gow [9], p. 119, for groups of type 2B 2 and 2G2 . (3) Assume that ρ is unipotent but not necessarily cuspidal. Then ρ occurs with non-zero multiplicity in the Harish-Chandra induction RLG (ψ) where L is an F -stable Levi complement in some F -stable proper parabolic subgroup of G and ψ is a cuspidal unipotent character of LF . In this situation, the Schur index of ρ equals that of ψ; see [6], Proposition 5.6. Thus, there are two remaining cases to be dealt with: • the characters E7 [±ξ ] in type E7 ; √ • the characters E8 [± −1] for a group of type E8 in characteristic 5. For each of these cases, we show that the determination of the Schur index can be reduced to the problem of computing explicitly the values of certain induced characters. One of these problems is discussed in [7].

2. On the rationality of generalized Gelfand–Graev representations The aim of this section is to show that certain generalized Gelfand–Graev representations of a finite group of Lie type can be realized over Q. For this purpose, we have to recall in some detail the construction of these representations. We shall freely

On the Schur indices of cuspidal unipotent characters

89

use standard results and notations concerning (connected reductive) linear algebraic groups and their Lie algebras (see Carter [2] and the references there). Let G be a connected reductive group defined over the finite field Fq , with corresponding Frobenius map F : G → G. We fix an F -stable Borel subgroup B ⊆ G and write B = U.T where U is the unipotent radical of B and T is an F -stable maximal torus. Let X = Hom(T , k × ) be the character group of T and ⊆ X be the root system of G with respect to T . Then B determines a positive system + ⊆ and a corresponding set of simple roots ⊆ + . We have G = T , Xα | α ∈ and U = Xα α∈+

where Xα is the root subgroup corresponding to α ∈ . (Here, it is understood that the product is taken over some fixed order of the roots.) For each α ∈ , let α ∨ ∈ Y = Hom(k × , T ) be the corresponding coroot. Given λ ∈ X and µ ∈ Y , let λ, µ be the unique integer such that γ λ,µ = (λ ◦ µ)(γ ) for all γ ∈ k × . This defines a non-degenerate pairing , : X × Y → Z; the matrix C := α, β ∨ α,β∈ is the Cartan matrix of G. Now, for any α ∈ , there is an isomorphism of algebraic groups xα : k + → Xα such that txα (ξ )t −1 = xα (α(t)ξ )

for all t ∈ T and ξ ∈ k.

Since we will only need to consider this case, let us assume that F is of split type, such that F (xα (ξ )) = xα (ξ q ) for all α ∈ , ξ ∈ k, F (t) = t q for all t ∈ T . We consider the conjugacy classes of unipotent elements in G. For this purpose, we assume throughout that the characteristic p of Fq is a good prime for G. Recall that this means that p is good for each simple factor involved in G, and that the conditions for the various simple types are as follows. An : Bn , Cn , Dn : G2 , F4 , E6 , E7 : E8 :

no condition, p = 2, p = 2, 3, p = 2, 3, 5.

Then it is known (see Kawanaka [11] and the references there; see also Premet [21]) that one can naturally attach to each unipotent class of G a so-called weighted Dynkin diagram, that is, an additive map d : → Z such that

d(α) ∈ {0, 1, 2} for all α ∈ .

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The assignment from unipotent classes to additive maps as above is injective, but not surjective in general. Complete lists of weighted Dynkin diagrams for the various types of simple algebraic groups can be found in Carter [2], §13.1. Given such a weighted Dynkin diagram d, the corresponding unipotent class is determined as follows. We set Ld := T , Xα | α ∈ , d(α) = 0 and Ud,i := Xα α∈+ d(α)i

for i = 1, 2, 3, . . . . Then Pd := Ud,1 .Ld is a parabolic subgroup of G, with unipotent radical Ud,1 and Levi complement Ld . By Kawanaka [11], Theorem 2.1.1, there is a unique unipotent class C in G such that C ∩ Ud,2 is dense in Ud,2 ; furthermore, C ∩ Ud,2 is a single Pd -conjugacy class and we have CG (u) ⊆ Pd

for all u ∈ C ∩ Ud,2 .

Then C is the unipotent class attached to the given weighted Dynkin diagram d. In order to define the generalized Gelfand–Graev representation associated with F , we need to introduce some further notation. Let g be the Lie an element u ∈ C ∩ Ud,2 algebra of G over k = Fq . Then g is also defined over Fq and we have a corresponding Frobenius map F : g → g. Let t ⊆ g be the Lie algebra of T . We have a Cartan decomposition g=t⊕ keα where F (t) = t and F (eα ) = eα for all α ∈ . α∈

We shall set cα := κ(eα∗ , eα ) for any α ∈ + , where κ : g×g → k is a non-degenerate G-invariant, associative bilinear form and x → x ∗ is an opposition Fq -automorphism of g, that is, an automorphism such that t∗ = t and eα∗ ∈ Fq e−α for all α ∈ (see Kawanaka [11], §3.1). × Finally, we fix a non-trivial homomorphism χ : F+ q →C . F ; write Definition 2.1 (Kawanaka). Consider a unipotent element u ∈ C ∩ Ud,2 F u∈ xα (ηα ) · Ud,3 where ηα ∈ Fq .

(∗)

α∈+ d(α)=2

F → C× by With this notation, we define a map ϕu : Ud,2 ϕu xα (ξα ) = χ cα ηα ξα where ξα ∈ Fq . α∈+ d(α)2

α∈+ d(α)=2

Since κ(eα , eβ ) = 0 unless α = −β, this definition coincides with Kawanaka’s original definition in [11], (3.1.5); see also [5], Chapter 1. The map ϕu actually is a F . Inducing that character from group homomorphism, that is, a linear character of Ud,2

On the Schur indices of cuspidal unipotent characters

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F to GF , we obtain Ud,2 F

F F 1/2 (ϕu ) = [Ud,1 : Ud,2 ] · u IndG UF d,2

where u is the character of the generalized Gelfand–Graev representation associated F : U F ] is an even with u; see Kawanaka [11], (3.1.12). Note that the index [Ud,1 d,2 power of q, so the square root exists. F , Remark 2.2. In the above setting, let us choose another element u ∈ C ∩ Ud,2 F xα (ηα ) · Ud,3 where ηα ∈ Fq . u ∈ α∈+ d(α)=2

Then we can also apply the above constructions to u . Thus, we obtain a corresponding generalized Gelfand–Graev character u such that F

F F 1/2 (ϕu ) = [Ud,1 : Ud,2 ] · u . IndG UF d,2

Using the defining formula for u , it is straightforward to check that u = u if u and u lie in the same PdF -orbit. (Indeed, if u = gug −1 where g ∈ PdF , then F and so the induced characters are equal.) Now ϕu (x) = ϕu (g −1 xg) for all x ∈ Ud,2 let C be the G-conjugacy class of u. Then C F splits into orbits under the finite group GF , and it is well-known that these GF -orbits are parametrized by the F -conjugacy classes of CG (u)/CG (u)◦ . Since CG (u) ⊆ Pd , a complete set of representatives for F . the GF -orbits in C F can be found inside C ∩ Ud,2 Assume now that G is simple of adjoint type and write = {α1 , . . . , αl } where l is the rank of G. Then we have X = Z. Let {ω1 . . . , ωl } be the dual basis of Y , that is, we have αi , ωj = δij for 1 i, j l. Now we can state the following result which generalizes the arguments in Ohmori [19], Lemma 2 and Proposition 1(i). (See also Example 2.4 below.) Proposition 2.3. Let G be simple of adjoint type and assume that there exist integers nj ∈ Z (1 j l) such that l

nj α, ωj = 1 for all α ∈ + such that d(α) = 2 and ηα = 0,

j =1

where the coefficients ηα are defined as in Definition 2.1 (∗). Then the character F : U F ]1/2 · can be realized over Q. If, moreover, we also have [Ud,1 u d,2 l

nj α, ωj = 0 for all α ∈ + such that d(α) = 0,

j =1 F : U F ]1/2 · can be realized over Q, for every u ∈ C ∩ U F . then [Ud,1 u d,2 d,2

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Proof. Let ν be a generator for the multiplicative group of Fq . We claim that there exists an element t ∈ T F such that for all α ∈ + with d(α) = 2 and ηα = 0.

α(t) = ν

(†)

This is seen as follows. By [2], Proposition 3.1.2, the map h : k × × · · · × k × → T ,

(x1 , . . . , xl ) →

l

ωi (xi ),

i=1

l factors

is an Fq -isomorphism. In particular, we have T F = {h(x1 , . . . , xl ) | xi ∈ F× q }. Now set t = h(ν n1 , . . . , ν nl ) ∈ T F . Let α ∈ + be such that d(α) = 2 and ηα = 0. Then we have l l l α,ω α(t) = xj j = ν nj α,ωj = ν j =1 nj α,ωj = ν, j =1

j =1

as required. Thus, (†) is proved. F . Now let H := t. Then H is a group of order q − 1 which normalizes Ud,2 F to the semidirect product U F .H and denote the induced Let us induce ϕu from Ud,2 d,2 character by ψu . We will show that ψu is a rational-valued irreducible character. First F . Furthermore, by Mackey’s note that ψu has non-zero values only on elements in Ud,2 F is given by h h −1 formula, the restriction of ψu to Ud,2 h∈H ϕu where ϕu (x) = ϕu (hxh ) F for all x ∈ Ud,2 . Now let F x := xα (ξα ) ∈ Ud,2 for any ξα ∈ Fq . α∈+ d(α)2

Then we have txt −1 =

txα (ξα )t −1 =

α∈+ d(α)2

xα (α(t)ξα )

α∈+ d(α)2

and so, using (†), ϕut (x) = χ

cα ηα α(t)ξα = χ ν cα ηα ξα .

α∈+ d(α)=2

α∈+ d(α)=2

Similarly, for any 1 i q − 1, we obtain i

ϕut (x) = χ(ν i γx )

where γx :=

α∈+ d(α)=2

cα ηα ξα ∈ Fq .

On the Schur indices of cuspidal unipotent characters

93

i

This shows, first of all, that the characters ϕut (1 i q − 1) are pairwise different and, hence, the character ψu is seen to be irreducible. Furthermore, we obtain that q−1 q−1 i q − 1 if γx = 0, ϕuh (x) = ϕut (x) = χ (ν i γx ) = ψu (x) = −1 if γx = 0. h∈H i=1 i=1 F . Thus, ψ is a rational-valued In particular, this shows that ψu (x) ∈ Z for all x ∈ Ud,2 u F .H . In order to show that ψ can be realized over Q, we irreducible character of Ud,2 u note that (again by Mackey’s formula), the restriction of ψu to H is the character of the regular representation of H . Hence the trivial character 1H occurs with multiplicity 1 in that restriction. By Frobenius reciprocity, ψu itself occurs with multiplicity 1 in F .H obtained by inducing 1 . Since the latter character clearly is the character of Ud,2 H realized over Q, a standard argument on Schur indices shows that ψu can be realized over Q (see Isaacs [10], Corollary 10.2). Finally, by the transitivity of induction, the character F

F F 1/2 (ϕu ) = [Ud,1 : Ud,2 ] · u IndG UF d,2

can be realized over Q, as claimed. Now assume that the integers nj satisfy the additional equations described above F . Thus, we have u = gug −1 where and consider an arbitrary element u ∈ C ∩ Ud,2 g ∈ Pd . Let F xα (ηα ) · Ud,3 where ηα ∈ Fq . u ∈ α∈+ d(α)=2

We claim that if ηα = 0, then α = α1 + β where d(β) = 0, d(α1 ) = 2 and ηα1 = 0. To see this, it is enough to consider the special case where g ∈ T or where g ∈ Xβ with d(β) 0. The assertion is clear if g ∈ T . On the other hand, if g ∈ Xβ where d(β) 0, then Chevalley’s commutator relations show that gug −1 will lie in a product of root subgroups Xiβ+α1 where d(α1 ) 2 and i = 0, 1, 2, . . . , and the assertion also follows. We can now conclude that the integers nj automatically have the property that l j =1

nj α, ωj = 1 for all α ∈ + such that d(α) = 2 and ηα = 0.

Thus, we can apply the above arguments to u as well and conclude that the character F : U F ]1/2 · can be realized over Q. [Ud,1 u d,2 Example 2.4 (cf. Ohmori [19], p. 151). Assume that G is simple of adjoint type. The weighted Dynkin diagram corresponding to the conjugacy class of regular unipotent elements in G is the map d0 : → Z such that d0 (αi ) = 2 for all 1 i l. In this

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case, we have Pd0 = B, Ld0 = T and Ud0 ,1 = Ud0 ,2 = U ;

furthermore, u = li=1 xαi (ηi ), where ηi ∈ F× q , is a representative in that class. Now the system of equations in Proposition 2.3 is simply given by l

nj αi , ωj = 1 for all 1 i l.

j =1

Thus, we have nj = 1 for 1 j l. Consequently, u (which is just an ordinary Gelfand–Graev character) can be realized over Q. Example 2.5. Assume that G is simple of adjoint type and that d is a weighted Dynkin diagram of the following special type: There exists some i0 ∈ {1, . . . , l} such that d(αi0 ) = 2

and d(αi ) = 0

for i ∈ {1, . . . , l} \ {i0 }.

We claim that, in this case, the generalized Gelfand–Graev character u can be realized F . This is seen as follows. First note that U over Q, for any u ∈ C ∩ Ud,2 d,1 = Ud,2 . Now let ηα be defined as in Definition 2.1(∗) and consider the two sets of equations in Proposition 2.3. The second set of equations reads l

nj αi , ωj = 0

for all i ∈ {1, . . . , l} \ {i0 }.

j =1

This yields ni = 0 for all i = i0 . Then the first set of equations reads ni0 α, ωi0 = 1

for all α ∈ + such that d(α) = 2 and ηα = 0.

However, writing any such α as a linear combination of simple roots, we see that αi0 always occurs with coefficient 1 and so the above equations hold with ni0 = 1. Thus, the systems of equations in Proposition 2.3 have a unique solution. We close this section with a general remark concerning the Schur indices of unipotent characters. This remark shows that it will be enough to consider the unipotent characters of simple groups of adjoint type. Remark 2.6. Assume that G is simple of adjoint type. Let G1 be an algebraic group over Fq such that G1 /Z(G1 ) is simple of the same type as G. We denote the Frobenius map on G1 again by F . Then we have a surjective homomorphism of algebraic groups ϕ : G1 → G which is defined over Fq and such that ker(ϕ) = Z(G1 ). Let χ be a unipotent character of GF . Denote by χ1 the character of GF1 which is obtained by first restricting χ to ϕ(GF1 ) and then pulling back to GF1 via ϕ. By [4], Proposition 7.10, the map χ → χ1 defines a bijection between the unipotent characters of GF and GF1 , respectively. One can express this by saying that the unipotent characters are “insensitive” to the center of G.

On the Schur indices of cuspidal unipotent characters

95

For a unipotent character χ ∈ Irr(GF ), denote by Q(χ ) = Q(χ (g) | g ∈ GF ) the character field of χ and by mQ (χ) the Schur index of χ ; we use similar notations for a unipotent character of GF1 . We claim that Q(χ) = Q(χ1 )

and

mQ (χ ) = mQ (χ1 ).

Proof. First note that we clearly have Q(χ1 ) ⊆ Q(χ ). To show that we have equality, let E ⊇ Q be a finite Galois extension such that any unipotent character of GF and of GF1 can be realized over E. Given any τ ∈ Gal(E/Q) and any unipotent character χ ∈ Irr(GF ), we denote by χ τ the irreducible character obtained by algebraic conjugation; it is still a unipotent character of GF (see the remarks in [6], (5.1)). Now the map χ → χ1 certainly is compatible with field automorphisms. Thus, we have χ1τ = (χ τ )1 . Since the map χ → χ1 is a bijection, we conclude that χ1τ = χ1 if and only if χ τ = χ . This implies that Q(χ) = Q(χ1 ), as required. Now consider the Schur indices. First note that we certainly have mQ (χ1 ) mQ (χ ). (Indeed, if χ can be realized over some extension field of Q, then χ1 is automatically realized over the same field.) To show the reverse inequality, we argue as follows. The character χ1 can be realized over an extension field E1 ⊇ Q such that [E1 : Q] = mQ (χ1 ). We now regard χ1 as a character of ϕ(GF ). By Frobenius reciprocity, χ occurs with multiplicity 1 in the character obtained by inducing χ1 from ϕ(GF ) to GF . Since that induced character can also be realized over E1 , a standard argument on Schur indices ([10], Corollary 10.2) shows that mQ (χ ) mQ (χ1 ), as required.

3. Simple groups of type G2 , F4 and E8 Throughout this section, let G be simple of (adjoint) type G2 , F4 or E8 , defined over the finite field Fq with corresponding Frobenius map F : G → G. Then F is of split type, in the sense of the previous section. The cuspidal unipotent characters of GF and the previously known results on their Schur indices are listed in Table 1. Now let us assume that G is defined over a field of good characteristic. All the cuspidal unipotent characters of GF have the same “unipotent support” in the sense of Lusztig [14]; see also [8], Proposition 4.2. Hence there is a unique unipotent class C0 in G such that all the cuspidal unipotent characters of GF have unipotent support C0 . Let e0 = dim Bu (the dimension of the variety of Borel subgroups containing an element u ∈ C0 ). Then, by [8], Theorem 3.7, q e0 is the exact power of q dividing the degrees of the unipotent characters of GF having unipotent support C0 . By inspection of the tables in [2], Chapter 9, we find that e0 = 1, 4 or 16 for G of type G2 , F4 or E8 , respectively. Furthermore, for G of type G2 or F4 , the class C0 is uniquely determined by this condition. In type E8 , we use the additional information that CG (u)/CG (u)◦ must be non-trivial (see once more the formula in [8], Theorem 3.7). Then C0 is uniquely determined. The weighted Dynkin diagram d0 associated to C0 is specified

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Table 1. Schur indices of cuspidal unipotent characters in type G2 , F4 and E8 (notation of Carter [2], §13.9; see also [6], Table 1)

Type G2 F4

E8

Character G2 [1], G2 [−1] G2 [θ], G2 [θ 2 ] F4I [1], F4I I [1], F4 [−1] F4 [i], F4 [−i] F4 [θ], F4 [θ 2 ] E8I [1], E8I I [1], E8 [−1] E8 [−θ], E8 [−θ 2 ], E8 [θ], E8 [θ 2 ] E8 [i], E8 [−i] E8 [ζ ], E8 [ζ 2 ], E8 [ζ 3 ], E8 [ζ 4 ] i :=

√ −1,

Schur index 1 (see Lusztig [16]) 1 (see [6]) 1 (see Lusztig [16]) ? (1 in char. 2, 3; see [6]) 1 (see [6]) 1 (see Lusztig [16]) ? (1 in char. 2, 3, 5; see [6]) ? (1 in char. 2, 3, 5; see [6]) ? (1 in char. 2, 3; see [6]) 1 (see [6])

θ := exp(2π i/3),

ζ := exp(2π i/5).

in Table 2. Furthermore, we have CG (u)/CG (u)◦ ∼ = S3 , S4 or S5 for G of type G2 , F4 or E8 , respectively.

Table 2. Weighted Dynkin diagrams for the unipotent supports of cuspidal unipotent characters in type G2 , F4 and E8

0 w < α1

G2

E8

0 w α1

2 w α2

0 w α1

F4

0 w α3

α4

0 w

2 w α5

2 w > α2 0 w α6

0 w α7

0 w α3

0 w α3 0 w α8

0 w α2 Now let us fix a cuspidal unipotent character ρ of GF . By Lusztig, [14], Theorem 11.2, there exists some u ∈ C0 ∩UdF0 ,2 such that ρ occurs with “small” multiplicity in u . In the present situation, Kawanaka actually obtained an explicit multiplicity formula. To state that formula, we need to recall some facts about the parametrization of unipotent characters; see Lusztig [12], Main Theorem 4.23.

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97

Let F0 be the unique family of unipotent characters which contains the cuspidal unipotent characters. Let G0 be the finite group attached to that family; we have G0 ∼ = CG (u)/CG (u)◦ in the present situation. Let M0 be the set of all pairs (x, σ ) where x ∈ G0 (up to conjugacy) and σ ∈ Irr(CG0 (x)). Then we have a bijection M0 → F0 ,

(x, σ ) → ρ(x,σ ) .

(which satisfies some further conditions which we do not need to recall here). On the other hand, there is a well-defined “split” element u0 ∈ C0 ∩ UdF0 ,2 ; see Shoji [22], Remark 5.1. Having fixed that element, there is a canonical parametrization of the GF -conjugacy classes contained in C0F by the conjugacy classes of G0 . We shall denote by uy the generalized Gelfand–Graev character with respect to a unipotent element uy ∈ C0 ∩ UdF0 ,2 in the GF -class parametrized by y ∈ G0 . Having fixed this notation, we can now state the following result. Theorem 3.1 (Kawanaka). Recall that G is a group of type G2 , F4 or E8 in good characteristic. Let ρ(x,σ ) , uy be the multiplicity of ρ(x,σ ) in the generalized Gelfand– Graev character uy , where x, y ∈ G0 and σ ∈ Irr(CG0 (x)). Then we have σ (1) if x = y (mod G0 ), ρ(x,σ ) , uy = 0 otherwise. Proof. Kawanaka obtained an explicit formula for the values of all the generalized Gelfand–Graev characters of GF ; see [11], Corollary 3.2.7 and Lemma 3.3.10. Furthermore, in [11], §4.2, he obtained explicit formulas for the values of the unipotent characters on unipotent elements. (In the latter reference, it is actually assumed that the characteristic is large. But this hypothesis is only used in reference to results on the Green functions of GF , which are now known to hold in general; see Shoji [23], Theorem 5.5.) The above multiplicity formula is obtained by a simple formal rewriting of Kawanaka’s formulas. In large characteristic, that formula could also be deduced from Lusztig’s more general results in [14]. Corollary 3.2. Recall that G is a group of type G2 , F4 or E8 in good characteristic. Then every cuspidal unipotent character of GF has Schur index 1. Proof. By inspection of the tables in [2], §13.9, all cuspidal unipotent characters of GF are labelled by pairs (x, σ ) ∈ M0 where σ (1) = 1. Thus, by Theorem 3.1, every such character occurs with multiplicity 1 in some generalized Gelfand–Graev character attached to an element in C0 ∩ UdF0 ,2 . Now we notice that the weighted Dynkin diagrams in Table 2 are of the type considered in Example 2.5. Thus, all the generalized Gelfand–Graev characters associated with unipotent elements in C0 ∩UdF0 ,2 can be realized over Q. Hence, the assertion follows by a standard argument on Schur indices (see Isaacs [10], Corollary 10.2).

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The above result also covers some of the cases (at least in good characteristic) already dealt with by Lusztig [16] or the author [6]. In combination with the previously known results listed in Table 1, we see that there is only one case left that remains to be considered: the characters E8 [±i] for a group of type E8 in characteristic 5. We shall now outline a strategy for dealing with this case. That strategy is analogous to that in [6], (6.5). For the remainder of this section, let G be a simple group of type E8 over Fq where q is odd and p ≡ 1 mod 4. Let us label the simple roots of G as in Table 2 and let {ω1 , . . . , ω8 } be the dual basis of Y . Then we have T = {h(x1 , . . . , x8 ) | xi ∈ k × } as in the proof of Proposition 2.3. Now let ν be a generator of the multiplicative group of Fq and consider the semisimple element s := h(1, 1, 1, 1, 1, ν (q−1)/4 , 1, 1) ∈ T F . Thus, s is an element of order 4. Its centralizer is given by G1 = T , Xα | α ∈ 1 where 1 = {α ∈ | α(s) = 1}. It is readily checked that a system of simple roots of 1 is given by 1 = ( \ {α6 }) ∪ {−α0 }, where α0 = 2α1 + 3α2 + 4α3 + 6α4 + 5α5 + 4α6 + 3α7 + 2α8 is the unique highest root in . The root system 1 is of type D5 × A3 . Now, by Lusztig [13], Proposition 21.3, there is a well-defined unipotent class C1 ⊆ G1 which supports a “cuspidal local system”. That unipotent class is determined explicitly by [17], Corollary 4.9, as far as the D5 -factor is concerned. (We necessarily have the class of regular unipotent elements in the A3 -factor; see [13], §18.) Using [2], §13.9, we see that the weighted Dynkin diagram d1 of C1 is given as follows: D5 ×A3

2 w α1

2 w α3

α4

0 w

2 w α5

2 w α7

2 w α8

2 w −α0

2 w α2 By [1], Theorem 11.3.2, we can explicitly identify an adjoint group of type D5 with a 10-dimensional orthogonal group. Determining the Jordan normal form of the matrix corresponding to the element u1 := xα1 (1)xα5 (1)xα2 (1)xα3 +α4 (1)xα4 +α5 (1) · xα7 (1)xα8 (1)x−α0 (1), ◦ (u ) ∼ Z/4Z; see [6], we see that u1 ∈ C1 ∩ UdF1 ,2 . Now, we have CG1 (u1 )/CG 1 = 1 Table 2, and the references there. Let ψ be one of the two faithful irreducible characters of this group and define (C1 ,ψ) as in [14], (7.5). Thus, (C1 ,ψ) is a certain linear combination (where the coefficients involve the values of ψ) of the generalized Gelfand–Graev characters of GF1 associated with the various unipotent elements in C1 ∩ UdF1 ,2 . Now we can state the following result.

On the Schur indices of cuspidal unipotent characters

99

Proposition 3.3. In the above setting, assume that the following conditions hold: (a) The values of (C1 ,ψ) are given by Lusztig’s formula [14], Corollary 7.6. (This is known to be true if the characteristic is “large”.) (b) The assertion in [15], Theorem 0.8, holds for cuspidal character sheaves in G. (This is known to be true if the characteristic is good; see [23], Theorem 4.1.) Then the Schur index of the cuspidal unipotent character E8 [±i] is 1. Proof. We begin by showing that the generalized Gelfand–Graev characters of GF1 which are associated with the unipotent elements in C1 ∩ UdF1 ,2 can be realized over Q. For this purpose, we use an argument analogous to that in Proposition 2.3. We shall consider a system of equations as in Proposition 2.3 (but with respect to the unipotent element u1 ∈ G1 specified above): 8 1 for α ∈ {α1 , α5 , α2 , α3 + α4 , α4 + α5 , α7 , α8 , −α0 }, nj α, ωj = 0 for all α ∈ + such that d1 (α) = 0. j =1 One readily checks that this system has a unique solution: (n1 , n2 , . . . , n8 ) = (1, 1, 1, 0, 1, −5, 1, 1). Thus, t := h(ν, ν, ν, 1, ν, ν −5 , ν, ν) ∈ T F is a semisimple element of order q − 1. Now let u ∈ C1 ∩ UdF1 ,2 . Then, as in the proof of Proposition 2.3, we see that α(t) = ν (if α is involved in the expression of u as a product of root subgroup elements) or α(t) = 1 (if d1 (α) = 0). Let H = t and denote by ψu the character obtained by inducing ϕu from UdF1 ,2 to UdF1 ,2 .H . Following exactly the same kind of arguments as in the proof of Proposition 2.3, we see that ψu is an irreducible character which can actually be realized over Q. Consequently, the generalized Gelfand–Graev character u of GF1 can also be realized over Q, as claimed. We now follow the line of argument in [6], (6.5). First note that the character values of E8 [±i] generate the field Q(i); see [6], Table 1. Thus, we must show that E8 [±i] can be realized over Q(i). Let be the G-conjugacy class of su1 . Then supports a “cuspidal local system” E ; see [6], Table 2 and the references there. Using [12], Main Theorem 4.23, the explicit description of the Fourier matrix for type E8 , and our assumption (b), we see that E8 [±i], χ(,E ) =

1 ε 4

where ε ∈ C, |ε| = 1;

here, χ(,E ) is a characteristic function of (, E ) and , denotes the usual hermitian product on the space of complex-valued class functions on GF . On the other hand, ◦ (u ) ∼ Z/4Z, the set C ∩ U F splits into four classes in GF , since CG1 (u1 )/CG 1 = 1 1 d1 ,2 1 with representatives u1 , u2 , u3 , u4 ∈ C1 ∩UdF1 ,2 say. Then, our assumption (a) implies

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that we have 4 χ(,E ) =

4

F

r=1 λ∈Irr(Z1 )

ξr,λ IndG (λ ϕur ) Z ×U F d1 ,2

1

where Z1 := Z(G1 ) = s and ξr,λ ∈ Z[i]; see the analogous equation in [6], (6.5), and note that Z(G1 )◦ = {1}. Now, since Z1 ∩ H = {1}, it is easily shown that (Z1 ×UdF ,2 ).H

IndZ

1

F 1 ×Ud ,2 1

UdF ,2 .H

(λ ϕur ) = λ IndU F1

d1 ,2

(ϕur ) = λ ψur

can be realized over Q(i). Thus, we have 4 χ(,E ) = α rα ρα where rα ∈ Z[i] and ρα are characters of GF which can be realized over Q(i). Since E8 [±i], 4χ(,E ) = ε has absolute value 1, we conclude that the greatest common divisor of the multiplicities E8 [±i], ρα must be 1. Hence, a standard argument on Schur indices ([10], Corollary 10.2) implies that E8 [±i] can be realized over Q(i), as desired. In principal, the hypothesis on (C1 ,ψ) might be verified by an explicit computation for a group of type D5 ×A3 (in the same way as Kawanaka [11] verified that hypothesis for groups of exceptional type).

4. Simple groups of type E7 Let G be a simple group of adjoint type E7 in good characteristic. There are precisely √ two cuspidal unipotent characters of GF denoted by E7 [±ξ ] where ξ = −q; see the table in [2], §13.9. By [6], Table 1, the character values of E7 [±ξ ] lie in the field Q(ξ ) and the Schur index of E7 [±ξ ] is 1, 2 or 4. Furthermore, by [6], Example 6.4, we already know that the Schur index is 1 if p ≡ 1 mod 4 or if q is not a square, where p is the characteristic of Fq . Thus, the remaining task is to determine the Schur index when q is a square and p ≡ 1 mod 4. Let C0 be the common unipotent support of the two cuspidal unipotent characters. Arguing as in the previous section (that is, checking the exact power of q dividing the degree of E7 [±ξ ] and comparing with the tables in [2], Chapter 9), we find that the corresponding weighted Dynkin diagram d0 is given as follows. E7

1 w α1

0 w α3

α4 0

1 w

w α2

0 w α5

1 w α6

0 w α7

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Lemma 4.1. Assume that q is an even power of p. Let u ∈ C0 ∩ UdF0 ,2 and recall the definition of the linear character ϕu from Section 2. Then there exists a field K ⊇ Q F (ϕu ) can be realized over K. such that [K : Q] 2 and IndG UF d0 ,2

Proof. The following argument is inspired by Ohmori [19], Proposition 1. By Mizuno [18], Lemma 28, a representative u0 ∈ C0 ∩ UdF0 ,2 is given by u0 = x20 (1)x21 (1)x24 (1)x28 (1)x30 (1), where the subscripts correspond to the following roots in + : 21 : α1 + α3 + α4 + α5 , 20 : α1 + α2 + α3 + α4 , 28 : α2 + α3 + 2α4 + α5 , 24 : α2 + α4 + α5 + α6 , 30 : α3 + α4 + α5 + α6 + α7 . We consider a set of equations similar to that in Proposition 2.3: 7 2 for α ∈ + labelled by 20, 21, 24, 28 and 30, nj α, ωj = 0 for all α ∈ + such that d0 (α) = 0. j =1 One readily checks that this system has a unique solution: (n1 , n2 , . . . , n7 ) = (1, 0, 0, 1, 0, 1, 0). Now let ν be a generator of the multiplicative group of Fp ⊆ Fq . Since q is an even power of p, we can find a square root of ν in Fq , which we denote by ν 1/2 ∈ Fq . Then the element t := h(ν n1 /2 , . . . , ν n7 /2 ) = h(ν 1/2 , 1, 1, ν 1/2 , 1, ν 1/2 , 1) ∈ T F has order 2(p − 1) and, as in the proof of Proposition 2.3, we see that α(t) = ν for all α ∈ + such that d0 (α) = 2 and ηα = 0 where the coefficients ηα are defined as in Definition 2.1(∗) (with respect to the unipotent element u ∈ C0 ∩ UdF0 ,2 ). Now let H = t. Then H is a group of order 2(p − 1) which normalizes UdF0 ,2 . Let us induce ϕu from UdF0 ,2 to the semidirect product UdF0 ,2 .H and denote the induced character by ψu . Arguing as in the proof of Ohmori [19], Proposition 1(ii), we see that ψu is the sum of two rational-valued irreducible characters ψ1 and ψ2 , where one of them, ψ1 say, can be realized over Q. By the Brauer–Speiser theorem (see [3], 74.27), the character ψ2 has Schur index at most 2. Thus, there exists a field K ⊇ Q such that [K : Q] 2 and ψ = ψ1 + ψ2 can be realized over K. Consequently, by the transitivity of induction, the character F

F

d0 ,2

d0 ,2 .H

(ϕu ) = IndG IndG UF UF can also be realized over K.

(ψu )

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Remark 4.2. We have remarked at the beginning of this section that it is enough to compute the Schur index of E7 [±ξ ] in the case where q is a square √ and p ≡ 1 mod 4. In this case, the character values of E7 [±ξ ] lie in the field Q( −1). Now Ohmori [19], p. 154, points out that the character ψ2 arising in the above proof has non-trivial local√Schur indices at ∞ and at the prime p. This implies √ that p ≡ 1 mod 4. Indeed, if we had K ⊆ Q( K is not contained in Q( −1) if √ √ −1), then ψ2 could be realized over Q( −1). Since p ≡ 1 mod 4, the field Q( −1) is contained in the field of p-adic numbers. Hence, ψ2 could be realized over the latter field, contradicting the fact that the local Schur index at p is not one. Corollary 4.3. Recall that G is a group of type E7 in good characteristic. Then the two cuspidal unipotent characters of GF have Schur index at most 2. Proof. We have already remarked at the beginning of this section that the Schur index of E7 [±ξ ] is 1 if q is not a square. So let us now assume that q is an even power of p. Let u0 ∈ C0 ∩ UdF0 ,2 . Then it is known that CG (u0 )/CG (u0 )◦ ∼ = Z/2Z (see F the tables in [2], §13.1). Thus, C0 splits into two classes in the finite group GF . Let u1 ∈ C0 ∩ UdF0 ,2 be a representative of the second GF -conjugacy class contained in C0F . Combining [8], Theorem 3.7 and Remark 3.8, we have that E7 [±ξ ], u0 + E7 [±ξ ], u1 = 1. (This could also be deduced from Kawanaka’s explicit multiplicity formulas [11].) Thus, E7 [±ξ ] occurs with multiplicity 1 in u for some u ∈ C0 ∩UdF0 ,2 . Consequently, we have F (ϕu ) = [UdF0 ,1 : UdF0 ,2 ]1/2 =: m0 , E7 [±ξ ], IndG UF d0 ,2

where the number m0 on the right hand side is odd since we are in good characteristic. Let K ⊇ Q be as in Lemma 4.1. Then a standard result on Schur indices (see Isaacs [10], Lemma 10.4) shows that the Schur index of E7 [±ξ ] divides [K : Q] m0 . However, as we have already remarked at the beginning of this section, the Schur index of E7 [±ξ ] must be 1, 2 or 4. Thus, since m0 is odd, we can now conclude that the Schur index of E7 [±ξ ] divides [K : Q] 2. Remark 4.4. The Schur index of E7 [±ξ ] equals 1 or 2, according to whether F E7 [±ξ ], IndG (ψ1 ) is odd or even, U F .H d0 ,2

where ψ1 is defined in the proof of Lemma 4.1. Thus, the problem is reduced to the computation of the above scalar product, and this will be discussed in [7]. Acknowledgements. I wish to thank the organizers of the “Finite Groups 2003” conference, C. Y. Ho, P. Sin, P. H. Tiep and A. Turull, for the invitation and the University of Gainesville for its hospitality.

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103

References

[1]

R. W. Carter, Simple groups of Lie type, Wiley, New York 1972; reprinted 1989 as Wiley Classics Library Edition.

[2]

R. W. Carter, Finite groups of Lie type: Conjugacy classes and complex characters, Wiley, New York 1985; reprinted 1993 as Wiley Classics Library Edition.

[3]

C. W. Curtis and I. Reiner, Methods of representation theory, vol. II, Wiley, New York 1987; reprinted 1994 as Wiley Classics Library Edition.

[4]

P. Deligne and G. Lusztig, Representations of reductive groups over finite fields, Ann. of Math. 103 (1976), 103–161.

[5]

M. Geck, Verallgemeinerte Gelfand-Graev Charaktere und Zerlegungszahlen endlicher Gruppen vom Lie-Typ, Dissertation, RWTH Aachen 1990.

[6]

M. Geck, Character values, Schur indices and character sheaves, Represent. Theory 7 (2003), 19–55.

[7]

M. Geck, The Schur indices of the cuspidal unipotent characters of the finite Chevalley groups E7 (q), Osaka J. Math., to appear.

[8]

M. Geck and G. Malle, On the existence of a unipotent support for the irreducible characters of finite groups of Lie type, Trans. Amer. Math. Soc. 352 (2000), 429–456.

[9]

R. Gow, Schur indices of some groups of Lie type, J. Algebra 42 (1976), 102–120.

[10] I. M. Isaacs, Character Theory of Finite Groups, Academic Press, New York 1976; corrected reprint by Dover Publ., New York 1994. [11] N. Kawanaka, Generalized Gelfand–Graev representations of exceptional algebraic groups I, Invent. Math. 84 (1986), 575–616. [12] G. Lusztig, Characters of reductive groups over a finite field, Ann. of Math. Stud. 107, Princeton University Press, Princeton 1984. [13] G. Lusztig, Character sheaves, IV, Adv. Math. 59 (1986), 1–63. [14] G. Lusztig, A unipotent support for irreducible representations, Adv. Math. 94 (1992), 139–179. [15] G. Lusztig, Remarks on computing irreducible characters, J. Amer. Math. Soc. 5 (1992), 971–986. [16] G. Lusztig, Rationality properties of unipotent representations, J. Algebra 258 (2002), 1–22. [17] G. Lusztig and N. Spaltenstein, On the generalized Springer correspondence for classical groups, in Algebraic groups and related topics, Adv. Stud. Pure Math. 6, Kinokuniya and North-Holland, Tokyo and Amsterdam, 289–316. [18] K. Mizuno, The conjugate classes of unipotent elements of the Chevalley groups E7 and E8 , Tokyo J. Math. 3 (1980), 391–459. [19] Z. Ohmori, On the Schur indices of certain irreducible characters of reductive groups over finite fields, Osaka J. Math 25 (1988), 149–159.

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[20] Z. Ohmori, The Schur indices of the cuspidal unipotent characters of the finite unitary groups, Proc. Japan Acad. Ser. A Math. Sci. 72 (1996), 111–113. [21] A. Premet, Nilpotent orbits in good characteristic and the Kempf–Rousseau theory, J. Algebra 260 (2003), 338–366. [22] T. Shoji, Green functions of reductive groups over a finite field, Proc.Symp. Pure Math. 47, Amer. Math. Soc., Providence, RI 1987, 289–302. [23] T. Shoji, Character sheaves and almost characters of reductive groups II, Adv. Math. 111 (1995), 314–354. Meinolf Geck, Institut Girard Desargues, Université Lyon 1, 21 av Claude Bernard, 69622 Villeurbanne cedex, France E-mail: [email protected]

An extension of Thompson’s Replacement Theorem by algebraic group methods George Glauberman

1. Introduction Thompson’s Replacement Theorem was proved in 1969. Starting with two abelian subgroups A, B of a finite p-group, one constructs another abelian subgroup A∗ , “better” than A, that can replace A for some applications. While the proof is relatively short, the existence of A∗ seems somewhat mysterious. The purpose of this expository article is to interpret the theorem by methods from algebraic group theory and to give some extensions and open problems suggested by these methods. It is a pleasure to have this opportunity to thank John Thompson for forty years of help, encouragement, and inspiration. Throughout this paper, p denotes an arbitrary prime and S an arbitrary finite pgroup. We let (S) be the set of abelian subgroups A of S of maximal order, i.e., such that |A| ≥ |B| for every abelian subgroup B of S. We let e (S) be the set of elementary abelian subgroups of maximal order (and thus of maximal rank) in S. Finally, for each subgroup T of S, let |T ∩ N |, f (T ) = NS

i.e., the product of the orders |T ∩ N | as N ranges over all the normal subgroups of S. (This function is convenient for stating results, although not for the actual proofs.) The following is a combination of part of Thompson’s Replacement Theorem, its proof, and its corollaries ([HB] III, pp. 19–22): Theorem 1 (Thompson). Suppose A is in (S) or e (S), B is an abelian subgroup of S, and A normalizes B, but B does not normalize A. Then there exists an abelian subgroup A∗ of AB such that (a) |A∗ | ≥ |A|; (b) f (A∗ ) > f (A); (c) A∗ normalizes A; and (d) if p is odd and A is elementary abelian, then A∗ is elementary abelian.

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Moreover, there exists an element x in B such that [x, A] normalizes A, but x does not normalize A. For such an element x, let A1 = [x, A]. Then we may take A∗ = A1 CA (A1 ). For p = 2, there is a counterexample to part (d) of the theorem in which S is a dihedral group of order 16, A is a noncyclic subgroup of order 4, and B is the cyclic subgroup of order 8 in S. Here A∗ must be the cyclic subgroup of order 4 in S. Corollary to Theorem 1. Assume A is chosen in (S) with f (A) maximal. Then whenever A normalizes an abelian subgroup B of S, then B normalizes A. One might suspect in the corollary that A is normal in S. However, J. Alperin showed in 1965 ([H], p.349; [G1], pp. 324–325) that for each choice of p there is an example in which no element of (S) or of e (S) is normal in S.

2. Extensions of the Replacement Theorem About ten years ago, I attempted to extend Theorem 1 by allowing B to have nilpotence class two. A counterexample for p = 2 shows that one must assume p is odd. I found a natural candidate for A∗ , but also an example where the candidate is not abelian. However, detailed study of this putative counterexample revealed an awkward gap, namely, an unnatural candidate for A∗ that worked. This led to the following result: Theorem 2 (1997, [G1], Theorems 1 and 2). Assume p is odd. (1) Suppose A is in (S), B is a subgroup of S of nilpotence class at most two; and A normalizes B, but B does not normalize A. Take x as in Theorem 1. Let i Aˆ = Ax | all integers i ,

A1 = [[A, x], x],

ˆ A2 = a[a , x]2 | a, a ∈ A and x centralizes a[a , x] (mod Z(A)) and A∗ = A1 A2 . Then A and A∗ satisfy conditions (a)–(c) of Theorem 1. (2) Suppose p ≥ 5. Take A in (S) with f (A) maximal. Then A AS . Remark. Here AS denotes the normal closure of A in S, i.e., the subgroup of S generated by all the conjugates Ax as x ranges over S. The group A∗ appeared “unnatural” because of the appearance of the squared terms [a , x]2 . After Theorem 2 was proved, Alperin mentioned to me an analogous result for complex Lie algebras (discussed in Section 4 below) that had been proved by using algebraic group theory. This led to an extension of Theorem 2 for elementary abelian subgroups, and later for all abelian subgroups:

An extension of Thompson’s Replacement Theorem by algebraic group methods

107

Theorem 3 (1998, 2004). (1) ([G2], Theorem D) Suppose A is an abelian subgroup of S; B is a subgroup of S of nilpotence class at most p − 1; and A normalizes B, but B does not normalize A. Then there exists an abelian subgroup A∗ of AB that satisfies conditions (a)–(d) of Theorem 1. (2) ([AG], Theorem 5.5) Suppose p ≥ 5. Take A in e (S) with f (A) maximal. Then A AS . Remark. Part (1) was first proved in [AG] for the case in which A is elementary abelian. The case in Part (1) in which p is 2 or 3 follows from Theorems 1 and 2. Theorems 2 and 3 suggest the following open problems: Problem 1. Suppose p is 2 or 3. Does there exist A in (S) such that A is normal in AS ? Problem 2. Suppose p = 3. Does there exist A in e (S) such that A is normal in AS ? We do not allow p = 2 in Problem 2 because there is no A when S is a dihedral group of order 2n for n ≥ 5.

3. Algebraic group methods in proofs To illustrate the use of algebraic group methods in the proofs of Theorems 2 and 3, we consider the special case in which B is abelian. This gives us a new proof and a new interpretation of Thompson’s Replacement Theorem. Unfortunately, we have to assume that p is odd. Assume the hypothesis of the theorem, and suppose p is odd. It is easy to find an i x as in the theorem. Take Aˆ = Ax | all i , as in Theorem 2. Then [[A, x], x] ≤ [[A, B], B] ≤ [B, B] = 1

and

[[A, x], [A, x]] = 1.

(3.1)

and Aˆ has nilpotence class at most 2 (since A, It is now easy to show that Aˆ = x ˆ A A ). Since p is odd, we may apply an idea of R. Baer and M. Lazard ([HB], ˆ let y+z = y 1/2 zy 1/2 and [y, z] = y −1 z−1 yz pp. 347–349; [Kh], p. 126). For y, z in A, (as usual). Then Aˆ is a Lie ring with respect to the operations + and [ , ]. AAx

Moreover, let p e be the exponent of A and let R be the ring Z/pe Z. Then Aˆ is a Lie algebra over R. Now let t be an indeterminate over R and form the polynomial ring R[t]. Then the tensor product L = Aˆ ⊗R R[t] becomes a Lie algebra over R[t]. ˆ it induces an automorphism α on Aˆ by conjugation. Since x normalizes the group A, Then α respects the operations + and [ , ], and hence is an automorphism of Aˆ as a Lie algebra. By letting α act trivially on the polynomial ring R[t], we extend α to an automorphism of L.

108

George Glauberman

Let δ = α − 1. By (3.1), δ2 = 0

and

ˆ Aδ] ˆ = [Lδ, Lδ]. 0 = [Aδ,

(3.2)

For every integer r, α r = (1 + δ)r = 1 + rδ. Now we “raise α to the power t” in some sense by defining τ = 1 + tδ. By (3.2) and some calculation, we may show that τ is an automorphism of the Lie algebra L over R[t]. As A is an abelian subgroup of S, we have [A, A] = 0. Since τ is an automorphism of L, [τ (A), τ (A)] = 0.

(3.3)

Let L0 be τ (A)R[t], the R[t]-submodule of L spanned by τ (A). By (3.3), [L0 , L0 ] = 0. For any nonzero elements f and g of L0 , consider the leading terms, say, f = an t n + · · ·

and g = bm t m + · · · .

Since 0 = [f, g], we obtain [an , bm ] = 0. Finally, we let A∗ consist of the zero element, together with the leading coefficients of all of the nonzero elements of L0 . Then a few calculations show that this group is the same as the group A∗ constructed in the Replacement Theorem. This proof is generalized to prove Theorems 2 and 3. We take x and Aˆ as in Theorem 2, so that i i Aˆ = Ax | all i and Ax Aˆ for all i. Since B has class at most p − 1, we can eventually show that Aˆ has class at most p − 1. Therefore, Aˆ becomes a Lie algebra by a beautiful result of Lazard ([Kh], p. 124). We take R, t, L, and α as before. Here (α − 1)p = 0. Let δ = log α =

p−1 i=1

(α − 1)i (−1)i+1 . i

Then p

δ =0

and α = exp δ =

p−1 δi i=0

Let τ = α t = exp tδ. Now define L0 and A∗ from τ as above.

i!

.

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109

4. Other open problems Theorem 3 was originally inspired by a result on nilpotent complex Lie algebras by B. Kostant that uses methods of algebraic group theory ([AG], p. 534). Kostant’s method also easily gives the following result: Suppose G is a unipotent algebraic group over an algebraically closed field K of characteristic 0. Let (G) be the set of all closed abelian subgroups of G of maximal dimension. Then some element of (G) is normal in G. I do not know whether the analogous proposition for K of prime characteristic is valid. The examples of Alperin mentioned above show that the analogous proposition for finite p-groups is false. Thus, we cannot generally regard finite p-groups as analogues of algebraic groups of characteristic 0. Rather, it appears that the finite analogues of unipotent algebraic groups of characteristic 0 form a proper subclass of the finite p-groups. What is this subclass? We may regard the question as an imprecise open problem, since it is a matter of taste as well as proofs. Lazard’s work strongly suggests that this subclass includes all finite p-groups of class at most p − 1, since there is a Lie algebra associated to each of these groups and to each algebraic group. Thompson’s Replacement Theorem and many other results concerning J (S) or Je (S) are proved by showing that certain large subgroups of S lie in a given normal subgroup of S. They suggest the following imprecise conjecture: If p is odd, then S has a “large” characteristic subgroup N that is analogous to a unipotent algebraic group of characteristic 0. As a step in this direction, we ask: Problem 3. (a) Which p-groups S can be written as, or uniquely associated with, Lie algebras? (b) Can S be so written or associated if there is an algebraic group G defined over Fp such that S is the set of all Fp -rational points of G? (c) Which p-groups arise as in (b)? Thompson’s Replacement Theorem is actually broader than Theorem 1; it remains valid if [B, A], rather than B, is abelian. Problem 4. Extend the proof in Section 3 to cover the cases in which [B, A] is abelian or p = 2. (Here one should keep in mind the counterexample to part (d) of Theorem 1 for p = 2.)

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References [AG] J. Alperin and G. Glauberman, Limits of abelian subgroups in finite p-groups, J. Algebra 203 (1998), 533–566. [G1]

G. Glauberman, Large abelian subgroups of finite p-groups, J. Algebra 196 (1997), 301–338.

[G2]

G. Glauberman, Large subgroups of small class in finite p-groups, J. Algebra 272 (2004), 128–153.

[H]

B. Huppert, Endliche Gruppen I, Springer-Verlag, Berlin 1967.

[HB] B. Huppert and N. Blackburn, Finite Groups II, III, Springer-Verlag, Berlin 1982. [Kh]

E. I. Khukhro, p-Automorphisms of Finite p-Groups, London Math. Soc. Lecture Note Ser. 246, Cambridge University Press, Cambridge 1998.

[T]

J. G. Thompson, A replacement theorem for p-groups and a conjecture, J. Algebra 13 (1969), 149–151.

George Glauberman, Department of Mathematics, University of Chicago, 5734 S. University Ave., Chicago, IL 60637, U.S.A. E-mail: [email protected]

Simple connectedness of the geometry of nondegenerate subspaces of a symplectic space over arbitrary fields Ralf Gramlich∗

Abstract. K. M. Das [4], [5] has shown that the geometry of nondegenerate subspaces of a symplectic space over a finite field is simply connected. The purpose of the present article is to provide a short, direct and general proof of that result for arbitrary fields. Some standard amalgam-theoretic consequences for the groups Sp2n (F) are given as well.

1. Introduction Let F be a field and let V be a 2n-dimensional vector space over F endowed with a nondegenerate alternating bilinear form. Let (n, F) be the geometry of rank n − 1 whose elements are the nondegenerate proper subspaces of V with natural incidence. Then the following holds. Theorem 1. Let n ≥ 4 and let F be a field. Then the geometry (n, F) is 2-simply connected. The rank of (3, F) is two, whence it cannot be simply connected (the incidence graph of (3, F) is not a tree), so the above result is sharp. The group Sp2n (F) acts flag-transitively on (n, F). Choose a canonical basis e1 , f1 , …, en , fn of V . Then e1 , f1 , e1 , f1 , e2 , f2 , …, e1 , f1 , . . . , en−1 , fn−1 is a chamber (i.e., a maximal flag) of (n, F). The maximal parabolics Pi , 1 ≤ i ≤ n − 1 with respect to this chamber are of the form Sp2i (F) × Sp2n−2i (F). A consequence of Theorem 1 is the following. Theorem 2. Let n ≥ 4 and let F be a field. Then Sp2n (F) is the universal completion of the amalgam of at least three maximal parabolics Pi . Denote by A2 (n, F) the amalgam of 1- and 2-parabolics of Sp2n (F) with respect to the chamber e1 , f1 , e1 , f1 , e2 , f2 , …, e1 , f1 , . . . , en−1 , fn−1 . That is, A2 (n, F) is the union of all intersections i∈J Pi where J is a subset of {1, . . . , n − 1} of co-cardinality at most two. Then we have the following. ∗ The author would like to express his gratitude to the organizers for the funding of his trip to the USA and his stay at the conference.

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Theorem 3. Let n ≥ 4 and let F be a field. Then Sp2n (F) is the universal completion of the amalgam A2 (n, F). For finite fields Theorem 1 has been proved independently first by K. M. Das ([4] and [5], for all finite fields) and later by C. Hoffman, S. Shpectorov and the author ([7], for all finite fields of size at least four). While Das achieved this result as a consequence of his study of Quillen complexes of finite symplectic groups based on work of Aschbacher [1] and Quillen [14], the proof by Hoffman, Shpectorov and the author makes use of the simple connectedness of another finite geometry that is closely related to (n, F). The purpose of this paper is to present a short, direct and general proof of the simple connectedness of (n, F) for an arbitrary field F. The investigation of geometries like (n, F) is motivated by the desire for a systematic approach to Phan-type theorems similar to the results of [12] and [13]. For an outline of the idea, using chamber systems and buildings, as well as precise definitions of flips and flipflop geometries refer to [3]. (Note that those concepts are not relevant for the understanding of the present article.) Example 1b and Example 2b of [3] give a description of similar geometries. However one big difference remains: while the examples of [3] can be described using flips and opposite chamber systems, one cannot obtain the geometry (n, F) in this way; for, a flip induced by a symplectic form does not admit a chamber that is mapped onto its opposite, so the resulting flipflop geometry is empty. The most remarkable property of (n, F) is that the simple connectedness holds and can be shown completely independent of the field. Neither the geometry from Example 1b nor the geometry from Example 2b of [3] have this property; they are not simply connected for low rank over the field of two elements. Theorem 2 and Theorem 3 dealing with the presentation of Sp2n (F) by certain amalgams nowadays are standard corollaries of Theorem 1. A classification of those amalgams is not included in the present article. For a closely related characterization of Sp2n (F) refer to Theorem 4.4.25 of [6] or to the results of [8].

2. Connectedness properties of Let F be a field and let V be a 2n-dimensional vector space over F endowed with a nondegenerate alternating bilinear form. Perpendicularity with respect to that form is denoted by ⊥. Let = (n, F) be the pregeometry of rank n − 1 whose elements are the nondegenerate proper subspaces of V with symmetrized containment as incidence. The nondegenerate subspaces of V of dimension two are called points, those of dimension four lines and so on. Evidently, is a geometry (i.e., maximal flags are chambers, which means they have size n − 1). Moreover, by induction on n using Witt’s theorem, the geometry is flag-transitive with the group Sp2n (F) as a flag-transitive group of automorphisms of ; see Lemma 7.3 of [7] for a proof. Lemma 2.1. Let n ≥ 2 and let l, m be nondegenerate two-dimensional subspaces of V with l ∩ m = {0} and dim(l ⊥ ∩ m) ≥ 1. Then l, m is nondegenerate.

Nondegenerate subspaces of a symplectic space

113

Proof. The statement is obvious, if m ⊆ l ⊥ . If m ⊆ l ⊥ , then m ∩ l ⊥ is a onedimensional subspace of V , say p. Choose a canonical basis a, b of l and choose q ∈ m, such that p, q form a canonical basis of m. Then the Gram matrix of the symplectic form restricted to l, m with respect to the basis a, b, p, q has the shape 0 1 0 ∗ −1 0 0 ∗ 0 0 0 1 , ∗ ∗ −1 0 which evidently has full rank, whence l, m is nondegenerate. Proposition 2.2. Let n ≥ 3. Then the collinearity graph of has diameter two. In particular, is connected. Proof. Let l, m be nondegenerate two-dimensional subspaces of V . Let p be an arbitrary one-dimensional subspace of l, m⊥ and let q be an arbitrary one-dimensional subspace of V that intersects p ⊥ trivially. Then p, q is a point of (as q ⊆ p ⊥ ) which is collinear to l and m (by Lemma 2.1). Corollary 2.3. Let n ≥ 3. Then the geometry is residually connected. Recall the definition of the fundamental group of a connected geometry .1 A path of length k in the geometry is a sequence of elements x0 , . . . , xk of such that xi and xi+1 are incident, 0 ≤ i ≤ k − 1. We do not allow repetitions; hence xi = xi+1 . A cycle based at an element x is a path in which x0 = xk = x. Two paths are homotopically equivalent if one can be obtained from the other via the following operations (called elementary homotopies): inserting or deleting a return (i.e., a cycle of length 2) or a triangle (i.e., a cycle of length 3). The equivalence classes of cycles based at an element x form a group under the operation induced by concatenation of cycles. This group is called the fundamental group of and denoted by π1 (, x). A geometry is called simply connected if its fundamental group is trivial.2 Notice that in order to prove that is simply connected it is enough to prove that any cycle based at x is homotopically equivalent to the cycle of length 0. A cycle with this property is called null-homotopic, or homotopically trivial. Let us go back to the geometry (n, F). We pick the base element x to be a point of . Lemma 2.4. Let n ≥ 4. Every cycle of based at x is homotopically equivalent to a cycle passing only through points and lines. 1 Note that there are two concurrent notions of a fundamental group. One notion of a fundamental group

considers the (incidence graph of the) geometry as a one-dimensional simplicial complex (triangles are not null-homotopic), the other notion considers the (incidence graph of the) geometry as a two-dimensional simplicial complex (triangles are null-homotopic). In the present paper we use the latter notion. 2 Standard topology shows that a geometry is simply connected if and only if it does not admit any proper simplicial cover. For combinatorial proofs see [9], [10] or [15]; alternatively, see [16].

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Proof. This follows immediately from the residual connectedness of . See Lemma 5.1 of [7] for a proof; alternatively, see [2]. We can therefore restrict our attention to the point-line incidence graph of . However, as is not a partially linear geometry (there exist nondegenerate twodimensional subspaces of V that are contained in more than one nondegenerate fourdimensional subspace of V , so a pair of collinear points does not necessarily admit a unique joining line), it is not immediately clear that we can restrict ourselves to the collinearity graph of . The following lemma takes care of that problem. Lemma 2.5. Let n ≥ 4. Any digon p, l, q, m of consisting of points p = q and lines l = m is homotopically trivial. Proof. The dimension of l, m equals five, because l and m intersect in a threedimensional space. The space x = l ⊥ ∩ l, m is the radical of l, m. Choosing any vector y ∈ l ⊥ \x ⊥ , we obtain a nondegenerate six-dimensional space l, m, y that contains the digon p, l, q, m. Therefore the digon is homotopically trivial. The above lemma allows us to restrict our attention to the collinearity graph of , since the homotopy type of a path γ passing through points and lines is independent of the particular choice of the joining lines of the pairs of collinear points occurring in γ . In order to determine the fundamental group of we have to distinguish between good and bad triangles, i.e., triangles a, b, c whose points span a nondegenerate subspace

a, b, c of V and triangles a, b, c whose points do not span a nondegenerate subspace

a, b, c of V . Obviously, a good triangle is null-homotopic as it spans a line or a plane of the geometry . For bad triangles we have the following lemma. Lemma 2.6. Let n ≥ 4. Any triangle in the collinearity graph of can be decomposed into good triangles. Proof. Let a, b, c be the nondegenerate two-dimensional subspaces of V that constitute the three points of some triangle. Choose some line l of containing a and b. The space l ⊥ is nondegenerate and has dimension at least four. Therefore, to any onedimensional subspace p ⊆ l, c⊥ , we can find a one-dimensional subspace q of l ⊥ such that p, q =: d ⊥ l ⊇ a, b is nondegenerate and intersects c trivially. By Lemma 2.1, the space c, d is nondegenerate, and we have decomposed the triangle a, b, c into triangles which admit two points that are perpendicular. So now assume we have a triangle a, b, c with a ⊥ c. If a ∩ b = ∅ and b ∩ c = ∅, then b ⊂ a, c, and there is nothing to prove. Hence, by symmetry, we can assume that b ∩ c = ∅. As above choose some line l of containing a and b. The space l ⊥ is nondegenerate and has dimension at least four. To any one-dimensional subspace p ⊆ l, c⊥ , we can find a one-dimensional subspace q of l ⊥ such that p, q =: d ⊥ l ⊇ a, b is nondegenerate and intersects b, c trivially. This is possible as the dimension of b, c ∩ l ⊥ is at most two (since ∅ = b ∩ b⊥ ⊃ b ∩ l ⊥ ). Notice that the triangles a, b, d and a, c, d are good (indeed, d ⊥ l ⊃ a, b and a ⊥ c, d). Moreover,

Nondegenerate subspaces of a symplectic space

115

for hyperbolic bases xb , yb of b and xc , yc of c and p, yd of d, we obtain the following Gram matrix for the alternating form on b, c, d with respect to xb , yb , xc , yc , p, yd : 0 1 ∗ ∗ 0 0 −1 0 ∗ ∗ 0 0 ∗ ∗ 0 1 0 ∗ ∗ ∗ −1 0 0 ∗ . 0 0 0 0 0 1 0 0 ∗ ∗ −1 0 Evidently, this matrix has full rank if and only if its submatrix consisting of the first four rows and columns has full rank. But that submatrix is the Gram matrix of the form restricted to the nondegenerate space b, c. Therefore we have decomposed the original triangle into good triangles. Lemma 2.7. Let n ≥ 4. Any quadrangle in the collinearity graph of can be decomposed into triangles. Proof. Let a, b, c, d be the nondegenerate two-dimensional subspaces of V that constitute the four points of some quadrangle in the collinearity graph of . Consider W := a, b⊥ and U := c, d⊥ . If W ⊥ U , then c, d⊥ = U ⊆ W ⊥ = a, b. Since the dimension of c, d⊥ is at least four and the dimension of a, b at most four, equality holds. But this implies that a ⊥ c and b ⊥ d, yielding an immediate decomposition of the quadrangle into triangles. If W ⊥ U , we can find one-dimensional subspaces p ⊆ W , q ⊆ U with p ⊥ q. The resulting nondegenerate two-dimensional space e := p, q is collinear to a, b, c, d by Lemma 2.1. (Notice that e cannot intersect a, b, c, or d nontrivially since a, b ⊥ W ⊆ p and c, d ⊥ U ⊆ q, but p ⊥ q.) Lemma 2.8. Let n ≥ 4. Any pentagon in the collinearity graph of can be decomposed into triangles and quadrangles. Proof. Let a, b, c, d, e be the nondegenerate two-dimensional subspaces of V that constitute the five points of some pentagon in the collinearity graph of . Choose a line l of passing through c and d. Then l ⊥ is nondegenerate and has dimension at least four. Let p ⊆ a, l⊥ be an arbitrary one-dimensional space. There exists a onedimensional subspace of l ⊥ that together with p spans a nondegenerate space f that intersects a trivially. We have f ⊥ l ⊇ c, d. Moreover, f also spans a nondegenerate space with a, by Lemma 2.1, decomposing the pentagon. Any cycle of greater length decomposes by the bound on the diameter of the collinearity graph of from Lemma 2.2. Therefore is simply connected. An induction argument as in [7] implies that is 2-simply connected, and Theorem 1 follows. The proof of Theorem 2 is identical to the proof of Theorem 4 of [7]. Theorem 3 follows by Tits’ lemma (Corollaire 1 of [17]) and a standard induction argument as in the proof of Theorem 1 of [7].

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References [1]

M. Aschbacher, Simple connectivity of p-group complexes, Israel J. Math. 82 (1993), 1–43.

[2]

C. D. Bennett and S. Shpectorov, A new proof of a theorem of Phan, J. Group Theory 7 (2004), 287–310.

[3]

C. D. Bennett, R. Gramlich, C. Hoffman and S. Shpectorov, Curtis–Phan–Tits theory, in Groups, Combinatorics and Geometries (Durham, 2001), ed. by A. A. Ivanov, M. W. and J. Saxl, World Sci. Publishing, River Edge, NJ, 2003, 13–29.

[4]

K. M. Das, Some results about the Quillen complex of Sp2n (q), J. Algebra 209 (1998), 427–445.

[5]

K. M. Das, The Quillen complex of groups of symplectic type: the characteristic 2 case, J. Algebra 223 (2000), 556–561.

[6]

R. Gramlich, On graphs, geometries, and groups of Lie type, PhD thesis, TU Eindhoven 2002.

[7]

R. Gramlich, C. Hoffman and S. Shpectorov, A Phan-type theorem for Sp(2n, q), J. Algebra 264 (2003), 358–384.

[8]

R. Gramlich, On the hyperbolic symplectic geometry, J. Combin. Theory Ser. A 105 (2004), 97–110.

[9]

R. Gramlich, G. W. Hofmann, K.-H. Neeb, Semi-edges, reflections, and Coxeter groups, preprint.

[10] R. Gramlich, Phan theory, Habilitationsschrift, TU Darmstadt, 2004. [11] B. Mühlherr, On the simple connectedness of a chamber system associated to a twin building, preprint. [12] K. W. Phan, On groups generated by three-dimensional special unitary groups, I. J.Austral. Math. Soc. Ser. A 23 (1977), 67–77. [13] K. W. Phan, On groups generated by three-dimensional special unitary groups. II, J. Austral. Math. Soc. Ser. A 23 (1977), 129–146. [14] D. Quillen, Homotopy properties of the poset of nontrivial p-subgroups of a group, Adv. Math. 28 (1978), 101–128. [15] H. Seifert, W. Threlfall, Lehrbuch der Topologie, Chelsea Publishing Company, NewYork 1934. [16] D. B. Surowski, Covers of simplicial complexes and applications to geometry, Geom. Dedicata 16 (1984), 35–62. [17] J. Tits, Ensembles Ordonnés, immeubles et sommes amalgamées, Bull. Soc. Math. Belg. Sér. A 38 (1986), 367–387. Ralf Gramlich, TU Darmstadt, FB Mathematik / AG 5, Schloßgartenstraße 7, 64289 Darmstadt, Germany E-mail: [email protected]

Classification of 2F -modules, II Robert M. Guralnick and Gunter Malle∗

Abstract. We classify the irreducible 2F -modules for the quasi-simple groups of Lie type with the modules in the natural characteristic (leaving three open cases). In concert with the results of our first paper, this gives the complete classification of such modules that arise in the context of the Aschbacher-Smith classification of quasithin groups. As a direct application we also obtain the list of irreducible F -modules of finite quasi-simple groups. 2000 Mathematics Subject Classification: 20G05, 20D06

1. Introduction and statement of results In this paper we study the following situation: Let G be a finite group such that F ∗ (G) is quasi-simple. For an absolutely irreducible faithful KG-modules V over a field K of characteristic and a subgroup A ≤ G of G we let f (A) := f V (A) := |A|2 · |CV (A)|. We want to classify those modules V for which G has a nontrivial elementary abelian -subgroup A satisfying |V | ≤ f (A).

(1.1)

Modules satisfying this and similar conditions have been studied for quite a while – going back to Thompson, Glauberman, Mason and others (see the remarks and references in [6]). Usually, there are conditions imposed in addition to (1.1) involving an assumption of quadratic or cubic action of the subgroup A on V . These modules play a crucial role in a new approach to the revision of the classification of finite simple groups. In particular, for = 2, the results of this paper and [6] are used by Aschbacher and Smith in the proof of the classification of quasithin simple groups [2]. The Aschbacher–Smith result is the final paper in the original proof of the classification of finite simple groups and is also part of the second generation proof. These modules and variations also come up in Meierfrankenfeld’s program to study finite simple groups. ∗ The first author was partially supported by NSF grant DMS 0140578.

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In [6], we essentially found all examples of such modules with F ∗ (G) quasi-simple aside from the case that F ∗ (G) is a group of Lie type and V is a module in the natural characteristic for G. In this paper, we consider this case and classify all but three cases. In particular, we classify all such modules that arise in [2]. We expect to complete the classification of the remaining situations in a future article. We feel that it is important to have the results needed for the quasithin classification in the literature in a timely manner. Theorem A. The absolutely irreducible 2F -modules for quasi-simple groups of Lie type in their defining characteristic are among those given in Table 1. We also shows that all modules in Table 1 except possibly those in the last three rows actually satisfy the 2F -condition. Here, for the natural module Yn of Spinn (a ), has to be odd when n is odd. Table 1. 2F -modules in defining characteristic

G SLn (a ) SLn (a ) SLn (a ) SLn (2a ) SL6 (a ) SUn (a ) Sp2n (a ) Sp2n (a )

d n n

2 n+1 2

n2 20 n 2n 2n 2 − 1 − δ,n

V Yn 2 ∧ (Yn )

2 (Yn ) (a) Yn ⊗ Y n ∧3 (Y6 ) Yn Y2n 2 ˜ ∧ (Y2n )

f a a a a a 2a a a

(a)

Sp4 (2a ) (±) Spinn (a ) Spin2n+1 (a ) a Spin+ 2n ( ) a Spin− 2n ( ) 2B (22a+1 ) 2 G2 (a ) F4 (2a ) E6 (a )sc

16 n 2n 2n−1 2n−1 4 7 − δ,2 26 27

Y4 ⊗ Y4 Yn Zn Zn−1 Zn−1 M(λ1 ), M(λ2 ) M(λ2 ) M(λ1 ), M(λ4 ) M(λ1 ), M(λ6 )

a a a a 2a 2a + 1 a a a

F4 (a ) 2E (a ) 6 sc E7 (a )sc

26 − δ,3 27 56

M(λ4 ) M(λ1 ), M(λ6 ) M(λ7 )

a 2a a

conditions n≥4 ≥3

n = 2, 3 or = 2, n = 4 n≥6 n = 3, 4, 5 n = 4, 5, 6 n = 4, 5 a>1

>2

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In this table, the modules are labeled by their highest weights, and the fundamental weights λi are labeled as in Table 4, §3.1. Furthermore, Yn denotes the natural module (of dimension n), Zn is the 2n -dimensional spin module of the spin group Spin2n+1 (q) with highest weight λ1 (this includes the case of symplectic groups in even characteristic), and Zn−1 denotes either one of the two 2n−1 -dimensional half-spin modules of (±) spin groups Spin2n (q) with highest weight λ1 or λ2 . Next, ∧2 , ∧3 , and 2 denote the alternating square, the alternating cube, and the symmetric square, respectively, and M (a) is the a -Frobenius twist of the module M. Also, recall that for a given type X of finite groups of Lie type there are several isogenous variants which share the same non-abelian composition factor and the same order but have slightly different structures. In particular, the adjoint group Xad is almost simple, whereas the simplyconnected type group Xsc is quasi-simple (for example, 2E6 (q)sc is quasi-simple with center of order (3, q + 1)). The three open cases are E7 (q)sc , 2E6 (q)sc and F4 (q) on their smallest modules (the last only for odd characteristic). Note that in the first case f (A) = q 55 = |V |/q for A an elementary abelian subgroup of order q 27 . There are two other open cases remaining from the first part — the two largest Conway groups on their smallest modules in characteristic 2. Again, in those two cases there are elementary abelian subgroups A with f (A) = |V |/2. Our approach is to classify the modules satisfying (1.1) and not impose other conditions (of course, one obtains corollaries to our main results by then imposing extra conditions). Given the results in [6], it thus remains to consider groups of Lie type G in the same characteristic . By a basic lemma, we know log (|G|) grows quadratically with the rank. The irreducible KG-modules of this dimension are known. We quote a recent paper of Lübeck [11] for this (which depends upon a result of Premet [15]). This follows from the beautiful theory of representations of Chevalley groups in the natural characteristic (see for example [5, Sec. 2.8] for background on this). Liebeck [10] has obtained some weaker bounds than those given in [11], but these results are not sufficient for our purposes. We should point out that the background material that we use is a bit different than that of Aschbacher–Smith [2] or Gorenstein–Lyons–Solomon [5]. In the first part [6], we assumed the results from the modular Atlas [8] about lower bounds for the sporadic groups, of James about representations of alternating groups and of Landazuri–Seitz–Zalesskii for cross characteristic representations. As we remarked above, in the equicharacteristic case, we are using the results of Premet and Lübeck. One could produce independent proofs of these results for the cases of interest in [2] to make that result more self-contained. For the exceptional groups we also use results of Lawther [9] on Jordan blocks of exceptional groups on certain small modules and some information on the order relation on unipotent classes, see Spaltenstein [16]. This does not concern any of the modules considered in [2]. In the present paper we treat the absolutely irreducible representations of finite quasi-simple groups of Lie type G in their defining characteristic . In fact, we will

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not consider here the non-simple groups Sp4 (2) and G2 (2) in characteristic 2, and 2G (3) in characteristic 3, since these were considered as the automorphism groups 2 S6 , U3 (3).2 and L2 (8).3 of simple groups in non-defining characteristic in the first part [6, Props. 3.1, 4.3, 4.1]. On the other hand, we do treat here the following pairs (G, ) left open in the first part: (A5 ∼ = SL2 (4), 2),

(2.A5 ∼ = SL2 (5), 5),

(A8 ∼ = SL4 (2), 2),

(L2 (7) ∼ = SL3 (2), 2),

(2.U4 (2) ∼ = Sp4 (3), 3),

(S4 (3) ∼ = SU4 (2), 2),

(2.A6 ∼ = SL2 (9), 3), (2.L3 (2) ∼ = SL2 (7), 7) (G2 (2) ∼ = SU3 (3), 3),

as well as the ‘sporadic group’ (2F4 (2) , 2). Note that exceptional Schur multipliers of groups of Lie type are always of order a power of the defining characteristic, so we need not worry about them and just have to consider quotients of simply connected groups of Lie type by central subgroups. By a slight abuse of notation, in the sequel we shall weaken the faithfulness-requirement a little and say that V is a 2F -module for G if it is so for some quotient by a central subgroup contained in G . The explicit description of representations in the defining characteristic via their highest weight makes it easy to work out the kernel of such a representation in any individual case. Since F -modules are, by definition, also 2F -modules we also obtain a classification of F -modules of finite quasi-simple groups as a rather easy application of our results on 2F -modules: Theorem B. The absolutely irreducible F -modules of finite quasi-simple groups are given in Table 2. Table 2. F -modules for quasi-simple groups

G SLn (a ) SLn (a ) SUn (a ) Sp2n (a ) (±) Spinn (a ) Spin7 (a ) a Spin+ 2n ( ) G2 (2a ) A2n 3.A6 A7

d n n 2

n 2n n 8 2n−1 6 2n − 2 3 4

V Yn 2 ∧ (Yn ) Yn Y2n Yn Z3 Zn−1 M(λ2 )

2 2 2

f a a 2a a a a a a 1 2 1

conditions n≥4 n≥4 n≥6 n = 4, 5

Here, for the natural module Yn of Spinn (a ), has to be odd when n is odd.

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121

This result is already in the literature. The earlier papers used very heavily the fact that by the Thompson replacement lemma, one can assume that the subgroup A acts quadratically. In particular, see Aschbacher [1], Meierfrankenfeld-Stroth [12], [13] for alternating, sporadic and odd characteristic Lie-type groups in characteristic 2, and Meixner [14] for odd characteristic Lie-type groups in odd characteristic. Our proof is independent of these references. We now give an outline of the contents of the paper. In order to study modules in defining characteristic and determine precisely which of them satisfy (1.1), very frequently we need to deal with tensor products, symmetric squares and exterior powers of smaller representations. Upper bounds for centralizers on such modules are derived in Section 2.2. In Section 3.1 we recall some basic results about representations in defining characteristic. In 3.3 we determine the maximal size of a subgroup of a group of Lie type consisting only of long root elements, and in 3.4 we collect information on the -ranks of classical groups. In Sections 4–8 we treat the families of linear and unitary groups, of symplectic groups, of orthogonal groups, and of exceptional groups of Lie type, with Section 5 containing some results on the particularly difficult case of spin representations. Note that, in contrast to the situation in the first part [6], we don’t reach a complete classification of 2F -modules for non-perfect almost quasi-simple groups. Still, we prove the following partial result: if F ∗ (G) is quasi-simple but not an orthogonal group in even dimension, then all composition factors of the restriction to F ∗ (G) of a 2F -module for G are again 2F -modules. We hope to come back to the case of the orthogonal groups and more generally to the complete determination of 2F -modules for almost quasi-simple groups in a future paper.

2. Auxiliary results 2.1. 2F -modules. Let K be a finite field of characteristic , G a finite group and V a finite-dimensional KG-module. For x ∈ G we call the codimension of CV (x) the rank of x (on V ) (i.e., the rank of x − 1). For a subgroup A of G write f (A) := f V (A) := |A|2 · |CV (A)|. V is called a 2F -module for G with offender A = 1 if dim(V ) > 1 and |V | ≤ f (A). Similarly we define f1 (A) := f1V (A) := |A| · |CV (A)| and say that V is an F -module for G with offender A = 1 if dim(V ) > 1 and |V | ≤ f1 (A). We also say that V is an F+ -module for G if f1 (A) > |V | for some A ≤ G. The following well-known observation will be useful (see [3], [6, Lemma 2.1]): Lemma 2.1. Let A, B ≤ G with f (B) maximal among all subgroups of G and let C := A, B. Then f (C) ≥ f (A).

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In particular, if V is a 2F -module for G with offender A = 1, and G is the normal closure of A then f (A) ≤ f (G), Moreover, if V is irreducible then f (A) ≤ |G|2 . Also it follows that: Lemma 2.2. Suppose C ≤ G is an elementary abelian -subgroup containing A and B, with f (A) = f (B) maximal among all subgroups of C. Then f (AB) = f (A). This shows that when searching for examples satisfying (1.1) we may sometimes assume that A is maximal elementary abelian. The following can be used to obtain upper bounds on the centralizer of an subgroup: Lemma 2.3. Let B ≤ A ≤ G with dim CV (A) = dim CV (B). Then CV (Ag | g ∈ CG (B)) = CV (B) Proof. Let g ∈ CG (B). Then CV (A) = CV (B) = CV (B g ) = CV (Ag ). We will often use this in the following form. Let B be an elementary abelian -subgroup, g an -element in CG (B) \ B and A := B, g. If Ah | h ∈ CG (B) has smaller centralizer in V than B, then A has smaller centralizer than B. For example, if CG (B) is an extension of B by a simple group H , then all classes of -elements in CG (B) outside B generate H . But H contains -elements and their centralizer can be read off from the Brauer character of V . The following result is very useful for inductive arguments along parabolic subgroups (see [6, Prop. 2.4]): Proposition 2.4. Let V be a 2F -module for G. Let H ≤ G such that H contains an offender. The either there exists an offender A ≤ U := O (H ), or CV (U ) is a 2F -module for H /U . The next result is used for treating automorphism groups (see [6, Prop. 2.7]): Proposition 2.5. Let G be a finite group with a normal subgroup H ≤ G of index . Let W be an absolutely irreducible Ff H -module of dimension dim W ≥ 2 whose induction V = IndG H W to G is again irreducible. If V is a 2F -module for G with offender A, then one of the following holds: (i) W is a 2F -module for H with offender B := A ∩ H ; (ii) = f = 2, dim CW (B) = 1, dim W ≤ r where |A| = 2r ; (iii) |A| = = 2, dim W = 2, f ≤ 2. Note that mutatis mutandis all the results above remain true for F -modules and f1 (A). The following result will be applied to parabolic subgroups. Lemma 2.6. Let P = U L be a semidirect product with U a normal elementary abelian -subgroup of maximal size in P , with complement L. Then U is not an F+ -module for L. Moreover, the following are equivalent:

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(i) U is not an F -module for L; (ii) U is the unique elementary abelian subgroup of P with order |U |. Proof. If A ≤ L is an elementary abelian -subgroup, then |A||CU (A)| ≤ |U | (because U is the maximal order of an elementary abelian -subgroup), whence U is not an F+ -module. The same argument shows that if U is an F -module, then ACU (A) has order |U | for any offender A (offender in the sense of F -modules). Thus, U an F -module implies that U is not unique in P . Conversely, if W is another elementary abelian subgroup of P of order |U |, then take A = W U/U = 1 and B = A ∩ U ≤ CU (W ). So |A||CU (A)| = |U | and A is an offender. 2.2. Centralizers on tensor products. It will become obvious from the description of the irreducible modules of groups of Lie type in defining characteristic in Section 3.1 that in order to identify possible 2F -modules we need some information on centralizer spaces of unipotent subgroups on tensor products and alternating and symmetric powers. We start by considering indecomposable representations of cyclic groups C of prime order over a field K of characteristic . Here the following result of Hughes and Kemper [7] will be useful to estimate dimensions of centralizer spaces on symmetric and alternating squares: Lemma 2.7. Let G = C be the cyclic group of order and V an indecomposable KG-module of dimension n (so n ≤ ). Then dim(∧k (V ))G =

1 2

ζ kn−2i1 −···−2ik ,

(1 + ζ ) ζ ∈M2

0≤i1 <···

where the first sum runs over the set M2 of all 2th complex roots of unity. This follows for example from [7, Th. 2.5 and Prop. 3.2]. Lemma 2.8. Let G = C be the cyclic group of order and V , W be indecomposable KG-modules of dimension n, m respectively. Then (a) dim(V ⊗ W )G = min{n, m}, when > 2, (b) dim 2 (V )G = n+1 2 (c) dim ∧2 (V )G = n2 . Proof. By assumption both V , W are uniserial. Thus the centralizer of G on V ⊗ W has dimension dim(V ⊗ W )G = dim HomG (1, V ⊗ W ) = dim HomG (V ∗ , W ) = min{n, m}. The third part follows easily by evaluation of the formula in Lemma 2.7. If > 2 then V ⊗ V is the direct sum of ∧2 (V ) and 2 (V ), so the second part is immediate from (a) and (c).

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Let us next consider arbitrary modules for the cyclic group. Lemma 2.9. Let G = C and V , W be KG-modules of dimension n, m respectively, with z := dim V G , y := dim W G . Then we have (a) dim(V ⊗ W )G ≤ min{ny, zm, (n − z)(m − y) + zy}, n−z z+1 for > 2, (b) dim 2 (V )G ≤ min z(n+1) 2 , 2 + 2 n−z+1 + 2z . (c) dim ∧2 (V )G ≤ min zn 2 , 2 Proof. Part (a) has been proved in Lemma 2.8 when V , W are indecomposable. Now assume for example that V = V1 ⊕ V2 as KG-module, so that V ⊗ W = V1 ⊗ W ⊕ V2 ⊗ W. Since both taking fixed points as well as our stipulated bounds are clearly additive in both components, we get the desired result by induction. In part (b), our formula holds again for indecomposable V by Lemma 2.8. If V = V1 ⊕ V2 then

2 (V ) ∼ = 2 (V1 ) ⊕ V1 ⊗ V2 ⊕ 2 (V2 ). Writing ni := dim Vi , zi := dim ViG , this shows that the centralizer space has dimension at most equal to z1 (n1 + 1)/2 + (z1 n2 + z2 n1 )/2 + z2 (n2 + 1)/2 = (z1 + z2 )(n1 + n2 + 1)/2 by induction respectively by part (a), and similarly for the second bound. In case (c), assume that V = V1 ⊕ V2 . Then we have

∧2 (V ) ∼ = ∧2 (V1 ) ⊕ V1 ⊗ V2 ⊕ ∧2 (V2 ). In the notation introduced in the previous part this gives a centralizer space of dimension at most z1 n1 /2 + (z1 n2 + z2 n1 )/2 + z2 n2 /2 = (z1 + z2 )(n1 + n2 )/2. The second bound is proved analogously by using the second type of inequality from part (a). We are now ready to deal with arbitrary elementary abelian -groups G. Assume that G is generated by g1 , . . . , gr say. Let V be a KG-module. Since G is abelian, gi stabilizes CV (gj ) for all j , and this allows to apply induction on the number r of generators. Lemma 2.10. Let G = g1 , . . . , gr be an elementary abelian -group and V , W finite g dimensional KG-modules. Define V0 := V , Vi := Vi−1i (1 ≤ i ≤ r), ni := dim Vi (0 ≤ i ≤ r), and similarly Wi of dimension mi . Then

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Classification of 2F -modules, II

(a) dim(V ⊗ W )G r

≤

min{(ni−1 − ni )(mi−1 − mi ), (ni−1 − ni )mi , ni (mi−1 − mi )} + nr mr , i=1

(b)

dim 2 (V )G

r nr + 1 (ni−1 − ni ) min{ni , ni−1 − ni − 1} + ≤ for > 2, 2 2 i=1

(c)

dim ∧2 (V )G

r

≤ i=1

nr (ni−1 − ni + 1) . min{ni , ni−1 − ni } + 2 2

Proof. The proof will be by induction on r. The case r = 1 is Lemma 2.9. First consider part (a). By induction, for G := g1 , . . . , gr−1 we have dim(V ⊗ W )

G

r−1

≤

(ni−1 − ni )(mi−1 − mi ) + nr−1 mr−1 . i=1

On the other hand, clearly V G ⊗ W G ≤ (V ⊗ W )G , and this subspace is stabilized by gr . Thus we obtain an upper bound for dim(V ⊗ W )G as r−1

G

dim(V ⊗ W ) ≤

(ni−1 − ni )(mi−1 − mi ) + dim(V G ⊗ W G )gr

i=1 r−1

≤

(ni−1 − ni )(mi−1 − mi ) + (nr−1 − nr )(mr−1 − mr ) + nr mr i=1

by the case r = 1 in Lemma 2.9, as claimed. In (b) and (c) we may argue in precisely the same way, using that 2 (V G ) ≤ 2

(V )G and ∧2 (V G ) ≤ ∧2 (V )G . The previous result allows to conclude that Lemma 2.9 in fact holds for arbitrary elementary abelian -groups. Corollary 2.11. Let G be an elementary abelian -group, V , W two KG-modules of dimension n, m, with centralizer spaces of dimension z := dim V G , y := dim W G respectively. Then (a) dim(V ⊗ W )G ≤ min{ny, zm, (n − z)(m − y) + zy}, n−z z+1 for > 2, (b) dim 2 (V )G ≤ min z(n+1) 2 , 2 + 2 n−z+1 + 2z . (c) dim ∧2 (V )G ≤ min zn 2 , 2

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Proof. We have dim(V ⊗ W )G = dim HomG (1, V ⊗ W ) = dim HomG (V ∗ , W ). Now any G-homomorphism from V ∗ to W is determined by the image of the head of V ∗ , isomorphic to the socle V G of V . So dim HomG (V ∗ , W ) ≤ dim V G dim W = zm. The second bound in (a) follows by symmetry, the third with Lemma 2.10. In the notation there we have n = n0 , m = m0 , z = nr , y = mr , and r

G

dim(V ⊗ W ) ≤

(ni−1 − ni )(mi−1 − mi ) + nr mr i=1 r

≤

(n0 − nr )(mi−1 − mi ) + nr mr = (n − z)(m − y) + zy. i=1

Parts (b) and (c) follow in precisely the same way from Lemma 2.10 (b) and (c). For convenient reference, let us reformulate this result in terms of the codimension c := n − z of V G (resp. d := m − y of W G ): codim(V ⊗ W )G ≥ max{nd, mc, (n − c)d + (m − d)c}, # "c for > 2, (2.1) codim 2 (V )G ≥ max (n + 1), c(n − c + 1) " 2n # (c − 1), c(n − c − 1) . codim ∧2 (V )G ≥ max 2 It would be possible with a little bit more work to obtain similar results as above for centralizers on 3 (V ) and ∧3 (V ). But in the present work we need such results only for ∧3 of at most 10-dimensional modules, where ad-hoc arguments can be used.

Table 3. Ranks on exterior squares and cubes

dim V J (x)

3

21n−2 22 1n−4 23 1n−6 31n−3

1 − − 2

4

5

6

7

8

4

5

∧2 (V ) 2 2 − 4

3 4 − 6

4 5 6 6 8 10 6 9 12 8 10 12

6

7

8

9 10

∧3 (V ) 1 2 − 2

3 4 − 6

6 10 15 21 8 14 22 32 10 16 25 37 12 20 30 42

28 44 52 56

For the convenience of the reader we have collected the ranks, (i.e., the codimension of the centralizer, as defined in Section 2.1) of some unipotent elements x with simple Jordan types, on exterior square and cube of low dimensional vector spaces V in Table 3. The first column describes the Jordan type J (x) of x on V , the next six

Classification of 2F -modules, II

127

columns give the rank on ∧2 (V ) for 3 ≤ dim V ≤ 8, and the last seven columns give lower bounds for the rank on ∧3 (V ), with 4 ≤ dim V ≤ 10. The following result gives the centralizer dimension for elements interchanging the factors of a tensor product: Lemma 2.12. Let G be a group, V a finite dimensional KG-module and σ ∈ Aut(G). Let V σ := {vσ | v ∈ V } denote a copy of V with KG-module structure defined by vσ.g σ := (v.g)σ for g ∈ G, so V ⊗ V σ becomes a Gσ -module in the obvious way. Then

dim V + 1 σ σ dim(V ⊗ V ) = . 2 Proof. Choose a basis B of V . Then σ acts asa permutation on the basis {v ⊗ wσ | v, w ∈ B} of V ⊗ V σ , with precisely dim 2V +1 orbits.

3. Groups of Lie type in defining characteristic In this section, we first recall the description of the representations of groups of Lie type in their defining characteristic. In the second part we use orbit closure arguments to show that most of the time long root elements have largest centralizer on a given module. We next determine the maximal size of subgroups consisting of long root elements only. In the fourth part we prove results on centralizers of subspaces in the natural representation of classical groups. Finally, we outline the approach to the classification of 2F -modules in the individual cases. 3.1. Representations in defining characteristic. We start by recalling the wellknown description of absolutely irreducible representations of groups of Lie type in their defining characteristic, due to Steinberg, as expounded for example in [5, 2.8]. ¯ be a simple simply-connected algebraic group over the algebraic closure Let first G K of a finite field Fa and its weight lattice. To any dominant weight λ ∈ there is ¯ The modules M(λ), for λ associated an irreducible highest weight module M(λ) for G. running over the dominant weights, form a complete set of non-isomorphic irreducible ¯ ¯ →G ¯ be a Frobenius morphism G-modules over K (see [5, 1.14.6]). Now let F : G defining a rational structure over Fa with corresponding finite group of fixed points ¯ F . Then M(λ) remains irreducible upon restriction to G if λ is a restricted G := G weight for G, that is, all coefficients of λ when expressed as a linear combination of fundamental weights are less than a . Moreover, the (restrictions of the) M(λ), where λ is restricted, constitute a complete set of non-isomorphic irreducible G-modules over K [5, 2.8.2]. These can be described in more detail by Steinberg’s tensor product theorem. A highest weight representation M(λ) is called basic if λ is a basic weight for G, that is, all coefficients of λ when expressed as a linear combination of fundamental weights

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¯ ¯ →G ¯ is any endomorphism, are less than . If V is a G-module over K and F : G ¯ we adopt the following notation for the G-module obtained by composing F with the representation on V . If F is the untwisted Frobenius morphism defining a rational ¯ is of type B2 , G2 , F4 in characteristic structure over Fi , then we write V (i) . If G = 2, 3, 2 respectively, and F is an exceptional Frobenius morphism of order two with ¯ F defined over F2a+1 then we write V (a+1/2) . corresponding group of fixed points G d a Now first assume that G = Gn ( ) is not a Suzuki group. Let M(λ) be an or Ree i λ , where the λ are basic. irreducible KG-module with highest weight λ = a−1 i i i=0 Then M(λ) is isomorphic to the tensor product (1)

V0 ⊗ V1

(a−1)

⊗ · · · ⊗ Va−1

(3.1)

of different Frobenius twists of basic modules Vi ∼ = M(λi ), 0 ≤ i ≤ a − 1, for G. 2 2a+1 ) is a Suzuki or Ree group then any irreducible KG-module is If G = Gn ( isomorphic to a tensor product (1/2)

V0 ⊗ V1

(a)

⊗ · · · ⊗ V2a

(3.2)

of different Frobenius twists of basic modules Vi , 0 ≤ i ≤ 2a, for G. In both cases this tensor decomposition is unique [5, 2.8.6]. We next describe the fields of definition for these representations (see [5, 2.8.8]). First assume that G is untwisted. Then the field of definition of a tensor product as in (3.1) has degree s over F , where s denotes the index of the centralizer of (V0 , V1 , . . . , Va−1 ) in the cyclic permutation group generated by (0, 1, . . . , a − 1). In particular, it has index at most r in Fa if the tensor product involves precisely r ≥ 1 nontrivial factors. Thus all non-trivial basic modules have field of definition Fa . The corresponding statement, with a replaced by 2a + 1, holds for the Suzuki and Ree ¯ of groups. If G is a Steinberg twist, with corresponding graph automorphism ρ of G order t, then the preceding description remains true if all factors Vi , that is, their basic weights, are invariant under ρ. Otherwise the field of definition is the extension of degree t of the field described above. An effective tool in the investigation of groups of Lie type in their defining characteristic will be Proposition 2.4 in conjunction with the following result of Smith and Timmesfeld (see [5, Thm. 2.8.11] and the subsequent remark): ¯ be a simple algebraic group over the algebraic closure K of a Theorem 3.1. Let G ¯ F a corresponding finite group of Lie type, P ≤ G a parabolic finite field, G = G subgroup with Levi decomposition P = U L. If V is an irreducible KG-module then ¯ then the L acts irreducibly on CV (U ). Moreover, if V has highest weight λ for G highest weight for L on CV (U ) is the restriction of λ. We label the fundamental weights for classical groups according to the convention given in Table 4.

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Classification of 2F -modules, II Table 4. Labeling of Dynkin diagrams of classical type

An

n≥1

Cn

n≥2

1 t

2 t

2 1 t> t

3 t 3 t

p p p

p p p

n t n t

Bn

n≥3

Dn

n≥4

1 2 t< t t1 @

3 @ @t

t

3 t

p p p

n t

4 t

p p p

n t

2

3.2. Orbit closures and centralizers. In this section we show that root elements are the non-trivial unipotent elements with largest centralizer dimension on any given module. ¯ be a connected reductive algebraic group over an algebraically Lemma 3.2. Let G ¯ and T¯ ≤ B¯ a maximal torus. Then the closed field K. Let B¯ be a Borel subgroup of G, closure of the T¯ -orbit of any unipotent element 1 = u ∈ B¯ contains a root element. ¯ with respect to (T¯ , B) ¯ and + ⊂ the set Proof. Let be the root system of G + of positive roots. Then there exist tα ∈ K (α ∈ ) with u = α uα (tα ), where ¯ Assume that tα , tβ = 0 for at least two {uα (t) | t ∈ K} are the root subgroups in B. + roots α = β. Since no two roots in are proper multiples of each other, there exists a subtorus T¯ of T¯ of co-dimension 1 centralizing uβ (t) but acting non-trivially on the root subgroup for α. In particular the closure of the T¯ -orbit of u contains an element u whose product decomposition involves at least one factor less than the one for u, but which involves uβ (tβ ). Induction now shows the claim. ¯ be a connected reductive algebraic group over an algebraically Corollary 3.3. Let G closed field K. Then the closure of any non-trivial unipotent class contains a long root element, respectively a long or short root element if G has a factor Cm or F4 in characteristic 2, or a factor G2 in characteristic 2 or 3. Proof. If all roots of G have the same length, this follows immediately from the previous lemma. Now assume that G has two different root lengths, that is, G has a factor of type Bm , Cm , m ≥ 2, G2 or F4 . Direct computation with the commutator relations shows that the closure of the class of short root elements contains long root elements in B2 = C2 , G2 and F4 , except in the characteristics given in the statement. Thus we may conclude with Lemma 3.2. Corollary 3.4. Let G be a finite group of Lie type in characteristic and V an F Gmodule over a field F of characteristic . Then the centralizer dimension on V of any non-trivial unipotent element is bounded above by the centralizer dimension of a long root element (respectively of a long or short root element if G is of type Cm or F4 in characteristic 2 or of type G2 in characteristic 2 or 3).

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¯ denote the algebraic group corresponding to G over the algebraic closure Proof. Let G K of the ground field. Then the representation of G on V extends to a representation ¯ on V¯ := V ⊗ K, see Section 3.1. Thus it suffices to prove the assertion for G. ¯ of G Now note that the property of u having centralizer dimension z on V¯ is equivalent to the fact that all (n − z + 1) × (n − z + 1)-minors of a matrix representing u − 1 vanish, where n = dim V . This closed condition is clearly inherited by elements in the closure of the conjugacy class of u. Thus the assertion follows from Corollary 3.3. We extend the previous result to outer elements in extensions of algebraic groups ¯ be a connected reductive algebraic group over by graph automorphisms. For this let G ¯ of an algebraically closed field of characteristic and σ a graph automorphism of G ¯ and order . More precisely, fix a Borel subgroup B¯ and a maximal torus T¯ ≤ B¯ of G ¯ assume (as we may) that σ stabilizes the set of simple roots with respect to (T¯ , B). ¯ σ be as above. Then σ lies in the closure of any unipotent class Lemma 3.5. Let G, ¯ in the coset σ G. ¯ any unipotent class in σ G ¯ has a representative σ u in σ B. ¯ Proof. Since σ stabilizes B, + ¯ ¯ ¯ ¯ Let T ≤ T be the centralizer in T of σ , a subtorus. If α ∈ vanishes on all of T , then o(σ )−1

α(t) = −

i

α(t σ )

for all t ∈ T ,

i=1

o(σ )−1 i whence i=0 α σ = 0. But σ stabilizes the set of positive roots + , hence this is not possible. We may now copy the argument from the proof of Lemma 3.2 to conclude that the ¯ T -orbit of σ u contains σ . Since representations of groups of Lie type extended by a graph automorphism extend to the algebraic group, we obtain as above: Corollary 3.6. Let G be a finite group of Lie type in characteristic , σ a graphautomorphism of G of order and V an F G, σ -module over a field F of characteristic . Then the centralizer dimension on V of any non-trivial unipotent element in σ G is bounded above by the centralizer dimension of σ on V . The following result is well known. Proposition 3.7. Let G be a finite group of Lie-type of simply connected type and P < G a proper parabolic subgroup, P − the opposite parabolic subgroup. Let U and U − denote the unipotent radicals. Then G = U, U − . Proof. Let L be the common Levi subgroup of P and P − . Set H = U, U − . Then L normalizes U and U − and so H . So P and P − each normalize H . Clearly G = P , P − (since they are contained in no common proper parabolic subgroup and the only overgroups of P are parabolic subgroups). If G is simple (modulo its center),

Classification of 2F -modules, II

131

the result follows (since the center is contained in the Frattini subgroup). Note that in the cases that G is not simple, it is still true that G is the normal closure of its Sylow -subgroup. In particular, this shows that H = G if G is simple or G is rank one (for then U is a Sylow -subgroup). This leaves only the cases 2F4 (2), Sp4 (2) and G2 (2). One checks directly that the unipotent radical of a maximal parabolic subgroup is not contained in the derived subgroup (for example, see [4]). 3.3. Long root elements. We also need an upper bound for abelian subgroups consisting entirely of long root elements. Lemma 3.8. Let G be a simple algebraic group over an algebraically closed field. Let α, γ be two different long roots (with respect to the choice of a maximal torus T of G) with α + γ not a root. If the product Xα Xγ of the corresponding root subgroups contains a long root element outside of Xα ∪ Xγ , then γ − α is a root. Proof. After conjugation with a suitable element of the Weyl group we may assume that γ is the highest root. Then Xγ is normalized by the corresponding Borel subgroup B = U T . Let 1 = x ∈ Xα , 1 = y ∈ Xγ with v := xy a long root element. Since G is simple, all long root subgroups are conjugate and there exists g ∈ G with v g ∈ Xγ . Write g = uwb with u ∈ U , w in the Weyl group and b ∈ B according to the Bruhat decomposition, then v uw ∈ Xγ , so v u ∈ Xγ˜ with γ˜ w = γ . On the other hand v ∈ Xα Xγ , so v u ∈ Xαu Xγ ⊂ U . By the Chevalley commutator relations all non-trivial elements in Xαu have lowest term xα (t) for some t = 0. On the other hand v u = xγ˜ (t ) for some t = 0, so γ˜ = α. Note that Xαu = Xα . But then again by the commutator relations there must exist a root β with α + β = γ . ¯ be a simple algebraic group defined over Fq , G the set of fixed Proposition 3.9. Let G points under the corresponding Frobenius map, not of Suzuki- or Ree-type. Let A ≤ G be an abelian subgroup all of whose non-identity elements are long root elements. Let X be a long root subgroup. (a) Then logq |A| is bounded above by the values given in Table 5; and (b) A is conjugate to a subgroup of the unipotent radical of the parabolic subgroup NG (X). Proof. Let x, y be two commuting long root elements and assume that v = xy is again a long root element. Let X, Y be long root subgroups containing x, y respectively. Then [X, Y ] = 1 by [5, Prop. 3.2.9]. Moreover, since both lie in a Borel subgroup, and any two Borel subgroups have a maximal torus in common, we may assume that both are standard root subgroups: X = Xγ , Y = Xα . If X = Y then by Lemma 3.8 the difference γ − α is again a long root. Thus an upper bound for A is obtained by determining the maximal size of a set of long roots such that no sum of two of them is a root, but any difference is. This is easily done. It turns out that the maximal rank is realized by the maximal type A-subsystem consisting of long roots.

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Moreover, we can conjugate and assume that X = Xγ with γ the highest root. Write NG (X) = LQ with Q the unipotent radical and L the Levi complement. Suppose that Y = Xα . If Y is not contained in Q, then Y ≤ L and also X−α ∈ L. Thus, X−α commutes with X and so γ − α is not a root, a contradiction to Lemma 3.8. Table 5. Maximal ranks of subgroups of commuting long root elements

G logq |A|

An n

Bn (n ≥ 2) Cn n−1 1

Dn (n ≥ 4) n−1

G2 2

F4 3

E6 5

E7 7

E8 8

3.4. -ranks and centralizers. For the investigation of possible 2F -modules we sometimes need quite precise information on the -ranks of subgroups of classical groups with given centralizer on the natural module. This is a straightforward generalization of results in [5, Thm. 3.3.3]: Proposition 3.10. Let A ≤ SLn (a ) be an elementary abelian -subgroup with centralizer dimension dim CYn (A) ≥ n − c on the natural module Yn . Then |A| ≤ min{ac(n−c) , an

2 /4

}.

Moreover, for each c there is a subgroup A for which this bound is attained. In particular the -rank of SLn (a ) equals an2 /4. Proof. Since SLn (a ) acts transitively on the set of subspaces of Yn of fixed dimension we may assume that CYn (A) is spanned by the first n − c standard basis vectors. Thus I 0 A ≤ R := . ∗ ∗ The subgroup R is generated by full root subgroups of G. The result now follows precisely as in the proof of [5, Thm. 3.3.3], where the case c = n was treated. An example of an elementary abelian -subgroup of maximal rank is given by the group of lower unitriangular matrices with non-zero entries only in the lower left n/2×n/2-block if n is even (respectively (n+1)/2×(n−1)/2-block if n is odd). For the other classical subgroups of GLn (q) we make use of the invariant form on the natural module Yn : Proposition 3.11. Let A ≤ SUn (a ), n ≥ 2, be an elementary abelian -subgroup with centralizer dimension dim CYn (A) ≥ n − c on the natural module Yn . Then n 2 , 2 n even, ac arn |A| ≤ min{ , }, with rn := 2 2 n−1 + 1, n odd. 2

In particular the -rank of SUn

(a )

is at most arn .

Classification of 2F -modules, II

133

Proof. The precise -rank of SUn (a ) is given in [5, Table 3.3.1], and a short examination shows that it is bounded above by arn , proving the second bound. If c > n/2 then the first bound is not smaller than the second, so we may now assume c ≤ n/2. We will argue by induction on n, the induction base being given by the trivial case n = 2. Let W = R ⊕ X be the decomposition of the centralizer of A into its radical and the nonsingular part. Then c := dim R ≤ c and X has dimension n − c − c . Now A acts faithfully on X ⊥ of dimension c + c . Since X ⊥ ∩ W = R has dimension c the centralizer of A on X⊥ has dimension at most n − c. If c + c < n 2 then |A| ≤ ac by induction. Thus it remains to consider the case c + c = n, so c = c = n/2. In particular, n is even. Then the second bound gives the desired conclusion. We next consider the maximal -rank of elementary abelian -subgroups of Sp2n (q) with given centralizer on the natural module Y2n : Proposition 3.12. Let A ≤ Sp2n (a ) be an elementary abelian -subgroup with centralizer dimension dim CY2n (A) ≥ 2n − c on the natural module Y2n . Then c+1 n+1 |A| ≤ min a ( 2 ) , a ( 2 ) , and for each c there is a subgroup A for which this bound is attained. In particular the -rank of Sp2n (a ) equals a n+1 2 . n Moreover, if c = n and |A| > a (2) then CY (A) is maximal totally isotropic. 2n

Proof. Let A be an elementary abelian unipotent subgroup of G := Sp2n (q), q := a , trivial on a space of dimension 2n − c. If c ≥ n, then already a maximal abelian unipotent subgroup of G fixes a subspace of dimension n ≥ 2n − c, so assume that c ≤ n. Let W be the centralizer of A and R the radical of W . Then W = R ⊕ X as A-module with X nonsingular. Since A is trivial on X, it leaves X⊥ invariant. Suppose that R has dimension c – note c ≤ c (since R is contained in W ⊥ ). Now X has dimension 2n − c − c and so X⊥ has dimension c + c . But A is faithful on X⊥ and so by the result [5, Thm. 3.3.3] on maximal abelian subgroups, |A| ≤ qd(d+1)/2 where d = (c + c )/2 ≤ c, whence |A| ≤ q c(c+1)/2 – of course we may have c = c and obtain equality. Finally, if c = n and W is not totally isotropic, then c < c, d ≤ n−1, and thus |A| ≤ q n(n−1)/2 . Lemma 3.13. Assume that n ≥ 2. Let A be an elementary abelian unipotent subgroup of G := Sp2n (q) of order q n(n+1)/2 . Then A is the unipotent radical of the stabilizer of some totally singular subspace of dimension n in the natural representation for G unless q is even and n = 2. In that case then the result is true up to a graph automorphism. Proof. Let P be the stabilizer of a maximal totally singular subspace with Q the unipotent radical of P . We may assume that A ≤ P . By Lemma 2.6 it suffices to

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show that Q is not an F -module for L. For q odd, Q is the symmetric square of the natural module of GLn (q), and the claim follows from (2.1) in conjunction with Proposition 3.10 (see also Theorem 9.5). Suppose now that q is even. Then Q contains an L-submodule R isomorphic to the wedge square of the natural module for L. It follows from Proposition 3.14 that R is not an F + module for n > 3, whence Q is not an F -module. If n = 3, we still see that Q is not an F -module since its composition factors are the dual of the natural module and the Frobenius twist and the same argument applies. If n = 2, the result follows by an easy inspection (or by considering abelian sets of roots). Finally, the same investigation for orthogonal groups: (±)

Proposition 3.14. Let A ≤ SOn (a ), n ≥ 4, be an elementary abelian -subgroup with centralizer dimension dim CYn (A) ≥ n − c on the natural module Yn . Then n−1 2 + 1, n ≥ 9 odd, > 2, 2 c n − 2, n = 5, 7, > 2, |A| ≤ min{a (2) , arn }, with rn := n 2 , n ≥ 8 even, 2 n2 + 1, n = 4, 6. 2 (±)

In particular the -rank of SOn (a ) is at most arn . (±)

Proof. The precise -rank of SOn (a ) is given in [5, Table 3.3.1], and a short examination shows that it is bounded above by arn , proving the second bound. If c > n/2 then the first bound is not smaller than the second, so we may now assume c ≤ n/2. We will argue by induction on n, the induction base being given by the trivial case n = 2. As above let W = R ⊕ X be the decomposition of the centralizer of A into its radical and the nonsingular part. Then c := dim R ≤ c and X has dimension n − c − c . Our group A acts faithfully on X⊥ of dimension c + c . centralizer of A on X⊥ has dimension at Since X ⊥ ∩ W = R has dimension c the a (2c ) most n − c. If c + c < n then |A| ≤ by induction. Thus it remains to consider the case c + c = n, so c = c = n/2, R totally isotropic. If n = 4, 6 then again the second bound gives the desired conclusion. For n = 4, 6 it is easy to write out explicit matrices for the centralizer of a maximal isotropic subspace in orthogonal group the and verify (with the argument from loc.cit.) that its -rank is n/2 2 . The following result will allow to deal with the 27-dimensional modules for E6 (q). Lemma 3.15. Use the notation of 3.3.4 in [5]. Suppose that V has a unique tame subalgebra I of maximal order and moreover V = V (A) where A is the unipotent radical of a parabolic subgroup P . Then A is the unique elementary abelian subgroup of U of order |A|.

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Proof. Suppose that B is an elementary abelian unipotent subgroup of U of order |A|. By [5, 3.3.4] it follows that V (B) has the same order and so by hypothesis, V (B) = V (A). Let be the set of fundamental roots corresponding to U . Now A is just the product of the root subgroups, Uλ for λ ∈ \ where is the set of roots corresponding to the Levi subgroup L of P . Let = ∩ . Choose the linear functional f (as in loc. cit.) so that f (α) > f (γ ) where α ∈ \ and γ ∈ . With this ordering, it is clear that V (B) ≤ V (A) implies that B ≤ A. In particular, V (B) = V (A) implies that B = A, whence the result. Lemma 3.16. Let A be an elementary abelian unipotent subgroup of Spin+ 14 (q) of order |A| > q 20 . Then A is contained in the unipotent radical of the stabilizer of a totally singular subspace of maximal dimension. Proof. By Proposition 3.14 the tame abelian homogeneous subalgebra I (A) must belong to an abelian set of roots of maximal size. Direct computation shows that such sets correspond to unipotent radicals as in the claim, so I (A) ≤ I (Q) for such a radical Q. Since these roots correspond to a radical, we see that for some appropriate ordering of the roots, A ≤ U (I (Q)) = Q. 3.5. The general strategy. We can now describe the general approach to the determination of 2F -modules for groups of Lie type in their defining characteristic. Let ˜ = t G(q), q = a , a quasi-simple group of Lie ˜ be a finite group with G := F ∗ (G) G ˜ type, and (G : G) of -power order. Note that G is a central quotient of the group of fixed points under a Frobenius map of a simple simply connected algebraic group over the algebraic closure of F , or G = 2F4 (2) . Let V be a faithful absolutely irreducible ˜ Thus, the restriction of V to G is a direct sum of absolutely irreducible F¯ G-module. F¯ G-modules as described in Section 3.1, or G = 2F4 (2) . The 2-modular irreducible representations of the latter are contained in [8]. We distinguish two cases. If V remains irreducible upon restriction to G, then by Steinberg’s tensor product theorem it is a product V = V1 ⊗ · · · ⊗ Vr of r ≤ a Frobenius twists of nontrivial basic representations. Furthermore, by the description of fields of definition, it is defined over a subfield of Fq t of index at most rt. So by Lemma 2.1 we arrive at the necessary condition r ! i=1

dim Vi ≤ 2r logq |Aut(G)|

(3.3)

for V to be a 2F -module. If V is basic this reduces to dim V ≤ 2 logq |G|, and all such modules have been classified by Lübeck [11]. For any given type we can thus reduce to a short list of possibilities which then have to be investigated further. Alternatively, if V is a product of at least two non-trivial factors, by Lemma 2.10 the centralizer on V of a possible subgroup A has dimension at most dim V /d1 where d1 is the minimum of the {di := dim Vi | 1 ≤ i ≤ r}. If b denotes the -rank of G,

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we arrive at the inequality d2 . . . dr ≤ 2rb/a.

(3.4)

For small values of d1 this may be better than (3.3). If A involves outer automorphisms then we may replace b by the -rank of the centralizer of outer -elements, which gives even stronger conditions. Similarly, if G is twisted, then the field of definition of V may be larger and we get better inequalities. On the other hand, if V restricted to G is reducible, that is, if V is induced, then Proposition 2.5 applies. Moreover, any outer -element has centralizer space of dimension at most dim(V )( − 1)/ on V . Since dim V ≥ 3, A has to contain nontrivial inner elements as well, which have at least d2 . . . dr -dimensional commutator on the centralizer of outer elements. Together this gives the condition r . (3.5) d2 . . . dr (d1 − d1 + 1) ≤ 2 rb + a

4. Linear and unitary groups 4.1. The case SLn (q). In this subsection we consider the case of Ln (q), q = a . Let Yn denote the natural module for SLn (q) of dimension n, with right SLn (q)-action. Let λ1 , . . . , λn−1 denote the fundamental weights for SLn , ordered along the Dynkin diagram of type An−1 . Clearly, an SLn (q)-module V is a 2F -module if and only if its dual V ∗ is. Similarly, if V is a 2F -module, then so are all of its Frobenius-twists V (a) . It thus suffices to consider representations up to taking duals and up to Frobenius-twists. Now if V a λ has highest weight i i , then the contragredient representation has highest weight i a λ . We therefor only need to consider highest weights up to the symmetry n−i i i induced by the graph automorphism. It turns out to be convenient to treat the 2-dimensional linear groups separately. This will also serve as an induction base for the investigation of the other groups of Lie type. Proposition 4.1. Let V be an absolutely irreducible 2F -module in defining characteristic for a group G with F ∗ (G) = SL2 (a ). Then all composition factors of the restriction V |F ∗ (G) occur in Table 7. In particular, their highest weights are one of i λ1 ,

2i λ1 ( > 2),

(i + i+a/2 )λ1 (if 2 | a)

where 0 ≤ i < a. Proof. The basic SL2 (a )-modules, with highest weights tλ1 , 0 ≤ t ≤ − 1, are the symmetric powers t (Y2 ) of the natural module Y2 . On these, -elements of SL2 (a ) act indecomposably. 1. We first consider G = SL2 (a )-modules V . Let A be an offender on V . Then A is contained in the unipotent radical U of a Borel subgroup B of G. Since the maximal

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torus of B has only gcd( − 1, 2) non-trivial orbits on U , Lemma 2.1 shows that U itself is an offender. Hence we may assume A = U . By the Theorem 3.1 of Smith the torus then acts irreducibly on CV (A), so dim CV (A) = 1. If V is defined over the field with f elements this gives the condition |V | = df ≤ |A|2 |CV (A)| = 2a f , where d = dim V , hence f (d − 1) ≤ 2a. By the description in Section 3.1 V is a tensor product V1 ⊗ · · · ⊗ Vr of Frobenius twists of basic modules Vi of dimension di = dim Vi ≥ 2, with field of definition of index at most r in Fa . Thus we have f ≥ a/r and hence 2r − 1 ≤ 2r. This immediately forces r ≤ 2. In the case r = 1, that is, V is a Frobenius twist of a basic module, our inequality shows that d = d1 ≤ 3. This leads to Examples 4.1 and 4.3 with highest weights λ1 and 2λ1 . If r = 2 the assumption f = a violates the first inequality, since d ≥ 4. Hence f = a/2, the two factors are conjugate and we arrive at Example 4.4 with highest weight (1 + a/2 )λ1 . 2. The only outer automorphisms of S = L2 (a ) of order are field automorphisms, and these exist if and only if | a. So for the rest of the proof let G = SL2 (a )., where | a. The -rank of the centralizer in S of a field automorphism of order is a/. We first consider extendable representations V = V1 ⊗ · · · ⊗ Vr of G. Then necessarily | r and our standard inequality (3.4) reads d1 . . . dr−1 (dr − 1) ≤ 2r/ + 2r/a. This is satisfied only when di = 2 and either r = = a = 3 or r = = 2, a ≤ 2. But for SL2 (27) the 4-dimensional centralizer of the outer automorphism of order 3 is not centralized by an inner 3-element, so this case is out. The second possibility gives an example for the highest weight (1 + a/2 )λ1 . 3. Finally assume that V is induced from a representation W of SL2 (q) which is not invariant under the field automorphism of order . The field automorphism has centralizer of dimension dim(W ) on the induced module V = W ↑G S , and on this centralizer, non-trivial inner -elements have at least d1 . . . dr−1 (dr − 1)-dimensional commutator when W = V1 ⊗ · · · ⊗ Vr with di = dim Vi . Since V is defined over a subfield of index at most r of Fa , we arrive at the condition d1 . . . dr−1 (dr − 1) ≤ 2r + 2r/a. This forces r ≤ 2, = 2, di = 2. The case r = 1 occurs only when a = 2, 4 and leads to an example from the natural module. When r = 2 then moreover a = 2. But the group L2 (4) does not have non-extendable representations of degree 4. Note that the conclusion of Proposition 4.1 even holds for the solvable groups SL2 (2) and SL2 (3). For the remaining groups of Lie type we make use of the results of Liebeck [10] classifying the basic modules of Chevalley groups G in defining characteristic of small dimension (see also the more extensive lists of Lübeck [11]).

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Lemma 4.2. Let n ≥ 3, M(λ) a basic highest weight module for SLn (a ) of dimension dim M(λ) ≤ n2 +1. Then M(λ) is ( possibly the dual of ) one of the modules in Table 6. Table 6. The basic SLn (a )-modules of dimension at most n2 + 1

λ λ1 λ2 2λ1 λ1 + λn−1

M(λ) Yn ∧2 (Yn )

2 (Yn ) ˜ n) A(Y

λ3 3λ1 λ1 + λ 2

∧3 (Yn )

3 (Y3 )

dim M(λ) n n 2 n+1

n≥4 >2 n≥3

2

n2 − 1 − δ,n n 3

10 16

6≤n≤8 n = 3, > 3 n = 4, = 3

This is Theorem 2.2 in [10]. Thus generically (i.e., for n ≥ 9) only four types of modules occur. Moreover, the two smallest non-trivial dimensions are n and n2 . We need an extension of Lemma 4.2 giving lower bounds for the dimension of modules with symmetric highest weight for small n, as well as for modules with weights supported on the end nodes only; both follow for example from [11, Thm. 4.4]: Lemma 4.3. Let λ = t1 (λ1 + λn−1 ) + t2 (λ2 + λn−2 ) + · · · with 0 ≤ ti < be a symmetric basic weight for SLn (a ), n ≥ 3. Then dim M(λ) ≥ 2n2 unless λ = λ1 + λn−1 , Lemma 4.4.

or λ = λn/2 for n = 4, 6, 8,

or λ = 2λ2 for n = 4.

(a) We have dim M(λ1 + 2λn−1 ) ≥ 2n2 for n ≥ 4.

(b) We have dim M(2λ1 + 2λn−1 ) ≥ 2n2 for n ≥ 3. Lemma 4.5. Let n ≥ 3, λ = t1 λ1 + t2 λn−1 with t1 , t2 ∈ {i , 2i , i + a/2+i | 0 ≤ i < a/2}. Then the highest weight module M(λ) is not a 2F -module for SLn (a ). Proof. 1. For λ one of λ1 +i tλn−1 , 2λ1 +i tλn−1 or t (λ1 +i λn−1 ) with t = 1+a/2 the corresponding highest weight module M(λ) is a tensor product of dimension at least n(n2 − 2), n3 , (n2 − 2)2 , respectively. Only in the latter case M(λ) is defined over a proper subfield of Fq , of index 2. In any case the dimension exceeds the bound (2n2 − 1)a/f . Thus we may assume that t1 , t2 ∈ {i , 2i }, and then, by taking a suitable Frobenius twist, that t1 ∈ {1, 2}, t2 = 2i if t1 = 2. 2. Now let V = M(λ) for λ = λ1 + i λn−1 . Thus V is the tensor product ˜ n ) for Yn ⊗ (Yn∗ )(i) , except when i = 0 in which case V is the adjoint module A(Y a SLn ( ), a submodule of possibly a quotient by a 1-dimensional subspace of Yn ⊗ Yn∗ .

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Now first note that the assertion of Lemma 2.10 (a) for the centralizer remains true ˜ n ). Indeed, if A is cyclic, then all indecomposable modules are uniserial, so for A(Y the centralizer on a quotient cannot increase. The general statement was derived by considering successive actions of cyclic groups on previous centralizers, so the proof remains valid. Returning to V , if the centralizer of A on the natural module or its dual has dimension less than (n − 1)/2 then the first bound from Corollary 2.11 (a) gives a contradiction. If A contains non-trivial elements with distinct centralizers, then Lemma 2.10 (a) shows that the centralizer on V is too small. If A contains no transvections, then its rank is again too small. Thus A consists of transvections all fixing the same hyperplane in Yn , and all fixing the same hyperplane in (Yn∗ )(i) . But this forces |A| = q, and now Corollary 2.11 (a) gives the final contradiction. 3. Thus, if t2 = 2 then M(λ) is a tensor product defined over Fq of dimension at 2 least n n+1 2 , which is larger than 2n −1. So we are left with (t1 , t2 ) ∈ {(1, 2), (2, 2)}. For λ = 2λ1 +2λn−1 , the corresponding highest weight module has dimension at least 2n2 by Lemma 4.4 (b), hence this is also impossible. Also, the weight λ1 + 2λn−1 for n ≥ 4 is excluded by Lemma 4.4 (a). 4. So finally we consider λ = λ1 + 2λ2 in the case of G = SL3 (q). For this, we investigate possible offenders. Let + = {α, β, α + β} denote the set of positive roots for G with respect to the choice of a maximal torus and a Borel subgroup B containing it. For a root ν ∈ + let Xν = {xν (t) | t ∈ Fq } be the corresponding root subgroup. If there exists an offender A in the unipotent radical U of a maximal parabolic subgroup, then we may assume A = U by Lemma 2.1. But U together with the unipotent radical of the opposite parabolic subgroup generate G, by Lemma 3.7, so the centralizer dimension of U on M(λ) is at most dim(M(λ))/2. This implies that dim(M(λ)) ≤ 8, which is not the case by Lemma 4.2. Hence A contains elements involving xα (t) and xβ (u) for some non-zero t, u ∈ Fq . Then A also contains an element involving the product xα (t)xβ (u). But the normal closure of such an element in the Borel subgroup B is the whole unipotent radical U of B. Thus again by Lemma 2.1 we may and will assume that A = U . We have |A| = 3a , dim CV (A) = 1 by Theorem 3.1, and so dim(M(λ)) ≤ 6 if A is an offender on V . This is again a contradiction to Lemma 4.2. Lemma 4.6. The exterior cube ∧3 (Y7 ) of the natural module, with highest weight λ3 , is not a 2F -module for SL7 (q). Proof. 1. Let U be the unipotent radical of a Borel subgroup of SL7 (q) consisting of all lower unitriangular matrices. We may assume that a possible offender A on V = ∧3 (Y7 ) is contained in U . By Table 3 any transvection has at most 25-dimensional centralizer on V , so by Lemma 3.4 the same is true for any non-trivial unipotent element. Moreover, if A contains more than q elements, then dim CV (A) ≤ 21, by arguments similar to those in the proof of Lemma 2.10. Thus, |A| ≥ q (36−21)/2 > q 7 . Let P1 = U1 L1 be the end-node parabolic above U containing GL6 (q) on the first six basis vectors. Its unipotent radical U1 is elementary abelian of order q 6 . Thus, by our previous argument, we can’t have A ≤ U1 .

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2. Now assume that A ∩ U1 is trivial. Then A is isomorphic to its image B in the Levi subgroup of P1 of type GL6 (q). The composition factors of V for GL6 (q) are ∧3 (Y6 ) and ∧2 (Y6 ). Any unipotent element of GL6 (q) has rank at least 6 on ∧3 (Y6 ) and at least 4 on ∧2 (Y6 ). Moreover, any unipotent subgroup of order larger than q has centralizer codimension at least 9 on ∧3 (Y6 ) and at least 7 on ∧2 (Y6 ) by Lemma 2.10. Thus, |B| ≥ q 8 . Then either B is contained in the unipotent radical of a GL4 (q)-parabolic of GL6 (q), which can be ruled out by the standard argument, or it contains elements with at least three different centralizers, which decreases both centralizer dimensions at least by 1, and |B| = q 9 . Then by Proposition 3.10 B is the unipotent radical of a GL3 (q) × GL3 (q)-parabolic of GL6 (q), whose centralizer is too small by Theorem 3.1. Hence A ∩ U1 is non-trivial, so the centralizer of A on ∧2 (Y6 ) has codimension at least 10, and since A ≤ U1 , the centralizer on ∧3 (Y6 ) has codimension at least 6. Thus |A| ≥ q 8 . 3. Next assume that |A ∩ U1 | ≤ q. Since B := AU1 /U1 can have order at most q 9 by Proposition 3.10, we have q 8 < |A| ≤ q 10 . Let P2 = U2 .GL5 (q) be the end-node parabolic of L1 containing GL5 (q) on the first five basis vectors. By Theorem 3.1, CV (U1 ) is ∧3 (Y6 ) for GL6 (q), and on this the Levi factor GL5 (q) has composition factors ∧3 (Y5 ) and ∧2 (Y5 ). If |B ∩ U2 | ≤ q then the image of B in the Levi factor GL5 (q) has maximal possible order q 6 . But then its centralizer is only 3-dimensional on ∧3 (Y5 ) by Theorem 3.1, and we get a contradiction. On the other hand, if |B ∩ U2 | > q, then the centralizer of A ∩ U1 U2 has codimension at least 10 + 9 = 19 on V , whence |A| > q 9 . Then the image of A in GL5 (q) has order larger than q 3 , and thus a centralizer of codimension at least 5 on ∧3 (Y5 ). Then |A| ≥ q 12 , so A is the unipotent radical of a GL3 (q) ◦ GL4 (q)-parabolic, which has at most 2-dimensional centralizer by Theorem 3.1, impossible. 4. We may thus assume that |A ∩ U1 | > q. Then A ∩ U1 has centralizer of codimension at least 14 on V by direct computation, and the image B has centralizer of codimension at least 6 on CV (U1 ) = ∧3 (Y6 ). Thus |A| ≥ q 10 . But then necessarily B ∩ U2 = 1. The case that A is contained in U1 U2 was ruled out before, so the image C of A in GL5 (q) is non-trivial. Then we obtain a centralizer of codimension at least 20+3 = 23 on V , whence |A| > q 11 . First assume that |B ∩U2 | ≤ q. Then |C| > q 4 , which has centralizer of codimension at least 5 on ∧3 (Y5 ), contradicting the bound. On the other hand, if |B ∩ U2 | > q, then the centralizer of A has codimension at least 14 + 9 + 3 = 26 on V , again impossible. As a first step we treat the case where G does not involve outer automorphisms of S = Ln (q). Proposition 4.7. The absolutely irreducible 2F -modules in defining characteristic for SLn (q), q = a , n ≥ 3, are as given in Table 7. In particular, their highest weights are (up to Frobenius twists and taking the dual) one of λ1 ,

2λ1 ,

(1 + a/2 )λ1 ,

λ2 ,

λ3 (for n = 6).

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Proof. We proceed by induction, the induction base being given by the case n = 2 in Proposition 4.1. Note that for basic 2F -modules, (3.3) shows that dim V ≤ 2n2 − 1. 1. Let V be a 2F -module for G = SLn (q) of dimension d, defined over Ff . Let P = U.L = q n−1 .GLn−1 (q) be a maximal parabolic subgroup of G corresponding to an end node on the Dynkin diagram. First assume that there exists an offender A ≤ U for V . Since U is the natural module for L, hence irreducible, Lemma 2.1 allows to conclude that in this case U itself is an offender. On the other hand, by Proposition 3.7 the centralizer dimension of U on V is at most d/2. Thus the 2F condition gives df/2 ≤ 2(n − 1)a, hence d ≤ 4(n − 1)a/f . By Steinberg’s theorem, V is a tensor product V = V1 ⊗ · · · ⊗ Vr of Frobenius twists of non-trivial basic modules Vi of dimension di := dim Vi ≥ n, and defined over a subfield of Fq of index at most r. We obtain nr ≤ d ≤ 4(n − 1)a/f ≤ 4(n − 1)r. Since n ≥ 3 this forces r ≤ 2. Moreover, if r = 2 then di = n, and V1 , V2 are Galois (a/2) conjugate, so V = Yn ⊗ Yn up to Frobenius twists and taking the dual. This is Example 4.4. When r = 1 then d ≤ 4(n − 1), so by Lemma 4.2 we are in one of ˜ n ) with highest weight λ1 + λn−1 , the cases of Example 4.1–4.3 or 4.5, or V = A(Y which was excluded in Lemma 4.5. 2. Thus, by Proposition 2.4 we may assume that CV (U ) is a 2F -module for L (hence for L = SLn−1 (q)) for both end node parabolics. Hence by induction the highest weight λ of V or its dual is one of t1 λ1 + t2 λn−1

(t1 , t2 ∈ {0, i , 2i , i + a/2+i }),

i λ2 , tλ2 i λ3

(t ∈ {2i , i + a/2+i }, n = 4), (n = 6, 7).

Indeed, if λ has more than one non-zero coefficient, then there are precisely two, and these lie at the two ends, since by induction all 2F -weights have support only on one node. Moreover the entries are at most those for SL2 (q). Similarly for the cases with just one non-zero coefficient. The weights of the form t1 λ1 + t2 λn−1 not occurring in the conclusion were ruled out in Lemma 4.5; the possibility λ = λ3 for n = 7 was excluded in Lemma 4.6. 3. For n = 4, λ = (1 + a/2 )λ2 , the module M(λ) is the tensor product of V1 with (a/2) V1 , where V1 is the natural module for SO+ 6 (q) (namely, the exterior square of the natural module for G). The precise bounds for the rank of a subgroup of SO+ 6 (q) with given centralizer in Proposition 3.14 together with Corollary 2.11 show that this case does not occur. 4. Finally, the highest weight module M(λ) of G = SL4 (q) with highest weight λ = 2λ2 is the heart of 2 (∧2 (Y4 )), the symmetric square of the alternating square of the natural module. By Lemma 2.10 and Corollary 3.4 any non-trivial unipotent

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element of G has centralizer of dimension at most 4 on ∧2 (Y4 ). But then the centralizer ˜ 4 ) in the proof of Lemma 4.5 we on 2 (∧2 (Y4 )) has dimension at most 11. As for A(Y conclude that the same is true on M(λ). Hence an offender has order at least q 4 , so is equal to the unipotent radical of a GL2 (q)◦GL2 (q)-parabolic. But by Theorem 3.1 the latter only has centralizer dimension 3 on M(λ). Thus M(λ) is not an example. Proposition 4.8. Let V be an absolutely irreducible 2F -module in characteristic for a group G with F ∗ (G) = SLn (q), q = a , n ≥ 3. Then all composition factors of the restriction V |F ∗ (G) are 2F -modules for F ∗ (G). Proof. Let V be a 2F -module for G. 1. First assume that the restriction of V to F ∗ (G) is not irreducible. Let W be one of the irreducible constituents. By Proposition 2.5 either W itself is a 2F -module for a proper subgroup of G containing F ∗ (G), or = 2, W is defined over F22 and dim W ≤ k, where k denotes the -rank of an offender A. If a ≤ 2 this forces dim W ≤ n2 /4 + 2, so W is a twist of a basic module, and Lemma 4.2 shows that W = Yn , or n = 4, W = ∧2 (Y4 ), up to twists and duals. These are Examples 4.1 and 4.2. If a ≥ 3 then W has to be a product of at least a/2 factors, of dimension at least n, which gives na/2 ≤ an2 /8 + 2. This does not lead to new cases. Hence it remains to consider modules V with irreducible restriction to F ∗ (G), that is, highest weight modules whose highest weight is invariant under G. 2. Assume that G = SLn (a ). involves a field automorphism of order (so in particular | a). We may assume that A is not contained in F ∗ (G), so neither of the unipotent radicals of end node parabolics contains an offender. Theorem 3.1 yields that W := CV (U ) is an example for SLn−1 (a ).. By induction (with induction base SL2 (q)) this forces V or its dual to have highest weight λ ∈ {tλ1 , t (λ1 + λn−1 ), tλ2 (n = 4) | t = i + i+a/2 }. In particular = 2 and 2 | a. The first case is Example 4.4. The second one is a ˜ n ) ⊗ A(Y ˜ n(a/2) ), hence has dimension at least (n2 − 2)2 . This Frobenius twist of A(Y violates the upper bound 2(2n2 −1). The third module is a Frobenius twist of the tensor (a/2) product ∧2 (Y4 ) ⊗ ∧2 (Y4 ) for n = 4. In particular, the centralizer of an offender A has codimension at least 2 24 , so the 2F -condition reads 4(4 − 1) ≤ 2(42 /4 + 1), contradiction. 3. Next consider the case that = 2 and G = SLn (2a ).2 is obtained by adjoining the graph automorphism γ . The centralizer in SLn (2a ) of a graph automorphism of order 2 is contained in a symplectic group Sp n (2a ). Thus by Proposition 3.12 its 2 2-rank equals an(n + 2)/8 for n even, a(n2 − 1)/8 for n odd. By step 1 we may restrict to modules V with highest weight invariant under the graph automorphism. 3A. First consider modules with highest weight of the form t (λ1 + λn−1 ). Since ˜ n ). If = 2 these are tensor products of Frobenius twists of the adjoint module A(Y there are at least two factors, the dimension becomes too large. Now assume that V is ˜ n ). Let A be an offender on V for G. If B := A ∩ SLn (q) has centralizer a twist of A(Y

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of codimension c on the natural module, then Corollary 2.11(a) Lemma 2.12 show and that the centralizer on V has codimension at least 2c(n−c)+ n−c 2 . This is minimal for c = 1, and comparing with the rank of A we deduce n ≤ 4. If n = 4 then necessarily B has rank at least 3a/2, thus contains elements with distinct centralizers. This gives a contradiction with Lemma 2.10 (a). In the case n = 3 we have that Y3 ⊗Y3∗ = V ⊕1, that is, the centralizer on V is strictly smaller than on the tensor product Y3 ⊗ Y3∗ . This leads to a stronger inequality and thus to a contradiction. Since for n = 3 the only invariant weights are of the form t (λ1 + λ2 ), we may conclude that n ≥ 4. 3B. By Proposition 2.4 we may assume that W := CV (U ) is an example for the Levi complement, extended by γ , of any γ -stable parabolic subgroup of G. For n odd, induction starting from the case n = 3 in step 3A shows that only weights of the ˜ n ). form t (λ1 + λn−1 ) are possible, that is, tensor products of Frobenius twists of A(Y But these were ruled out in step 3A. For n even we first study basic modules. For n = 4 by step 3A the only basic highest weight to consider is λ1 + λ2 + λ3 . But this belongs to the Steinberg module for SL4 (2) (by [5, Thm. 2.8.7]), so has dimension 64, bigger than 2n2 − 1. For n = 6, our induction argument together with step 3A only leaves the weight λ1 + λ3 + λ5 , which is ruled out by Lemma 4.3. Now induction shows that there are no cases for even n ≥ 8. Still for n even, if V is an invariant tensor product of r ≥ 2 factors, then necessarily all of its factors have to be invariant. We obtain the condition n(n + 2) 2r d2 . . . dr ≤ r + . 4 a 4 This leads to r = 2, n = 4, d = 2 . Moreover, the two factors have to be Galois conjugate, so 2 | a. The subgroup A ∩ SL4 (q) has centralizer of dimension at most 4 on ∧2 (Y4 ), so its centralizer on V has dimension at most 20 by Corollary 2.11 (a). This forces a = 1, but we just saw that a has to be even in this case. This completes the consideration of the extension by the graph automorphism. 4. Next consider the case that = 2 and G = SLn (2a ).2 is obtained by adjoining a graph-field automorphism γ (so a is even). Here, the centralizer of γ of order 2 2 in SLn (2a ) is SUn (2a/2 ). So by Proposition 3.11 it has 2-rank n2 a/2 if n is even, 2 + 1 a/2 if n is odd. Basic representations do not extend, so respectively n−1 2 V is a non-trivial tensor product. The only possibility for invariant tensor products is V = Yn ⊗ Yn∗ (a/2) . But this is defined over Fa , so Corollary 2.11(a) yields a contradiction. 5. Finally assume that = 2, 2 | a, G = SLn (2a ).22 contains both the graph and the field automorphism of order 2. Here again basic representations are not invariant. The standard inequality (3.5) shows that the case of tensor products does not lead to examples in this situation. 4.2. The case SUn (q). The reasoning for the unitary groups is very similar to the one for linear groups.

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Proposition 4.9. Let G = SU6 (q) and V the exterior cube of the natural module. Let A be a nontrivial elementary abelian unipotent subgroup of G, then f (A) < |V |. Proof. Let A be an elementary abelian unipotent subgroup with f (A) maximal among such subgroups and assume that f (A) ≥ |V |. We actually work in GU6 (q). Let Q be the unipotent radical of the parabolic subgroup P fixing a totally singular three dimensional space in the natural module for G. So |Q| = q 9 . Write P = QL where L ∼ = GL3 (q 2 ) is the Levi complement. We note that f (Q) = q 19 by Theorem 3.1. In particular, since Q is an irreducible P -module, it follows that f (A) < |V | if A ≤ Q. Set B = A ∩ Q. We note that if 1 = x ∈ A, then CV (A) is at most of dimension 14 and moreover if x is not a long root element, then CV (A) has dimension at most 12, by Table 3. Since the only long root elements in A must be contained in Q, it follows that CV (A) has dimension at most 12. In particular, |A| ≥ q 4 . Consider A/B (as a subgroup of L). Note that as a P -module, we have a filtration of V : 0 < C = CV (P ) < W < M < V where W/C and M/W are 9-dimensional Fq L-modules, with W/C isomorphic to the natural module for L tensor its Frobenius twist and M/W the dual of W/C. First suppose that A/B contains a regular unipotent element of L (note that this implies that > 2). Choose x ∈ A with xB this regular unipotent element. It follows that CV (A) is at most 8-dimensional, whence |A| ≥ q 6 . In particular, |B| ≥ q 2 and does not consist of long root elements. Thus, CV (B)/CW (B) has dimension at most 3. This implies that dim CV (A) ≤ 7. In particular, |B| > q 2 and is contained in CQ (x). It follows by inspection that B contains a rank 3 element in Q. Thus, CV (B)/CW (B) is at most 1-dimensional. Thus, CV (A) has dimension at most 1 + 3 + 1 = 5. So f (A) ≤ q 19 , a contradiction. So A/B consists of long root elements in L. Suppose that A/B embeds in a long root subgroup X of L (this is not a long root subgroup of G). If |A/B| > q, then |CS (A/B)| ≤ q 4 where S = Q or S is one of the two nontrivial L-composition factors on V . If B = 1, this implies that f (A) ≤ q 14 . If B = 1, then CV (A) ≤ W and so CV (A) has dimension at most 9. Then |A| > q 5 and so |B| > q 3 . This allows to improve the estimate on CV (A) to be of dimension at most 8, whence |A| = q 6 (since that is an upper bound). In particular, this implies that B has order q 4 and so must be CQ (X). A straightforward computation shows that CV (B) is 4-dimensional. This implies that f (A) ≤ q 16 . The remaining possibility is that A/B is not contained in a long root subgroup but is contained in the unipotent radical of either the stabilizer of 1-space or 2-space in the natural representation of L. In the latter case, we observe that CW/C (A/B) is 1-dimensional. If |B| ≤ q, then CV (A) has dimension at most 2 + 5 = 7, a contradiction. If |B| > q, then CV (A) has dimension at most 2 + 3 = 5 and |A| ≤ q 2 |CQ (A/B)| ≤ q 6 , whence f (A) < |V |. So A/B centralizes a 2-space in the natural representation for L but is not contained in a root subgroup. It follows that |CQ (A/B)| = q 4 , |CW/C (A/B)| = q 4 and

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CM/W (A/B) is 1-dimensional. Thus, CV (A) is at most 7-dimensional and so |A| ≥ q 6 . This implies that B = 1 and so CV (A) is at most 6-dimensional. Thus, |A| = q 7 (since it has order at most q 7 ). In particular, we see that B = CQ (X) where X is the full unipotent radical containing A/B. We then compute that CV (B) is 4-dimensional, whence f (A) ≤ q 18 . Theorem 4.10. The absolutely irreducible 2F -modules in defining characteristic for SUn (q), q = a , n ≥ 3, are as given in Table 7 (see p. 147). Proof. 1. The unitary group SUn (q) is a subgroup of the linear group SLn (q 2 ) defined over the field of q 2 elements. By the general results presented in Section 3.1, the irreducible representations of SUn (q) are the restrictions of those of SLn (q 2 ). We distinguish two cases. 2. First assume that the restriction V to SUn (q) of the SLn (q 2 )-representation is defined over a proper subfield (necessarily of index 2). Then its highest weight has to be invariant under the graph automorphism γ . In particular, each tensor factor is itself invariant γ , so has dimension at least d0 := n2 − 2 for n = 4, 6, respectively nunder d0 := n/2 for n = 4, 6, by Lemma 4.2. We first show that in this case V is a twist ˜ n ), or of ∧k (Yn ) for n = 2k ∈ {4, 6, 8}. of A(Y 2A. If V is a tensor product V1 ⊗ · · · ⊗ Vr of at least two factors, of dimensions di = dim(Vi ) ≥ d0 , then our fundamental inequality (3.3) shows that d0r ≤ r(2n2 −1). With the lower bound for d0 mentioned above this only leaves the case n = 4, r = 2, Vi Galois conjugate twists of ∧2 (Yn ). This is the natural module for O− 6 (q), and Proposition 3.14 in conjunction with Corollary 2.11(a) allows to exclude this. 2B. Hence we may now assume that V is (a twist of) a basic module, of dimension at most 2n2 − 1. By Lemma 4.3 the only possibility to consider is n = 4, λ = 2λ2 , with dim M(λ) ≥ 19. Let P1 = U1 .L1 be a maximal parabolic subgroup with Levi complement L1 of type GU2 (q), so U1 is special of order q 1+4 . First suppose that there exists an offender A ≤ U1 . Since both Z(U1 ) and U1 /Z(U1 ) are irreducible L1 -modules, we may assume by Lemma 2.1 that either A = Z(U1 ) or A = U1 . In the first case, A is a root subgroup, 7 conjugates of which generate G, so the 2F -condition gives d/7 ≤ 2, which is not the case. In the second case A is a unipotent radical, which by Theorem 3.1 has centralizer of dimension 3 on M(λ), too small. So we may assume that A is not contained in U1 . Let P2 = U2 .L2 be the maximal parabolic subgroup with Levi complement L2 = GL2 (q 2 ). Then L2 acts irreducibly on the elementary abelian unipotent radical U2 of order |U2 | = q 4 . Thus, if A is contained in U2 , we may assume that A = U2 . Since two conjugates of U2 generate SU4 (q) by Proposition 3.7, we get dim M(λ) ≤ 16, which is not the case. We may thus assume that A is not contained in a conjugate of U1 or U2 . But then the normal closure of A in the unipotent radical U of a Borel subgroup is U itself, of order q 6 , and with centralizer of dimension 1. This yields the final contradiction for M(2λ2 ).

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˜ n ) and ∧k (Yn ), n = 2k ≤ 8. The case k = 2 is 2C. It remains to deal with A(Y ˜ Example 4.7. For V = A(Yn ) let A be an offender with centralizer of codimension c on the natural module Yn . Then by Proposition 3.11 the 2F -condition becomes 2c(n − c) − 2 ≤ 2c2 . This shows that c ≥ (n − 1)/2, so the stronger inequalities from Lemma 2.10 (a) apply and we obtain a contradiction. The wedge cube ∧3 (Y6 ) is not a 2F -module by Proposition 4.9. Let P = U L be a maximal parabolic subgroup of SU8 (q) with Levi complement of type GU6 (q) and U special of order q 1+12 . If ∧4 (Y8 ) is a 2F -module for SU8 (q), then the previous result implies that an offender must be contained in U . But this is ruled out immediately by Theorem 3.1 and Proposition 3.9. 3. Now assume that the representation of SLn (q 2 ) is defined over the same field as the restriction to SUn (q). Then any example for SUn (q) is a fortiori an example for SLn (q 2 ), hence one of the cases in Table 7. Of these, all except ∧2 (Y4 ), ∧3 (Y6 ), and the fourth example are defined over Fq 2 . The first example, which is the natural module, is Example 4.6. For V = ∧2 (Yn ), n ≥ 5, let A be an offender with centralizer of codimension c on the natural module. With Proposition 3.11 and (2.1) we obtain the inequality 2c(n − c − 1) ≤ 2c2 , which forces c ≥ (n − 1)/2. But this can be ruled out with the stronger assertion from Lemma 2.10 (c). The symmetric square is excluded in the same way. Proposition 4.11. Let V be an absolutely irreducible 2F -module in characteristic for a group G with F ∗ (G) = SUn (q), q = a , n ≥ 3. Then all composition factors of the restriction V |F ∗ (G) are 2F -modules for F ∗ (G). Proof. The case of 2F -modules V with reducible restriction to F ∗ (G) may be dealt with precisely as in the proof of Proposition 4.8. So now assume that V is irreducible for SUn (q). 1. First assume that is odd and G = SUn (a ). involves a field automorphism of order . Since basic representations are not invariant under field automorphisms of odd order, V is a tensor product of at least 2 factors. If all factors Vi are already defined over Fq , then dim Vi ≥ n2 − 2 for n = 4, 6, respectively dim Vi ≥ 6, 20 when n = 4, 6. Our standard inequality (3.3) shows that in this case V must be the product of two 6-dimensional representations for SU4 (q). But then the field of definition does not drop, and the inequality is not satisfied. If at least one factor is defined over Fq 2 then the improved inequality again rules out this case. 2. It remains to consider the case that = 2 and G is obtained from SUn (2a ) by adjoining the graph automorphism of order 2. Since V restricts irreducibly to SUn (2a ), all factors must have symmetric highest weight. If V is basic, then all coefficients of ˜ n ). For the highest weight are at most 1, since = 2. For n = 3 this only leaves A(Y ˜ n ), ∧2 (Yn ) and the Steinberg module with weight λ1 +λ2 +λ3 . The n = 4 we find A(Y dimension 64 of the latter is too large. By Theorem 3.1 we may assume that CV (U ) is an example for the Levi complement of any stable parabolic subgroup P = U.L. ˜ n ). For n ≥ 6 Induction now proves that for n = 5 we only have to consider A(Y n/2 ˜ Lemma 4.3 only leaves A(Yn ), or ∧ (Yn ) with n = 6, 8.

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˜ n ). Indeed, if A is an offender, First note that V cannot be the adjoint module A(Y a with B = A ∩ SUn (2 ) with centralizer of dimension n − c on the natural module, then

c+1 2c(n − c) ≤ 2 + 2/a 2 by Proposition 3.12. This implies c ≥ 2/3n − 1, but then the upper bound for the right hand side from Proposition 3.12 eliminates this case. Let V := ∧3 (Y6 ) for the extension G of SU6 (q) with the graph automorphism. The centralizers in SU6 (q) of both classes of outer involutions are contained in Sp6 (q), for which V has two natural composition factors Y6 (exchanged by outer elements of G) and one spin module Z3 . In particular the centralizer of B := A ∩ Sp6 (q) is contained in a module with composition factors Y6 and Z3 , and outer involutions have rank at least 6. Thus |A| ≥ q 3 , and |B| ≥ q 2 . By Propositions 3.12 and 3.14 such subgroups centralize at most a 4 + 6-dimensional subspace of Y6 ⊕ Z3 , whence |A| ≥ q 5 , |B| ≥ q 4 . Again, such a subgroup has at most 3 + 4-dimensional centralizer on Y6 ⊕ Z3 , forcing |B| = q 6 maximal possible. But then B is conjugate to the stabilizer of a totally isotropic subspace on the natural module, by Lemma 3.13, which has just 1-dimensional centralizer on Y6 . This would imply |B| > q 6 , impossible. The case of ∧4 (Y8 ) is now ruled out by induction. If V is a product of at least two factors, then the usual argument only leaves the possibility n = 4, d1 = d2 = 6, that is, the Vi are natural modules for SO− 6 (q). This can be ruled out with Proposition 3.14 and (2.1). 4.3. The examples for SLn (q) and SUn (q). We proceed to verify that the cases left open in the previous proofs do in fact all lead to examples of 2F -modules. Table 7 contains the examples of 2F -modules V for SLn (q) and SUn (q) of dimension d Table 7. 2F -modules for linear and unitary groups in defining characteristic

G SLn (a ) SLn (a ) SLn (a ) SLn (2a ) SL6 (a ) SUn (a ) SU4 (a )

d n n

V f Yn a 2 ∧ (Yn ) a 2 n+1 a

2 (Yn ) 2 (a) 2 n Yn ⊗ Y n a 3 20 ∧ (Y6 ) a n Yn 2a 2 6 ∧ (Yn ) a

conditions n≥4 ≥3

log |A| a (n − 1)a (n − 1)a 2n2 /4a 5a a a

defined over Ff , up to Frobenius twists and taking duals, together with the size of one possible offender A.

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Case 4.1: On its natural module Yn , SLn (q) acts as a transvection group, so we get an example with |A| = q. Case 4.2: When n ≥ 4 the alternating square ∧2 (Yn ) of the natural module is irreducible for SLn (q). The subgroup 1 1 (4.1) A := ai ∈ Fq 1 a1 . . . an−1 1 on ∧2 (Yn ), so this of order q n−1 visibly has centralizer of dimension at least n−1 2 yields an example. Case 4.3: For > 2 the symmetric square 2 (Yn ) is irreducible. Again the subgroup A in (4.1) gives an example: its centralizer has dimension at least n2 . (a) Case 4.4: Let q = 2a and Yn the a-th Frobenius twist of the natural module. (a) Then Yn ⊗ Yn is irreducible for SLn (q) and defined over the subfield Fa of Fq of index 2. Choosing for A a maximal elementary abelian -subgroup, condition (1.1) reads 2 n −1 2 n a − za ≤ 2 · 2a 4 where z ≥ 1 denotes the dimension of the centralizer of A. Thus this gives an example. Case 4.5: For n = 6 the alternating cube ∧3 (Y6 ) gives an example for SLn (q) with the offending subgroup A from (4.1): its rank equals 5a, and it centralizes at least the alternating cube of a 5-dimensional subspace, of dimension 10. Case 4.6: The special unitary group SUn (q) acts as a transvection group on its natural module Yn over Fq 2 , thus we get an example with |A| = q. Case 4.7: The 4-dimensional unitary group SU4 (q) is isomorphic to the 6-dimensional orthogonal group of minus-type. The latter is a bi-transvection group, so we get an example with |A| = q.

5. Spin modules In order to treat the symplectic and orthogonal groups we now collect some facts on the 2n -dimensional spin module Zn of the orthogonal group Spin2n+1 (q), with highest weight λ1 (this includes the case of symplectic groups in even characteristic), and of (±) the two 2n−1 -dimensional half-spin modules Zn−1 of orthogonal groups Spin2n (q) with highest weight λ1 or λ2 . 5.1. Ranks on spin modules. For the following lemma we make use of the known classification and closure relation on the set of unipotent classes, which is described

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for example in the book of Spaltenstein [16]. Let K be the algebraic closure of a finite field Fq . Lemma 5.1. Let n ≥ 5, 1 = x ∈ G := Spinn (K) be unipotent, V a (half ) spin module for G. Then (a) dim CV (x) ≤ (3/4) dim V , and (b) dim CV (x) ≤ (5/8) dim V unless n = 8 or x is a long root element. Proof. Since the spin module for Spin2k−1 (K) is the restriction of the half-spin module for Spin2k (K) and the class of root elements of Spin2k (K) intersects Spin2k−1 (K) in the class of long root elements, it is sufficient to prove the assertion for n even. Clearly root elements x of Spin4 (K) ∼ = SL4 (K) have dim CV (x) = (3/4) dim V on the natural module V for SL4 (K). Now the half-spin module of Spin2k (K) splits into the two different half-spin modules of Spin2k−2 (K) upon restriction, so the first claim follows. For the remaining part let us assume that q is even (the case of odd characteristic is similar). Then the unipotent classes of GO2k (K) are parameterized by pairs (λ, ) of a partition λ of 2k in which each even part occurs an even number of times (giving the Jordan blocks in the natural representation) and a map from the parts of λ to {0, 1}. Furthermore, if the class parameterized by (λ1 , 1 ) lies in the closure of the class parameterized by (λ2 , 2 ) then λ1 ≤ λ2 in the dominance order on partitions, plus a condition on the i (see [16, I.2.5]). Now long root elements lie in the class parameterized by 202 12k−2 , and there are just two unipotent classes directly above this one, parameterized by 22 12k−4 and 204 12k−8 . Let us first consider k = 4. Then the class 204 splits into two SO2k (K)-classes, and all three classes are fused under triality. By our remarks above, the class 22 14 has rank 2 on the natural module while the other two classes have rank 4. Via triality this implies that 22 14 has rank 4 on the half-spin modules, while the other two classes have rank 2 on one of the half-spin modules, rank 4 on the other. Since all other unipotent classes of SO2k (K) lie above all three of these classes, the assertion is proved for k = 4. For k ≥ 5 the two classes above 202 12k−2 don’t split in SO2k (K). The rank can be computed inductively by restricting to SO2k−2 (K), and we obtain that 22 12k−4 has rank (1/2) dim V while 204 12k−8 has rank (3/8) dim V . Lemma 5.2. Let V be a half-spin module for Spin+ 2n (q) with n ≥ 3. If |A| > q, then dim CV (A) ≤ (5/8) dim V . Proof. By Lemma 5.1 it suffices to consider A consisting of root elements. By Lemma 3.9 we see that A ≤ U1 , the unipotent radical of NG (X) where X is a root subgroup (this is the stabilizer of a two dimensional totally singular subspace in the natural representation). We may assume that there is a nontrivial element x ∈ A ∩ X. Since |A| > q, we may choose y ∈ A \ X. Then x, y is trivial on a space of + codimension 4 in the natural representation and so embed in Spin+ 8 (q) × Spin2n−8 (q).

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Thus, it suffices to consider n = 4 (since the spin module for G is a direct sum of copies of the spin module when restricted to Spin+ 8 (q)). We see by Proposition 3.14 that the fixed space of x, y on the spin module is at most 5 dimensional. The key to studying the spin modules is the existence of maximal parabolic subgroups whose unipotent radical has large centralizer space. Indeed, let G = (±) Spinn (q), and P = U L a maximal parabolic subgroup with U elementary abelian (±) of order |U | = q n−2 and L of type Spinn−2 (q). Then upon restriction the (half-) spin module V for G splits into two (distinct half-)spin modules for L. Moreover, W := CV (U ) is a (half-)spin module for L. Let A ≤ P be a unipotent subgroup of G. Then we have f1 (A) = |A| |CV (A)| ≤ |AU/U | |CW (AU/U )| |A ∩ U | |CV (A ∩ U )/W | = f1W (AU/U )|A ∩ U | |CV (A ∩ U )/W |,

(5.1)

and the corresponding equation for f (A). 5.2. Spin modules for even-dimensional orthogonal groups 9 Proposition 5.3. On the half spin modules V of Spin+ 10 (q) we have |A| |CV (A)|< |V | for all unipotent elementary abelian subgroups A = 1. 7

9 Proof. Let 1 = A ≤ Spin+ 10 (q) with f0 (A) := |A| |CV (A)| maximal and assume 7

36

that f0 (A) ≥ |V |. Root elements have rank 4 on V , so |A| ≥ q 7 > q 5 . In particular A contains non-root elements, of rank at least 6, whence |A| > q 7 . Let P = U L be a maximal parabolic subgroup containing A with U abelian of order q 8 and L of type Spin+ 8 (q). Since CV (U ) has dimension 8 we have A ≤ U . Assume first that AU/U consists only of root elements, so |AU/U | ≤ q 3 and |A ∩ U | > q 4 . Then CV (A) 72 has codimension at least 8 by (5.1), forcing |A| ≥ q 7 > q 10 , larger than possible. Thus |AU/U | > q 3 . Via triality we may interpret the spin module W := CV (U ) for L as the natural module, and then by Proposition 3.14 the centralizer of AU/U has codimension at least 4 on W . Since A ∩ U = 1, its centralizer has codimension at least 4 as well, whence |A| > q 10 , the final contradiction. Proposition 5.4. The half spin modules M(λ1 ), M(λ2 ) of Spin+ 2n (q) are not F modules for n ≥ 6. Proof. Let V denote one of the two half-spin modules for Spin+ 12 (q), A a possible offender on V , and P = U L a maximal parabolic subgroup containing A with U abelian of order q 10 , L of type Spin+ 10 (q). The ranks in Lemma 5.1 and Proposition 3.9 show that |A| ≥ q 12 . Thus neither A ≤ U nor A ∩ U = 1. Assume first that |A ∩ U | > q 5 . Then A ∩ U contains not only root elements, whence CV (A) has codimension at least 4 + 12 = 16 by (5.1) and Lemma 5.1. Thus |A| ≥ q 16 , too large. Hence we have that |A ∩ U | ≤ q 5 , and |AU/U | ≥ q 7 . Then (5.1) gives centralizer codimension at least 6 + 8 = 14. We conclude that |A| ≥ q 14 , so |AU/U | ≥ q 9 . But

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now Proposition 5.3 applied to AU/U ≤ L on W shows that |CW (AU/U )| < q 9 , so |A| > q 15 , again too large. The case of n ≥ 7 follows by induction. Proposition 5.5. The half spin modules M(λ1 ), M(λ2 ) of Spin+ 2n (q) are not 2F modules for n ≥ 7. Proof. Let V denote one of the two half-spin modules for Spin+ 14 (q), A a possible offender on V , and P = U L the stabilizer of a singular vector containing A with U of order q 12 , L of type Spin+ 12 (q). We first claim that for any subgroup B ≤ U with |B| > q we have f1 (B) ≤ q 44 . Indeed, choose such a subgroup with f1 (B) maximal and subject to that with |B| maximal. Since q < |B| we have |CV (B)| ≤ q 40 by Lemma 5.2, so f1 (B) ≤ q 43 if |B| ≤ q 3 . By the maximality assumption, B is a vector space over Fq (this will not change CV (B)), so |B| ≥ q 4 . Note that either B contains a totally singular 2-space (considering B as an L-module) or B contains a nondegenerate 2-space. Let g ∈ L stabilize this 2-space. We see that f1 (BB g ) ≥ f1 (B) (since |B ∩ B g | ≥ q 2 ) and so by the choice of B, B g = B for any such g. Witt’s theorem implies that either B = U or the 2-space is the radical of B — now choose a different 2-space to see that B = U . Thus, f1 (B) = f1 (U ) = q 12+32 = q 44 . Now let W := CV (U ) and set B := A ∩ U . As usual we see from Lemma 5.1 that |A| ≥ q 12 . Proposition 3.14 implies that B = 1. If A = B, then f (A) = f (U ) = q 56 < |V |. So B = 1 and AU/U = 1. Now f (A) ≤ |A| |AU/U | |CW (AU/U )| |A ∩ U | |CV (A ∩ U )/W | = |A|f1W (AU/U ) f1 (B)/|W |, so Proposition 5.4 implies that f (A) < |A|f1 (B) ≤ |A|q 44 . Thus, |A| > q 20 . This implies by Lemma 3.16 that A ≤ Q where Q is the unipotent radical of the stabilizer of a maximal totally isotropic space. Thus, f (A) ≤ f (Q). However, CV (Q) has dimension either 1 or 7 (depending upon which class of maximal totally isotropic subspaces we take). Thus, f (Q) ≤ q 49 , a contradiction. The case of n ≥ 8 follows by induction. Proposition 5.6. The half spin modules M(λ1 ), M(λ2 ) of Spin− 2n (q) are not F modules for n ≥ 4. Proof. The half-spin module V for Spin− 6 (q) is the natural module for SU4 (q), defined over Fq 2 . Proposition 3.11 shows that an offender A with f1 (A) ≥ |V | necessarily has |A| = q 4 maximal possible and thus f1 (A) = |V |. Now let G = Spin− 8 (q). Long root elements have rank 2 on the half-spin module V , 2 so an offender A has |A| ≥ q 4 . Application of Lemma 5.2 to the overgroup Spin+ 8 (q ) 6 shows that dim CV (A) ≤ 5, whence |A| = q is maximal possible. Consider a maximal parabolic subgroup P = U L containing A with U abelian of order q 6 and Levi complement L of type Spin− 6 (q). If A ≤ U , then A = U , and Theorem 3.1 gives

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a contradiction to the F-condition. Otherwise, W := CV (U ) is a half-spin module for L, so from (5.1) and the first part of the proof we have f1 (A) < q 6 q 2 f1W (AU/U ) ≤ q 16 . The case of n ≥ 5 now follows by induction using Theorem 3.1. Proposition 5.7. The half spin modules M(λ1 ), M(λ2 ) of Spin− 2n (q) are not 2F modules for n ≥ 6. Proof. The half-spin modules V of dimension 32 for Spin− 12 (q) are defined over Fq 2 . As usually we see from Lemma 5.1 that |A| ≥ q 12 . Let U be the unipotent radical of a maximal parabolic subgroup containing A and with Levi subgroup of type Spin− 10 (q). Note that AU/U = 1, so dim CW (AU/U ) ≤ 12. If A ∩ U contains not only long root elements then dim CV (A ∩ U )/W ≤ 4 and |A| ≥ q 16 , too large. Thus |A ∩ U | ≤ q 4 , so |AU/U | ≥ q 8 , dim CW (AU/U ) ≤ 10,

and

dim CV (A ∩ U )/W ≤ 8,

whence |A| ≥ q 14 . But then |AU/U | = q 10 is maximal possible and only centralizes half of W . This forces |A| ≥ q 16 , again too large. The case of n > 6 follows by induction using Theorem 3.1. 5.3. Spin modules for odd-dimensional orthogonal groups Proposition 5.8. The spin module M(λ1 ) is not an F -module for Spin2n+1 (q), n ≥ 4, q odd. Proof. On the spin module for Spin9 (q), q = a odd, long root elements have rank 4, and all other non-trivial elements have rank at least 6 by Lemma 5.1. Thus an offender A has |A| ≥ q 6 . Consider a maximal parabolic subgroup P = U L with U abelian of order q 7 and L of type Spin7 (q). The case that A ≤ U is excluded by Theorem 3.1. Let W := CV (U ), the spin module for L. If A∩U consists of long root elements only, then |AU/U | ≥ q 3 by Table 5. But such a group has |CW (AU )| ≤ q 5 by Lemma 5.1, while |CV (A ∩ U )/W | ≤ q 4 . So (5.1) gives |A| ≥ q 7 , whence |AU/U | ≥ q 4 . This has centralizer codimension at most 4 on W , forcing |A| ≥ q 8 , too large. Otherwise, |CW (AU/U )| ≤ q 6 and |CV (A ∩ U )/W | ≤ q 2 , which is again impossible. By induction this also excludes the spin module for Spin2n+1 (a ), n ≥ 5. Proposition 5.9. The spin module M(λ1 ) is not a 2F -module for Spin13 (q). Proof. The half-spin modules of Spin+ 14 (q) restrict irreducibly to the spin module of the subgroup Spin13 (q). Thus the result follows from Proposition 5.5.

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5.4. Spin modules for symplectic groups. Our Lemma 5.1 also applies to the spin module of Sp2n (2a ), but note that long roots of Spin2n+1 (2a ) become short roots of Sp2n (2a ). Proposition 5.10. The spin module M(λ1 ) is not an F -module for Sp2n (q), q even, when n ≥ 4. Proof. Let V = M(λ1 ) be the spin module for Sp8 (q) of dimension 16. From Lemma 5.1 we find that |A| ≥ q 4 . By Proposition 3.9 (viewing our group as a group of type Bn where the long roots are short roots in type Cn ), A can’t consist of short root elements only, so there are elements of rank at least 6 and |A| ≥ q 6 . Let P = U L be a maximal parabolic subgroup containing A with U abelian of order q 7 and L of type Sp6 (q), the normalizer of a long root subgroup X ≤ U . If A ≤ U then f1 (A) ≤ f1 (U ) = q 7 q 8 < |V |. On the other hand, if B := A ∩ U = 1 then |AU/U | = |A| = q 6 is maximal possible. By Lemma 3.13 A is the centralizer in Sp6 (q) of a maximal isotropic subspace on Y6 , which has 1-dimensional centralizer on the spin module Z3 , a contradiction. So B is a nontrivial proper subgroup of A. Let W := CV (U ), the spin module for L. This embeds Sp6 (q) into Spin+ 8 (q) on W , which may be viewed as the natural (q) by triality. In particular Proposition 3.14 gives upper bounds for module for Spin+ 8 unipotent subgroups of L with given centralizer on W . Suppose that |B| ≤ q. Then |AU/U | ≥ q 5 and so CW (AU/U ) has dimension at most 4 and CV (A)/CW (A) has dimension at most 4. Thus, |A| ≥ q 8 and |A/B| ≥ q 7 , too large. So |B| > q. Then Lemma 5.2 implies that |CW (AU/U )| ≤ q 6

and

|CV (A ∩ U )/W | ≤ q 2 ,

so |A| ≥ q 8 and |AU/U | ≥ q. But U/X is the natural module for L, so a subgroup of order q centralizes at most a subspace of size q 6 of U . Thus |A ∩ U | ≤ q 6 , implying |AU/U | ≥ q 2 . But this only centralizes a 5-dimensional subspace, whence |AU/U | ≥ q 3 . This in turn has at most 5-dimensional centralizer on W , so |A| ≥ q 9 . This forces |AU/U | ≥ q 4 , with at most 4-dimensional centralizer on W . Hence |A| ≥ q 10 and |AU/U | ≥ q 5 , but then |A ∩ U | ≤ q 4 , a contradiction. The case of n ≥ 5 follows by induction applying Theorem 3.1 to the maximal parabolic subgroup stabilizing a 1-space.

6. Symplectic groups Let Y2n denote the natural 2n-dimensional symplectic module for Sp2n (q) and Zn the 2n -dimensional spin module. We need some information on small basic modules from Lübeck [11, Thm. 4.4]:

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Lemma 6.1. Let M(λ) be a basic highest weight module for Sp2n (a ), n ≥ 2, of dimension dim M(λ) ≤ 4n2 + 2n + 1. Then M(λ) is one of the modules in Table 8. Here the fundamental weights are numbered according to Table 4. Table 8. The basic Sp2n (a )-modules of dimension at most 4n2 + 2n + 1

λ λn

M(λ) Y2n

λn−1

∧˜ (Y2n )

2λn

2 (Y2n )

λ1 λ1 + λ 2 2λ1 3λ2 λ1 λ1 λ2 λ3

Zn

2

3 (Y

2)

dim M(λ) 2n 2n 2 − 1 − δ,n 2n+1 2 2n

16 − 4δ,5 14 − δ,5 20 14 42 − δ,3 48 − 8δ,3 110 − 10δ,2

>2 n ≤ 8, n=2 n = 2, n = 2, n = 3, n = 4, n=4 n=5

=2 >2 >3 >2 >2

6.1. The case Sp4 (q). It is convenient to handle the case of Sp4 (q) ∼ = Spin5 (q) separately. (Note that the isomorphism interchanges the two fundamental weights λ1 and λ2 .) Proposition 6.2. Let V be an absolutely irreducible 2F -modules in characteristic for a group G with F ∗ (G) = Sp4 (q), q = a = 2. Then the composition factors of V |F ∗ (G) are as given in Table 9 (see p. 163). Proof. 1. We start by considering modules for G = Sp4 (q). The two end-node parabolic subgroups P1 = U1 .Sp2 (q)(q − 1) and P2 = U2 .GL2 (q) have special respectively abelian unipotent radical of order |Ui | = q 3 . Let U0 denote the invariant 1-dimensional subspace of U1 . 1A. If there exists an offender A ≤ U0 , then by Lemma 2.1 we may assume that A = U0 . This is a full long root subgroup, five of which generate G. Hence the 2F -condition for a module V of dimension d defined over Ff reads df/5 ≤ 2a, that is, d ≤ 10a/f . On the other hand, if A is contained in U1 but not in U0 , then we may choose A = U1 . Now two conjugates of A generate G, so we arrive at the condition df/2 ≤ 6a. The same argument applies when A ≤ U2 . Hence in any case d ≤ 12a/f . If V = V1 ⊗ · · · ⊗ Vr then using that di = dim(Vi ) ≥ 4 and f ≥ a/r we

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get (4)r ≤ 12r. This is only satisfied when r = 2, d1 d2 ≤ 24 and V1 , V2 are Galois conjugate, or when r = 1, d ≤ 12. In the first case necessarily d1 = d2 = 4, which is Example 6.4. If r = 1 and d = 12 then A ∈ {U1 , U2 } and dim CV (A) = 6. But Lemma 6.1 shows that V = M(λ1 + λ2 ), whence dim CV (A) ≤ 4, too small. On the other hand, if d ≤ 11, then V is one of the modules in Lemma 6.1 which we will consider in 1B. 1B. If there exists an offender contained in none of U1 , U2 , then it must involve root elements corresponding to both simple roots. But the normal closure of such a subgroup inside the Borel subgroup is the whole unipotent radical. Then |A| = q 4 , dim CV (A) = 1, so (d − 1)f ≤ 8a. This forces d ≤ 9, and again we are in one of the cases of Lemma 6.1. So V has highest weight λ1 , λ2 or 2λ2 . The first two are Examples 6.1 and 6.2. The highest weight module for 2λ2 , = 2, is the symmetric square of Y4 . Let A be an offender with centralizer of codimension c on Y4 . Then the centralizer on 2 (Y4 )hascodimension at least c(5 − c) by (2.1). Since the rank of A 3a is bounded above by c+1 2 a by Proposition 3.12, we conclude that c = 2, |A| = . But then the stronger inequality in Lemma 2.10 (b) gives a contradiction. 2. Now assume that G = Sp4 (a ). involves a field automorphism, so | a. Let first V be a module with irreducible restriction to Sp4 (a ). Since basic representations are not invariant this implies that V is a product of r ≥ 2 factors. Then (3.4) yields that r = = 2 and d1 = d2 = 4. Let A be a candidate offender with centralizer of dimension 4−c on the natural module. For these, by Lemma 2.10 and Proposition 3.12 we obtain c = a = 2. But in this case the centralizer of an outer involution is isomorphic to S6 × 2, so has 2-rank 4, and our inequality is violated. If V˜ is induced from V , then (3.5) yields that r = 1. If d = 4 then by Proposition 3.12 we have d( − 1) + c ≤ c2 + c + 2/a for a subgroup A with centralizer dimension 4 − c. This is satisfied if d = 4, the natural module. Otherwise A has at most d − 2-dimensional centralizer on V˜ , and we have d( − 1) ≤ 6, which by Lemma 6.1 gives no further example. 3. Finally we assume that = 2, 2 a and G = Sp4 (2a ).2 involves a graphautomorphism. Since we treated Sp4 (2) = S6 in the first part [6, Prop. 3.1], we may assume that a ≥ 3. Our standard inequalities allow to rule out this case immediately.

6.2. Some fundamental representations. We first recall properties of the fundamental representations of the symplectic group. In characteristic 0, the fundamental representations occur as direct summands of exterior powers of the natural representation as follows:

∧n+1−k (Y2n ) = M(λk ) ⊕ M(λk+2 ) for 1 ≤ k ≤ n as Sp2n (C)-modules (where M(λk+2 ) is the zero-module for k + 2 > n). In positive characteristic, the fundamental representations still occur inside the exterior powers of the natural representation. If the dimensions of the M(λk ) are known, then the composition factors of ∧k (Y2n ) can be worked out inductively, using the restriction

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formula

∧k (Yn )|GLn−1 = ∧k (Yn−1 ) ⊕ ∧k−1 (Yn−1 ). Proposition 6.3. The highest weight module M(λ1 ) of Sp2n (q), q odd, is not a 2F module for n ≥ 3. Proof. Let V = M(λ1 ) be the irreducible Sp6 (q)-module of dimension 14. Note that ∧3 (Y6 ) = V ⊕ Y6 . Thus, by Table 3, long root elements have rank 5, elements in Sp4 (q) which are not long root elements have rank 6, and elements with 3 Jordan blocks of size 2 have rank 7. Let P be the stabilizer of a totally isotropic 3-dimensional space in the natural module. So P = QL where L = GL3 (q) and |Q| = q 6 . Consider an L-composition series of V : 1 < F < W < M < V where F and V /M are trivial 1-dimensional modules, W/F is the symmetric square of a natural module for L and M/W is its dual. Moreover, Q ∼ = (W/F )∗ as L-modules. In particular, if 1 = x ∈ L, then dim CW/F (x) ≤ 3 by (2.1). Let A be a nontrivial elementary abelian unipotent subgroup with f (A) maximal and assume that f (A) ≥ |V |. If A ≤ Q then f (A) ≤ f (Q) = q 13 < |V |, impossible. Thus both B := A ∩ Q = 1 and AU/U = 1. Since Q ∼ = (W/F )∗ this forces |B| ≤ q 3 . Moreover, since CV (B) ≤ M we find that dim CV (A) ≤ 7, and |A| > q 3 . In particular, |B| > q, and B contains short root elements. But then dim CV (A)/CW (A) = dim CM (A)/CW (A) ≤ 2, so dim CV (A) ≤ 6 and |A| ≥ q 4 . We note also that we can view A as a vector space over Fq (since we may replace the nilpotent part of any element of A by a scalar multiple – these still generate an elementary abelian unipotent subgroup and centralize the same space). In particular, |A| is a power of q. First assume that AU/U doesn’t consist of root elements of L only. Since CV (B) ≤ M this implies that CV (A) has dimension at most 5. Thus |A| > q 4 , |B| > q 2 , contradicting the fact that AU/U only centralizes a 2-dimensional subspace of Q. So A/B consists of root elements of L. If |B| = q 3 , we see that CV (B) is at most 6-dimensional (because CV (B)/CW (B) has dimension at most 2 and CW (B) has dimension at most 4). Suppose that CV (A) = CV (B). Then B := Ag | g ∈ Q also centralizes CV (A). Since [A, Q] = CA (Q) (since A does not act quadratically on Q), B = B. The fact that the centralizers are the same on W implies that they generate the same Fq -subspace of Q and so CW (B) has dimension less than 4. Thus, CV (A) has dimension at most 5 and |A/B| = q 2 . It follows that A/B is the full unipotent radical of a maximal parabolic subgroup of L (and the one that centralizes a 3-dimensional subspace of Q). Thus, A/B centralizes a 1-dimensional subspace on W/F and so CV (A) has dimension at most 4. Thus, CV (A) has dimension 4 and |A| = q 5 . We leave this case for the moment and return to it later. So |B| < q 3 and |A| < q 5 , whence CV (A) has dimension at least 6 and so CW (A) has dimension at least 4. So |B| = q 2 and |A/B| = q 2 . It follows that CV (A) = CV (B) is six dimensional – a contradiction as above.

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We go back to the remaining case and consider the other parabolic U J = R = NG (X) where X is a long root subgroup. We see that B does contain a long root subgroup and so we may take X ≤ A < R. Let B1 = A ∩ U . If |B1 | > q 2 , then A ≤ C(B1 ) is conjugate to Q, a case we have already dealt with. Otherwise, |B1 | ≤ q 2 and so since |A| = q 5 , we have |B1 | = q 2 and A/B1 of order q 3 in J . Taking a composition series for J consisting of the orthogonal module, the natural module and the orthogonal module, we see that since X ≤ A, the centralizer of A is contained in the first two. Since |A/B| = q 3 this is the unipotent radical of the stabilizer of a totally singular 2-space in the natural module of J ; this fixes a 2-space in the natural module and a 1-space in the orthogonal module. It follows that CV (A) is 3-dimensional. Thus, f (A) = q 13 . Thus, V is not an 2F -module. The case of n ≥ 4 follows by induction. Proposition 6.4. The highest weight module M(λ2 ) of Sp8 (q) is not a 2F -module. Proof. In characteristic different from 3 we have ∧3 (Y8 ) = M(λ2 ) ⊕ Y8 . Thus from Table 3 we see that transvections have rank 14 on V , and bitransvections (and hence by [16, p. 237] any other non-transvections) have rank at least 20. Since A can’t consist of transvections only by Proposition 3.9, this forces |A| = q 10 , maximal possible. But this means that A is the unipotent radical of a GL4 (q)-parabolic, which leads to a contradiction with Theorem 3.1. In characteristic 3 the wedge cube ∧3 (Y8 ) has composition factors M(λ2 ) and two copies of Y8 . Let P = U L be a maximal parabolic subgroup with Levi factor of type Sp6 (q), and special unipotent radical U of order q 1+6 . Then the restriction of V to the Levi factor has composition factors M(λ1 ) and two copies of M(λ2 ) (the heart of the exterior square ∧2 (Y6 )) for Sp6 (q). Clearly the centralizer of any element of Sp6 (q) on V is at most as large as on the semisimplification of V . For the latter, Table3 shows that transvections have rank at least 13. Hence |A| > q 6 and A has to contain non-transvections. Any other non-trivial unipotent elements have rank at least 18, whence |A| ≥ q 9 . Then |A ∩ U | ≥ q 3 , so this group contains non-transvections. These contribute rank at least 18 on V . On the other hand the image of A in L, of order at least q 5 , has centralizer codimension at least 4 on the ˜ 2 (Y6 ). This forces |A| ≥ q 11 , impossible. Sp6 (q)-module CV (U ) = ∧ Proposition 6.5. The highest weight module M(λ3 ) of Sp10 (q) is not a 2F -module. Proof. Let first q be odd. Then ∧3 (Y10 ) has composition factors V := M(λ3 ) and M(λ1 ) = Y10 . As Y10 is self-dual for Sp10 (q) we have ∧3 (Y10 ) = M(λ3 ) ⊕ Y10 . Thus from Table 3 and Corollary 3.4 we see that transvections have rank 27 on V , and any other non-trivial unipotent element has rank at least 42. With Proposition 3.12 this implies that an offender on V has to consist of long root elements, but this is a contradiction with Proposition 3.9.

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In characteristic 2 the wedge cube ∧3 (Y10 ) has composition factors V := M(λ3 ) and two copies of Y10 . In order to determine the centralizers of transvections and bitransvections on V , we restrict to Sp8 (q). As an Fq Sp8 (q)-module, V has composition factors M(λ2 ) and two copies of M(λ3 ). Clearly the centralizer of any element of Sp8 (q) on V is at most as large as on the semisimplification. Using the results in the proof of Proposition 6.4 we find that transvections have rank at least 26 on V , and bitransvections (and hence by [16, p. 237] any other non-transvections) have rank at least 40. We may now conclude as before. Let G := Sp2n (q), n ≥ 2. Let En denote the codimension 1 submodule of the exterior square of the natural module for G. This is either an irreducible module or contains a 1-dimensional trivial module (the latter occurring precisely when | n). Let V be the nontrivial irreducible composition factor of En . In particular, dim V = 2n2 − n − δ where δ = 1 unless | n in which case δ = 2. Let P be the parabolic subgroup of G fixing a 1-space in the natural representation. Then P = NG (X) where X is a long root subgroup. Let U be the unipotent radical of P . Write P = LU where L is the Levi complement. Note that U/X is the natural L-module and that U is special if is odd and is abelian if q is even. Let W = CV (U ), W = CV (U − ) and M = CV (X). We see that W and W are natural L-modules and that M/W is isomorphic to ∧2 (W ) as an [L, L]-module if does not divide n and is isomorphic to En−1 otherwise. By (2.1) any nontrivial unipotent element of G has rank at least 2n − 2 on V . Lemma 6.6. Let B be a nontrivial subgroup of U . Then (CV (B) + M)/M has dimension at most 1. If B ∩ X = 1 or |B| > q, then CV (B) ≤ M. In particular, if A is a subgroup of P and A ∩ U = 1, then dim CV (A)/CM (A) ≤ 1. Proof. Note that if 1 = x ∈ X, then CV (x) = CV (X) = M and the result is clear. Note also that all elements in U \ X are conjugate in P . So we can choose any such element x. Choose a basis for the natural module for G of the form e, f, v1 , . . . , vn−2 where e generates the space fixed by P and the vi are in e⊥ . Now choose x ∈ U with xf = f + vn−2 . Let z ∈ CV (B) \ M. So z = f ∧ v + m where m ∈ M. Thus, z = xz = xf ∧ v + m + e ∧ v , where v ∈ e⊥ . Thus, z = (f + vn−2 ) ∧ v + m. This implies that vn−2 ∧ v = 0. Thus, v is a multiple of vn−2 (modulo e) and the result follows. If |B| > q, the same argument shows that CV (B) ≤ M. This proves the first statement. The second statement follows for CV (A)/CM (A) ∼ = (CV (A) + M)/M ⊆ (CV (A ∩ U ) + M)/M. Lemma 6.7. Let A be an elementary abelian subgroup of U . If |AX| ≥ q d+1 , then CM (A)/W has dimension at most c(c − 1)/2 where c = 2n − 2 − d. Proof. Fix y ∈ U \ X. Let W1 be the fixed hyperplane of y on W . Fix w ∈ W not in W1 with yw = w + e. Identifying M/W with the exterior square of W , we see that

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y − 1 maps w ∧ w to e ∧ w. Thus, the image of y − 1 is precisely e ∧ W1 and has dimension 2n − 3. Moreover, we see that the fixed space of y on M/W is the exterior square of W1 . So we see that if y1 , . . . , yd are linearly independent elements in U/X, then the centralizer of the subgroup generated by them (modulo W ) is isomorphic to the exterior square of a c-dimensional subspace of W , whence the result. Lemma 6.8. Let A be an elementary abelian unipotent subgroup of G := Sp2n (q). Assume that n ≥ 3 and |A| ≤ q n(n−1)/2 . Then f (A) < |V |. Proof. The fundamental inequality from (2.1) and Proposition 3.12 for an offender A on V with centralizer codimension c on the natural module reads

c+1 c(2n − c − 1) ≤ 2 , 2 so c ≥ n − 1 and |A| ≥ q n(n−1)/2 . In particular A contains non-trivial elements with different centralizers, and the stronger inequality from Lemma 2.10(c) forces |A| > q n(n−1)/2 . Lemma 6.9. Let G = Sp6 (q). If V is the 14 dimensional irreducible (indecomposable in characteristic 3) with highest weight λ2 , then (1) V is not an F -module; and (2) f (A) ≤ q 15 for any elementary abelian unipotent subgroup 1 = A of G. Proof. By the previous result, f (A) < |V | if |A| ≤ q 3 . Note that f (Q) = q 15 . So this inequality holds for any subgroup A of Q as well (since either |A| ≤ q 3 or the weak closure of A in Q is Q and so f (A) ≤ f (Q)). Similarly any subgroup A of U satisfies f (A) ≤ f (U ) = q 14 . Indeed, if q is odd, then |A| ≤ q 3 and so f (A) < q 14 . Moreover, we can replace A by the Fq -subspace it generates (this will not change the centralizer in V ) and so f (A) ≤ q 13 for q odd. The same argument applies if q is even and |A| ≤ q 3 . Let A be an abelian unipotent subgroup of G with f (A) maximal. So |A| > q 3 . Set B = A ∩ U . If B = 1, then A embeds in L and so |A| ≤ q 3 , a contradiction. So B = 1. By Lemma 6.6 this implies that we may assume that X ≤ B (for either A ∩ X = 1 and CV (A)/CV (AX) has dimension at most 1 or CV (A) = CV (AX)). Similarly, we may assume that B is an Fq -subspace of U . If |B| > q 3 , then q is even and CL (B) has order at most q (the subspace generated by B will contain a nondegenerate space). Thus, f (A) ≤ q 2 f (B). Now CW (A) has dimension at most 3 and CW (B) = W , so in fact f (A) ≤ qf (B) ≤ q 15 . The same argument applies unless B/X is totally isotropic (which is always the case for q odd). So B/X is totally isotropic. In particular, f (B) ≤ q 13 . If |A| ≤ q, this implies that f (A) ≤ q 2 f (B) ≤ q 15 . So AU/U centralizes at most a 2-dimensional space on W . So f (A) ≤ |A/B|2 f (B)/q 2 ≤ q 15 unless |A/B| > q 2 . In the latter case, we see that the Fq -subspace generated by A/B in L is the unipotent radical of the

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stabilizer of a totally singular 2-space (in the natural module of L). It follows that A/B centralizes only a 1-dimensional space in the orthogonal module M/W . So CV (A) has dimension at most 3, whence f (A) ≤ q 12 q 3 = q 15 . This proves the second statement. Since f1 (A) = f (A)/|A|, the only possibility for f1 (A) ≥ |V | is that |A| ≤ q and centralizes a hyperplane. Since G contains no transvections on V , this cannot occur. Proposition 6.10. Let A be an elementary abelian unipotent subgroup of G := Sp2n (q). Assume that n ≥ 4. Then f (A) < |V | unless possibly f (A) = |V | when n = 4 and q is even. Proof. Choose A with f (A) maximal and assume the contrary. We keep notation as above. In particular, W and W are the spaces centralized by U and U opp and M = CV (X). Set V = M/W and as we have remarked V is either the exterior square or the irreducible constituent of the exterior square of the natural module for L. We induct on n. We may assume that A ≤ P , and, by Lemma 6.8, that |A| > q n(n−1)/2 . Set B = A ∩ U . Suppose that A centralizes some nondegenerate subspace of the natural module or more generally A ∩ U = 1. There is no loss of generality in assuming that A ≤ L. Thus, f (A) ≤ |A|2 |CV (A)||W |2 /q 2 . This is less than |V | by induction if n > 4 and by the previous lemma if n = 4. Suppose that A ≤ U . If A ≤ X, then f (A) ≤ f (X) < |V |, a contradiction. Otherwise, f (A) ≤ f (U ) = |U |2 q 2n−2 = q 6n−4 < |V |, also a contradiction. If |B| ≤ q, then CV (A)/CM (A) is at most 1-dimensional by Lemma 6.6. So |AU/U |2 |CV (A)| ≤ q|AU/U |2 |CV (AU/U )||W |/q and so f (A) ≤ |AU/U |2 |CV (AU/U )||W |q 2 < |V | (we are using induction if n > 4 and the previous lemma if n = 4). So assume that |B| > q. In particular, CV (A) = CM (A). If |B| > q n or more generally the space generated by BX/X contains a nondegenerate subspace (of dimension 2), then |A/B| ≤ q (n−2)(n−3)/2 . Thus, |A| ≤ q n+1 q (n−2)(n−3)/2 ≤ q n(n−1)/2 , a contradiction (since n ≥ 4). So BX/X is totally singular. On the other hand, |A| > q n(n−1)/2 , whence |A/B| > n(n−3)/2 . It follows that the fixed space of A/B on W is totally singular and so has q dimension at most n. Plugging this into our formula above yields f (A) ≤ |AU/U |2 |CV (AU/U )|q|B|2 q n−1 < |V |, unless |B|2 ≥ q 3n−4 . However, |B| ≤ q n , a contradiction unless n = 4. If q is even, this still gives the desired conclusion that f (A) ≤ |V |. The remaining case is n = 4, |B| = q 4 and q is odd. So we see that B/X is a maximal totally isotropic subspace of U/X. Thus, A ≤ CP (B) is contained in a conjugate of Q. Since f (Q) < |V |, the result follows.

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6.3. 2F -modules for Sp2n (q) Theorem 6.11. The absolutely irreducible 2F -modules in defining characteristic for Sp2n (q), q = a , n ≥ 3, are as given in Table 9 (see p. 163). Proof. Let P1 = U1 .Sp2n−2 (q)(q −1) and P2 = U2 .GLn (q) denote the two end-node n+1 parabolic subgroups, with unipotent radicals of order |U1 | = q 2n−1 and |U2 | = q ( 2 ) .

1. First assume that there exists an offender A ≤ U1 . If A lies in the invariant 1-dimensional Sp2n−2 (q)-subspace U0 of U1 (that is, in the center of U1 if is odd), then by Lemma 2.1 we may assume that A = U0 . This is a full long root subgroup. But 2n + 1 conjugates of it generate Sp2n (q), so the 2F -condition for a module V of dimension d defined over Ff reads df/(2n + 1) ≤ 2a, that is, d ≤ 2(2n + 1)a/f . Otherwise, if A is not contained in U0 , then again by Lemma 2.1 we may assume A = U1 . Now two conjugates of A generate Sp2n (q), so we arrive at the condition df/2 ≤ 2(2n − 1)a. Hence in any case d ≤ 4(2n − 1)a/f . If V = V1 ⊗ · · · ⊗ Vr then using that di = dim(Vi ) ≥ 2n and f ≥ a/r we get (2n)r ≤ 4(2n − 1)r.

This is only satisfied when either r = 2, n = 4, or r = 1. In the first case, we necessarily have f = a/r, so V1 , V2 are Galois conjugate, and d = 2n. Inspection shows that f (U0 ) < |V | and f (U1 ) < |V |. If r = 1 then we saw that d ≤ 4(2n−1), so V is a twist of one of the modules in Table 8. The ones not occurring in the conclusion are the spin module for n = 6, which was already ruled out in Proposition 5.9, and the modules with highest weights 2λn or λ1 for n = 3, > 2. Explicit computations show that U0 is not a centralizer. Similarly, computing the centralizer of U1 via Theorem 3.1 shows that U1 is not an offender on M(2λn ). On the other hand, the non-abelian group U1 is an offender on M(λ1 ) for n = 3; this case was excluded in 6.3. 2. From now on we may assume that there is no offender in U1 , hence by Proposition 2.4 the centralizer CV (U1 ) is a 2F -module for Sp2n−2 (q). These are known by induction starting with the case of Sp4 (q) in Proposition 6.2. In particular, the restriction of the highest weight of V to the first n − 1 nodes is known. If V is basic, then it occurs in Table 8. The symmetric square does not lead to an example, as can be seen by using Lemma 2.10(b), as in the proof of the previous proposition. The other modules are either excluded in Proposition 5.9, Propositions 6.3-6.5 or in Proposition 6.10 or occur in the conclusion. The case of tensor products V = V1 ⊗ · · · ⊗ Vr , r ≥ 2, of twists Vi of basic modules of dimension di leads to the inequality (3.4) d2 . . . dr ≤ rn(n + 1). Since d ≥ 2n this forces r ≤ 3. The case r = 3 can only hold when n = 3, d = 2n = 6, and the three modules form a Galois orbit. Moreover, A necessarily has d − 1-dimensional centralizer on Vi by Lemma 2.10. But then the rank of A is too small by Proposition 3.12.

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3. Thus r = 2. First assume that V1 , V2 are Galois conjugate, hence of the same dimension d. If Vi are not twists of the natural module, then any non-trivial unipotent element has rank at least 2 on Vi , so we obtain the sharper inequality d ≤ n(n + 1). This is only satisfied when = 2, n ≤ 4, and Vi are spin modules of dimension d = 2n . If n = 3 then (2.1) together with Lemma 5.1 shows that short and long root elements have rank at least 24 on V1 ⊗ V2 , so an offender has size |A| ≥ q 6 , maximal possible. So A is the stabilizer of a maximal isotropic subspace, with centralizer of dimension 1, too small. The case n = 4 is ruled out by induction. Thus V1 , V2 are Galois conjugate twists of the natural module. By Proposition 3.12 and Corollary 2.11(a) we obtain the inequality

c+1 max{2nc, 2c(2n − c)} ≤ 4 2 for the codimension c of the centralizer on the natural module of a possible offender 2 (a) A on V := Y2n ⊗ Y2n . Thus c ≥ (2n − 1)/2 > n − 1 and |A| ≥ q n /2 . Moreover, n+1 if c > n then |A| = q ( 2 ) is maximal, so A equals the unipotent radical U2 of P2 .

Similarly, if c = n then Proposition 3.12 shows that CY2n (A) must be maximally isotropic and again A ≤ U2 . But the centralizer of U2 on V is only of dimension n2 . 4. So we are left with the case that V is the tensor product of r = 2 factors which are not conjugate. Then d2 ≤ n(n + 1), so either d1 = d2 = 2n, or = 2, n ≤ 4 and d2 = 2n . The first case is certainly not an example, since even in the Galois conjugate case it wasn’t. In the cases involving a spin module, dim V is larger than the bound (3.3).

Proposition 6.12. Let V be an absolutely irreducible 2F -module in characteristic for a group G with F ∗ (G) = Sp2n (q), q = a , n ≥ 3. Then all composition factors of the restriction V |F ∗ (G) are 2F -modules for F ∗ (G). Proof. Field automorphisms are the only outer automorphisms of Sp2n (q), n ≥ 3, so assume that G = Sp2n (a ). involves a field automorphism. If V has reducible restriction to F ∗ (G) then the standard argument shows that the claim holds. So now let V be a module with irreducible restriction to Sp2n (a ). Since basic representations are not invariant this implies that V is a product of r ≥ 2 factors. Then (3.4) yields that r = = 2 and d1 = d2 = 2n. Let A be a candidate subgroup with centralizer of dimension 2n − c on the natural module. For these, by Lemma 2.10 and Proposition 3.12 we obtain n = c = a = 2, which is excluded here. 6.4. The examples for Sp2n (q). The conventions for the following table are as for Table 7. Case 6.1: The symplectic group Sp2n (q) acts as a transvection group on its natural module Y2n over Fq , thus we get an example with |A| = q. Case 6.2: The irreducible representation with highest weight λn−1 of Sp2n (a ) is the heart of the exterior square of the natural representation. Consider the maximal

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Classification of 2F -modules, II Table 9. 2F -modules for symplectic groups in defining characteristic

G Sp2n (a ) Sp2n (a )

d 2n 2n 2 − 1 − δ,n

Sp2n (2a ) Sp4 (2a )

2n 16

V f Y2n a 2 ∧˜ (Y2n ) a Zn (a) Y4 ⊗ Y4

a a

n = 2, 3 or = 2, n = 4 n = 3, 4, 5

log |A| a n+1 2 a 10a (2n − 1)a 3a

parabolic subgroup of type GLn . The centralizer of its unipotent radical U has weight λn−2 , hence dimension n2 . The unipotent radical U is elementary abelian of -rank a n+1 2 . Thus we find an example when n ≤ 3. In the case n = 2 we have an alternative interpretation: The 4-dimensional symplectic group is isomorphic to the 5-dimensional orthogonal group. The latter is a bi-transvection group on its natural module, so we get an example for Sp4 (q) in dimension 5, when = 2. When = 2 then this representation is reducible, and its irreducible constituent of dimension 4 yields an example. Case 6.3: The spin module Zn for Sp2n (2a ) has highest weight λ1 . Consider the maximal parabolic subgroup of type Sp2n−2 . The centralizer of its unipotent radical U is again the spin module for Sp2n−2 (2a ), hence has dimension 2n−1 . But U is elementary abelian of rank (2n − 1)a. Thus U is an offender on Zn if 2n − 2n−1 ≤ 2(2n − 1), so for n ≤ 5. Case 6.4: The tensor product of the natural module Y4 for Sp4 (2a ) with its twist (a) Y4 is defined over Fa . A maximal elementary abelian -subgroup A of order 3a has (a) 2-dimensional centralizer on Y4 , hence at least 4-dimensional centralizer on Y4 ⊗Y4 . Thus A is an offender.

7. Orthogonal groups (±)

In this section we consider orthogonal groups Spinn (q) of dimension n ≥ 7, defined over fields of odd order if n is odd. By the results on linear, unitary and symplectic groups in the previous sections and well-known isomorphisms this suffices to cover all finite quasi-simple orthogonal groups. 7.1. The case Spin2n+1 (q). We start by recording the basic modules of small dimension from [11, Thm. 4.4].

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Lemma 7.1. Let n ≥ 3, > 2, and M(λ) be a basic highest weight module for Spin2n+1 (a ) of dimension dim M(λ) ≤ 4n2 + 2n + 1. Then M(λ) is one of the modules in Table 10. Table 10. The basic Spin2n+1 (a )-modules, n ≥ 3, > 2, of dimension at most 4n2 + 2n + 1

λ λn λn−1 2λn λ1 2λ1 λ1 + λ3

M(λ) Y2n+1 ∧2 (Y2n+1 ) ˜ 2 (Y2n+1 )

Zn

dim M(λ) 2n + 1 2n+1 2n+2 2

2

− 1 − δ,2n+1 2n 35 40

n≤8 n=3 n = 3, = 7

Theorem 7.2. The absolutely irreducible 2F -modules in defining characteristic > 2 for Spin2n+1 (q), q = a , n ≥ 3, are as given in Table 12 (see p. 167). Proof. Let V be a 2F -module for G = Spin2n+1 (q), with offender A. 1. Let P1 = U1 .L1 denote the end node parabolic subgroup of G with Levi factor L1 of type Spin2n−1 (q). Then U1 is abelian of order q 2n−1 and irreducible under L1 . Thus, if A ≤ U1 then by Lemma 2.1 we may assume that A = U1 . By Proposition 3.7, the 2F -condition for V implies dim V ≤ 4(2n − 1)a/f if V is defined over Ff . By Lemma 7.1 any basic representation of G has dimension at least 2n + 1. This shows that V cannot be a tensor product of at least two factors. But otherwise, V is (a Frobenius twist of) one of the modules in Table 10 of dimension at most 4(2n − 1), and all these occur in the conclusion. 2. Thus, by Proposition 2.4 we may assume that CV (U1 ) is a 2F -module for Spin2n−1 (q). First assume that V is a basic representation, hence its highest weight ˜ 2 (Y2n+1 ) the basic inequality from (2.1) and Propois contained in Table 10. For sition 3.14 gives a contradiction. Similarly for V = ∧2 (Y2n+1 ) it gives n = 3, and |A| ≥ q 4 . But then A contains non-trivial elements with at least two different centralizers and Lemma 2.10(c) yields a contradiction. The spin module Z6 for n = 6 is not an example by Proposition 5.9. By step 1 this also rules out the spin modules for n ≥ 7. The module with highest weight 2λ1 for n = 3 is not an example since the centralizer of U1 , with the same highest weight, is not an example for Spin5 (q) ∼ = Sp4 (q) by Proposition 6.2. Explicit construction of the 40-dimensional module with highest weight λ1 + λ3 as a submodule of the tensor product of the natural module with the spin module shows that long root elements have 18-dimensional commutator space, hence by Corollary 3.4 and Proposition 3.14 this is not an example.

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3. Next assume that V = V1 ⊗ · · · ⊗ Vr is a tensor product of r ≥ 2 factors of dimensions d = d1 ≤ d2 ≤ · · · ≤ dr . Since G is not a transvection group, the commutator of any non-trivial element in any non-trivial representation is at least 2-dimensional. Then (3.4) shows that 1 r(n2 − n + 4). 2 This forces r = 2, V1 , V2 are Galois conjugate, and either d1 = d2 = 2n + 1, or n = 3, d1 = d2 = 8. Both cases are eliminated by the inequalities from (2.1) together with Lemma 3.14. d r−1 ≤

Proposition 7.3. Let V be an absolutely irreducible 2F -module in characteristic > 2 for a group G with F ∗ (G) = Spin2n+1 (q), q = a , n ≥ 3. Then all composition factors of the restriction V |F ∗ (G) are 2F -modules for F ∗ (G). Proof. We only have to care about field automorphisms. For induced modules the assertion is immediate from Proposition 2.5(i) since = 2. If V has irreducible restriction to F ∗ (G), then its highest weight is invariant under the field automorphism, in particular it is not basic. So V is a tensor product of r ≥ 2 factors. The standard inequality (3.4) then shows that r = 2 and = 2, which is excluded here. 7.2. The case Spin+ 2n (q). Again we first collect the basic modules of small dimension from [11, Thm. 4.4]. a Lemma 7.4. Let n ≥ 4, let M(λ) be a basic highest weight module for Spin+ 2n ( ) of 2 dimension dim M(λ) ≤ 4n − 2n + 4. Then M(λ) is one of the modules in Table 11.

2 Table 11. The basic Spin+ 2n (q)-modules of dimension at most 4n − 2n + 4

λ λn λn−1 2λn λ1 , λ2 λ1 + λ2 , λ1 + λ4 , λ2 + λ4

M(λ) Y2n ∧2 (Y2n ) ˜ 2 (Y2n )

Zn−1

dim M(λ) 2n 2n − (n, 2)δ,2 2 2n+1 − 1 − δ,n 2 2n−1 56 − 8δ,2

>2 n≤9 n=4

Theorem 7.5. The absolutely irreducible 2F -modules in defining characteristic for a Spin+ 2n (q), q = , n ≥ 4, are as given in Table 12 (see p. 167).

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Proof. 1. Let us first handle some particular cases. The natural module Y2n has highest weight λn and gives Example 7.1. The highest weight λ = (1 + a/2 )λn belongs to (a/2) the tensor product Y2n ⊗ Y2n . The 2F -condition for an offender with centralizer of codimension c on Y2n then reads 2c(2n − c)/2 ≤ 2 2c by (2.1) and Lemma 3.14, which implies c > n, but then the upper bound for the rank in Lemma 3.14 is violated. ˜ 2 (Y2n ) with highest weight 2λn we similarly For the heart of the symmetricsquare

c obtain c(2n − c + 1) − 2 ≤ 2 2 , again not possible. The highest weight module for λ = λn−1 is the heart ofthe exterior square of Y2n . Lemma 3.14 and (2.1) show that c(2n − c − 1) − 2 ≤ 2 2c . This forces c ≥ n, A of maximal rank. But then A contains elements with different centralizers, and Lemma 2.10 (c) gives a contradiction. The half spin modules for n ≤ 6 with highest weight λ1 or λ2 are Example 7.3. The half-spin modules Z6 for Spin+ 14 (q) were excluded in Proposition 5.5. 2. Let P1 = U1 .L1 be a maximal parabolic subgroup of G corresponding to the last node of the Dynkin diagram, that is, with Levi factor of type Spin+ 2n−2 (q). Then U1 is the natural module for L1 , hence irreducible. If A is contained in U1 , we may take A = U1 . Two conjugates generate G, hence df/2 ≤ 2(2n − 2)a for a 2F -module V defined over Ff with offender A. If V is the tensor product of r factors Vi , of dimensions di = dim Vi ≥ 2n, then we get (2n)r ≤ 8(n − 1)r. This shows that r = 1, and thus d < 2n2 . Hence V is a twist of one of the basic modules in Table 11 all of which were treated in the first part. 3. If U1 does not contain an offender, we may assume by Proposition 2.4 that V is an example for the Levi factor L1 , hence for Spin+ 2n−2 (q). Let first n = 4, so + ∼ L1 contains the linear group Spin6 (q) = SL4 (q). Then we may repeat the above argument for all three end nodes of the Dynkin diagram. By Proposition 4.7 this only leaves the possibilities λ4 , λ3 , 2λ4 , (1 + a/2 )λ4 and their twists and images under the graph automorphisms for λ. These were investigated in the first part. 4. Now let n ≥ 5 and P2 = U2 .L2 a maximal parabolic subgroup with L2 = GLn (q), U2 the exterior square of the natural module for L2 . If A ≤ U2 , then d ≤ 4 n2 a/f by the standard argument. This forces V to be a twist of a basic module, of dimension d < 2n2 , hence one of the cases in step 1. Otherwise, V is an example for SLn (q). This, together with induction starting from step 3 shows that λ is one of tλn (t = 1, 2, 1 + a/2 ), λ1 (n ≤ 7), λn−1 up to twists and the graph automorphism of order 2. These were considered in the first part of the proof. 7.3. The case Spin− 2n (q) Theorem 7.6. The absolutely irreducible 2F -modules in defining characteristic for a Spin− 2n (q), q = , n ≥ 4, are as given in Table 12.

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(a/2) ˜ 2 (Y2n ) Proof. 1. The natural module is Example 7.1. The cases Y2n ⊗ Y2n and are excluded precisely as for orthogonal groups of plus-type. The spin modules for n ≤ 5 are Example 7.4. The half-spin modules for Spin− 2n (q), n ≥ 6, were excluded 2 ˜ is excluded using Proposition 3.14 in in Proposition 5.7. The adjoint module ∧ conjunction with (2.1). 2. Let V be a 2F -module for G = Spin− 2n (q) and A an offender on V . Let P1 = U1 .L1 be the end node parabolic with Levi complement of type Spin− 2n−2 (q). (q) we find that either V is one of the Then arguing as in step 2 of the proof for Spin+ 2n modules treated in step 1, or CV (U1 ) is an example for L1 . 3. Assume that V is a basic module M(λ). Then dim V ≤ 4n2 − 2n + 4 by (3.3), and dim V ≤ 2n2 − n + 2 if the highest weight λ is not invariant under the graph automorphism exchanging nodes 1 and 2. Thus

λ ∈ {λn , λn−1 , 2λn },

or

λ = λ1 , λ2 for n ≤ 7,

or

λ = λ1 + λ2 for n = 4

by Proposition 7.4. All except the last case were considered in step 1. The weight λ1 + λ2 does not lead to an example by step 2, since CV (U1 ) is not an example for ∼ Spin− 6 (q) = SU4 (q) by Theorem 4.10. 4. If V is a tensor product of r ≥ 2 factors, then (3.4) shows that r = 2, both factors have dimension 2n and are Galois conjugate. Exactly the same calculation as for Spin+ 2n (q) allows to rule out this case. 7.4. The examples for Spin(±) n (q). In the first line of the following table, is odd when n is odd. Table 12. 2F -modules for orthogonal groups in defining characteristic

G

(±) Spinn (a ) Spin2n+1 (a ) a Spin+ 2n ( ) − a Spin2n ( )

d

V

f

n 2n 2n−1 2n−1

Yn Zn

a a a 2a

Zn−1 Zn−1

(±)

log |A| n≥7 n = 3, 4, 5 n = 4, 5, 6 n = 4, 5

a (2n − 1)a (2n − 2)a (2n − 2)a

Case 7.1: The orthogonal group Spinn (q) acts as a bi-transvection group on its natural module, hence we obtain an example with offender of order q. Case 7.2: The spin module Zn for Spin2n+1 (a ) has highest weight λ1 . Consider the maximal parabolic subgroup of type Spin2n−1 . The centralizer of its unipotent radical U is again the spin module for Spin2n−1 (a ), hence has dimension 2n−1 . But U is elementary abelian of rank (2n − 1)a. Thus U is an offender on Zn if 2n − 2n−1 ≤ 2(2n − 1), so for n ≤ 5.

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Case 7.3: For the half-spin modules Zn−1 of dimension 2n−1 of G = Spin+ 2n (q) let A be the unipotent radical of the maximal parabolic subgroup with Levi complement of + type Spin+ 2n−2 (q). Then the centralizer CV (A) is a half-spin module for Spin2n−2 (q), of dimension 2n−2 , while A is elementary abelian of order q 2n−2 , so we find an example whenever n ≤ 6. Case 7.4: The half-spin modules Zn−1 of dimension 2n−1 of G = Spin− 2n (q) are defined over Fq 2 . Let A be the unipotent radical of the maximal parabolic subgroup with Levi complement of type Spin− 2n−2 (q). Then the centralizer CV (A) is a spin modn−2 , while A is elementary abelian of order q 2n−2 , (q), of dimension 2 ule for Spin− 2n−2 so we find an example whenever n ≤ 5.

8. Exceptional groups For exceptional groups we make use of the table of -ranks m in [5, Table 3.3.1], which we reproduce here as Table 13 for the convenience of the reader, and the determination of small basic modules in [11, Thm. 4.4]. Moreover, we employ the tables in Lawther [9] on Jordan shapes of unipotent elements of exceptional groups on small modules in conjunction with information on the closure relation on the set of unipotent classes as can be found in Spaltenstein’s book [16], for example. Table 13. -ranks of exceptional groups

G

m (G)

G

2B (22a+1 ) 2

2a + 1

F4 (a )

2(2a + 1) 3a = 3 4a = 3

(a )

2G (32a+1 ) 2

G2 (a ) 3D 2F

4

(a )

4 2a+1 (2 )

5a 5(2a + 1)

E6 2E

6 (

a)

(a )

E7 E8 (a )

9a

m (G) >2

11a

=2

12a

16a >2

13a

=2 27a 36a

Theorem 8.1. Let V be an absolutely irreducible 2F -modules in defining characteristic for a group G with F ∗ (G) an exceptional group of Lie type. Then the composition factors of V |F ∗ (G) are as given in Table 17 (see p. 175). The proof will be given in the following subsections. We denote by G a finite group such that F ∗ (G) is a quasi-simple exceptional group of Lie-type in characteristic , and let S be the unique non-abelian simple composition factor of G. By our general

Classification of 2F -modules, II

169

considerations we may assume that F ∗ (G) is the universal covering group of S, quotiented out by the -part of the Schur multiplier. Thus, F ∗ (G) is the group of fixed points under a Frobenius morphism of a simple algebraic group of simply-connected type. We indicate this by a subscript sc. The corresponding simple group is written without this subscript. First we rule out the adjoint modules: Proposition 8.2. Let G be one of G2 (q), F4 (q), E6 (q)sc , 2E6 (q)sc , E7 (q)sc , E8 (q). Then the adjoint representation of G does not give rise to a 2F -module for G. Proof. Consider first the case of G = E8 (q), q = a . Let V be the adjoint representation of G on its Lie algebra. By [9, Tab. 9] root elements have rank 58 on V , so by Corollary 3.4 any offender A has order at least q 29 . Moreover, if x = 1 is not a root element, then its rank is at least 92, too large compared to the -rank 36a. Thus A consists of long root elements only. But then Proposition 3.9 shows that A has size at most q 8 , too small. For G = E7 (q)sc , an offender A on the adjoint representation V can only contain elements from classes labeled A1 , 2A1 , 3A 1 by [9, Tab. 8]. (Using the table of the closure relation of unipotent classes in [16, p. 248] and the argument in the proof of Corollary 3.4 we only have to know the centralizer dimensions of classes 3A 1 and 4A1 to verify this.) First assume that A contains an element x from class 3A 1 , with rank at least 53 on V . Then A is contained in the centralizer of x, with semisimple part of type F4 , which lies in the maximal parabolic subgroup P = U.L of type E6 , with U elementary abelian of order 27. By Theorem 3.1 we don’t have A ≤ U , and since E6 (q)sc has -rank 16a by Table 13, we neither have U ∩ A = 1. But any unipotent element of F4 (q) has commutator of dimension at least 16 on the Lie algebra of F4 , so A has centralizer of codimension at least 69, which is much too large. The case of elements of type 2A1 can be excluded similarly. So finally assume that A only consists of root elements, of type A1 , with centralizer codimension 34 on the Lie algebra. Then Proposition 3.9 gives a contradiction. For G = E6 (q)sc , long root elements have rank 22 on the Lie algebra, by [9, Tab. 6], so |A| ≥ q 11 . Thus A cannot consist of long root elements only by Proposition 3.9. By [9, Tab. 6] the only other non-trivial unipotent elements possible in A are those of type 2A1 , with rank 32. Their centralizer has semisimple part of type B3 , so is contained in the maximal parabolic subgroup P = U.L of type D5 , with elementary abelian radical of order q 16 . As usually we see that A is not contained in U , but no non-trivial unipotent element from L centralizes the Lie algebra of B3 , thus the centralizer has codimension larger than 32, which is impossible by Table 13. For G = 2E6 (q)sc , essentially the same argument applies. Now let G = F4 (q). By [9, Tab. 4] root elements have rank at least 16 on the 52-dimensional adjoint module, thus an offender A has |A| ≥ q 8 . Let P = U L denote a maximal parabolic subgroup containing A with Levi complement L of type Sp6 (q). Then the composition factors of U as L-module have dimensions 1, 14 for 2 q, respectively 1, 6, 8 for q even by [5, p. 99]. By the previous estimate, A is

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not contained in a proper L-submodule of U . Thus A ≤ U by Lemma 2.1 and Theorem 3.1. Since L has -rank 6a we can’t have A ∩ U = 1 either. Now adding up centralizer codimensions of a pair of elements 1 = x ∈ A ∩ U and y ∈ A \ U gives a contradiction to the 2F -condition. Finally consider the adjoint module V for G = G2 (a ), = 3. Let first = 2. Long root elements have rank 6 on V , and any other nontrivial unipotent element has rank at least 7, by [9, Tab. 2]. By Table 13, any elementary abelian subgroup of G has order at most q 3 , so it follows that either f (A) < |V | or A consists of long root elements. By Lemma 3.9, this implies that |A| ≤ q 2 and so f (A) ≤ q 12 < V . Suppose that = 2. As above, any nontrivial x ∈ A must have a fixed space of dimension 8. The only such elements are long or short root elements. This implies that the centralizer of x in G is [P , P ] where P is a maximal parabolic subgroup and the fixed space of x on V is the Lie algebra of [P , P ]. So if |A| > q, dim CV (A) < 8 and f (A) < |V |. If |A| ≤ q, then f (A) ≤ q 10 . 8.1. The Suzuki-groups Lemma 8.3. If F ∗ (G) = 2B2 (22a+1 ) then any 2F -module for G is as in the statement of Theorem 8.1. Proof. Let F ∗ (G) = 2B2 (q), q = 22a+1 . Since Suzuki groups have no outer automorphisms of even order or multiplier of odd order we in fact have G = 2B2 (22a+1 ). The basic representations have highest weights λ1 , λ2 , λ1 + λ2 . The first two are the natural representations, Examples 8.1. The Steinberg representation has dimension 16, larger than 10 ≥ 2 logq |G|, so cannot occur. For tensor products V of r ≥ 2 factors (of dimension at least 4) we reach the inequality 4r /r ≤ 10. Hence r = 2 and both factors have dimension 4. But then V is defined over Fq , q = 22a+1 , and the inequality is not satisfied. 8.2. Groups of type G2 . From [11, A.49] we find: Lemma 8.4. Let M(λ) be a basic highest weight module for G2 (q) of dimension d = dim M(λ) ≤ 29. Then M(λ) is one of the modules in Table 14. Lemma 8.5. If F ∗ (G) = 2G2 (32a+1 ) or F ∗ (G) = G2 (q) then any 2F -module for G is as in the statement of Theorem 8.1. Proof. Let first F ∗ (G) = 2G2 (q), q = 32a+1 and first consider the simple group. Here 2 logq |G| ≤ 14. The basic representations of dimension at most 14 are those with highest weights λ1 , λ2 , of dimension 7 both, by Table 14. The group G has two classes of elements of order 3, both of which have representatives already inside 2G (3) = Aut(L (8)). For the latter group, such elements have Jordan blocks of sizes 2 2 3, 2, 2 respectively 3, 3, 1 on the 7-dimensional module. In particular, the centralizer of any offender is at most 3-dimensional. But then A must have order q 2 , which is

Classification of 2F -modules, II

171

Table 14. The basic G2 (q)-modules of dimension at most 29

λ λ2 λ1 2λ2 2λ1

dim M(λ) 7 − δ,2 14 − 7δ,3 27 − δ,7 27

= 2 =3

maximal possible. In this case A intersects both conjugacy classes non-trivially. Now the sizes of Jordan blocks show that the centralizer has at most dimension 2, so we get no example. The case of tensor products of at least two factors is ruled out by the standard inequality (3.3). Now assume that 3 | 2a + 1 and G = 2G2 (32a+1 ).3 is the extension by the corresponding field automorphism. Let W be a basic representation. Since basic representations are not invariant, the induction V of W to G is irreducible. Outer elements have at most d/3-dimensional centralizer, where d = dim V . The standard inequality shows that this does not lead to examples. Similarly, tensor products can be ruled out. Let now F ∗ (G) = G2 (a ), a = 2 (the case a = 2 was treated in the first part [6, Prop. 4.3] as U3 (3)). First consider the simple group G = G2 (a ). The 6- and 7-dimensional basic modules are Examples 8.2 and 8.3. The adjoint module was handled in Proposition 8.2. The 26- and 27-dimensional representations are the heart of the symmetric square of the 7-dimensional representation(s) V0 . Since G2 (q) is not a transvection group, non-trivial unipotent elements have at most 5-dimensional centralizer on V0 , hence at ˜ 2 (V0 ) by (2.1). Thus they are not examples by most 15-dimensional centralizer on Table 13. The inequality (3.3) excludes non-trivial tensor products except for the case of two factors of dimension at most 7. Again, this does not lead to examples since non-trivial unipotent elements are not transvections. Now assume that G is G2 (a ) extended by a field automorphism of order , so | a. Let V be a 2F -module for G. The basic representations of G2 (a ) do not extend to G. Then (3.5) shows that dim V = 12 (and = a = 2), so V |F ∗ (G) indeed consists of 2F -modules. The case of tensor products is ruled out by (3.3). Finally note that graph-automorphisms (of order 2) only occur in characteristic 3, so need not be considered here. 8.3. The groups 3D4 (q) Lemma 8.6. If F ∗ (G) = 3D4 (q) then any 2F -module for G is as in the statement of Theorem 8.1. Proof. Let V be a 2F -module for G and A an elementary abelian unipotent subgroup of G with f (A) maximal.

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We note as usual that |V | ≤ q 56 . It follows by Table 11 that (up to quasiequivalence) V is either the natural 8-dimensional module Y8 (over Fq 3 ) or the adjoint ˜ 2 (Y8 ) for the 8-dimensional orthogonal group. The latter has dimension module ∧ (over Fq ) 28 if is odd and 26 if = 2. Consider the second case. If is odd, then dim CV (x) = 18 for x a long root element and dim CV (y) < 18 for 1 = y a unipotent element that is not a long root element by Table 3. If A consists of long root elements, then f (A) ≤ q 4 q 18 < |V | by Proposition 3.9. Otherwise f (A) ≤ q 10 q 17 < |V | by Table 13. If = 2, then dim CV (x) = 16 for x a long root element and dim CV (y) ≤ 14 for any involution y not a root element. The argument of the previous paragraph applies. Finally consider the case that V is 8-dimensional (over Fq 3 ). Let X be a long root subgroup centralizing A. Let P = NG (X) be a maximal parabolic subgroup with radical Q. Note dim CV (x) = 6 (over Fq 3 ) for x a long root element (and is no bigger for any other unipotent element). Thus, |A| ≥ q 3 . Note that the composition factors for the Levi complement L ≥ SL2 (q 3 ) have dimension 2, 2 and 4. The latter is a tensor product of two twists of the natural module. In particular, it follows that for any unipotent element y ∈ P \ Q, |CV (y)| ≤ q 12 . So if A is not contained in Q, then f (A) ≤ q 12 q 10 < |V |. Thus, we may assume that A ≤ Q. Note that AX is the normal closure of A in Q. Then f (AX) ≥ f (A) and since AX is abelian, there is no harm in assuming that X < A (and is proper). If CV (A) has dimension less than 5, then f (A) ≤ q 22 < |V |. Since A properly contains X and Q/X acts faithfully on W := CV (X) (a 6-dimensional space), it must be the case that CV (A) is 5-dimensional. Since L acts irreducibly on Q/X, Q is the normal closure of A in P and so f (A) ≤ f (Q) = |V |. So the only possibility is that f (A) = |V | and f (B) ≤ |V | for any subgroup B of Q. In particular, this implies that |A| = q 9/2 (and so q is a square). Choose g ∈ L with Ag = A. Then A ∩ Ag is trivial on W and so A ∩ Ag = X. Thus, AAg has order q 8 and fixes a 4 dimensional subspace of W . So f (AAg ) = q 28 > f (Q), a contradiction. Now assume that G is obtained by extending 3D4 (q) by a field automorphism of order . Basic invariant modules only exist when = 3, and the only one below the dimension bound is the adjoint module of dimension 28. But as before any inner element has rank at least 10. But the outer 3-rank is controlled by the centralizer G2 (a ), hence at most 4a/3 + 1. Thus we necessarily have a = 3, but then the centralizer of inner elements has dimension at most 16, and we obtain a contradiction. Induced basic modules can be ruled out by Proposition 2.5, tensor products can’t occur by (3.3). 8.4. Groups of type F4 . From [11, A.50] we find: Lemma 8.7. Let M(λ) be a basic highest weight module for F4 (q) of dimension d = dim M(λ) ≤ 105. Then M(λ) is one of the modules in Table 15.

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Table 15. The basic F4 (q)-modules of dimension at most 105

λ λ4 λ1 λ1

dim M(λ) 26 − δ,3 26 52

=2 = 2

Lemma 8.8. If F ∗ (G) = 2F4 (22a+1 ) or F ∗ (G) = F4 (q) then any 2F -module for G is as in the statement of Theorem 8.1. Proof. First assume F ∗ (G) = 2F4 (q) , q = 22a+1 . We only have to consider G = 2F (22a+1 ) since there is no odd multiplier and the outer automorphism of order 2 of 4 2 F (2) does not have representatives of order 2. We have 2 log |G| ≤ 52(2a + 1), 4 q so by Table 15 only the basic highest weight modules of dimension 26 might occur. The 2-rank of G is 5(2a + 1) by Table 13. Both classes of involutions 2A, 2B have representatives already inside 2 F4 (2) . Since these invert elements of order 5 respectively 13 we may conclude that their centralizers on the 26-dimensional modules are at most 16- respectively 14-dimensional. Hence a possible offender A must be 2A-pure of order 25(2a+1) . But the center of the centralizer of a 2A-element clearly has a smaller 2-rank. Thus we get a contradiction with Lemma 2.3. For a tensor product of r factors we need 26r ≤ 52r, which does not hold when r ≥ 2. Now let G = F4 (q), q = a . The small basic modules were collected in Table 15. The adjoint module was excluded in Proposition 8.2. We have to leave open the case of the 25- respectively 26-dimensional module. For tensor products, the inequality (3.3) leads to a contradiction. For groups G obtained from F4 (q) by extension with a field automorphism, the standard estimates rule out 2F -modules. The 2-rank of the centralizer of an exceptional graph automorphism of order 2 in characteristic = 2 is 5a + 1, and again the standard estimates rule out this case. 8.5. Groups of type En . From [11, A.51] we find: Lemma 8.9. Let M(λ) be a basic highest weight module for E6 (q)sc of dimension d = dim M(λ) ≤ 158. Then M(λ) is one of the modules in Table 16. Table 16. The basic E6 (q)sc -modules of dimension at most 158

λ λ1 , λ6 λ2

dim M(λ) 27 78 − δ,3

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Lemma 8.10. If F ∗ (G) = En (q)sc , n = 6, 7, 8, or F ∗ (G) = 2E6 (q)sc , then any 2F -module for G is as in the statement of Theorem 8.1. Proof. 1. First let G = E6 (q)sc . The unipotent radical U of a maximal parabolic subgroup of type Spin+ 10 (q) is a spin module for the Levi subgroup, hence irreducible. If A ≤ U is an offender on a 2F -module V defined over Ff then our usual argument gives df/2 ≤ 32a. In particular, V must be a twist of a basic module, of dimension at most 64. Hence by Table 16 it is one of the 27-dimensional modules. On the other hand, if U contains no offender, then CV (U ) is an example for Spin+ 10 (q). Applying this to both classes of D5 -parabolics and using the result for the orthogonal groups in Theorem 7.5 we see that up to twist V is one of the basic modules one Table 16. The 27-dimensional modules are Example 8.4, the adjoint modules were treated in Proposition 8.2. Now assume that G = E6 (q)sc . involves a field automorphism of order . If V is a tensor product, then 27r a/r ≤ 2 log |G|. This is only possible for r = 1. But basic modules are not invariant, so V is induced from a twist of a basic module W . Furthermore, a 2F -module for G is either already an example for the derived group, or every proper parabolic subgroup leads to an example. The results for SL6 and Spin10 show that this implies that W is (a twist of ) the 27-dimensional or the adjoint module. Using that outer elements have centralizer at most dim W on V and the -rank 8a + 1 we arrive at = 2, dim W = 27, which occurs in Table 17. Next assume that = 2 and G is obtained from E6 (q)sc by extension with the graph automorphism of order 2. The 27-dimensional module is not invariant, hence can be excluded by Proposition 2.5. Since the graph automorphism centralizes an F4 (q), outer involutions don’t act as bi-transvections, so B := A ∩ G = 1, and |A| ≤ 2q 11 by Table 13. The table in [9] shows that B has to consist of long root elements, but then |B| ≤ q 5 by Table 5, while long root elements have rank 22, contradiction. Non-trivial tensor products do not satisfy the basic inequality 3.4. If G involves a graph-field automorphism of order 2, then no basic representations is invariant, and no case arises. The same happens for the extension of E6 (2a )sc by the full group of graph and field automorphisms of order 2. 2. Now let G = 2E6 (q)sc . Arguing as above, only the basic representations of dimensions 27 and 78 − δ,3 need to be considered. The adjoint one is ruled out by Proposition 8.2. At present, we can’t decide whether the 27-dimensional module gives an example. The basic representations do not extend to the extension of 2E6 (q)sc by a field automorphism of order . Thus, as before, this case can be ruled out by the standard arguments. 3. Now let G = E7 (q)sc . By [11] the only small basic modules are the one of dimension 56 and the adjoint one of dimension 133 − δ,2 . The latter was ruled out in Proposition 8.2. We leave open the case of the 56-dimensional module. Tensor products do not lead to examples, using that G has -rank 27a (see Table 13). For extensions by field automorphisms of order the standard estimates allow to conclude. 4. Finally let G = E8 (q). Here [11] shows that the only possible basic module is the adjoint one of dimension 248. This was ruled out in Proposition 8.2. Tensor

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products do not give rise to examples by our basic inequality, using that the -rank of G is 36a (see Table 13). Similarly, the case of field automorphisms can be excluded by standard estimates. 8.6. The examples for exceptional groups. Case 8.1: Let A be the center of a Sylow 2-subgroup of the Suzuki group 2B2 (22a+1 ), of order 22a+1 . Then it is well known that A has 2-dimensional centralizer on the 4-dimensional modules, hence we obtain an example. Case 8.2: The group G2 (2a ) has 2-rank 3a, so we find an example on the 6dimensional module. Case 8.3: The group G2 (a ) has -rank at least 3a, so we find an example on the 7-dimensional module(s). Case 8.4: The Chevalley group F4 (2a ) contains the classical group SO9 (2a ) ∼ = Sp8 (2a ). The 26-dimensional module for F4 (2a ) restricts to the heart of the exterior ˜ 2 (Y8 ) for Sp8 (2a ), as can be seen on the extended square of the natural module ∧ Dynkin diagram. Since the latter is an example for the symplectic group by Table 9, it is a fortiori an example for F4 (2a ). Case 8.5: Let U be the abelian unipotent radical, of order q 16 , of a maximal parabolic subgroup of type D5 of E6 (q)sc . Clearly this gives an example on the 27-dimensional modules. Table 17. 2F -modules for exceptional groups in defining characteristic

G 2B (22a+1 ) 2 G2 (2a ) G2 (a ) F4 (2a ) E6 (a )sc F4 (a ) 2E (a ) 6 sc a E7 ( )sc

d 4 6 7 26 27 26 − δ,3 27 56

λ f λ1 , λ2 2a + 1 λ2 a λ2 a λ1 , λ4 a λ1 , λ6 a λ4 a λ1 , λ6 2a λ7 a

a>1 >2

|A| 2a + 1 3a 3a 10a 16a

>2

9. F -modules for quasi-simple groups In this section we obtain the list of all absolutely irreducible F -modules of finite quasisimple groups, as an easy application of our classification of 2F -modules in this paper and the previous part [6].

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Lemma 9.1. The 56-dimensional module for G := E7 (q)sc is not an F -module. Proof. Let P = U L where U is elementary abelian of order q 27 and L is of type E6 . Then V has an L-composition series F < W < M < V where F and V /M are 1-dimensional and W/F ∼ = U. = U ∗ and M/W ∼ Let A be a nontrivial elementary abelian unipotent subgroup of G with f1 (A) maximal and assume that f1 (A) ≥ |V |. Set B = A ∩ U . If B = 1, the result follows since the 27-dimensional module U for E6 (q) is not an F -module. If B does not consist solely of long root elements, then CV (B) has dimension at most 36 by [9, Table 7]. Since CW (y) is 27-dimensional (note that U acts trivially on W/F , whence [y, W ] = F for each nontrivial y ∈ U ), this implies that CV (A)/CW (A) has dimension at most 9. Thus, f1 (A) ≤ q 9 |A/B||CW (A/B)||B| < q 37 |B| since W/F is not an F -module for E6 (q). Thus, |B| > q 19 . By loc. cit. this implies that A/B consists of long root elements. Now we observe that if |A/B| > q, then |CB (A/B)| ≤ q 18 . (sketch: a long root element centralizes a space B0 of dimension 21 on B. Now work in the centralizer of this long root subgroup X in C := CE6 (X) = q 1+20 SL6 (q). The module B has three composition factors of dimensions 6, 15, 6 for SL6 (q). Suppose that y is in the unipotent radical Q of C outside X. Then y −1 y g is a nontrivial element of X for some g ∈ Q. Since an element of X has rank 6 on B0 , it follows that y has rank at least 3 on B0 , whence CB0 (y) = CB (y, x) has dimension at most 18. If y is not in U , then it fixes a 5-dimensional space at most on each of the 6-dimensional composition factors and a 13-dimensional space on the 15-dimensional composition factor). So in this case, |A| ≤ q|B|. Note that the action of U on W gives an identification of U with a subspace of W ∗ (y → y −1 and y −1 is a map from W to F ). In particular, any subspace R of U of dimension r centralizes a space of dimension 28 − r on W . In particular, |B||CW (B)| ≤ |W |. Thus, |CV (A)||A| ≤ q|W ||CV (A)/CW (A)| ≤ q 10 |W | < |V |. So B consists solely of long root elements. It follows from Table 5 that |B| ≤ q 7 . Thus, |A| ≤ q 7 |A/B|. Moreover, f1 (A) ≤ q 7 |A/B||CW (A)||CV (A)/CW (A)|. By the E6 -result, |A/B||CW (A)| < |W |. Even for y ∈ B a long root element, we have CV (y)/CW (y) of dimension at most 18, whence f1 (A) < q 53 , a contradiction. For the properties of M24 , Co1 and Co2 needed in the following proofs we refer to [5, Tab. 5.3e, 5.3k and 5.3l]. Lemma 9.2. Let V be an irreducible F2 -module of dimension 11 for G = M24 . Then V is not an F -module. Proof. We first note that the centralizer dimensions for involutions on V are 7 for a 2-central involution and 6 for the other class of involutions.

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Suppose that A is an offender with f1 (A) maximal. Let z be an involution of class 2A. We work with the subgroup A ≤ P with U = O2 (P ) elementary abelian of order 16 and P /U = A8 . Note that this contains D := CG (z) (after conjugating) where Q = O2 (D) is extraspecial of order 27 and D/Q = L3 (2). We claim that A must contain a conjugate of z. If not, then A ∩ U = 1 (since U intersects only one class of involutions). Then A embeds in P /U and so has order at most 16, but CV (A) has dimension at most 6, a contradiction. So we may assume that z ∈ A ≤ D. Set W = CV (z) of dimension 7 and M = [z, V ] of dimension 4. Note that W/M is an irreducible D-module. Suppose that A ∩ Q has order 2. Then |A| ≤ 8, a contradiction. So let z = y ∈ A ∩ Q be an involution. Since z = y −1 y g for some g ∈ Q, it follows that [y, W ] has dimension at least 2. Thus, CV (y) has dimension at most 5, whence exactly 5 and |A| = 26 . Thus, A is not contained in Q and so does not centralize M. Then CV (A) has dimension at most 4, a contradiction. Lemma 9.3. Let V be the 22-dimensional F2 -module for G = Co2 . Then V is not an F -module. Proof. Let A be an offender with f1 (A) maximal. Note that CV (x) has dimension at most 16 for any involution x. Thus, |A| ≥ 26 . Let z be a 2-central involution and set D = CG (z). Set U = O2 (D). Then D/U = Sp6 (2). Since |A| ≥ 64, A is not contained in U (the largest abelian subgroup of U has order 32). If A contains an involution not conjugate to z, then |CV (A)| ≤ 214 and so |A| ≥ 28 . We first show that A contains a conjugate of z. If not, then as we saw |A| ≥ 28 and B := A ∩ U = 1. Set W = CV (z) and M = [z, V ]. So dim W = 16 and dim M = 6. Now U centralizes M (since it is irreducible for D). If A/B has order 2, then f1 (A) ≤ f1 (B) < |V |. If A/B has order at least 4, then it follows that CV (A) has codimension at least 2 in CV (B) and so dim CV (A) ≤ 12, whence |A| = 210 . Since that is the maximal rank possible, it follows that A contains a 2-central involution. So we may assume that z ∈ A and B is a nontrivial proper subgroup of A. Note that CV (A) is properly contained in CV (B) as B = CM (A). If A/B has order 2, then f1 (A) ≤ f1 (B) < |V |. If A/B has order at least 4, it follows that CM (A) has dimension at most 4 and so CV (A) has codimension at least 2 in CV (B). If y ∈ B is an involution other than z, we see that CW (y) has codimension at least 3 in W . Thus, CV (B) has dimension at most 13, whence by the remark above, CV (A) has dimension at most 11 and f1 (A) < |V |. So B must have order 2 and |A| ≤ 27 . Then W = CV (z) is 16-dimensional and so CV (A) has dimension at most 14, whence f1 (A) ≤ 221 . Lemma 9.4. Let V be the 24-dimensional F2 -module for G = Co1 . Then V is not an F -module.

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Proof. Suppose that A is a nontrivial elementary abelian subgroup of G with f1 (A) ≥ |V | and maximal. Note that |A| ≤ 211 . Consider D = CG (z) with z an involution in class 2A. Set U = O2 (D) and recall that D/U = O+ 8 (2). Note that CV (A) has dimension at least 13, whence all involutions in A are conjugate to z (by centralizer dimensions). We assume that z ∈ A. Note that D has 3 composition factors on V , each of dimension 8. Set W = CV (z) (of dimension 16). Moreover, if x ∈ A \ U , then CW (x) has dimension at most 12 (since x does not induce a transvection on any composition factor). So if A is not contained in U , then f1 (A) ≤ |A|212 < |V |. If A is contained in U , then |A| ≤ 25 and so f1 (A) ≤ 221 . Theorem 9.5. The absolutely irreducible F -modules of finite quasi-simple groups are given in Tables 18 and 19. Proof. By definition, F -modules are in particular 2F -modules, so we just have to go through our list of 2F -modules and determine which of them satisfy the F-condition. We start with the groups of Lie type in their defining characteristic whose 2F -modules were classified in this paper. The symmetric square 2 (Yn ) of the natural module for SLn (q) is not an F -module by (2.1) in conjunction with Proposition 3.10. On the wedge cube V := ∧3 (Y6 ) of the natural module for SL6 (q), transvections have rank 6, so a possible offender A has |A| ≥ q 6 . Consider an end-node parabolic P = U L containing A. If A ≤ L = GL5 (q), then it is maximal possible, centralizing a 2- or 3-space on the natural module. But then the centralizer on V |GL5 (q) = ∧3 (Y5 ) ⊕ ∧2 (Y5 ) is just 4-dimensional. Thus A ∩ U = 1, and the centralizer of A is at most 11-dimensional, whence |A| = q 9 is maximal possible, the unipotent radical of a GL3 (q) × GL3 (q)-parabolic. This is ruled out by Theorem 3.1. (a) For the tensor product module V := Yn ⊗ Yn of SLn (2a ) we use induction. For n = 2 an offender A is contained in the unipotent radical of a Borel subgroup, but the latter violates the F-condition. For n > 2, induction and Theorem 2.4 show that a possible offender must lie inside the unipotent radical U of a maximal parabolic subgroup q n−1 GLn−1 (q). But U has centralizer dimension (n − 1)2 on V , thus we don’t get an example. For SU3 (q), the maximal size of -subgroups centralizing subspaces of the natural module of given dimension is bounded in Proposition 3.11. This shows that the Fcondition is not satisfied. ˜ 2 (Y2n ) of the natural module for Sp2n (q), n ≥ 2, The heart of the exterior square ∧ (a) is not an example by (2.1) in conjunction with Proposition 3.12. The case of Y4 ⊗ Y4 2a a for Sp4 ( ) is ruled out by the same argument. The spin modules for Sp2n (2 ), n ≥ 4, are not F -modules by Proposition 5.10. The spin module for Spin2n+1 (q), n ≥ 4, q = a odd, was excluded in Proposition 5.8. − The half-spin modules for Spin+ 2n (q), n ≥ 6, respectively for Spin2n (q), n ≥ 4, were excluded in Propositions 5.4 and 5.6.

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The Suzuki-group 2B2 (22a+1 ) has 2-rank 2a+1, but does not contain transvections, so the 4-dimensional module over F22a+1 is not an F -module. On the 7-dimensional module for G2 (a ), > 2, long root elements have rank 2, all other non-trivial unipotent elements have rank at least 4. Thus this is not an F module for > 3. For = 3 we obtain that the only possibility is that |A| = q 4 and CV (A) is 3-dimensional. Thus the corresponding tame abelian subalgebra I (A) has order q 4 , so must correspond to four abelian roots. The only possibility for this is the set of all non-simple roots. These roots are the ones of maximal weight in any reasonable ordering on the roots, so A = U (I (A)) is the subgroup generated by the root subgroups corresponding to the four positive nonsimple roots. Let X be any short root subgroup in A. Since CV (X) = CV (A), it follows that CV (A) is invariant under NG (X). Since there are two different short root subgroups in A and since NG (X) is a maximal parabolic subgroup of G, it follows that G leaves CV (A) invariant, a contradiction. For F4 (q) on the 25- or 26-dimensional module, long root elements have rank 6, all other non-trivial unipotent elements rank at least 10, by [9, Tab. 3]. So by Table 13 this is not an F -module when > 2. For = 2, the 26-dimensional module occurs as a direct summand in the restriction of the 27-dimensional module for E6 (q). Since the latter is not an F -module (see below), this case is excluded as well. Let P = U L be a maximal parabolic subgroup of E7 (q)sc with Levi factor L of type E6 . Then U is one of the 27-dimensional modules for E6 (q) ≤ L. Now U is the unique abelian subgroup of maximal order in a Sylow -subgroup of P by Lemma 3.15. Thus, by Lemma 2.6 the module U is not an F -module for E6 (q). The other 27-dimensional module for E6 (q) is the image under the graph automorphism of order 2, hence isn’t an F -module either. On the 27-dimensional modules for 2E6 (q) over Fq 2 long root elements have rank 6, all other non-trivial unipotent elements have rank at least 10, by [9, Tab. 5]. Thus Table 13 shows that the F-condition cannot be satisfied. In non-defining characteristic we just have to consider the 2F -modules determined in the first part [6] in Tables 6.3, 6.5, and 6.7. Subgroups of order k in Sn contain elements with at most n − k fixed points, so of rank ( − 1)k on the permutation module. Passing to the deleted permutation module we see that this is not an example when > 2. For = 2, subgroups of order 2k fixing n − 2k points necessarily contain odd elements. Thus if A ≤ An has size |A| = 2k then it contains elements of rank k + 1. If n is odd, the permutation module is a direct sum of the n − 1-dimensional irreducible module with the trivial one. Thus the deleted permutation module is not an F -module for An , n odd, in characteristic = 2. For the 8-dimensional representations of A9 over F2 not contained in the permutation module, explicit computation shows that elementary abelian subgroups of order 8 have at most 4-dimensional centralizer and those of order 16 have 2-dimensional centralizer. The 2-dimensional representation of 2.A5 over F9 does not give an example since the Sylow 3-subgroup has only order 3. On the 8-dimensional representation of

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2.A9 over F3 elements of order 3 have rank 4. Since the 3-rank is 3, we don’t get an example. The 6-dimensional F2 -representation of U3 (3) is the restriction from G2 (2). There long root elements have rank 2, so A has to contain other non-trivial unipotent elements by Table 5. But these have rank at least 3, while U3 (3) only has 2-rank 2, so there are no offenders. The 6-dimensional F3 -module for G := 2.L3 (4) is the restriction of the natural module for SO− 6 (3). Since G only has 3-rank 2, this is not an F -module for G by Proposition 3.14. The 7-dimensional F3 -module for S6 (2) is orthogonal. On restriction to the sub ∼ ∼ ∼ group SO− 6 (2) = SU4 (2) = S4 (3) = SO5 (3) of 3 -index we find a copy of the natural module for SO5 (3). This is the heart of the exterior square of the natural module for Sp4 (3), hence not an F -module by what we showed above. The 8-dimensional F3 -module for 2.S6 (2) is the restriction of the spin module for Spin7 (3). On restriction to the 3 -index subgroup SO5 (3) we obtain two copies of the spin module, the natural module for Sp4 (3). Now Proposition 3.12 shows that this is not an example. The 6-dimensional F4 -module V of 31 .U4 (3) is of orthogonal type, so involutions have rank at least 2. Hence |A| ≥ 24 , but then A centralizes at most a 3-dimensional subspace by Proposition 3.14. This contradicts the fact that 31 .U4 (3) has 2-rank 4. On the 8-dimensional F3 -module for 2.O+ 8 (2), elements of order 3 have rank 2 or 4 and subgroups of order 9 have centralizer dimension at most 4. So if there exists an offender, the maximal torus of order 34 is also an offender. In particular we may assume that A contains elements from all classes of 3-elements. Now the Weyl group of type E8 contains the direct product of two Weyl groups of type A4 (visible on the extended Dynkin diagram), as well as the product of four Weyl groups of type A2 (visible in E6 × A2 ). Thus there are 3-elements with Jordan type 32 12 and with Jordan type 24 . Clearly no subgroup containing these two types can have centralizer dimension 4. The 6-dimensional F4 -module V of 3.M22 is the restriction of the natural module for SU6 (2). Since 3.M22 does not contain transvections the centralizer of a 5dimensional subspace is trivial, and the centralizer of a 4-dimensional space has size less than 24 by Lemma 3.11. Since 3.M22 has 2-rank 4, this shows that V is not an F -module. The 6-dimensional F4 -module V of J2 is the restriction of the 6-dimensional module for G2 (4). By [9, Table 1] long root elements have rank 2, the other involutions have rank 3. Since J2 has 2-rank 4, an offender would have to consist of long root elements only. But such subgroups in G2 (4) have size at most 4 by Table 3.9. On the 10-dimensional F2 -module of M12 , involutions have rank 4, but the 2-rank equals 3. On the 10-dimensional F2 -module of M22 , involutions have rank 4, Klein four groups have centralizer codimension at least 5, but the 2-rank equals 4.

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On the 11-dimensional F2 -module of M23 , involutions have rank 4, Klein four groups have centralizer codimension at least 5, but the 2-rank equals 4. The 11-dimensional F2 -module of M24 , the 22-dimensional F2 -module of Co2 and the 24-dimensional F2 -module of Co1 were discussed in Lemmas 9.2–9.4. On the 5-dimensional F3 -module of M11 , elements of order 3 have rank 3, so this is no example. On the 6-dimensional F3 -module of M12 , elements of order 3 have rank 3 or 4, so this is not an example. 9.1. The examples for F -modules. The following two tables contain the F -modules V for finite quasi-simple groups. In the last column we give the size of one possible (±) offender on V . For the natural module Yn of Spinn (a ) in the fifth line, has to be odd when n is odd. Table 18. F -modules for groups of Lie type in defining characteristic

G SLn (a ) SLn (a ) SUn (a ) Sp2n (a ) (±) Spinn (a ) Spin7 (a ) a Spin+ 2n ( ) G2 (2a )

d n n 2

n 2n n 8 2n−1 6

V f Yn a 2 ∧ (Yn ) a Yn 2a Y2n a Yn a Z3 a Zn−1 a M(λ2 ) a

conditions n≥4 n≥4 n≥6 n = 4, 5

log |A| a (n − 1)a 4a a 3a 5a (2n − 2)a 3a

Case 9.1: On its natural module Yn , SLn (q) acts as a transvection group, so we get an example with |A| = q. Case 9.2: Thesubgroup of SLn (q) of order q n−1 in (4.1) has centralizer of dimension at least n−1 on ∧2 (Yn ), so this yields an F -module. 2 Case 9.3: For the special unitary group SUn (q), n ≥ 4, on its natural module Yn over Fq 2 Proposition 3.11 shows that there exists a subspace of codimension 2 whose centralizer contains an elementary abelian subgroup of order q 4 . Case 9.4: The symplectic group Sp2n (q) acts as a transvection group on its natural module Y2n , thus we get an example with |A| = q. (±) Case 9.5: For the orthogonal group Spinn (q), n ≥ 6, on its natural module Proposition 3.14 shows that there exists a subspace of codimension 3 whose centralizer contains an elementary abelian subgroup of order q 3 . Case 9.6: The spin module Z3 for Spin7 (q) has highest weight λ1 . Consider the maximal parabolic subgroup of type Spin5 (q). The centralizer of its unipotent radical

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U is again the spin module for Spin5 (q), hence has dimension 4. But U is elementary abelian of order q 5 . Thus U is an offender on Z3 . Case 9.7: For the half-spin modules V := Zn−1 of dimension 2n−1 of Spin+ 2n (q) let A be the unipotent radical of a maximal parabolic subgroup with Levi complement of + type Spin+ 2n−2 (q). Then the centralizer CV (A) is a half-spin module for Spin2n−2 (q), of dimension 2n−2 , while A is elementary abelian of order q 2n−2 , so we find an example whenever n ≤ 5. Case 9.8: The 6-dimensional module V embeds G2 (q), q = 2a , into Sp6 (q). Let B be a Borel subgroup of G2 (q). Then B fixes a totally singular 3-space W of V . This defines a homomorphism into GL3 (q), with kernel the centralizer A := CB (W ) of W . Thus |A| ≥ q 3 , and A is an offender. Table 19. F -modules in non-defining characteristic

G A2n 3.A6 A7

d f 2n − 2 1 2 3 2 2 4 1 2

|A| 2n−1 4 4

Case 9.9: Let V be the 2n − 2-dimensional heart of the permutation module of A2n in characteristic 2. Let A be the intersection with A2n of a maximal 2-subgroup of S2n generated by commuting transpositions. Then |A| = 2n−1 , A centralizes a subspace of dimension n on the permutation module, and by direct computation, still an n − 1-dimensional subspace of V . Thus A is an offender. (For A6 ∼ = Sp4 (2) this is ∼ the natural module, for A8 = SL4 (2) it is the exterior square of the natural module.) Case 9.10: The group 3.A6 on its 3-dimensional F4 -module gives an example with |A| = 4. Case 9.11: The group A7 on its 4-dimensional F2 -module (the natural module for SL4 (2) ∼ = A8 ) contains A6 ∼ = Sp4 (2) in its natural representation. The latter already is Example 9.9. Note added in proof: The open cases in Table 1 have now been settled in joint work with R. Lawther. The only case that is a 2F -module is F4 (3a ).

References [1]

M. Aschbacher, GF(2)-representations of finite groups. Amer. J. Math. 104 (1982), 683–771.

[2]

M. Aschbacher, S. Smith, The classification of quasithin groups, vol 1: Structure of strongly quasithin K-groups, vol 2: Main theorems: the classification of simple QTKE groups, Math. Surveys Monogr. 111, 112, Amer. Math. Soc., Providence, 2004.

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[3]

A. Chermak, A. Delgado, A measuring argument for finite groups, Proc. Amer. Math. Soc. 107 (1989), 907–914.

[4]

J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, R. A. Wilson, Atlas of finite groups, Clarendon Press, Oxford 1985.

[5]

D. Gorenstein, R. Lyons, R. Solomon, The classification of the finite simple groups, Number 3, Math. Surveys Monogr. 40.3, Amer. Math. Soc., Providence 1998.

[6]

R. M. Guralnick, G. Malle, Classification of 2F -modules, I, J. Algebra 257 (2002), 348–372.

[7]

I. Hughes, G. Kemper, Symmetric powers of modular representations, Hilbert series and degree bounds, Comm. Algebra 28 (2000), 2059–2088.

[8]

C. Jansen, K. Lux, R. Parker, R. Wilson, An atlas of Brauer characters, Clarendon Press, Oxford 1995.

[9]

R. Lawther, Jordan block sizes of unipotent elements in exceptional algebraic groups, Comm. Algebra 23 (1995), 4125–4156. Correction: ibid. 26 (1998), 2709.

[10] M. W. Liebeck, The affine permutation groups of rank three, Proc. Lond. Math. Soc. (3) 54 (1987), 477–516. [11] F. Lübeck, Small degree representations of finite Chevalley groups in defining characteristic, LMS J. Comput. Math. 4 (2001), 135–169. [12] U. Meierfrankenfeld, G. Stroth, On quadratic GF(2)-modules for Chevalley groups over fields of odd order. Arch. Math. (Basel) 55 (1990), 105–110. [13] U. Meierfrankenfeld, G. Stroth, Quadratic GF(2)-modules for sporadic simple groups and alternating groups. Comm. Algebra 18 (1990), 2099–2139. [14] Th. Meixner, Failure of factorization modules for Lie-type groups in odd characteristic, Comm. Algebra 19 (1991), 3193–3222. [15] A. Premet, Weights of infinitesimally irreducible representations of Chevalley groups over a field of prime characteristic, Math. USSS-Sb. 61 (1988), 167–183. [16] N. Spaltenstein, Classes unipotentes et sous-groupes de Borel, Lecture Notes in Math. 946, Springer-Verlag, Berlin–New York 1982. [17] G. Stroth, 2F -modules with quadratic offender for the finite simple groups, in: Groups and combinatorics—in memory of Michio Suzuki, Adv. Stud. Pure Math. 32, Math. Soc. Japan, Tokyo 2001, 391–400. Robert M. Guralnick, Department of Mathematics, University of Southern California, Los Angeles CA 90089-1113, U.S.A. E-mail: [email protected] Gunter Malle, FB Mathematik/Informatik, Universität Kassel, Heinrich-Plett-Str. 40, 34132 Kassel, Germany E-mail: [email protected]

On the real Schur indices associated with infinite Coxeter groups Allen Herman and Barry Monson∗

1. Introduction This article is motivated by an attempt to explain, using character theory, the three geometric formulae proved in [6] that give, for finite abstract regular polytopes, basic relationships between the arithmetic properties of their automorphism groups, their essential Wythoff space dimensions, and the dimensions of their realization cones. We will show that, while two of the three formulae are fairly easy consequences of character theory, the other cannot be explained using character theory as such. Indeed, this formula indicates a relationship between the geometric realizations of an abstract regular polytope and the realizability of the complex irreducible representations of its automorphism group over the field of real numbers.

2. Finite string C-groups A C-group is a group generated by a distinguished finite set S = {s0 , s1 , . . . , sn−1 } of involutions satisfying the intersection property with respect to S, i.e. for all subsets I , J ⊆ S, sk : k ∈ I ∩ sk : k ∈ J = sk : k ∈ I ∩ J . A string C-group is a C-group having the further property that (sj si )2 = 1, whenever |i − j | > 1. Thus is a rather special homomorphic image of some (possibly infinite) string Coxeter group, say W = ρ0 , ρ1 , . . . , ρn−1 , whose distinguished generators ρi satisfy the defining relations ρi 2 = 1, for all i ∈ {0, . . . , n − 1} (ρi−1 ρi )mi = 1 for all i ∈ {1, . . . , n − 1}, and (ρj ρi )2 = 1

if |i − j | > 1,

for m1 , . . . , mn−1 ∈ {∞, 2, 3, 4, . . . }. (The third set of relations implies that the Coxeter diagram for W has the shape of a string.) It is quite special that the homomor∗ The authors acknowledge the support of NSERC.

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phism induced by the mapping ρi → si should preserve the intersection condition, which does hold in any Coxeter group W [4, §5.5]. String C-groups are precisely the automorphism groups of abstract regular polytopes [7, §2B and §2E], which we discuss in § 4 below. In theory, string C-groups are quite plentiful. This is because any infinite Coxeter group W is a finitely generated linear group via the standard (faithful) reflection representation [4, §5.4]. Malcev’s theorem, concerning finitely generated linear groups, shows that W is residually finite; that is, for any finite subset {w1 , . . . , wm } of W \ 1, there exists a surjective homomorphism f : W → such that is finite and 1 ∈ {f (w1 ), . . . , f (wm )}. Using this property, one can (in a non-constructive way) prove the existence of several infinite families of finite abstract regular polytopes, each with its own finite string C-group of automorphisms [7, §4C]. −1 −1 For example, if m1 , m2 ≥ 3 are integers satisfying m−1 1 + m2 < 2 , then there are infinitely many finite regular maps of type {m1 , m2 }. (The latter notation indicates a regular map having only m1 -gonal faces, with m2 of them around each vertex.) On the other hand, while there are some explicit methods for producing plentiful examples of finite string C-groups, such methods typically involve very special constructions, such as the reduction of linear groups over a number field, modulo some prime ideal. Some other efforts to construct string C-groups involve taking the quotient of the Coxeter group W , say by prescribing the order of the Coxeter element ρ0 ρ1 . . . ρn , or by prescribing the orders of certain “holes” in the polytope. Several finite regular maps of these types, with group order up to 21 504, appear in Table 8 of [1].

3. Real Schur indices for finite Coxeter groups Let χ be an irreducible complex character of a finite group G, that is, the trace of an irreducible linear representation X : G → GLn (C) for some positive integer n. If K is a subfield of C, then χ is said to be realizable over K if there is a linear representation X which is similar to X such that X (G) ⊆ GLn (K). In order for χ to be realizable over K, it is necessary that K contain the field Q(χ ) := Q({χ (g) : g ∈ G}) generated by the character values over the rational field Q. In other words, it is necessary that K(χ ) = K. However, this condition is not sufficient for χ to be realizable over K. The Schur index of χ over the field K is the positive integer mK (χ) := min{[L : K(χ)] : χ is realizable over L}. There are three possibilities concerning realizability of complex irreducible characters over the field of real numbers R, which were classically explained by Frobenius

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and Schur using what is now known as the Frobenius–Schur indicator 1 ν2 (χ) := χ (g 2 ). |G| g∈G

The Frobenius–Schur indicator takes only the values 1, 0, and −1. It is 1 exactly when χ is realizable over R; it is 0 when χ is not realizable over R and R(χ ) = C; and it is −1 when χ is not realizable over R, R(χ ) = R, and χ has Schur index 2 over R. It has been known since the 70s that all irreducible representations of Weyl groups (i.e. finite Coxeter groups satisfying the crystallographic condition) are realizable over the field of rational numbers (see [3, §6.3]). In other words, all such representations have rational Schur index 1. On the other hand, there is one example of an irreducible representation of a finite Coxeter group with rational Schur index 2. (All others have rational Schur index 1.) The exception is an irreducible character of degree 48 of the√group of type H4 , which has rational character values but is only realizable over Q( 5) (see [3, §6.3]). Since this is a real subfield, it follows that every irreducible representation of a finite Coxeter group can be realized over the field of real numbers. For such groups then, all real irreducible representations are absolutely irreducible. On the other hand, there are examples of string C-groups that are known to possess irreducible characters with non-real fields of character values. For example, the groups PSL(2, 11) ∼ = [5, 5]5 := [5, 5]/(ρ0 ρ1 ρ2 )5 and PSL(2, 19) ∼ = [9, 9]3 := [9, 9]/(ρ0 ρ1 ρ2 )3 both possess a complex conjugate pair of irreducible characters of minimal nontrivial degree having imaginary quadratic fields of character values. (The value of the Frobenius–Schur indicator of such irreducible characters is 0.) Therefore, not all irreducible representations of finite string C-groups are realizable over the reals. However, no example is known of a finite string C-group that possesses an irreducible character with Frobenius–Schur indicator −1 (i.e. real Schur index 2).

4. Realizations of abstract regular polytopes An abstract regular n-polytope P is a partially ordered set having certain key properties of the face lattice of a classical convex regular n-polytope. (We refer to [7] for details.) The elements of P are called faces. P is equipped with a strictly monotone rank function on these faces, with range {−1, . . . , n}. Faces of rank 0 are vertices; those of rank 1 are edges; and there are unique minimal and maximal (improper) faces of ranks −1 and n, respectively.

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It is instructive to think of the vertices, edges and square facets of an ordinary cube P , along with ∅ and P itself, all ordered by inclusion. Note that the Euclidean symmetry group of P is the Coxeter group W = B3 generated by the ordinary reflections ρ0 , ρ1 , ρ2 . If abcd is a particular square facet of P , then we may suppose that ρ0 interchanges vertices a and b; that ρ1 interchanges edges ab, ad (while fixing a); and that ρ2 interchanges the two square facets which share edge ab. Note that the stabilizer in W of vertex a is ρ1 , ρ2 . Returning now to P , suppose that Fj is the face of rank j in some maximal chain F0 , . . . , Fn−1 of proper faces. Such a chain is called a flag; the polytope P is said to be regular if its automorphism group is transitive on the set of all flags. We may therefore fix F0 , . . . , Fn−1 as base flag. It is now a consequence of the axioms that for each j , 0 ≤ j ≤ n − 1, there exists a unique involutory automorphism sj moving only element Fj in the base flag. Furthermore,

= s0 , . . . , sn−1 is generated as a string C-group by these involutions; and Stab (F0 ) = 0 := s1 , . . . , sn−1 . Consequently, the vertices of P can be identified with the cosets of the standard subgroup 0 . We emphasize that, in general, P need not be a lattice, nor have any sort of familiar geometric structure. Despite this essentially combinatorial nature, however, it is naturally interesting to attempt to construct P as a ‘real’ object. When P is finite, we may modify the definitions in [5] as follows. For a finite P , let be any orthogonal representation of , of degree d. The Wythoff space for this representation is the subspace of points in Rd fixed by 0 . A realization of P is any pair P = [, x], where the base point x lies in the Wythoff space for . Let P0 denote the set of vertices of P . It follows that the map β : P0 → Rd given by β(g(F0 )) := (g)(x), g ∈ , is well-defined. Moreover, each g induces an isometric permutation of the geometric vertex set V := {(g)(x) | g ∈ }. This is the beginning of an inductive construction, wherein the representation is used to transfer the combinatorial structure of P to some sort of object in Rd (see [7, 5A]). We omit details, noting only that it may be inevitable that some of the structure of P is lost in the realization. In the extreme case, it can happen that the Wythoff space is trivial, so that x = o. We then obtain the trivial realization of P , with V = {o}.

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5. Character-theoretic proofs of some formulae concerning the geometric realizations of regular polytopes In [5], three equations were given concerning relationships between the dimensions and other key parameters for the realizations of finite, abstract regular polytopes. These formulae did not properly account for the possibility that some of the real irreducible representations involved might be reducible over the complex field. Corrected equations were later given by McMullen and Monson in [6]. The proofs of these results, in both the original and revised form, rely heavily on interesting convexity arguments. Here we show that two of the formulae can be viewed as consequences of charactertheoretic information concerning real representations of the automorphism group of the abstract polytope. Let Irr R ( ) and Irr C ( ) denote the set of characters of irreducible representations of the finite group over the real and complex numbers, respectively. For all α ∈ Irr R ( ), there is a χ ∈ Irr C ( ) such that when we consider α as a character over C, we have that either (1) α = χ , (2) α = χ + χ, ¯ or (3) α = 2χ . For any characters ϕ, α of , let (ϕ, α) denote the usual inner product of class functions. If ϕ is a real character of , and α ∈ Irr R ( ), then define if α is of type (1), (ϕ, α) (ϕ, α)∗ = 21 (ϕ, α) if α is of type (2), 1 4 (ϕ, α) if α is of type (3). Lemma 1. Suppose X is a real orthogonal representation of a finite group , affording the complex character ϕ. Then the decomposition of ϕ into real irreducible characters is (ϕ, α)∗ α. ϕ= α∈Irr R ( )

Proof. The decomposition of ϕ into complex irreducibles is (ϕ, χ )χ . ϕ= χ∈Irr C ( )

If χ is involved in a real irreducible character of type (1), then (ϕ, χ )χ = (ϕ, α)α = (ϕ, α)∗ α since χ = α. If χ is involved in a real irreducible character α of type (2), then α = χ + χ¯ . Since ϕ is a real character we must have (ϕ, χ) = (ϕ, χ¯ ). Thus (ϕ, χ )χ + (ϕ, χ¯ )χ¯ = (ϕ, χ )(χ + χ¯ ) = 21 (ϕ, α)α.

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If χ is involved in a real irreducible character α of type (3), then α = 2χ, and (ϕ, χ )χ = 41 [(ϕ, 2χ)2χ ] = 41 (ϕ, α)α. Since each χ ∈ Irr C ( ) is involved in exactly one of these cases we have (ϕ, χ)χ = (ϕ, α)∗ α. ϕ= χ∈Irr C ( )

α∈Irr R ( )

Let us consider again the finite abstract regular polytope P with automorphism group = ρ0 , . . . , ρn−1 . Let be any orthogonal representation of and let ψ be the (possibly reducible) complex character afforded by . Thus has degree dψ = ψ(1). If 0 is a subgroup of , we denote the restriction of a character ψ of

to 0 by ψ 0 , and the induction of a character θ of 0 to by θ . (Restricted and induced characters are characters in a natural way. The restriction of a character is the character of the restriction of the corresponding representation to 0 . Also, if M is a left C 0 -module affording θ, then θ is by definition the character afforded by the left C -module C ⊗C 0 M.) If = 1 is irreducible, then dψ coincides with the affine dimension of the orbit of any point x = o in the Wythoff space for . However, for 1 such an orbit is a single point, with affine dimension 0, even though d1 0 = 1. Note that the dimension of the Wythoff space is wψ = (1 0 , ψ 0 ), which equals ((1 0 ) , ψ) by Frobenius reciprocity. The dimension of the essential Wythoff space of is defined to be wψ ∗ := ((1 0 ) , ψ)∗ . (We are here factoring out the effect of the centralizer of ( ) in the orthogonal group for Rd , since such centralizing isometries merely map one realization to a congruent realization. Thus, as explained in [6], wψ ∗ measures the extent to which one and the same representation can provide incongruent realizations.) Concerning the trivial realization, it is a useful convention to agree that w1∗ = 0, even though w1 = 1. Recall from § 4 that |P0 | = v = [ : 0 ]. By pairing the v vertices of P with the standard basis vectors of Rv , we obtain a v-dimensional orthogonal representation of

, as well as a realization of P . We conclude that this representation of is induced by the trivial representation on 0 . Therefore, the character of this representation is (1 0 ) . Now, the simplex realization of the polytope P is obtained by observing that the sum of all standard basis vectors of Rv will be fixed by the action of , and so there is a representation of on (1, . . . , 1)⊥ ∼ = Rv−1 which affords the (real-valued) character

σ = (1 0 ) − 1 . Applying the lemma results in the two consequences of interest to us. Theorem 1 ([6], Theorem 3 (a), (c)). Let σ = (1 0 ) − 1 denote the real-valued character associated simplex realization of P . Then with the ∗ wα dα = v − 1, and (a) σ (1) = α∈Irr R ( ) (b) wα ∗ wα = wσ . α∈Irr R ( )

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Proof. By the lemma, we have σ (1) =

(σ, α)∗ α(1).

α∈Irr R ( )

As long as α = 1 , we have dα = α(1); also (σ, α) = ((1 0 ) , α) − 0 = wα , so (σ, α)∗ = wα ∗ . On the other hand, (σ, 1 )∗ = (σ, 1 ) = ((1 0 ) , 1 ) − (1 , 1 ) = (1 0 , 1 0 ) − 1 = 0 = w1∗ . Since σ (1) = v − 1, part (a) follows immediately. Part (b) concerns the dimension of the Wythoff space for the simplex realization: wσ = ((1 0 ) , σ ) (σ, α)∗ ((1 0 ) , α) = α∈Irr R ( )

= (σ, 1 )∗ ((1 0 ) , 1 ) +

(σ, α)∗ ((1 0 ) , α)

α =1

= (σ, 1 )∗ ((1 0 ) , 1 ) + =

wα ∗ wα

α =1

∗

wα wα ,

α∈Irr R ( )

again since w1∗ = 0.

6. The realization cone and realizability of characters As described in [5], the realizations of an abstract regular polytope P can be naturally arranged into isometry classes, and the set P˜ of these classes becomes a closed convex cone under the operations of blending of realization classes and scaling by positive real numbers. (For definitions of these terms, see [5].) A third formula from [6] equates the dimension r of the realization cone P˜ to the sum of the dimensions of the realization cones P˜α , as α runs through Irr R ( ). Here P˜α is the cone of all realization classes for which the corresponding orthogonal representation X : → On (E) affords the real irreducible character α. Furthermore, for each real irreducible character α, the dimension of the realization cone P˜α was shown (by geometric methods) to be 1 ∗ ∗ 2 wα (wα + 1). The number r, which counts the number of non-trivial diagonal classes in P , can also be expressed group-theoretically as r = |{ 0 g 0 ∪ 0 g −1 0 : g ∈ \ 0 }|.

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Similarly, we also have from [6] that the total number of double cosets is wσ + 1. If we now let N be the number of self-inverse double cosets of 0 in , then r + 1 = |{ 0 g 0 ∪ 0 g −1 0 : g ∈ }| 1 = N + (wσ + 1 − N). 2 On the other hand, J. S. Frame in [2, Theorem A], established character-theoretic formulas for the number of self-inverse double cosets of a subgroup in any finite group. These formulas in this case are N=

1 (1 0 ) (g 2 ) = | | g∈

ν2 (χ)((1 0 ) , χ) =

χ∈Irr C ( )

ν2 (χ )wχ .

χ ∈Irr C ( )

Since σ = (1 0 ) − 1 , we have wσ + 1 = ((1 0 ) , (1 0 ) ) =

((1 0 ) , χ)2 =

χ∈Irr C ( )

wχ 2 .

χ ∈Irr C ( )

Therefore r +1=

χ∈Irr C ( )

=

χ∈Irr C ( )

1 ν2 (χ)wχ + (wχ 2 − ν2 (χ )wχ ) 2 1 wχ (wχ + ν2 (χ )). 2

Comparing this with the expression for r in [6], we see that there is a perfect match over the irreducible characters χ with ν2 (χ) = 1. As for those characters with ν2 (χ ) = 1, it is convenient to first express the general formula in terms of real irreducible characters. For those χ ∈ Irr C ( ) with χ + χ¯ = α ∈ Irr R ( ), we have wχ = wχ¯ = wα ∗ and ν2 (χ ) = ν2 (χ¯ ) = 0, so 1 1 wχ (wχ + ν2 (χ)) + wχ¯ (wχ¯ + ν2 (χ¯ )) = wχ 2 = (wα ∗ )2 . 2 2 For those χ ∈ Irr C ( ) with 2χ = α ∈ Irr R ( ), we have wχ = ν2 (χ ) = −1, so

1 2 wα

= 2wα ∗ and

1 wχ (wχ + ν2 (χ)) = wα ∗ (2wα ∗ − 1). 2 Now in this calculation we have not had to assume that is a string C-group. It therefore follows from Frame’s formula that |{ 0 g 0 ∪ 0 g −1 0 : g ∈ G}| can be expressed as below in terms of real irreducible characters.

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Corollary 1. For any finite group and any subgroup 0 , |{ 0 g 0 ∪ 0 g −1 0 : g ∈ G}| equals 1 wα ∗ (wα ∗ + 1) + (wα ∗ )2 + wα ∗ (2wα ∗ − 1), 2 α=χ α=χ+χ¯

α=2χ

where each sum runs over the real irreducible characters α of that have the indicated decompositions as complex characters. Comparing this to the formula for r + 1, we obtain an interesting consequence. Theorem 2. Let α be a real irreducible character of the finite string C-group , which is not irreducible as a complex character. Then wα∗ = 0 or 1. In particular, if wα = wα ∗ then the corresponding realization cone Pα is one dimensional. Proof. Since McMullen and Monson’s expression for r + 1 is equal to the expression in Corollary 1, we have 1 1 wα ∗ (wα ∗ +1)+ wα ∗ (wα ∗ +1). (wα ∗ )2 + wα ∗ (2wα ∗ −1) = 2 2 α=χ +χ¯

α=2χ

α=χ+χ¯

α=2χ

Therefore 1 1 [(wα ∗ )2 − wα ∗ (wα ∗ + 1)] = [ wα ∗ (wα ∗ + 1) − wα ∗ (2wα ∗ − 1)]. (∗) 2 2 α=χ +χ¯

α=2χ

Since wα ∗ is a non-negative integer for all real irreducible characters α, we have 1 1 (wα ∗ )2 − wα ∗ (wα ∗ + 1) = [(wα ∗ )2 − wα ∗ ] ≥ 0 2 2 and 3 1 ∗ wα (wα ∗ + 1) − wα ∗ (2wα ∗ − 1) = [wα ∗ − (wα ∗ )2 ] ≤ 0. 2 2 Therefore, both sides of the equation (∗) must be 0. This implies wα ∗ = 0 or 1 for all real irreducible characters α which are not complex irreducible, and the theorem follows. Note that the group [5, 5]5 described in [6] admits a real irreducible character α for which the corresponding ν2 (χ) = 0. Indeed, α is the unique real irreducible character with wα ∗ = wα , and it is true that wα ∗ = 1. The next corollary is immediate from the theorem. Corollary 2. If dim P˜α > 1 for some α ∈ Irr R ( ), then α is necessarily the character of a complex irreducible representation that is realizable over the field of real numbers. Note that the above result really says nothing about whether there actually exists an irreducible character of a finite string C-group with Schur index 2, so the existence of such characters remains a mystery.

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References [1] H. S. M. Coxeter and W. O. Moser, Generators and Relations for Discrete Groups, SpringerVerlag, Berlin, Heidelberg 1980. [2] J. S. Frame, The double cosets of a finite group, Bull. Amer. Math. Soc. 47 (1941), 458–467. [3] M. Geck and G. Pfeiffer, Characters of Finite Coxeter Groups and Iwahori–Hecke Algebras, Oxford University Press, Oxford 2000. [4] J. E. Humphries, Reflection Groups and Coxeter Groups, Cambridge University Press, Cambridge 1990. [5] P. McMullen, Realizations of regular polytopes , Aequationes Math. 37(1989), 38–56. [6] P. McMullen and B. Monson, Realizations of regular polytopes II, Aequationes Math. 65 (2003), 102–112. [7] P. McMullen and E. Schulte, Abstract Regular Polytopes, Encyclopedia Math. Appl. 92, Cambridge University Press, Cambridge 2002. Allen Herman, Department of Mathematics and Statistics, University of Regina, Regina, SK, Canada S4S 0A2 E-mail: [email protected] Barry Monson, Department of Mathematics and Statistics, University of New Brunswick, Fredericton, NB, Canada E3B 5A3 E-mail: [email protected]

Some observations on products of characters of finite classical groups Gerhard Hiss and Frank Lübeck

Abstract. This is an extended version of the talk given by the first author at the conference. We report on the outcome of some experiments with the decomposition of products of (generalized) characters of some classical groups. In particular some results of the PhD-thesis of Dirk Mattig on the products of unipotent characters of general linear groups are presented and commented. In order to obtain similar patterns for other classical groups it seems necessary to replace unipotent characters by unipotent almost characters.

1. Introduction The unipotent characters of the general linear group GLn (q) are parametrized by partitions of n, independently of q. Let us fix n and consider three partitions λ, µ, and ν of n. For each prime power q, there are three associated unipotent characters q q q q χλ , χµ , and χν of GLn (q). We may consider the multiplicity of χλ in the product q q χµ · χν , as a function in q. Following Mattig, we show that this multiplicity function is a polynomial function over the rationals. Using CHEVIE, Mattig has computed the corresponding polynomials for all 1 ≤ n ≤ 8 (and all relevant triples of partitions). He found that in fact all these polynomials have non-negative integral coefficients. After a discussion of Mattig’s results, we present similar computations for the general unitary groups GUn (q). We also consider the problem of extending our observations to other series of classical groups. We suggest to replace unipotent characters by unipotent almost characters. Finally, we sketch some special results. For instance, we show that the constant q coefficient of the polynomial expressing the multiplicity of χν in the square of the Steinberg character, is equal to the degree of the irreducible character of the symmetric group Sn corresponding to ν. This fact has been observed by Lux and Malle on an example presented in the talk of the first author. Let us close this introduction by introducing two notational conventions. Let X be a subset of C and f : X → C, x → f (x) a function, and let S ⊆ C. We say that f is a polynomial in x over S, if there is a polynomial p ∈ C[X], with coefficients in S, such that f (x) = p(x) for all x ∈ X.

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If G is a finite group and if ϕ, ψ are class functions on G, we write 1 (ϕ, ψ)G = ϕ(g)ψ(g −1 ) |G|

(1)

g∈G

for the usual scalar product on the set of class functions of G.

2. Tensor product polynomials Let n be a positive integer, q a prime power and let G denote one of the groups GLn (q) or GUn (q). The unipotent characters of G form a distinguished subset of the set of absolutely irreducible ordinary characters of G (over the complex numbers). In case G = GLn (q) the unipotent characters of G are the irreducible constituents of the permutation character 1B G , where B denotes the Borel subgroup of upper triangular matrices of G. There is no elementary way to introduce the unipotent characters in case G = GUn (q). Here, they are constructed as characters on adic cohomology groups on Deligne–Lusztig varieties on which G acts (see, e.g., [2, Chapter 12]). There is, however, a common parametrization of the unipotent characters in both cases. Namely, there is a bijection between the unipotent characters of G and the partitions of n. In particular, the unipotent characters are parametrized independently q of q. Let us write χλ for the unipotent character of G labelled by the partition λ of n. q q In this parametrization χ(n) denotes the trivial character 1G and χ(1n ) the Steinberg character St G of G. If we fix n and three partitions λ, µ, and ν of n, we may consider the scalar products q

tλ,µ,ν (q) = (χλ , χµq · χνq )GLn (q)

and

q t¯λ,µ,ν (q) = (χλ , χµq · χνq )GUn (q)

(2)

as functions in q. Using the terminology introduced above, we can now state the first result. Proposition 2.1 (Mattig, [12, Propositon 3.1.6]). Let n be a positive integer and let λ, µ, ν be partitions of n. Then tλ,µ,ν (q) and t¯λ,µ,ν (q), as functions on the set of all prime powers q, are polynomials in q over Q. Let us sketch a proof of this result. First of all, the conjugacy classes of G can be grouped together into class types, and the class types can be classified independently of q. Let us write gs and gu for the semisimple and unipotent part of an element g ∈ G, respectively. Two elements g and h of G belong to the same class type, if CG (gs ) is conjugate to CG (hs ) and if gu is conjugate in CG (gs ) to a conjugate of hu lying in CG (gs ). (For a definition and more details on class types for general series of groups of Lie type see [6, Section 4].) Secondly, the unipotent characters of G are constant on class types, and thirdly, the numbers of conjugacy classes inside a class type, the lengths of conjugacy classes,

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and the values of the unipotent characters, viewed as functions of q, are polynomials in q over the rationals. Finally, the order of G is a polynomial in q over the integers, and hence, by (1), the scalar products (2) are rational functions in q. Since these take integer values for all prime powers q, they are polynomials in q over Q. Table 1 collects all the information that is needed to compute the tensor product polynomials for n = 2. The first column of that table gives the number of conjugacy classes of GL2 (q) in each of the four class types, the second column gives the length of each conjugacy class in a class type. The third and fourth columns give the values of the two unipotent characters of GL2 (q). Table 1. The generic unipotent character table of GL2 (q)

Nr. of Classes in Type

Length

χ(2)

χ(12 )

q −1 q −1 1 2 (q − 1)(q − 2) 1 2 (q − 1)q

1 q2 − 1 q(q + 1) q(q − 1)

1 1 1 1

q 0 1 −1

We call such a table a generic unipotent character table. The CHEVIE-system [6] contains the generic unipotent character tables of GLn (q) and GUn (q) for 2 ≤ n ≤ + 8, of CSp4 (q), CSp6 (q), SO− 8 (q), and Spin8 (q). Furthermore, CHEVIE includes programs to compute with these character tables. As examples we have computed the polynomials t(15 ),(15 ),ν (q) and t¯(15 ),(15 ),ν (q), where ν runs through the partitions of 5. The result is given in Table 2. Dirk Mattig has used CHEVIE to compute the polynomials tλ,µ,ν (q) for all triples of partitions of numbers up to 8. Let us write N for the set of non-negative integers. Observation 2.2. (Mattig, [12].) Let λ, µ, ν be partitions of the integer n with 1 ≤ n ≤ 8. Then the function tλ,µ,ν (q) is a polynomial in q over N for all prime powers q. The same statement does not hold for the polynomials t¯λ,µ,ν (q) but the calculations with CHEVIE lead to the following observation. Observation 2.3. Let λ, µ, ν be partitions of n, where n is an integer with 1 ≤ n ≤ 8. Then we have: (a) The functions t¯λ,µ,ν (q) are polynomials in q over the integers. (b) The difference functions tλ,µ,ν (q) − t¯λ,µ,ν (q) are polynomials in q over 2N, that is the coefficients are even non-negative integers. Remarks. There are formulae for the values of unipotent characters for GLn (q) and GUn (q) which are closely related by an operation called Ennola duality. Roughly speaking one has to substitute in the formulae for GL the parameter q by −q and to adjust some signs.

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But this does not lead to a simple relation between tλ,µ,ν (q) and t¯λ,µ,ν (q) (see the example in Table 2), since there is no such simple correspondence between the number of classes in class types (see the example in Table 3) and since the mentioned adjustment of signs depends of the class type. Note also that the polynomials do not specialize to the corresponding scalar products of characters of the symmetric group Sn at q = 1. For example, if n = 2 we have (St, St · St)GL2 (q) = 1. The Steinberg character of GL2 (q) corresponds to the alternating character of S2 , whose square equals the trivial character. Table 2. Some tensor products in GL5 (q) and in GU5 (q)

ν

(StG , StG ·χν ) for G = GL5 (q)

(5) 1 (4, 1) 4 (3, 2) q2 + 2 q + 5 (3, 12 ) q3 + 2 q2 + 4 q + 6 2 4 (2 , 1) q + q3 + 3 q2 + 5 q + 5 3 5 (2, 1 ) q + q 4 + 3 q 3 + 3 q 2 + 6 q + 4 q6 + q4 + 2 q3 + q2 + 3 q + 1 (15 )

(StG , StG ·χν ) for G = GU5 (q) 1 0 q2 + 1 q3 + 2 q + 2 4 q − q3 + q2 + q + 1 q5 + q3 − q4 − q2 6 q + q4 + q2 + q + 1

Table 3. Numbers of conjugacy classes in class types

GL3 (q)

GU3 (q)

q −1 q −1 q −1 (q − 1)(q − 2) (q − 1)(q − 2) 1 6 (q − 1)(q − 2)(q − 3) 1 2 2 q(q − 1) 1 3 q(q − 1)(q + 1)

q +1 q +1 q +1 (q + 1)q (q + 1)q 1 6 q(q − 1)(q + 1) 1 2 2 (q − 2)(q + 1) 1 3 q(q − 1)(q + 1)

3. Tensor product polynomials for other groups Now let {G(q)| q a prime power} be an arbitrary series of groups of Lie type. For example {Sp4 (q)}, the symplectic groups on a 4-dimensional vector space over Fq ,

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or {2 E6 (q)sc } the twisted groups of type E6 in a simply-connected algebraic group of this type. For a precise definition see [6, Section 4.1]. Again, each group G(q) has a distinguished set of unipotent characters which are parametrized for all groups in the series independently of q (see, e.g., [2, Sections 13.8,13.9]). We write for a set of parameters of the unipotent characters and q χλ for the character of G(q) corresponding to λ ∈ . The exact values of the unipotent characters are not yet known in all cases. But in those cases where they are known a certain generalization of Proposition 2.1 can be proved with the same arguments. Unlike the case {GLn (q)} it is in general no longer true that just one generic unipotent character table describes the values of the unipotent characters of G(q) for all prime powers q. The smallest example for this is the series {SL2 (q)}. For odd q these groups have three unipotent conjugacy classes while for even q there are only two. It turns out that for each series {G(q)} there is a number m ∈ N such that for any fixed 0 ≤ i ≤ m − 1 and all prime powers q ≡ i (mod m) the class types can be parametrized independently of q and the number of classes in a class type is given by a polynomial in q. Furthermore, the class length for all classes in a fixed class type is described by a polynomial in q. It is not clear a priori that the unipotent characters are constant on class types. But this is true in all known cases. Moreover, in the known cases all values of unipotent characters are given by polynomials in a square root q 1/2 of q with coefficients in some finite extension of Q which is independent of q within a congruence class of q modulo m as described above. Thus, fixing an integer 0 ≤ i ≤ m − 1 such that the generic unipotent character table of {G(q)| q ≡ i (mod m)} is known, we see as in Proposition 2.1 that the multiplicities q

tλ,µ,ν,i (q) = (χλ , χµq · χνq )G(q) for all prime powers q with q ≡ i (mod m) are polynomials in q over C. If we compute these polynomials in the case {Sp4 (q) | q even} (the character values were first computed by Enomoto [5]), we quickly find polynomials with non-integer coefficients. This shows that Observations 2.2 and 2.3 do not generalize directly to other series of groups. Almost Characters. To find such a generalization we have to look at the unipotent almost characters instead. These are certain C-linear combinations, with coefficients independent of q, of the unipotent characters. Almost characters were introduced by Lusztig when he described the parametrizations of unipotent characters. The transforming matrices from irreducible characters to almost characters are called Fourier transform matrices. For linear groups (like GLn (q) and SLn (q)) the unipotent characters coincide with the unipotent almost characters. For unitary groups (like GUn (q) and SUn (q)) they coincide up to sign. But for all other types of groups this is no longer true.

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It is conjectured by Lusztig and proved in many cases by Lusztig and Shoji that the almost characters are class functions associated to certain geometric objects called character sheaves, see [10], [11] and [14]. ˜ be a set of parameters for the unipotent almost characters of the groups Let ˜ we write χ˜ q for the corresponding almost character of G(q). For G(q); for λ ∈ λ q q q m as above and 0 ≤ i ≤ m − 1 set t˜λ,µ,ν,i (q) = (χ˜ λ , χ˜ µ · χ˜ ν )G(q) , viewed as a function on all prime powers q ≡ i (mod m). Then, in those cases where we know that the tλ,µ,ν,i (q) are polynomials in q the t˜λ,µ,ν,i (q) are also polynomials in q over the complex numbers. We have computed the polynomials t˜λ,µ,ν,i (q) for all 0 ≤ i ≤ m − 1 in the following cases.

Series

m

Series

m

{GLn (q)}, n ≤ 8 {SLn (q)}, n ≤ 8 {GUn (q)}, n ≤ 8 {SUn (q)}, n ≤ 8 {COn (q)}, n = 7, 9

1 n 1 n 2

{CSpn (q)}, n = 4, 6, 8 {Spn (q)}, n = 4, 6, 8 {Spin7 (q)} {Spin+ 8 (q)} {3D 4 (q)}

2 2 4 2 2

Note, that the case SLn (q) cannot easily be derived from the case GLn (q) although the sets of unipotent characters are in bijection via restriction. To see why, let us denote by ρ a linear character of GLn (q) generating the cyclic group GLn (q)/ SLn (q) of order q q − 1. The unipotent characters of SLn (q) are exactly the charactersχλ ↓SLn (q) , where λ runs through the partitions of n. Using Frobenius reciprocity and Clifford’s theorem we find q

(χλ ↓SLn (q) , χµq ↓SL

n (q)

q

· χνq ↓SLn (q) )SLn (q) = ([χλ ↓SLn (q) ]GLn (q) , χµq · χνq )GLn (q) =

q−2 q (χλ · ρ i , χµq · χνq )GLn (q) i=0

= tλ,µ,ν (q) +

q−2 q (χλ · ρ i , χµq · χνq )GLn (q) . i=1

q

q

q

If the center Z(GLn (q)) is in the kernel of ρ i , then (χλ · ρ i , χµ · χν )GLn (q) = 0 in general, and thus contributes to the tensor product polynomial for SLn (q). To give a specific example, t(13 ),(13 ),(13 ) = q +1, whereas in case 3 | q −1 the tensor product polynomial for SL3 (q) corresponding to ((13 ), (13 ), (13 )) equals 3q +1. This q q q reflects the fact that (χ(13 ) ·ρ i , χ(13 ) ·χ(13 ) )GL3 (q) = q for i = (q −1)/3 and 2(q −1)/3.

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Some of these tables of unipotent characters were available in CHEVIE and computed in [16], [5], [13], [4], [7], and [9]. The other non-linear group cases were computed by the second author using the results from [15]. Observation 3.1. In all cases mentioned above, the functions t˜λ,µ,ν,i (q) are polynomials in q over Z. Here, some negative coefficients occur in the cases GUn (q) with 5 ≤ n ≤ 8, SUn (q) with 3 ≤ n ≤ 8 (for some i), 3D 4 (q) and Spin7 (q) with q ≡ 3 (mod 4). This suggests that, if our observations are actually true for all series of groups of Lie type, one has to consider character sheaves for an explanation. Comparing the cases GL and GU it is interesting to know that the character sheaves leading to the values of unipotent characters are the same geometric objects for GL and GU. Maybe there is an interpretation of the coefficients of the tensor product polynomials using the same non-negative numbers which have to be added up with different signs. There does not seem to be an obvious notion of tensor products of character sheaves which correspond to the tensor products of the associated class functions.

4. Special results on the tensor product polynomials When the first author gave the talk on these results in Gainesville, Klaus Lux and Gunter Malle observed that the constant coefficient of the polynomial t(15 ),(15 ),ν in the first half of Table 2 equals ζ ν (1), where ζ ν is the irreducible character of the symmetric group S5 labelled by the partition ν. Gunter Malle observed furthermore, that the constant coefficient of the polynomial t¯(15 ),(15 ),ν in the second half of Table 2 equals 0, if χν does not lie in the principal series, and equals the degree of the corresponding character of the Weyl group of GU5 (q), otherwise. Here, the Weyl group is a dihedral group of order 8. During the conference a sketch of a proof for these facts emerged through discussions with Gunter Malle and Meinolf Geck. We give the details here. Proposition 4.1. Let n be a positive integer and let ν be a partition of n. Write ζ ν for the irreducible character of the symmetric group Sn labelled by ν. Put m := n/2. (a) The constant coefficient of t(1n ),(1n ),ν equals ζ ν (1). (b) The constant coefficient of t¯(1n ),(1n ),ν equals |ζ ν (σ )|, where σ ∈ Sn is an involution that is a product of m disjoint transpositions. The value |ζ ν (σ )| also has the following interpretation. Let κ and (α, β) denote the 2-core and the 2-quotient of ν, respectively. Then |ζ ν (σ )| equals 0, if |κ| > 1, and ζ (α,β) (1), otherwise. Here, ζ (α,β) denotes the irreducible character of the Weyl group of type Bm labelled by the bi-partition (α, β) of m. Proof. For 0 = f ∈ Q[X], f = Xl g with l ≥ 0 and X g, we write fX := Xl and fX := g = f/fX .

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The proof is given simultaneously for the general linear and unitary groups. Let ε be a parameter taking the values ±1. Let Gε denote the generic group giving rise to the series GLn (q), if ε = 1, and to GUn (q), if ε = −1. We write |Gε | for the order polynomial of Gε , i.e., the value of |Gε | at q is the order of the finite group Gε := Gε (q). Letting N := n(n − 1)/2, we have n (Xi − ε i ). |Gε | = X N

i=1

˜ label the class types ˜ be a set of labels for the class types of Gε and let ⊆ Let ˜ we of semisimple elements. Such sets can be chosen independently of ε. For ω ∈ write Stε (ω) and χν,ε (ω) for the value polynomials of the Steinberg character and the unipotent character χν , respectively. We also write nε (ω) and cε (ω) for the numbers of conjugacy classes of class type ω and the numbers of elements of a conjugacy class of class type ω, respectively. Finally, t ε denotes t, if ε = 1, and t¯, if ε = −1. We then have ε nε (ω)cε (ω) Stε (ω)2 χν,ε (ω). t(1 n ),(1n ),ν |Gε | = ˜ ω∈

This follows from Formula (1) with the above settings and the fact that χν,ε (ω) = χν,ε (ω ), where ω denotes the class type containing the inverses of the elements of class type ω. ˜ \ , then St ε (ω) = 0, and if ω ∈ , then St ε (ω) = ±(|Gε |/cε (ω))X (see If ω ∈ [2, Theorem 6.5.9]). Note that cε (ω) divides |Gε |, and that |Gε |/cε (ω) is the centralizer order polynomial corresponding to the class type ω. Let us write (|Gε |/cε (ω))X =: X mω,ε . Then, dividing both sides of the equation above by XN , we have ε t(1 n ),(1n ),ν |Gε |X =

nε (ω)cε (ω)X X mω,ε χν,ε (ω).

ω∈

Now mω,ε = 0 for ω ∈ , if and only if ω is a class type of regular semisimple elements, i.e., the generic centralizer is a torus. So, only regular semisimple classes ε contribute to the constant term of the polynomial t(1 n ),(1n ),ν . Finally, we have to consider the polynomials nε (ω) for class types of regular semisimple elements. Such class types are parametrized by the partitions of n as follows. Given a regular semisimple element in Gε , it has n pairwise different eigenvalues and the set of eigenvalues is permuted by raising these numbers to the εq-th power. All such sets of eigenvalues occur and each set of eigenvalues occurs for exactly one conjugacy class. The class type of the element is parametrized by the cycle type of this permutation on the eigenvalues. Using induction one can see that the number of sets of eigenvalues as above for a fixed cycle type is a polynomial in q which is divisible by q if there is: – any cycle of length greater than one, if ε = 1.

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203

– more than one cycle of length one or a cycle of length greater than two, if ε = −1. This shows that in each case there is only one class type, namely the one of the regular elements in a maximally split torus, which contributes to the constant term we are interested in. Let ω0 be the label of this class type. Then ε t(1 n ),(1n ),ν |Gε |X ≡ nε (ω0 )cε (ω0 )X χν,ε (ω0 )

(mod X).

Now, for ε = 1, i.e., for GLn (q), we find from the combinatorial interpretation of n1 (ω0 ) given above that n 1 (X − i). n1 (ω0 ) = n! i=1

Since c1 (ω0 )X =

n

i=1 (X

i

− 1) |Gε |X = , n (X − 1) (X − 1)n

it follows that the constant coefficient of t(1n ),(1n ),ν equals χν,1 (ω0 ). Now let ε = −1. In this case we have m 1 n−2m (X2 − X − 2i), n−1 (ω0 ) = m (X + 1) 2 m! i=1

and c−1 (ω0 )X =

n

(X

i i i=1 (X − (−1) ) . m + 1)n−2m i=1 (X 2 − 1)

It follows that the constant coefficient of t¯(1n ),(1n ),ν equals χν,−1 (ω0 ). To determine the value of χν,ε (ω0 ) we use the character formula for the values of Deligne–Lusztig generalized characters on semisimple elements (see [2, Proposition 7.5.3]), and the connection of unipotent characters with unipotent almost characters as presented in [1, p. 45]. Let ψν,ε denote the unipotent almost character of Gε corresponding to ζ ν . Then ψν,ε = χν,ε , if Gε = GLn (q), and ψν,ε = ±χν,ε , otherwise. Moreover, ψν,1 (ω0 ) = ζ ν (1), so we are done in case ε = 1. Let ε = −1. Then ψν,−1 (ω0 ) = ζ ν (σ ). Also, by [8, Corollary 2.7.33], |ζ ν (σ )| = 0, if |κ| > 1, and |ζ ν (σ )| = ζ (α,β) (1), if |κ| ≤ 1. It remains to show that the sign of ζ ν (σ ) equals the sign of ψν,−1 . By the definition of the almost characters, the degree of ψν,−1 is obtained from the degree of ψν,1 by replacing q by −q. Thus ψν,−1 = χν,−1 , if the generic degree polynomial has even degree, and ψν,−1 = −χν,−1 , otherwise. A formula for the generic degree polynomial can be found in [2, p. 466]. The sign of ζ ν (σ ) is determined in [8, pp. 80–82]. Using these descriptions and induction on the 2-weight of ν, it follows that ψν,−1 = −χν,−1 if and only if ζ ν (σ ) is negative.

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Remark. Note that the property that all but one of the numbers of classes in the class types of regular semisimple elements is divisible by q is a special property of the series GLn (q) and GUn (q). Thus there is no generalization of Theorem 4.1 to other series of groups. Let us now fix the positive integer n and put G = GLn (q) for the remainder of this section. The following result is also contained in Mattig’s PhD-thesis. Proposition 4.2 (Mattig, [12, Theorem 6.4.8]). For all partitions µ, ν of n, the polynomial t(n−1,1),µ,ν is constant (and hence a non-negative integer). In order to prove this, Mattig considers the permutation characters 1Pλ G on parabolic subgroups Pλ . Here, λ again is a partition of n and Pλ denotes the corresponding parabolic subgroup of G (whose Levi subgroup is the group of block diagonal matrices with blocks of sizes given by the parts of λ). Let us write Pn for the set of partitions of n. Given λ ∈ Pn , there are non-negative integers aλ,µ , independent of q, such that aλ,µ χµq (3) 1Pλ G = µ∈Pn

(see, e.g., [3, Theorem (70.24)]). Moreover, by [8, Section 2.2], the matrix of coefficients (aλ,µ )λ,µ∈Pn is invertible over the integers. In particular, the unipotent charq acters χλ can be expressed as Z-linear combinations of the permutation characters q 1Pµ G . For example, χ(n−1,1) = 1P(n−1,1) G − 1P(n) G = 1P(n−1,1) G − 1G . It follows from Proposition 2.1 and Equation (3) that there are polynomials rλ,µ,ν ∈ Q[X] such that rλ,µ,ν (q) = 1Pλ G , 1Pµ G · 1Pν G GL (q) n

for all prime powers q. Another interpretation of these polynomials is provided by G 1Pλ , 1Pµ G · 1Pν G GL (q) = |Pλ \G/Pµ ∩ xP ν |, (4) n

x∈Dµ,ν

where Dµ,ν denotes the set of distinguished double coset representatives corresponding to the parabolic subgroups Pµ and Pν . Note that Dµ,ν can be chosen, independently of q, as a subset of distinguished double coset representatives of the Weyl group Sn of G. Using (4), Mattig shows that r(n−1,1),µ,ν is constant for all partitions µ, ν of n. The remarks following (3) now prove Proposition 4.2. Of course, rλ,µ,ν ∈ N[X] if tλ,µ,ν ∈ N[X]. Let us consider a further special case, namely P(1n ) = B. We have q ζ λ (1) χλ . (5) 1B G = λ∈Pn

Moreover, by [2, Propositions 7.4.4 and 7.5.4] we have 1B G · St G = 1T G ,

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205

where T denotes the maximally split torus of G consisting of the diagonal matrices. Using Proposition 2.1, and Equation (5), we see that there is a polynomial sn ∈ Q[X] such that sn (q) = |B\G/T | = 1B G , 1T G G = 1B G , 1B G · St G G = ζ λ (1) ζ µ (1) tλ,µ,(1n ) (q). λ,µ∈Pn

Theorem 4.3 (Mattig, [12, Chapters 4 and 8]). The polynomials sn have integer coefficients for all n ≥ 1. The degree of sn equals (n − 1)(n − 2)/2, and the coefficient at X i is positive for all 0 ≤ i ≤ (n − 1)(n − 2)/2. Moreover, sn is monic except for n = 2. As examples, we give the polynomials sn for 1 ≤ n ≤ 6. s1 = 1 s2 = 3 s3 = X + 19 s4 = X3 + 6 X2 + 36 X + 211 s5 = X6 + 8 X5 + 35 X4 + 136 X3 + 410 X2 + 1253 X + 3651 s6 = X10 + 10 X9 + 54 X8 + 209 X7 + 685 X6 + 1969 X5 + 4951 X4 + 11592 X3 + 24415 X2 + 50547 X + 90921 Let us sketch the main arguments in the proof of Theorem 4.3. Let U denote the set of upper unitriangular matrices of G so that B = U T . Let W denote the Weyl group of G and w0 its longest element. For any w ∈ W , put Uw := U ∩ U w0 w . Then Uw is invariant under conjugation by T , and R := {wu | w ∈ W, u ∈ Uw } is a set of coset representatives of the right cosets of B in G. Moreover, for wu, w u ∈ R, we have BwuT = Bw u T if and only if w = w and u and u are in the same orbit of T on Uw , where T acts on Uw by conjugation. This shows that it suffices to determine, for each w ∈ W , the number of orbits of T on Uw as a function of q. Fix w ∈ W and write V := Uw . Let + denote the set of pairs {(i, j ) | 1 ≤ i < j ≤ n}, which corresponds to the set of positive roots of G. Then there is a subset ⊆ + such that | aii = 1 for 1 ≤ i ≤ n, and aij = 0 for i = j and (i, j ) ∈ / } V = {(aij ) ∈ Fn×n q (see [2, Proposition 2.5.16]). Choose an injective mapping ν : → N. A subset ⊆ can then be viewed as a weighted undirected graph, by associating ν(i, j ) to the edge (i, j ) ∈ . Since the weight function is injective, has a unique minimal spanning forest. Let X denote

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Gerhard Hiss and Frank Lübeck

the set of subsets of which arise as minimal spanning forests in this way. Then X contains a unique maximal element max . For a subset of let V∗ := {(aij ) ∈ V | aij = 0 if and only if i = j or (i, j ) ∈ }. For ∈ X, let V denote the union of the sets V∗ , where runs through the subsets of which have as their minimal spanning forest. Then, obviously, V is invariant under conjugation by T (since every V∗ is T -invariant), and V , V = ∈X

a disjoint union. The proof concludes by showing that the number of T -orbits on V equals q n , where n = |max | − ||.

References [1]

M. Broué, G. Malle, and J. Michel, Generic blocks of finite reductive groups, in Représentations unipotentes génériques et blocs des groupes réductifs finis, Astérisque 212 (1993), 7–92.

[2]

R. W. Carter, Finite Groups of Lie Type: Conjugacy Classes and Complex Characters, Pure Appl. Math., Wiley-Interscience , Chichester 1985.

[3]

C. W. Curtis and I. Reiner, Methods of Representation Theory I, Wiley, 1981; Methods of Representation Theory II, Wiley, New York 1987.

[4]

D. I. Deriziotis and G. O. Michler, Character tables and blocks of finite simple triality groups 3D 4 (q 3 ), Trans. Amer. Math. Soc. 303 (1987), 39–70.

[5]

H. Enomoto, The characters of the finite symplectic group Sp(4, q), q = 2f , Osaka J. Math. 9 (1972), 75–94.

[6]

M. Geck, G. Hiss, F. Lübeck, G. Malle, and G. Pfeiffer, CHEVIE—A system for computing and processing generic character tables, AAECC 7 (1996), 175–210.

[7]

M. Geck and G. Pfeiffer, Unipotent characters of the Chevalley groups D4 (q), q odd, Manuscripta Math. 76 (1992), 281–304.

[8]

G. D. James and A. Kerber, The Representation Theory of the Symmetric Group, Encyclopedia Math. Appl. 16, Addison-Wesley, Reading, MA 1981.

[9]

F. Lübeck, Charaktertafeln für die Gruppen CSp6 (q) mit ungeradem q und Sp6 (q) mit geradem q, Dissertation, Universität Heidelberg 1993.

[10] G. Lusztig, Character sheaves I to V, Adv. Math. 56, 57, 59, 61, 1985/1986. [11] G. Lusztig, Green functions and character sheaves, Ann. Math. 131 (1990), 355–408.

Some observations on products of characters in finite classical groups

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[12] D. Mattig, Tensor Products of Unipotent Characters of GL(n, q), Dissertation, Universität Heidelberg 1999. [13] K. Shinoda, The characters of the finite conformal symplectic group CSp(4,q), Comm. Algebra 10 (1982), 1369–1419. [14] T. Shoji, Character sheaves and almost characters of reductive groups I and II, Adv. Math. 111 (1995), 244–313, 314–354. [15] T. Shoji, Unipotent characters of finite classical groups, in Finite reductive groups: related structures and representations (Luminy, 1994), Prog. Math. 141, Birkhäuser, Boston 1997, 373–413. [16] B. Srinivasan, The characters of the finite symplectic group Sp(4, q), Trans. Amer. Math. Soc. 131 (1968), 488–525. Gerhard Hiss, Lehrstuhl D für Mathematik, RWTH Aachen, 52056 Aachen, Germany E-mail: [email protected] Frank Lübeck, Lehrstuhl D für Mathematik, RWTH Aachen, 52056 Aachen, Germany E-mail: [email protected]

The number of finite p-groups with bounded number of generators Andrei Jaikin-Zapirain ∗

Abstract. We establish new bounds for the number of finite p-groups generated by a fixed number of elements. 2000 Mathematics Subject Classification: Primary 20D15; Secondary 20E18

1. Introduction The investigation of the number of finite p-groups is a classical topic in group theory. Denoting by f (n) the number of groups of order n, we have the following result of G. Higman and C. C. Sims (see [3, 8]): Theorem 1.1 (Higman, Sims). 2

f (pk ) = p 27 k

3 +o(k 3 )

,

where the term o depends only on k and not on p. In this paper we consider the function f (n, d), which denotes the number of groups of order n which can be generated by d elements. By results of A. McIver, P. M. Neumann and A. Mann (see [4, 5]), we have Theorem 1.2 (McIver, Neumann and Mann). pc1 (d)k ≤ f (pk , d) ≤ pc2 (d)k , 2

2

for some positive constants c1 (d) and c2 (d). Thus the following question seems reasonable: Problem 1.3 (A. Mann, [6, Question 6]). Is there a constant c(d) such that 2 2 f (p k , d) = pc(d)k +o(k ) , and what is the value of this constant? We have not been able to find an exact answer to this question, but we give rather reasonable estimates for c(d): ∗ This work has been partially supported by the MCYT Grants BFM2001-0201, BFM2001-0180, FEDER and the Ramón y Cajal Program.

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Theorem 1.4. We have that p

d−1 2 2 4 k +o(k )

≤ f (pk , d) ≤ p

d−1 2 2 2 k +o(k )

.

We explain now the general strategy of the proof of these two bounds. The detailed proof is given in the following sections. Let Fd be a free pro-p group on d generators. Then every finite p-group G generated by d elements is a quotient of Fd : G ∼ = Fd /N for some N Fd . First of all we want to know how many such subgroups N there are. Of course, their number is less than or equal to the order of the group of automorphisms of G. On the other hand | Aut(G)| ≤ |G|d . Thus we obtain: Proposition 1.5. Let Npk (Fd ) be the number of normal subgroups of index pk in Fd . Then f (p k , d) ≤ Npk (Fd ) ≤ pkd f (pk , d). As we have seen in Theorem 1.2, f (p k , d) is like pk , and so it is much bigger than p kd when k is big. Hence the main problem is to estimate Npk (Fd ) (this function is called the normal subgroup growth of Fd ). Now, fix a normal subgroup N of index p t in Fd . We want to know how many normal subgroups of Fd of index p in N there are. Define this set by MN : 2

MN = {K | K Fd , K ≤ N, |N : K| = p}. We have an easy characterization of the set MN : if |N : K| = p then K ∈ MN if and only if [N, Fd ]N p ≤ K. Thus, we have to estimate the rank of N/[N, Fd ]N p . The following characterization of the rank of N/[N, Fd ]N p is well-known (see, for example, [7, Proposition 7.8.2] and [1, Lemma D6]): d(N/[N, Fd ]N p ) = rp (Fd /N ) + d − d(Fd /N ),

(1)

where d(G) and rp (G) denotes the minimum number of generators and the minimum number of defining pro-p relations respectively of a finite p-group G. We set g(p t , d) = max{d(N/[N, Fd ]N p ) | N Fd , |Fd /N| = p t }. g(p t ,d)

t

Then we have |MN | ≤ p p−1 −1 < pg(p ,d) . Now, we come back to the function Npt (Fd ). This number is clearly less than the number of chains Fd = N0 > N1 > N2 > · · · > Nt , where Ni is a normal t−1 s subgroup of Fd and |Ni : Ni+1 | = p, which is less than p s=0 g(p ,d) by the previous observation. In this way we have an upper bound for f (pt , d). We will see now that in the lower bound of f (p t , d) the function g(pt , d) also plays a crucial role. Let N be a normal subgroup of Fd of index ps such that d(N/[N, Fd ]N p ) = g(p s , d). We have that every subgroup K, satisfying [N, Fd ]N p ≤ K ≤ N, is normal in Fd . Hence the number of normal subgroups of Fd of index ps+k is at least the number of subspaces in N/[N, Fd ]N p of codimension k, which is greater than s p k(g(p ,d)−k) . Playing with k and s such that k + s = t, we obtain the lower bound for f (pt , d). In Section 3 we will modify this method in order to obtain a better bound.

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211

All this leads us to study the function g(pt , d). In view of (1), we should estimate the following function: h(pt , d) = max{rp (G) | d(G) = d, |G| = pt }. Our result in this direction is the following: Theorem 1.6. If d is fixed, then (d − 1)t (1 + o(1)) ≤ h(pt , d) ≤ (d − 1)t − as t goes to infinity.

d(d−3) 2

The proof of the upper bound is based on the following remark. Let G be a finite p-group. We can associate with G a Lie algebra L = L(G) = i γi (G)/γi+1 (G), where γi (G) is the ith term of the lower central series of G. Then it is well-known that the minimum number of generators of L (we denote it by d(L)) is equal to d(G) and that rp (G) is less than or equal to the minimum number of relations needed to define L. Let F be a free Lie Zp -algebra on d(G) generators and H an ideal of F such that L ∼ = F /H . Then the minimum number of relations needed to define L is equal to dF (H ) = logp |H /([H, F ] + pH )|. Hence the upper bound in the previous theorem follows from its analogue for Lie rings: Theorem 1.7. Let F be a free Lie Zp -algebra on d generators, H ≤ [F, F ] + pF an ideal of F and L = F /H a nilpotent Lie ring of order pt . Then dF (H ) ≤ (d − 1)t − d(d−3) 2 . Remark 1.8. If we remove the condition H ≤ [F, F ] + pF in the theorem then we have that the minimum number of relations needed to define L is equal to dF (H ) − d + d(L) and so the theorem gives d(L)(d(L) − 3) + d − d(L). 2 We prove this theorem in Section 2. The lower bound in Theorem 1.6 is obtained by constructing an explicit family of p-groups {Gt } such that |Gt | = p t and rp (Gt ) ≥ (d − 1)t + o(t). We present this family in Section 3. The notation is standard. If L is a Lie Zp -algebra and M is a right L-module then dL (M) = logp |M/(ML + pM)|. dF (H ) ≤ (d(L) − 1)t −

2. The upper bound in Theorem 1.4 In order to prove Theorem 1.7 we need to do some auxiliary work. First we consider the case when pF ≤ H . We need the following lemma. Lemma 2.1. Let F be a free Lie Zp -algebra on d generators, H an ideal of F , pF ≤ H and L = F /H a nilpotent Lie ring of order pt > 1. Suppose Z = T /H ≤ [L, L] is a minimal ideal of L. Then dF (H /pF ) ≤ dF (T /pF ) + (d − 1).

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Andrei Jaikin-Zapirain

Proof. Note that dF (H /pF ) = logp |H /([H, F ] + pF )| and dF (T /pF ) = logp |T /([T , F ] + pF )|. We put F¯ = F /([H, F ] + pF ), T¯ = T /([H, F ] + pF ), H¯ = H /([H, F ] + pF ) and K¯ = ([T , F ] + pF )/([H, F ] + pF ). Choose z ∈ T¯ \ H¯ . Then K¯ = [z, F¯ ]. Since [z, [F¯ , F¯ ]] = 0, we obtain that ¯ ¯ d−1 . |K| ≤ pd . Hence |H¯ | ≤ |T¯ /K|p Lemma 2.2. Let F be a free Lie Zp -algebra on d generators, pF ≤ H ≤ [F, F ]+pF an ideal of F and L = F /H a nilpotent Lie ring of order pt . Then dF (H /pF ) ≤ t (d − 1) − d(d−1) 2 . Proof. When L is abelian, the result is trivial. The general case can be proved by induction on the order of L using the previous lemma. We will need the following auxiliary result. For the reader’s convenience we include the proof. Lemma 2.3. Let L be a Lie Zp -Lie algebra and M a right L-module. Then, if N is a L-submodule of M, dL (M) ≤ dL (M/N) + dL (N ). Proof. Since N L + pN ≤ (ML + pM) ∩ N , then |M : (ML + pM)| = |M : (ML + pM + N )||(ML + pM + N) : (ML + pM)| = |M : (ML + pM + N )||N : (N ∩ (ML + pM)| ≤ |M : (ML + pM + N )||N : (N L + pN)|. Proof of Theorem 1.7. We prove Theorem 1.7 by induction on the order of L. When L is abelian, the result is trivial. By Lemma 2.3, we have that dF (H ) ≤ dF (H ∩pF )+dF (H /(H ∩pF )) ≤ dF (H ∩pF )+dF ((H +pF )/pF ). (2) Consider first dF ((H + pF )/pF ). Applying Lemma 2.2, we obtain that d(d − 1) . 2 Now, consider dF (H ∩ pF ). We can divide by p in pF . Hence dF ((H + pF )/pF ) ≤ (d − 1) logp (|F /(H + pF )|) − dF (H ∩ pF ) = dF (p−1 (H ∩ pF )). Note that |F : p−1 (H ∩ pF )| = |pF /(H ∩ pF )| = |(pF + H )/H |. Put M = F /p −1 (H ∩ pF ).

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213

Then, by induction, using Remark 1.8, we have that dF (H ∩ pF ) ≤ (d(M) − 1) logp (|(pF + H )/H |) −

d(M)(d(M) − 3) + d − d(M) 2

≤ (d − 1) logp (|(pF + H )/H |) + d. (The equality occurs only when M = 0.) dF (H )(d − 1) logp |F /H | − d(d−3) 2 .

Therefore from (2) it follows that

Proof of the upper bound in Theorem 1.4. As we have seen before, Theorem 1.7 implies the upper bound in Theorem 1.6. Hence we have g(pt , d) ≤ (d − 1)t + d. Now, as was shown in Introduction f (p t , d) ≤ p t

g(pi , d) ≤

i=1

t−1 i=o

(3)

g(p i ,d)

. From (3) we get

d −1 (t − 1)t + td. 2

3. The lower bound in Theorem 1.4 Now we prove the second part of Theorem 1.4. First we need the following technical lemma: Lemma 3.1. Let Md (n) = n1 k|n µ(k)d n/k , where µ(m) is the Möbius function which is defined for positive integers by the rules µ(1) = 1, and for n = p1e1 . . . pses ; p1 , . . . , ps being distinct primes, µ(n) = 0 if ei > 1, and µ(p1 . . . ps ) = (−1)s . Then when d is fixed and n goes to infinity the following holds: n (1) Md (n) = dn (1 + o(1)); d n+1 (1 + o(1)). (2) sd (n) = ni=1 Md (i) = n(d−1) n−1 n+1 d (3) i=1 sd (i) = n(d−1)2 (1 + o(1)). Proof. The first statement is well-known approximation of Md (n) (see [2, Chapter VIII]). The second and third follow from the next result. Suppose that an+1 − an = k. lim an = lim bn = ∞ and lim n→∞ n→∞ n→∞ bn+1 − bn Then limn→∞

an bn

= k.

Let F be a free pro-p group on d generators x1 , . . . , xd . The lower p-series Pi = Pi (F ) of F is defined inductively by P1 = F

and

p

Pi+1 = [Pi , F ]Pi .

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Andrei Jaikin-Zapirain

We know that logp |Pi /Pi+1 | = sd (i). Put Bn = F /Pn . Now, we calculate the number of defining relations of Bn . Lemma 3.2. We have that rp (Bn ) = (d − 1) logp |Bn |(1 + o(1)) as n goes to infinity. p

Proof. Note that [Pn , F ]Pn = Pn+1 . Hence the result follows from (1) and Lemma 3.1. Now, we want to construct a series {Pn,k } of normal subgroups of F , such that P1 = P1,d > P1,d−1 > · · · > Pn−1,1 > Pn = Pn,sd (n) > Pn,sd (n)−1 > · · · , the index of Pn,k in Pn,k+1 is p and rp (F /Pn,k ) = (d − 1) logp |F /Pn,k |(1 + o(1)) as n goes to infinity. Clearly this will give the lower bound in Theorem 1.6. Define inductively a basis Wi of Pi /Pi+1 in the following way: W1 = {w1,1 P2 , . . . , w1,d P2 },

w1,i = xi .

Now suppose we have a basis Wn = {wn,i Pn+1 |1 ≤ i ≤ sd (n)} of Pn /Pn+1 . Put p wn+1,i = wn,i , when 1 ≤ i ≤ sd (n). We know that wn+1,1 Pn+2 , . . . , wn+1,sd (n) Pn+2 are independent in Pn+1 /Pn+2 and we can complete this set to a basis Wn+1 of Pn+1 /Pn+2 by adding some vectors wn+1,sd (n)+1 Pn+2 , . . . , wn+1,sd (n+1) Pn+2 , where wn+1,k (when k ≥ sd (n) + 1) is of type [xi1 , . . . , xin+1 ]. Let Pn,k be the normal subgroup of F generated by Pn+1 and wn,i with i ≤ k. Lemma 3.3. Suppose that n ≥ 3 and k ≥ p wn,k+1

Proof. Since k ≥

d(d+1) 2 ,

d(d+1) 2 .

Then p

∈ [Pn,k , F ]Pn,k .

we have

wn,k+1 = [[xi1 , . . . , xis−1 ], xis ]p

n−s

for some 1 ≤ i1 , . . . , is ≤ d and s ≥ 3 which can be calculated explicitly from n−s+1 p , xis ]Pn+2 . Now, note that k and n. Hence wn,k+1 Pn+2 = [[xi1 , . . . , xis−1 ]p [xi1 , . . . , xis−1 ]p

n−s+1

p

∈ Pn,k and Pn+2 ≤ [Pn,k , F ]Pn,k .

Put Bn,k = F /Pn,k . In the next lemma we relate the number of defining relations of Bn,k+1 and Bn,k : Lemma 3.4. Suppose n ≥ 3. We have that rp (Bn,k ) ≤ rp (Bn,k+1 )+d−1 if k ≥ and rp (Bn,k ) ≤ rp (Bn,k+1 ) + d in general.

d(d+1) 2

The number of finite p-groups with bounded number of generators

215

Proof. Use a similar argument as in Lemma 2.1 bearing in mind Lemma 3.3. Corollary 3.5. We have that rp (Bn+1 ) ≤ rp (Bn,k ) + (d − 1)(logp |Bn+1 | − logp |Bn,k |) +

d(d + 1) . 2

Theorem 3.6. We have that rp (Bn,k ) ≥ (d − 1) logp |Bn,k |(1 + o(1)) as n goes to infinity. Proof. Suppose that there exist and an infinite number of pairs (n, k) with 1 ≤ k ≤ sd (n) such that rp (Bn,k ) ≤ (d − 1 − ) logp |Bn,k |. Hence, by Corollary 3.5, d(d + 1) . 2 Bearing in mind that logp |Bn,k | ≥ (logp |Bn+1 |)/d, we obtain that rp (Bn+1 ) ≤ (d − 1) logp |Bn+1 | − log |Bn,k | +

rp (Bn+1 ) ≤ (d − 1 − /d) logp |Bn+1 | +

d(d + 1) , 2

which contradicts Lemma 3.2. Corollary 3.7. If d is fixed, then h(pt , d) ≥ (d − 1)t (1 + o(1)) as t goes to infinity. Proof of the lower bound in Theorem 1.4. In order to prove the lower bound in Theorem 1.4 we have to show that for any α < 1 there exists T , such that f (pt , d) ≥ α(d−1) 2 p 4 t for any t ≥ T . a = β > α, and put Choose a natural number a such that pp(d−1)−1 a (d−1) a e = p (d − 1) + 1. Let F be a free pro-p group on d generators x1 , . . . , xd and pa H a normal subgroup of F generated by x1 , x2 , . . . , xd . Then F /H is a cyclic group of order p a . The group H is a free pro-p group on e generators pa

y1 = x1 ,

xk

y(i−2)pa +k+2 = xi 1 ,

2 ≤ i ≤ d, 0 ≤ k ≤ pa − 1.

Put Pi = Pi (H ). Note that x1 acts on P1 /P2 as an automorphism of order pa permuting the elements {yi P2 }, whence x1 also acts on Pi /Pi+1 as an automorphism of order p a . As before we can construct a series {Pn,k } of normal subgroups of H , such that P1 = P1,e > P1,e−1 > · · · > Pn−1,1 > Pn = Pn,se (n) > Pn,se (n)−1 > · · · , the index of Pn,k in Pn,k+1 is p and rp (H /Pn,k ) = (e − 1) logp |H /Pn,k |(1 + o(1))

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Andrei Jaikin-Zapirain

as n goes to infinity. Moreover, without loss of generality we can assume that Pn,k are x1 -invariant. Our purpose is to calculate the number of x1 -invariant subgroups of p Mn,k = Pn,k /[Pn,k , H ]Pn,k . This will give an estimate of the normal subgroup growth of F . For this we need to understand the structure of Mn,k as Fp [x1 ]-module. Note pa that x1 acts trivially on Mn,k , whence we can consider Mn,k as Fp [Cpa ]-module. Claim 1. The Fp [Cpa ]-module Mn,k is isomorphic to a direct sum An,k ⊕ Dn,k , where An,k is a maximal free Fp [Cpa ]-submodule and Dn,k is some complement. Moreover, the Fp [Cpa ]-rank of An,k is (d − 1) logp |F /Pn,k |(1 + o(1)) as n goes to infinity. Proof. Let An,k be a maximal free Fp [Cpa ]-submodule of Mn,k . It is clear that there exists a Fp [Cpa ]-submodule Dn,k such that Mn,k = An,k ⊕ Dn,k . Now, the Fp [Cpa ]p rank of Mn,k is equal to d(Pn,k /[Pn,k , F ]Pn,k ) and so, by Theorem 1.6, it is less than (d − 1) logp |F /Pn,k | + 1. Hence if r1 is the Fp [Cpa ]-rank of An,k and r2 is the Fp [Cpa ]-rank of Dn,k , then r1 + r2 ≤ (d − 1) logp |F /Pn,k | + 1.

(4)

By the construction of Pn,k , the Fp -rank of Mn,k is equal to (e − 1) logp |F /Pn,k |(1 + o(1)) when n goes to infinity. On the other hand, it is less than or equal to pa r1 + (p a − 1)r2 . Hence we obtain that pa r1 + (p a − 1)r2 ≥ pa (d − 1) logp |F /Pn,k |(1 + o(1))

(5)

Comparing (4) and (5) we obtain r1 = (d − 1) logp |F /Pn,k |(1 + o(1)). Claim 2. Let M be a free Fp [Cpa ]-module of rank m. Then there are at least submodules N of M such that M/N is a free Fp [Cpa ]-module of rank s.

a p p s(m−s)

Proof. The proof goes in the same lines as the count of the number of subspaces of codimension r in an m-dimensional Fp -space. We have that there are p (p

a −1)m(m−s)

(pm − 1)(p m − p) . . . (p m − p m−s+1 )

(m − s)-tuples, generating free Fp [Cpa ]-submodules of rank (m − s). On the other hand, every free Fp [Cpa ]-module of rank (m − s) has p (p

a −1)(m−s)2

(pm−s − 1)(p m − p) . . . (p m−s − p m−s−1 )

generating (m − s)-tuples. Hence the number of Fp [Cpa ]-submodules of rank (m − s) is equal to p (p

a −1)(m−s)s

This proves the claim.

ps+1 − 1 p m − 1 p m−1 − 1 . . . . p m−s − 1 p m−s−1 − 1 p−1

The number of finite p-groups with bounded number of generators

217

Hence, using Claim 1 and Claim 2, we obtain that the group Mn,k has at least pp

a s((d−1) log p

|F /Pn,k |(1+o(1))−s)

a

x1 -invariant subgroups of index pp s . Let s be the integer part of t/2pa . Consider L = Pn,k such that the order of a F /L is p t−sp . By the previous observation, the number of x1 -invariant subgroups of a L/[L, H ]Lp of index psp is at least p sp

a ((d−1)(t−sp a )−s)+o(t 2 )

=p

β(d−1) 2 t +o(t 2 ) 4

.

Since every x1 -invariant normal subgroup of H is a normal subgroup of F , the normal β(d−1) 2 2 subgroup growth of F is at least p 4 t +o(t ) . This implies that there exists T , such α(d−1) 2 that f (pt , d) ≥ p 4 t for any t ≥ T . Since α is an arbitrary number less than 1, (d−1) 2 we obtain that f (p t , d) ≥ p 4 t (1+o(1)) . Acknowledgments. I thank A. Mann for helpful conversations and D. Segal for reading a previous version of this work. A part of this work was done while I was visiting the Hebrew University of Jerusalem. I would like to take this opportunity to thank the Institute of Mathematics and personally A. Lubotzky for his hospitality.

References [1] J. D. Dixon, M. P. F. du Sautoy, A. Mann and D. Segal, Analytic pro-p groups, 2nd edition, Cambridge Stud. Adv. Math. 61, Cambridge University Press, Cambridge 1999. [2] B. Huppert and N. Blackburn, Finite groups II, Springer-Verlag, Berlin, Heidelberg 1982. [3] G. Higman, Enumerating p-groups I. Inequalities., Proc. London Math. Soc. 10 (1960), 24–30. [4] A. McIver and P. M. Neumann, Enumerating finite groups, Quart. J. Math. Oxford Ser. (2) 38 (1987), 473–488. [5] A. Mann, Enumerating finite p-groups and their defining relations, J. Group Theory 1 (1998), 59–64. [6] A. Mann, Some questions about p-groups, J. Austral. Math. Soc. 67 (1999), 356–379. [7] L. Ribes and P. Zalesskii, Profinite groups, Springer-Verlag, Berlin, Heidelberg 2000. [8] C. C. Sims, Enumerating p-groups, Proc. London Math. Soc. 15 (1965), 151–166. Andrei Jaikin-Zapirain, Departamento de Matemáticas, Facultad de Ciencias, Universidad Autónoma de Madrid, Cantoblanco, Ciudad Universitaria, 28049 Madrid, Spain E-mail: [email protected]

Enumerating highly non-soluble groups Benjamin Klopsch

Abstract. Let G be a finite group and 1 = G0 ≤ G1 ≤ · · · ≤ Gr = G any composition series of G. The Jordan–Hölder theorem shows that, if I denotes the set of all indices i such that Gi /Gi−1 is Abelian, then α(G) := i∈I |Gi /Gi−1 | is an invariant of G. Let A ∈ N. For n ∈ N let gA (n) denote the number of (isomorphism classes of) groups G such that α(G) ≤ A and |G| ≤ n. We determine the asymptotic behaviour of gA (n) as n → ∞; this provides a satisfactory answer to a question raised by Thompson [13]. 2000 Mathematics Subject Classification: 20D60, 20F69.

1. Introduction For every n ∈ N let g(n) denote the number of (isomorphism classes of) groups of 2 2 order at most n. It is known that g(n) = n( 27 +o(1))(log2 n) as n → ∞; see Higman [5] and Pyber [12]. There are also estimates for the number of groups with some given restrictions; e.g. see Holt [7] or Mann [10]. Here we are interested in the enumeration of groups which are far from soluble in the following precise sense. Let G be a finite group and 1 = G0 ≤ G1 ≤ · · · ≤ Gr = G any composition series of G. The Jordan– Hölder theorem shows that, if I denotes the set of all indices i such that Gi /Gi−1 is Abelian, then α(G) := i∈I |Gi /Gi−1 | is an invariant of G. For every A ∈ N and n ∈ N let gA (n) denote the number of (isomorphism classes of) groups G such that α(G) ≤ A and |G| ≤ n. A result of Camina, Everest, and Gagen [1], originally conjectured by Thompson [13], states that for every A ∈ N we have gA (n) = o(g(n)) as n → ∞. This can be interpreted as a measure of rarity (“most finite groups consist largely of solvable chunks”) and agrees well with our intuition. Indeed, Camina et al. get away with some fairly crude estimates. The purpose of this note is to pin down the precise growth rate of gA (n) as n → ∞, turning “rare” into “really rare”. The special case A = 1 corresponds to enumerating groups without any Abelian composition factors and was successfully completed in [8]; here we prove more generally Theorem A. For every A ∈ N there exist b, c ∈ R>0 such that for all n ∈ N≥60 , nb log2 log2 n ≤ gA (n) ≤ nc log2 log2 n .

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Of course, the lower bound is a direct consequence of the special case A = 1 and requires no further comment. To obtain the upper bound we follow closely the strategy laid out in [8]. A new feature, however, is the appearance of non-trivial second cohomology groups; in this context we employ ideas of Holt [7]. As in [8] the classification of finite simple groups (CFSG) is used in a mild sense. Notation. For every x ∈ R>0 we write log x := log2 x. In addition we define a modified logarithm log∗ : R≥0 → R by setting log∗ x := 0 for 0 ≤ x ≤ 1 and log∗ x := log x otherwise. Let G be a finite group. We denote by com(G) the composition length of G and we write comAb (G), respectively comnA (G), for the number of Abelian, respectively non-Abelian, composition factors of G counted with repetitions. The socle of G is denoted by soc(G) and we write d(G) for the minimal number of generators for G.

2. Generators and permutation actions In this section we derive estimates for the minimal number of generators and the number of (equivalence classes of) permutation actions of a finite group. Proposition 2.1. Let G be a group acting faithfully on a finite set . Then we have com(G) ≤ || and comnA (G) ≤ ||/5. Proof. Put n := ||. First suppose that G acts intransitively on . Then splits into G-orbits 1 , 2 , . . . , r where r ≥ 2 and ri=1 |i | = n. For each i ∈ {1, 2, . . . , r} let Gi ≤ Sym(i ) denote the permutation group induced by G on i . So induction yields com(G) ≤ Then r G is a subdirect r product of G1 , G2 , . . . , Gr . nA com(G ) ≤ | | = n and similarly com (G) ≤ n/5. i i i=1 i=1 Next suppose that G acts transitively, but imprimitively on . Choose a non-trivial, proper G-congruence ≈ on and denote by K the kernel of the induced action of G on /≈. Write d := |/≈| and note that K acts faithfully on each ≈-class. So induction gives com(G) ≤ com(G/K) + com(K) ≤ d + n/d ≤ n and similarly comnA (G) ≤ n/5. Finally suppose that G acts primitively on . Then M := soc(G) is isomorphic to T k , the k-th direct power of some simple group T . Put t := |T |. By the O’Nan–Scott Theorem (as stated in [9]) we have to consider the following cases. I: Affine type. In this case T ∼ = Cp is Abelian, n = p k , and G is a subgroup of the affine linear group AGL(k, p); the group M consists of all translations. First we prove that com(G) ≤ p k . Let ν(k, p) denote the number of prime factors of |GL(k, p)| counted with multiplicities. Certainly we have com(G) = com(G/M)+ com(M) ≤ ν(k, p) + k. If (k, p) ∈ {(2, 2), (3, 2), (4, 2)}, then ν(k, p) is equal to 2,5,10 respectively, and hence ν(k, p) + k ≤ pk . Otherwise the estimate ν(k, p) ≤ log|GL(k, p)| ≤ k 2 log p suffices to obtain ν(k, p) + k ≤ pk .

Enumerating highly non-soluble groups

221

Now we show that comnA (G) ≤ pk /5. If k = 1 then G is soluble and nothing remains to be proved. Otherwise we simply dispose of the obvious Abelian composition factors and obtain comnA (G) ≤ log60 |SL(k, p)| ≤ (k 2 − 1) log60 p ≤ pk /5. II: Almost simple type. In this case T is non-Abelian, k = 1, and G is an extension of T by some subgroup of Out(T ). Since T embeds into Sym(n), the inequalities n ≥ 5 and t ≤ nn hold. It is a consequence of CFSG that Out(T ) is soluble and |Out(T )| ≤ log t; see [2]. This gives com(G) ≤ 1 + log log t ≤ 1 + log n + log log n ≤ n and comnA (G) = 1 ≤ n/5. III. In this case T is non-Abelian and k ≥ 2. There are three subcases. III.1: Simple diagonal action. Here n = t k−1 and G is an extension of M = T k by some subgroup of Out(T ) × Sym(k). By CFSG, Out(T ) is soluble and |Out(T )| ≤ log t. Since t ≥ 60, induction yields com(G) = com(M) + com(G/M) ≤ 2k + log log t ≤ t k−1 and comnA (G) ≤ 6k/5 ≤ t k−1 /5. III.2: Product action. Here n = d l where d ≥ 5 and l ≥ 2. The group G is a subgroup of Sym(d) Sym(l), hence an extension of a permutation group of degree ld by a permutation group of degree l. By induction we find com(G) ≤ l(d + 1) ≤ d l and comnA (G) ≤ l(d + 1)/5 ≤ d l /5. III.3: Twisted wreath action. Here n = t k and G is a twisted wreath product of T and a permutation group of degree k. Since t ≥ 60, induction yields com(G) ≤ 2k ≤ t k and comnA (G) ≤ 6k/5 ≤ t k /5. The bounds in Proposition 2.1 are not unreasonable; this can be seen from Example 2.2. Let k ∈ N. Then Sylow 2-subgroups of Sym(2k ) are k-fold iterated wreath product of groups of order two and thus have composition length 2k − 1. Corollary 2.3. Let G be a finite group whose soluble radical is trivial. Then we have com(soc(G)) ≥ 56 comnA (G). Proof. The socle M := soc(G) is isomorphic to a product ri=1 Siki of non-Abelian r simple groups Si with multiplicities ki . Note that k := i=1 ki = com(M) and r Aut(S ) Sym(k ). Since M has trivial centre, the group Aut(M) ∼ = i i i=1 G acts faithfully by conjugation on M and we obtain a homomorphism r ϕ : G → Aut(M) → i=1 Sym(ki ) → Sym(k). The kernel K of ϕ contains M, and K/M → ri=1 Out(Si )ki is soluble by the validity of Schreier’s Conjecture. Proposition 2.1 yields comnA (G) − com(M) = comnA (G/K) ≤ k/5 and the claim follows. Proposition 2.4. Every finite group G satisfies d(G) ≤ comAb (G) +

13 5

log∗ comnA (G) + ε(G)

where ε(G) := 0 if G is soluble and ε(G) := 2 otherwise.

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Proof. Let G be a finite group and put C := 2/ log(12/7) ≤ 13/5. We prove the slightly stronger assertion d(G) ≤ comAb (G) + C log∗ comnA (G) + ε(G). First suppose that G possesses a non-trivial Abelian normal subgroup A. Then d(G) ≤ d(A) + d(G/A) ≤ comAb (A) + d(G/A) and induction yields the desired bound. Now suppose that soc(G) has trivial centre. Then an easy modification of [8, Proof of Proposition 3.1], calling upon Corollary 2.3 in place of [8, Lemma 2.2], proves our claim. Lemma 2.5. Let G be a finite group and k ∈ N0 . Then #{N G | com

nA

(N) = k} ≤ 2

(log α(G))2 comAb (G)

comnA (G) . k

Proof. If G contains a non-Abelian minimal normal subgroup, we can argue as in [8, Proof of Lemma 4.1]. Now suppose that M := soc(G) is Abelian. Put α := α(G), c1 := comAb (G), c2 := comnA (G), and N := {N G | comnA (N ) = k}. Note that M has at most α c1 = 2(log α)c1 different subgroups and that every non-trivial N ∈ N intersects M non-trivially. Induction shows that for every non-trivial subgroup 2 M0 ≤ M the number of N ∈ N with N ∩ M = M0 is bounded by 2(log α−1) c1 ck2 . 2 Thus |N | ≤ 2(log α) c1 ck2 , as claimed. Proposition 2.6. Let n ∈ N. Then every finite group G admits at most 2(log α(G))

2 com Ab (G) log n

· 27n(com

Ab (G)+3 log∗ com nA (G)+log n)

transitive permutation representations of degree n (up to equivalence). Proof. We indicate how the proof of [8, Proposition 5.1] can be modified to yield the present, more general claim. Let G be a finite group. By Proposition 2.1 and Lemma 2.5 there are at most

n/5 nA

2 Ab ∗ nA (log α(G))2 comAb (G) com (G) 2 ≤ 2(log α(G)) com (G) · 2n log com (G) k k=0

normal subgroups K G such that G/K → Sym(n). This effectively reduces the problem to counting faithful transitive actions. Indeed, it suffices to verify: Every finite group G admits (up to equivalence) at most 2F (G,n) faithful transitive permutation representations of degree n ≥ 2, where F (G, n) := (log α(G))2 comAb (G)(log n − 1) + 7n(comAb (G) +

20 7

log∗ comnA (G) + log n).

Let G be a finite group. The special cases com(G) ≤ 1 and n ≤ 7 are easily dealt with, because d(G) ≤ 7 implies that there are no more than n7n homomorphisms from G to Sym(n). Now suppose that com(G) ≥ 2 and n ≥ 8. Choose a minimal

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normal subgroup N of G and put H := G/N . By Proposition 2.4 we have d(H ) ≤ ∗ nA comAb (H ) + 13 5 log com (H ) + 2. Let Tintr (G, n), respectively Ttr (G, n), denote the number of (equivalence classes of) faithful transitive permutation representations G → Sym(n) with N acting intransitively, respectively transitively. We claim that both, Tintr (G, n) and Ttr (G, n) are bounded by 21 · 2F (G,n) ; this will finish the overall proof. Arguing by induction, similarly as in [8, Proof of Proposition 5.1]1 , we obtain Tintr (G, n) ≤ n · max{2f (n,r) | 2 ≤ r ≤ n/2} where ∗ nA 13 5 log com (H ) + 2) + (log α(N ))2 comAb (N) log nr + 7 nr (comAb (N ) + 3 log∗ comnA (N ) + log nr ) 2 Ab Ab ∗ nA

f (n, r) = r log nr (comAb (H ) + + (log α(H )) com

(H ) log r + 7r(com

(H ) + 3 log com

(H ) + log r).

(Here r corresponds to the number of N -orbits; for details see [8, Proof of Proposition 5.1].) Rearranging terms yields Tintr (G, n) ≤ 2f (n) where f (n) = (log α(G))2 comAb (G)(log n − 1) + max log n + 7( nr log nr + r log r) + 2r log nr 2≤r≤n/2

+ (comAb (G) + 3 log∗ comnA (G)) (7( nr + r) + r log nr ) .

It is a matter of routine to verify that for 2 ≤ r ≤ n/2 the inequalities log n + 7( nr log nr + r log r) + 2r log nr ≤ 7n log n − 1, 3 · (7( nr + r) + r log nr ) ≤

20 7

· 7n

hold. So we obtain Tintr (G, n) ≤ 21 · 2F (G,n) as wanted. It remains to bound Ttr (G, n), i.e. the number of faithful transitive representations G → Sym(n) with N acting transitively. If N is Abelian, then N acts regularly and its action is known. If N is non-Abelian, then at least it is characteristically simple, and the same trick as in [8, Proof of Proposition 5.1] shows that there are 20 ∗ nA at most 26n( 7 log com (N)+log n) possibilities for a faithful transitive representation N → Sym(n). Once the action of N is fixed, any extension to an action of G is determined by the images of a single reference point under a set of generators of G modulo N . So by Proposition 2.4 we find that Ttr (G, n) ≤ 2f (n) where ∗ nA Ab ∗ nA 13 f (n) = 6n( 20 7 log com (N)+log n)+(log n)(com (H )+ 5 log com (H )+2).

This yields Ttr (G, n) ≤

1 2

· 2F (G,n) , as wanted.

1 Incidentally, there is a small mistake in [8, Proof of Proposition 5.1]: the induced action of N on a single N -orbit is wrongly taken to be faithful. But this only affects the various constants which appear in the estimates.

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Corollary 2.7. Let n ∈ N. Then every finite group G admits at most 2n(2+(log α(G))

3 )/2

· 27n(log α(G)+3 log

∗ com nA (G)+log n)

permutation representations of degree n up to equivalence. Proof. Let G be a finite group. Then a choice of orbits for a representation G → Sym(n) corresponds to an additive partition of n and there are less than 2n−1 such partitions. Now the claim follows from Proposition 2.6.

3. Cohomology groups and proof of Theorem A In this section we establish an expedient bound for the size of second cohomology groups and prove our main theorem. Proposition 3.1. Let G be a finite group and M a finite G-module. Then |H 2 (G, M)| ≤ |G|(2+com

Ab (G)) log |M|

.

Proof. We follow closely the beginning of [7, Proof of Proposition 3.1]. For any submodule L ≤ M we have |H 2 (G, M)| ≤ |H 2 (G, L)||H 2 (G, M/L)|, so we may assume that M is an irreducible G-module. If G acts faithfully on M, then |H 2 (G, M)| ≤ |G|2 log |M| by [6, Theorem 1]. So now suppose that CG (M) = 1 and argue by induction on com(G). If G is simple, then the universal coefficient theorem yields H 2 (G, M) ∼ = Hom(M(G), M), where M(G) denotes the Schur multiplier of G, and a list of multipliers of the finite simple groups, such as in [4] or [2], gives |H 2 (G, M)| ≤ |G|log |M| . Now suppose that com(G) ≥ 2. Choose a minimal normal subgroup N of G which lies inside CG (M). We use the inequality |H 2 (G, M)| ≤ |H 2 (N, M)G/N | · |H 1 (G/N, H 1 (N, M))| · |H 2 (G/N, M N )|. (3.1) This can be regarded as a consequence of the famous Hochschild–Serre spectral sep,q quence E2 := H p (G/N, H q (N, M)) ⇒ E p+q = H p+q (G, M). Indeed, the p,2−p 0,2 1,1 2,0 , E∞ , E∞ and each E∞ is limit term E 2 = H 2 (G, M) is composed of E∞ p,2−p a subquotient of the initial term E2 ; see [11, Chapter II] for a comprehensible account. If N is non-Abelian, then |H 1 (N, M)| = |Hom(N, M)| = 1, and hence (3.1) and induction yield |H 2 (G, M)| ≤ |N|2 log |M| · 1 · |G/N|(2+com

Ab (G/N )) log |M|

≤ |G|(2+com

Ab (G)) log |M|

.

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Now suppose that N is Abelian. Then |H 1 (N, M)| = |Hom(N, M)| ≤ |M|com(N ) , and so (3.1), induction, and the trivial estimate d(G/N ) ≤ log |G/N| give |H 2 (G, M)| ≤ |N|(2+com(N)) log |M| · |M|com(N )d(G/N ) · |G/N|(2+com ≤ |G/N |(2+com ≤ |G|(2+com

Ab (G/N)+com(N)) log |M|

Ab (G)) log |M|

Ab (G/N )) log |M|

· |N|(2+com(N )) log |M|

.

We are now ready to prove the upper bound in Theorem A; recall that the lower bound was already established in [8]. Let A, n ∈ N. Then we claim that gA (n) ≤ n(log A)

3 /2+8 log A+5

· nmax{1+3 log A,32} log log n .

(3.2)

Observe that every group G with α(G) ≤ A and |G| ≤ n satisfies comnA (G) ≤ log60 n ≤ 15 log n, d(G) ≤ log A + 3 log∗ log n (by Proposition 2.4).

(3.3)

Next we introduce some ad-hoc terminology. A (composition) type is just a tuple of non-trivial finite groups each of which is either a direct product of non-Abelian simple groups or elementary Abelian. Two types are said to be isomorphic if they have the same length and corresponding components are isomorphic. Let M = (M1 , . . . , Mr ) be a type. We define α(M) := ri=1 α(Mi ) and |M| := ri=1 |Mi |. A group G realizes M, if it possesses a normal series 1 = G0 ≤ G1 ≤ · · · ≤ Gr = G such that for all i ∈ {1, 2, . . . , r} we have Gi /Gi−1 ∼ = Mi and Gi /Gi−1 = soc(G/Gi−1 ) if Mi is non-Abelian, Gi /Gi−1 is a minimal normal subgroup of G/Gi−1 if Mi is Abelian. Note that every finite group realizes at least one type. Returning to the proof of (3.2), we divide the argument into two steps. Step 1. The number of (isomorphism classes of) types M with |M| ≤ n is at most n4+log log n . Step 2. Any particular type M with α(M) ≤ A and |M| ≤ n is realized by at most nF (A)+a log log n groups where F (A) := (log A)3 /2 + 8 log A + 1 and a := max{3 log A, 31}. Step 1. There are n choices for the destined order m ≤ n of our type M. The number m has, counted with repetitions, at most log m prime factors and these can be rearranged and grouped in at most (log m)! · 2log m ≤ n1+log log n ways to obtain an ordered factorisation m = m1 · · · · · mr of m. How many types M = (M1 , . . . , Mr ) are there with |Mi | = mi for all indices i? Let i ∈ {1, 2, . . . , r}. If mi is a prime power, then Mi has to be elementary Abelian of order mi . Otherwise, according to [12, Proof of Lemma 2.3] the number of groups of order mi which are direct products of non-Abelian simple groups is less than or equal to m2i . So altogether there are at most m2 ≤ n2 choices for the components Mi of M.

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Benjamin Klopsch

Step 2. Fix a type M = (M1 , . . . , Mr ) with α(M) ≤ A and |M| ≤ n. If r ≤ 1, only one group realizes M. So let us assume that r ≥ 2, and write m := |M1 |. Suppose that G is a group realizing M. Then G has a normal subgroup M ∼ = M1 such that H := G/M realizes the type (M2 , . . . , Mr ). We can view G as an extension of M by H , and conjugation in G induces a coupling homomorphism χ : H → Out(M). The extension 1 → M → G → H → 1 is determined up to equivalence by the prescribed isomorphism class of M, i.e. M ∼ = M1 , and (1) the isomorphism class of H , (2) the coupling homomorphism χ : H → Out(M), (3) a suitable element of H 2 (G, Z(M)) where the centre Z(M) of M is regarded as a G-module via χ. Of course, the same data then determine the group G up to isomorphism. So we can bound the number of groups realizing M by estimating the number of possible choices in (1), (2), and (3). Note that (1) can be dealt with by induction. First suppose that M is a direct product of non-Abelian simple groups. Then Z(M) is trivial and so (3) becomes irrelevant. We estimate the number of coupling homomorphisms similarly as in [8, Section 6], with Corollary 2.7 and (3.3) replacing [8, Proposition 1.2 and (6.1)], and see that the number of groups realizing M is at most (n/m)F (A)+a log log(n/m) · 2(log m)(2+(log A)

3 )/2

· 27(log m)(log A+3 log log(n/m)+log log m) · mlog A+3 log log(n/m) 3 /2+8 log A+1+31 log log n

≤ (n/m)F (A)+a log log n · m(log A) ≤ nF (A)+a log log n .

Now suppose that M is elementary Abelian. Then |Out(M)| = |Aut(M)| ≤ mlog m and the number of choices in (2) can be estimated by (3.3). The size of the cohomology group in (3) is restricted by Proposition 3.1. Thus the number of groups realizing M is at most log(A/m)+3 log log(n/m) (n/m)F (A/m)+a log log(n/m) · mlog m · (n/m)(2+log(A/m)) log m = (n/m)F (A/m)+(2+log(A/m)) log m+a log log n · mlog(A/m) log m+3 log m log log n . It is a routine matter to verify that F (A/m) + (2 + log(A/m)) log m ≤ F (A); so the claim follows and the proof of (3.2) is complete. Acknowledgements. I thank Larry and Lorien Wilson for their hospitality during the conference week; it was at their home that I convinced myself of some of the allegedly routine matters.

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227

References [1]

A. R. Camina, G.R. Everest, and T. M. Gagen, Enumerating non-soluble groups – a conjecture of John G. Thompson, Bull. Lond. Math. Soc. 18 (1986), 265–268.

[2]

J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson, Atlas of finite groups, Claredon Press, Oxford 1985.

[3]

W. Gaschütz, Zu einem von B. H. und H. Neumann gestellten Problem, Math. Nachr. 14 (1956), 249–252.

[4]

R. L. Griess Jr., Schur multipliers of the known finite simple groups. II, in The Santa Cruz Conference on Finite groups, Santa Cruz, 1979, Proc. Symp. Pure Math. 37, Amer. Math. Soc., Providence 1980, 279–282.

[5]

G. Higman, Enumerating p-groups. I: Inequalities, Proc. Lond. Math. Soc. 10 (1960), 24–30.

[6]

D. F. Holt, On the second cohomology group of a finite group, Proc. Lond. Math. Soc. 55 (1987), 22–36.

[7]

D. F. Holt, Enumerating perfect groups, J. Lond. Math. Soc. 39 (1989), 67–78.

[8]

B. Klopsch, Enumerating finite groups without abelian composition factors, Israel J. Math. 137 (2003), 265–284.

[9]

M. W. Liebeck, C. E. Praeger, and J. Saxl, On the O’Nan-Scott Theorem for finite primitive permutation groups, J. Austral. Math. Soc. (Ser. A) 44 (1988), 389–396.

[10] A. Mann, Enumerating finite groups and their defining relations, J. Group Theory 1 (1998), 59–64. [11] J. Neukirch, A. Schmidt, and K. Wingberg, Cohomology of number fields, Grundlehren Math. Wiss. 323, Springer-Verlag, Berlin 2000. [12] L. Pyber, Enumerating finite groups of given order, Ann. Math. 137 (1993), 203–220. [13] J. G. Thompson, Finite Non-Solvable Groups, in Group Theory: essays for Philip Hall (eds. K. W. Gruenberg and J. E. Roseblade), Academic Press, London 1984, 1–12. Benjamin Klopsch, Mathematisches Institut, Heinrich-Heine-Universität, 40225 Düsseldorf, Germany E-mail: [email protected]

3-Signalizers in almost simple groups Inna Korchagina

The term q-signalizer (q being a prime) was originally introduced by John Thompson to denote a q -subgroup of a group G normalized by a Sylow q-subgroup of G. With time the term came to mean a q -subgroup of G normalized by a “large” q-subgroup of G, where “large” has various meanings (e.g., of q-rank at least 3, or containing the full centralizer of some q-element, etc.). Control of the size and embedding of such signalizers has been a central theme of the Classification of Finite Simple Groups. The goal of the q-signalizer analysis is to obtain information about the q-local structure of G. In this work we study certain 3-signalizers of G. Definition 0.1. Let G be a finite group and t ∈ Aut(G) of order 3. Take p to be a prime distinct from 3. Let X be a p-subgroup of G such that (1) [X, t] = X; and (2) CG (t) ≤ NG (X). Then X is called a CG (t)-signalizer. In this case we say that the triple (G, t, X) is of type (H ). As a result of study of such 3-signalizers in finite simple K-groups we obtain the following result. Theorem 0.2. Let G be a non-sporadic finite simple K-group. Suppose that G admits a nontrivial automorphism t of order 3 and X is a CG (t)-signalizer. Then either X = 1, or one of the following conclusions holds: (1) G ∈ Lie(r), r = 3 and there exists a parabolic subgroup P of G, such that X is contained in the unipotent radical of P and CG (t) is contained in its Levi complement; (2) G ∼ = PSp (r) with r ∈ {5, 7} and l ≥ 2, 2l

∼ Z3 × Sp ∼ t ∈ Inn(G) with CG (t) = 2l−2 (r) and X ≤ G with X = Q8 ; (3) G ∼ = L2 (r) with r ∈ {5, 7}, t ∈ Inn(G) and X ≤ G with X ∼ = E4 ; ∼ ∼ ∼ (4) G = A7 , t ∈ Inn(G) with CG (t) = E9 and X ≤ G with X = E4 ; or

(5) G ∼ = A8 , t ∈ Inn(G) with CG (t) ∼ = Z3 Z2 and X ≤ G with X ∼ = E24 . We anticipate, that this result may be of use in the revised proof of the Classification Theorem of Finite Simple Groups for dealing with the situation when G is a group of

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even type and e(G) = m2,3 (G) = 3 (a difficult subcase of the small even case). For that purpose we also formulate and prove the following theorem. Theorem 0.3. Let G be a finite sporadic simple K-group with e(G) ≤ 3. Suppose that G admits a nontrivial automorphism t of order 3 and X is a CG (t)-signalizer. Then either X = 1, or one of the following conclusions holds: (1) G ∼ = Z3 × A4 and X ∼ = E4 ; = M22 , t ∈ 3A with CG (t) ∼ (2) G ∼ = Z3 × A5 and X ≤ E16 ; = M23 , t ∈ 3A with CG (t) ∼ ∼ (3) G = M24

(a) t ∈ 3A with C(3A) ∼ = 3A6 and X ∼ = 26 ; (b) t ∈ 3B with C(3B) ∼ = 26 ; = Z3 × L3 (2) and X ∼ (4) G ∼ = Z3 × A4 and = J2 , t ∈ 3B with C(3B) ∼ 2+4 ∼ 2 .(Z3 × S3 ); XCG (t) ≤ N (J (S2 )) =

(5) G ∼ = 6M22 and XCG (t) ≤ Q68 (3M22 .2); = J4 , t ∈ 3A with C(3A) ∼ (6) G ∼ = E4 . = H e, t ∈ 3B with C(3B) ∼ = Z3 × L3 (2) and X ∼

We shall subdivide the proof of both theorems into a series of lemmas and propositions in Sections 2, 3 and 4. We begin with some important background results.

1. Preliminary lemmas and their corollaries Throughout this section r denotes a prime number distinct from 3. Let F¯ = F¯ r be an algebraic closure of the field Fr of r elements. We shall suppose that the algebraic ¯ groups we are dealing with are F-algebraic groups. We begin by proving the following result about the automorphisms of algebraic groups. ¯ be a simple algebraic group with fundamental root system Proposition 1.1. Let G = {α1 , . . . , αl } and corresponding Dynkin diagram G¯ . Suppose that t¯ is a ¯ of order 3. Denote C¯ = C ¯ (t¯). Then one of the nontrivial inner automorphism of G G following conditions holds: (E) t¯ is of equal-rank type (see definition in [GLS3]) and S¯ = C¯ 0 is a semisimple algebraic group, whose Dynkin diagram 0 can be obtained by erasing a node ¯ from the extended Dynkin diagram of G. (P) t¯ is of parabolic type (see definition in [GLS3]), S¯ = [C¯ 0 , C¯ 0 ] is a semisimple algebraic group and the Dynkin diagram 0 of S¯ can be obtained in one of the following ways: ¯ is one of the following groups: Bl , Cl , G2 , F4 , E7 , E8 , then we have (a) If G to erase precisely one node from G¯ ; ¯ is one of the following groups: Al , Dl , E6 , then we have to erase at (b) If G most two nodes from G¯ .

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Proof. By Theorem 4.1.9 [GLS3], t¯ is either of equal-rank type or of parabolic type. If t¯ is of equal-rank type, the result follows immediately from Definition 4.1.8 [GLS3]. Thus suppose that t¯ is of parabolic type. By Theorem 4.1.8 [GLS3], S¯ = [C¯ 0 , C¯ 0 ] is a semisimple algebraic group. Let 0 be its fundamental root system and 0 corresponding Dynkin diagram. Theorem 4.1.9 [GLS3] together with Definition 4.1.8 [GLS3] implies that we may suppose that 0 is a subset of . ¯ actually are, let us consider In order to determine what 0 , 0 (and therefore S) ¯ Since t¯ is a semisimple the action of t¯ on LG¯ , where LG¯ is the Lie algebra of G. ¯ So for every element of order 3, it must be an element of some maximal torus T¯ of G. α ∈ and the corresponding element eα of LG¯ , the following condition holds: t¯.eα = ωf (α) eα with f (α) ∈ {0, ±1},

(1)

where ω ∈ F¯ is the cube root of unity. Now that most of the notation has been established, let us see what actually happens. ¯ is one of the following groups: Bl , Cl , G2 , F4 , E7 or Lemma 1.2. Suppose that G E8 . Denote l = ||. Then |0 | = l − 1. Proof. Since as we remarked earlier 0 ⊆ , we must show that | − 0 | = 1. Assume the contrary. Since t¯ is of parabolic type, this means that | − 0 | ≥ 2. Then there exist two roots β1 , β2 such that βi ∈ − 0 for i = 1, 2 and without loss of generality β1 , β2 can be chosen in the following way : 1. β1 lies to the left of β2 on G¯ , where we use Dynkin diagrams as shown on p. 12 in [GLS3]; 2. If is the smallest connected subdiagram of G¯ containing both β1 and β2 , then all other nodes of are elements of 0 ; and 3. All the nodes to the right of β2 correspond to the elements of 0 . ˜ = {α˜ 1 , . . . , α˜ n } of , we may suppose Therefore for the fundamental system that α˜ 1 = β1 , α˜ n = β2 and α˜ i ∈ 0 for 1 < i < n. Since βi ∈ − 0 , we have that t¯.eβi = eβi for i = 1, 2 and t¯.eα = eα for every α ∈ 0 . So in terms of new notation we obtain t¯.eα˜ i = eα˜ i if i ∈ {1, n}, and t¯.eα˜ i = eα˜ i otherwise. Using formula (1), we discover that up to the choice of ω one of the following conditions holds: 1. t¯.eα˜ 1 = ωeα˜ 1 and t¯.eα˜ n = ω−1 eα˜ n ; or 2. t¯.eα˜ 1 = ωeα˜ 1 and t¯.eα˜ n = ωeα˜ n . The goal now is to reach a contradiction by producing a root vector which is fixed by t¯ and yet is linearly independent of 0 . Let us see how we can achieve it in each of the situations described above. Case 1. Since is a connected, not necessarily proper subdiagram of G¯ , and because of the minimal choice of , it can only be of one of the following types: An , ˜ e ˜ as listed in Table 1. Then t¯.eα˜ = ωf (α) Bn , Cn , G2 or F4 . Let α˜ be the root of () α˜

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with f (α) ˜ = ω · ω−1 = 1 and so t¯ fixes eα˜ . Since α˜ is linearly independent of 0 , eα˜ is not supposed to be fixed by t¯. This is a contradiction which proves the result. Table 1

Type of An Bn Cn G2 F4

α˜ α˜ 1 + α˜ 2 + · · · + α˜ n α˜ 1 + α˜ 2 + · · · + α˜ n α˜ 1 + 2α˜ 2 + · · · + 2α˜ n−1 + α˜ n α˜ 1 + α˜ 2 α˜ 1 + α˜ 2 + α˜ 3 + α˜ 4

Case 2. This time the situation becomes a bit more difficult. Because of the ¯ 1 , which is a subgroup of G ¯ defined by the choice of , we may have to consider G fundamental root system 1 , where 1 is a subset of which contains all the roots beginning with β1 through all the ones that lie to the right of β1 on G¯ . Denote ¯ 1. k = |1 |. Then n ≤ k ≤ l. Let 1 denote the corresponding Dynkin diagram of G ¯ is one of the following groups: Bn , Cn , G2 or F4 , let us immediately proceed If G ¯ is either E7 , or E8 , is of type An and from the complete root lists with Table 2. If G given in Tables 3 and 4 [AS], we may see that for every αi ∈ and every αj ∈ ¯ j with c + c¯ = 3. with i = j , there exists α˜ ∈ () such that α˜ = cαi + · · · + cα ¯ ¯ In both situations above, apply t to eα˜ . This time we obtain: t .eα˜ = ω·ω−1 eα˜ = eα˜ . Therefore t¯ fixes eα˜ , but α˜ is linearly independent of 0 , which is a contradiction. The result now follows. Table 2

Type of An if 1 ∼ Bk An if 1 ∼ Ck A2 if 1 ∼ F4 Bn Cn G2 F4

α˜ α˜ 1 + 2α˜ 2 + · · · + 2α˜ n + 2αk−n+1 + · · · + 2αk α˜ 1 + 2α˜ 2 + · · · + 2α˜ n + 2αk−n+1 + · · · + 2αk−1 + αk α1 + 2α2 + 2α3 + α4 α˜ 1 + 2α˜ 2 + · · · + 2α˜ n 2α˜ 1 + 2α˜ 2 + · · · + 2α˜ n−1 + α˜ n α˜ 1 + 2α˜ 2 α˜ 1 + 2α˜ 2 + 2α˜ 3 + 2α˜ 4

¯ = Al , Dl or E6 . Denote l = ||. Then |0 | ≥ l − 2. Lemma 1.3. Suppose that G Proof. Assume the contrary. Then there exists {β1 , β2 , β3 } ⊆ − 0 . Without loss of generality, we can make the following assumptions: βi = αji with αji ∈ , i = 1, 2, 3, and j1 < j2 < j3 under the “usual” enumeration of coming from G¯

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(see p. 12 [GLS3]). Since βi ∈ − 0 for i = 1, 2, 3, t¯.eβi = eβi for i = 1, 2, 3, while t¯.eα = eα for every α ∈ 0 . Using formula (1), we obtain that up to the choice of ω, one of the following conditions holds: 1. t¯.eβi = ωeβi for i = 1, 2, 3; or 2. t¯.eβi = ωεi eβi with εi ∈ {±1} and i εi = 1, i = 1, 2, 3. Let us see what happens in each of those situations: Case 1. Let be the smallest connected subdiagram of G¯ containing {β1 , β2 , β3 }. Let {α˜ 1 , . . . , α˜ n } be the fundamental system of . Since is a connected, not necessarily proper subdiagram of G¯ , it must be of one of the following types: An , Dn ˜ as listed in Table 3. Consider the action of t¯ on or E6 . Let α˜ be the root of (), f ( α) ˜ ¯ eα˜ with f (α) ˜ = ω · ω · ω = 1. Thus t¯ fixes eα˜ . Since α˜ is linearly eα˜ : t .eα˜ = ω independent of 0 , eα˜ should not be fixed by t¯. This is a contradiction, which proves the result. Table 3

Type of An Dn E6

α˜ α˜ 1 + α˜ 2 + · · · + α˜ n α˜ 1 + 2α˜ 2 + · · · + 2α˜ n−2 + α˜ n−1 + α˜ n α˜ 1 + α˜ 2 + α˜ 3 + α˜ 4 + α˜ 5 + α˜ 6

Case 2. Obviously there exist {i, j } ⊆ {1, 2, 3} such that εi = 1 and εj = −1. Choose i, j and ij so that εi + εj = 0, and ij is the smallest connected subdiagram of G¯ containing βi and βj , and βk ∈ ij for k ∈ {1, 2, 3} − {i, j }. Clearly ij is of the type An for some n. Let {α˜ 1 , . . . , α˜ n } be a fundamental root system of ij . Without loss of generality, we may suppose that α˜ 1 = βi and α˜ n = βj and α˜ s ∈ 0 ˜ α˜ = α˜ 1 + α˜ 2 + · · · + α˜ n . for s ∈ {1, n}. Take α˜ to be the following root of (): f ( α) ˜ −1 Then t¯.eα˜ = ω eα˜ with f (α) ˜ = ω · ω = 1. Therefore t¯ fixes eα˜ . Since α˜ is linearly independent of 0 , eα˜ cannot be fixed by t¯, which is a contradiction. The result follows. Combining Lemma 1.2 and Lemma 1.3, we obtain the desired result. Remark 1.4. The analogous result can be proven for all primes p. For p = 2 it is Theorem 4.3.3 in [GLS3]. For p ∈ {5, 7} the proof is similar to the proof of the proposition, but requires more calculations. For p ≥ 11, it is simply a linear algebra computation. Let us apply the result we just proved to a finite situation. Corollary 1.5. Let G ∈ Lie(r) with r = 3. Suppose that the untwisted rank of G is at least 4. Let t be a nontrivial inner diagonal automorphism of G of order 3. Then there exists Z ≤ CG (t) such that Z ∼ = Zr × Zr .

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¯ σ ) be a σ -setup of G satisfying the conclusion of Theorem 2.5.14 Proof. Let (G, ¯ which induces [GLS3]. By Lemma 4.1.1 [GLS3], there exists the unique t¯ ∈ Aut(G) the automorphism t on G and satisfies the hypotheses of Proposition 1.1. Let C = CG (t) and L = O r (C). By Theorem 4.2.2 (a) [GLS3], the following statement is true: L = L1 · · · Ls where s ≥ 0 and Li ∈ Lie(r), while part (e) of Theorem 4.2.2 [GLS3] asserts that Li ∼ = dii (q mi ). Since the ¯ untwisted rank of G is at least 4, Proposition 1.1 implies that the centralizer of t¯ in G has a nontrivial component and so s ≥ 1. If s ≥ 2, the result follows immediately. Suppose that s = 1. If L = L1 ∼ = A± 1 (r), mr (L) ≥ 2 by Theorem 3.3.3 [GLS3] and again the result follows. Assume that L = L1 ∼ = A± 1 (r). Then by part (e) of ¯ Theorem 4.2.2 [GLS3], S = A1 , which, together with the fact that the untwisted rank of G is at least 4, contradicts Proposition 1.1. The result now follows. Remark 1.6. Let us fix the notation: if (G, t, X) is a triple of type (H ), denote C := CG (t) and L := O r (C). Lemma 1.7. Let K = X t be a Frobenius group with kernel X and complement t where X is a 2-group and t 3 = 1 = t. Let R be an r-group where r is an odd prime distinct from 3. Suppose that K acts on R and (RK, t, X) is of type (H ). Then K does not act faithfully on R. Proof. Assume the contrary. Let R0 = [R, X] and V = [R0 / (R0 ), X]. Then K acts faithfully on V . By Lemma 9.12 [GLS2], CV (t) = 1 and so by Lemma 11.3 [GLS3], CR0 (t) = 1. Since CR0 (t) ≤ NRK (X), [CR0 (t), X] ≤ R ∩ X = 1, i.e., CR0 (t) centralizes X. Therefore CV (t) centralizes X and so CX (CV (t)) = X, which contradicts Lemma 9.12 [GLS2], proving the result. Lemma 1.8. Let p and r be distinct primes, and p = 3. Let X be a nontrivial pgroup, and t an automorphism of X of order 3 with [X, t] = X. Suppose that X t acts faithfully on an r-group R and G = RX t with (G, t, X) of type (H ). Then r = 3. Proof. Assume that r = 3. By Proposition 11.11 [GLS2], R contains a critical subgroup Q of class at most 2 and exponent 3. In fact, by Lemma [Be1], we may assume that Q is abelian. Since X t acts faithfully on Q and p = 3, Q = [Q, X] × CQ (X). Thus X acts fixed points free on [Q, X] and [Q, X] is t-invariant 3-group. But 1 = C[Q,X] (t) ≤ NG (X), hence 1 = C[Q,X] (t) ≤ C[Q,X] (X), which is a contradiction. Lemma 1.9. Let G be a group such that F ∗ (N) = Or (N ) for every r-local subgroup of G. Let p be an odd prime distinct from r. Let t ∈ Aut(G) and X ≤ G be such that (G, t, X) is of type (H ). Suppose that there exists a t-invariant r-local subgroup P of G. Then [X ∩ P , t] = 1.

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Proof. Let X0 = X ∩ P . Then X0 is t-invariant. Assume that [X0 , t] = 1. Then we may suppose that X0 = [X0 , t]. Since X0 acts non-trivially on Or (P ), we are in the conditions of Lemma 11.14 [GLS2], which asserts that COr (P ) (t) = 1. But COr (P ) (t) ≤ C ∩ P ≤ NG (X0 ) and so [COr (P ) (t), X0 ] ≤ Or (P ) ∩ X = 1, i.e., X0 centralizes COr (P ) (t). By Lemma 11.14 [GLS2], p = r, which is a contradiction. The result follows. This lemma has very important consequences. Corollary 1.10. Let G ∈ Lie(r) and p be a prime distinct from r. Let t ∈ Aut(G) and X ≤ G be such that (G, t, X) is of type (H ). If there exists Z ≤ C with Z ∼ = Zr × Zr , then p = 2 and X is a non-abelian 2-group. Proof. Since Z ≤ C, Z normalizes X. Since p = r, we are dealing with the coprime action. By 5.3.13 [G], we have that X = CX (u) | u ∈ Z # . Denote Xu = CX (u). Fix an arbitrary element u ∈ Z # with Xu = 1. Denote U = u and consider Pu = NG t (U ). By Corollary 3.1.4 [GLS3] to the Borel–Tits Theorem, Xu t acts faithfully on R = Or (Pu ). First let us show that X is a 2-group. Assume the contrary, i.e., suppose that p is odd. By Lemma 1.9, [X ∩ Pu , t] = 1. But Xu ≤ X ∩ Pu , and so [Xu , t] = 1. But u was an arbitrary element of Z # with the property that Xu = 1. Therefore X = Xu | u ∈ Z # commutes with t, which is a contradiction. So p = 2 and X is a 2-group. Assume that X is abelian. Since X = [X, t], Theorem 5.2.3 [G] implies that CX (t) = 1. Therefore X t is a Frobenius group and so is Xu t. This contradiction to Lemma 1.7 proves the result. Combining Corollaries 1.5 and 1.10, we obtain the following result. Lemma 1.11. Let G ∈ Lie(r) and p be a prime distinct from r. Let t ∈ Aut(G) and X ≤ G be such that (G, t, X) is of type (H ). Suppose that the untwisted rank of G is at least 4. Then p = 2 and X is a non-abelian 2-group.

2. Simple groups of Lie type We now begin the proof of Theorem 0.2 for the simple K-groups of Lie type. Proposition 2.1. Let G ∈ Lie(r). Let t ∈ Aut(G) and 1 = X ≤ G be such that (G, t, X) is of type (H ). Then r = 3. Proof. Assume that r = 3. If G is isomorphic to either A1 (33 ), or 2G2 (33 ) and t is a field automorphism, then X = 1 from the local structure of G. Otherwise, there exists U ≤ CG (t) with U ∼ = E9 . Since X is a CG (t)-signalizer, U acts on X and by Proposition 11.13 [GLS3], X = Xu = CX (u) | u ∈ U #

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Thus there exists a non-trivial element u ∈ U with Xu = 1. Therefore Xu t ≤ CG (u) t. Since G ∈ Lie(3), Borel–Tits Theorem implies that Xu t acts faithfully on F ∗ (CG (u)) = O3 (CG (u)). This contradiction with Lemma 1.8 proves the result. Proposition 2.2. Let G ∈ Lie(r), t ∈ Aut(G) and 1 = X ≤ G be such that (G, t, X) is of type (H ) and p = r. Then there exists a parabolic subgroup P of G, such that X is contained in the unipotent radical of P and CG (t) is contained in its Levi complement. Proof. The result follows immediately by an application of Borel–Tits Theorem 3.1.3 [GLS3] and Corollary 3.1.4 [GLS3]. Remark 2.3. For the remainder of this section, p = r = 3. Proposition 2.4. Let G ∈ Lie(r) and X be a p-subgroup of G. Suppose that t is a graph or a graph-field automorphism of G and (G, t, X) is of type (H ). Then X = 1. Proof. Since G admits a graph or a graph-field automorphism of order 3, either G ∼ = D4 (r a ), or G ∼ = 3D4 (r a ). By 4.7.3 and 4.9.1[GLS3], mr (C) ≥ 2. Hence by Corollary 1.10, p = 2 and X is a non-abelian 2-group. Thus by minimality, there exists an involution z ∈ X such that XCG (t) ≤ CG (X). Combining the information about the structure of 2- and 3-automorphisms of G from 4.5.1, 4.7.3 [GLS3], we obtain that X = 1. Proposition 2.5. Let G ∈ Lie(r) and X be a p-subgroup of G. Suppose that t is a field automorphism of G and (G, t, X) is of type (H ). Then X = 1. Proof. By Theorem 1 [BuGrLy1], if C ∈ {A1 (2), A1 (3),2 B2 (2)}, NG (C) is maximal in G. In particular, in this case X = 1. Thus if X = 1, G must be isomorphic to one of the following groups: A1 (8), A1 (27), 2B 2 (8). If G ∼ = S3 . Using the local structure of G, we get X = 1. = A1 (8), then CG (t) ∼ ∼ If G = A1 (27), then CG (t) ∼ = A1 (3). From the local structure of G, X = 1. If G∼ = F r20 which is a maximal subgroup of G. Hence X = 1. = 2B2 (8), then CG (t) ∼ The result now follows. Let us now deal with the inner diagonal automorphisms of groups of Lie type. We begin with the following definition. Definition 2.6. We say that a finite group K ∈ Li for i ∈ {1, 2, 3, 4}, if it is isomorphic to one of the following groups: (L1 ) A1 (2a ) with a ≥ 3 or L such that |L| = 2|A1 (r a )| where r is an odd prime, a ≥ 1 and L/Z(L) ∼ = PGL2 (r a ); a (L2 ) A± 2 (r ) where a ≥ 1 if r is odd and a ≥ 3 if r = 2;

(L3 ) B2 (2a ) with a ≥ 1 or L such that |L| = 2|B2 (r a )| where r is an odd prime, a ≥ 1 and L/Z(L) ∼ = PSO4 (r a );

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a a a (L4 ) A± 3 (r ), or G2 (r ), or B3 (r ) with a ≥ 1.

where we use the usual notation to mean all possible standard versions of L. Lemma 2.7. Suppose that G is isomorphic either to an element of L1 ∪ L2 , or a G∼ = A± 3 (r ) with a ≥ 1. Let t ∈ Aut(G) and X ≤ G be such that (G, t, X) is of type (H ). If X is non-abelian, then p = 2, i.e., X is a 2-group. Proof. Assume the contrary. Then X is a non-abelian p-group and p is odd. By 3.11 [Is], the degree of any faithful representation of X is at least p. Since p ≥ 5, G cannot be represented in less than 5 dimensions. In particular, the universal version of G cannot be represented in less than 5 dimensions. But all the universal versions of the groups we are dealing with have natural representations in at most 4 dimensions, which leads to a contradiction, proving the result. Lemma 2.8. Suppose that G ∈ L1 . Let t and X be such that (G, t, X) is of type (H ). Then X = 1. Proof. Since X is a p-group, there are two situations to consider: p is odd and p = 2. If p is odd, then by Lemma 2.7, X must be abelian. Since X = [X, t], Proposition 5.2.3 [G] asserts that CX (t) = 1. Thus X t is a Frobenius group. Since |Z(G)| ≤ 2, we may suppose that O 2 (G) = SL2 (r a ). This implies that a Frobenius group with complement of order 3 has a 2-dimensional faithful representation, which contradicts Theorem 5.16 [Is]. Therefore p = 2. Hence r is odd and G ∼ = Z2 , a ≥ 1 = A1 (r a ).T where T ∼ a ∼ and G/Z(G) = PGL2 (r ). Since t ∈ Inn(G), the situation depends upon the version of G: 1. G = Gu . Then X is a subgroup of the generalized quaternion group. Since X admits an automorphism of order 3, X ∼ = Q8 . 2. G = Ga . Then X is a subgroup of a dihedral group. Since X admits an automorphism of order 3, X ∼ = Z2 × Z2 . In both cases we may see that NG (X)/CG (X) is isomorphic to a subgroup of S3 . This means that C/CC (X) ∼ = Z3 and so 3 | (r a + ε), where ε ∈ {±1} depending on r. 1. Let G = Gu . Then C is an abelian group which contains a subgroup isomorphic to Zr a +ε . By 5.1.1 [G], |CG (X)| = 2 and so (r a + ε)/3 = 2. Thus r a ∈ {5, 7}. If ˜ = G/Z(G) ∼ ˜ = G/Z(G) ∼ r = 5, then G = PGL2 (7). In = S5 . If r = 7, then G ∼ both cases CG˜ (t˜) = Z3 × Z2 and so by 5.3.15 [G], 4 divides the order of C. Thus |CG (X)| = 4, which is a contradiction. 2. Suppose that G = Ga . Since C ∼ = Z(r a +ε)/2 is abelian and there exists an involution outside of X centralizing X, m2 (G) ≥ 3. On the other hand, by Theorem 4.10.5 (b) [GLS3], m2 (G) = m2 (PGL2 (r a )) = 2, which is a contradiction, proving the result. Lemma 2.9. Suppose that G ∈ L2 . Let t and X be such that (G, t, X) is of type (H ). Then X = 1.

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Proof. Assume the contrary. Combining Lemma 4.1.1 [GLS3] with Proposition 1.1 and Theorem 4.2.2 [GLS3], we obtain that either L = 1 or L ∼ = A1 (r a ). Suppose first that L = 1. Assume that X is abelian. Then X t is a Frobenius group. Assume that there exists a nontrivial element h ∈ CG (t) of order coprime to 3. If h acts fixed points freely on X, then X th is a Frobenius group with complement of order bigger than 3 represented in 3 dimensions, which is a contradiction. Therefore CX (h) = 1. Since CX (h) is t-invariant, CX (h) t is a Frobenius group which acts absolutely irreducibly on the natural module of G. Therefore by the Schur’s Lemma, h is a scalar matrix, i.e., h ∈ Z(G). Contradiction. If t is a genuine element of order 3, we may assume that G = Gu . If t is diagonalizable, then CG (t) ∼ = Zr a ε × Zr a ε where G ∼ = SLε3 (r a ), otherwise CG (t) ∼ = a a a GU1 (r ). In all the cases r ± 1 = 3, i.e., r ∈ {2, 4}, which is a contradiction, since ± neither A± 2 (2), nor A2 (4) are in L2 . Thus we may suppose that t is an actual element of G of order 9, which “cubes” into the center of G (i.e., |Z(G)| = 3). If 9 | (r a + ε), then there exists an element t1 of order 3 which acts as an outer automorphism on G and such that CG (t) = CG (t1 ). Thus we are reduced to the previous situation. Hence 9 does not divide r a + ε. This means, that t is in the Coxeter torus of G and so, by the Zigmondy’s Theorem, there exists element t1 of prime order s = 3 which centralizes t. Again, we are reduced to the original situation, leading to a contradiction. Therefore X is a non-abelian group. In particular, by Lemma 2.7, X is a 2-group. By minimality, there exists an involution z ∈ Z(X) such that XCG (t) ≤ CG (z). Invoking the results of 4.5.1, 4.5.2 [GLS3] and the previous lemma, we obtain that X = 1. Therefore L ∼ = A1 (r a ). Let u be a non-trivial r-element of L acting on X. Assume that X is abelian. If CX (u) = 1, consider CG (u). Since G is a group of r-type, F ∗ (CG (u)) = Or (CG (u)). If p is odd, then we obtain contradiction with Lemma 1.9, otherwise contradiction follows from an application of Lemma 1.7. Therefore X is non-abelian and thus by Lemma 2.7, is a 2-group. By minimality, we may suppose that t centralizes Z(X). Take an involution z ∈ Z(X) such that [C, z] = 1. Then XC ≤ CG (z). By Theorem 4.5.1 [GLS3], CG (z) ∈ L1 . Hence, by Lemma 2.8, X = 1. Lemma 2.10. Suppose that G ∈ L3 . Let t and X be such that (G, t, X) is of type (H ). Then X = 1. Proof. If G ∼ = B2 (2), the result follows immediately from the local structure of G. Suppose that G ∈ L3 − {B2 (2)}. Combining the results of Lemma 4.1.1 [GLS3], Theorem 4.2.2 (a,e) [GLS3] and Proposition 1.1, we obtain that L = L1 · · · Ls where Li ∈ Lie(r), s ∈ {1, 2}, and the following conditions hold: 1. If s = 1, then L = L1 ∼ = A1 (r a ); 2. If s = 2, then L = L1 L2 with Li ∼ = Ai (r a ) for i = 1, 2. In both cases there exists a nontrivial r-element u ∈ L acting on X. First assume that X is abelian. If CX (u) = 1, then Frobenius group CX (u) t acts faithfully on the r-group R = Or (CG (u)). Since this contradicts Lemma 1.9 if p is odd and Lemma 1.7 if p = 2, CX (u) = 1. So X u is a Frobenius group with a complement of order r

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represented in 4 dimensions. Therefore r = 2. Since a ≥ 2, there exists Z ∼ = Z2 × Z2 such that Z ≤ L and Z acts on X. Therefore X = CX (u) | u ∈ Z, u = 1, i.e., X = 1. Thus X is a non-abelian p-group. Since it is represented in 4 dimensions, p = 2. By minimality, we may suppose that, t centralizes Z(X). Take an involution z ∈ Z(X) such that [C, z] = 1. Then XC ≤ CG (z). By Theorem 4.5.1 [GLS3], CG (z) is isomorphic to one of the following groups: A1 (r a ).Z2 , (A1 (r a ) A1 (r a )).E4 , A1 (r 2a ).E4 and so we may reduce to the situation when X t is a subgroup of an element of L1 . By Lemma 2.8, X = 1. Lemma 2.11. Suppose that G ∈ L4 . Let t and X be such that (G, t, X) is of type (H ). Then X is a non-abelian 2-group. Proof. Combining Lemma 4.1.1 [GLS3], Theorem 4.2.2 (a,e) [GLS3] and Proposition 1.1, we obtain that L = L1 · · · Ls where Li ∈ Lie(r) and s ∈ {1, 2}. Moreover one of the following conditions must hold: a a 1. s = 1 and L = L1 ∈ {A1 (r a ), A± 2 (r ), B2 (r )}; or a 2. s = 2 and L = L1 L2 where Li ∼ = Ai (r ). If there exists Z ≤ L with Z ∼ = Zr × Zr , then the result follows immediately by Corollary 1.10. Suppose now that mr (L) = 1. Then L ∼ = A1 (r) and either G ∼ = A± 2 (r), or G ∼ = G2 (r). Assume that X is abelian. Since [X, t] = X, Proposition 5.2.3 [G] implies that CX (t) = 1. Let u ∈ L be a nontrivial r-element acting on X. If Xu = CX (u) = 1, then Xu t is a Frobenius group acting faithfully on an r-group R = Or (CG (u)), which contradicts Lemma 1.9 if p is odd and Lemma 1.7 if p = 2. Therefore CX (u) = 1. Assume first that G ∼ = A± 2 (r). Then X u is a Frobenius group with a complement of order r. Moreover, we may assume that X u is represented in 4 dimensions. ∼ ∼ Therefore r = 2, and so G ∼ = A± 3 (2). Since X = [X, t], if G = A3 (2) = A8 , the ∼ only way X = 1 would be if X = Z7 . There are two conjugacy classes of elements of order 3 in G. The centralizer of the first one, 31 , is isomorphic to Z3 × A5 , the centralizer of the other one, 32 , is isomorphic to Z3 × Z3 . Since neither of them is ∼ contained in a 7-local subgroup of G, X = 1. If G ∼ = A− 3 (2) = B2 (3), then X is a 2-group. But now we have a contradiction with the 2-local structure of G. Therefore X is non-abelian and by Lemma 2.7, X is a 2-group. Suppose now that G ∼ = A1 (r). Then X tu is a Frobenius = G2 (r) and L ∼ group with a complement of order 3r. Since G has a natural representation in 14 dimensions, 3r ≤ 14, i.e., r = 2. Therefore G ∼ = G2 (2), and so |X| = 7. In this case a contradiction follows from the 7-local structure of G. For K ∈ Lie(r), denote by r(K) the untwisted Lie rank of K. We are now ready to prove the following result. Proposition 2.12. Let G = G1 · · · Gl where l ≥ 1 and for all i ∈ {1, . . . , l} either r(Gi ) ≥ 4 and Gi ∈ Lie(r) − {Cn (r) | n ≥ 2}, or Gi ∈ L1 ∪ L2 ∪ L3 ∪ L4 . Suppose

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that t ∈ Inndiag(G) and X is a p-group with p = r and (G, t, X) of type (H ). Then X = 1. Proof. Since t ∈ Inndiag(G), we have that t|Gi = ti ∈ Inndiag(Gi ) for i = 1, . . . , l. If there exists an i with ti = 1, then since [X, t] = X, we shall have X|Gi = 1. Therefore we may suppose that ti = 1 for i = 1, . . . , l. Let us prove the result using the induction on n, where n = max{(r(Gi )) | i = 1, . . . , l}. For n ≤ 3 the result follows by Lemmas 2.8, 2.9 and 2.11. Suppose now that n ≥ 4. By Corollary 1.10 and Lemmas 2.9, 2.10 and 2.11, Xi = X|Gi is either trivial or a non-abelian 2-group for i = 1, . . . , l. Assume that there exists i with Xi = 1. Consider the corresponding triple (Gi , ti , Xi ). We may suppose that Gi is the adjoint group. By minimality, there exists a nontrivial involution z ∈ Z(Xi ) such that [z, ti ] = 1 and [z, CGi (ti )] = 1. So Xi CGi (ti ) ≤ CGi (z). By Theorems 4.5.1, 4.5.3 and 4.2.2 [GLS3], there exists C 0 ≤ CGi (z) such that CGi (z)/C 0 is a 2-group and C 0 = (L1 L2 )T satisfies the following conditions: 1. Either Li = 1, or Li satisfies the hypotheses of the proposition; and 2. T is an abelian group inducing the inner diagonal automorphisms on L1 L2 . Since Xi is non-abelian and ti ∈ Inndiag(Gi ), ti must induce an inner diagonal automorphism on L1 L2 . Moreover, Xi ≤ L1 L2 . Therefore we are reduced to the case when everything is happening in the group (L1 L2 )T . Since r(L1 L2 ) < r(Gi ), by induction, Xi = 1, proving the result. We have dealt with most of the groups of Lie type. But there are a few exceptions which we ignored. For example, what happens in the symplectic groups? Or, why a did we avoid talking about A± 2 (2 ) for a ≤ 2? Lemma 2.13. Suppose that G ∼ = A2 (2), p is an odd prime and t and X are such that (G, t, X) is of type (H ). Then X ∼ = Z3 . = Z7 and t ∈ G with CG (t) ∼ Proof. Since G ∼ = L2 (7), the local structure of G implies that if X = 1, then = A2 (2) ∼ ∼ X = Z7 . Moreover |G|3 = 3 and C ∼ = Z3 normalizes X, proving the result. ∼ A2 (4), p is an odd prime and (G, t, X) is of type (H ). Lemma 2.14. Suppose that G = Then the following conditions hold: (1) If G ∼ = SL3 (4), then X ∼ = Z7 and CG (t) ∼ = Z3 × Z3 ; (2) If G ∼ = L± 3 (4), then X = 1.

Proof. First suppose that G ∼ = L3 (4). Then |X| ∈ {5, 7}. Since X = [X, t], X ∈ Syl7 (G), and NG (X) is a Frobenius group of order 21. On the other hand, a centralizer of any inner automorphism of order 3 contains Z ∼ = Z3 × Z3 . Thus t ∈ Inn(G). Assume that t is a diagonal automorphism. There are two conjugacy classes of such automorphisms. The centralizer of the first one, 31 , in G is a group of order 60, which is not contained in any 7-local subgroup of G. The centralizer of the other one, 32 , is precisely the normalizer of X, which means that [X, t] = 1, which is a contradiction.

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If G ∼ = U3 (4), then using the p-local structure of G, we obtain that CG (t) is not contained in any p-local subgroup N of G with [Op (N ), t] = Op (N ). Therefore X = 1. Finally, suppose that G ∼ = SL3 (4). Things work the same way as far as the diagonal automorphisms are concerned. But if t is an inner automorphism, then CG (t) ∼ = Z3 × Z3 normalizes some Sylow 7-subgroup X of G. The result now follows. Finally, let us deal with the symplectic “situation”. Lemma 2.15. Suppose that G = Sp2 (r a ) with a ≥ 1, p a prime distinct from r and t and X are such that (G, t, X) is of type (H ). If X = 1, then a = 1, r ∈ {5, 7}, p = 2 and X ∼ = Q8 . Proof. Assume that X is abelian. Since [X, t] = X, it follows that X t is a Frobenius group with a complement of order 3 represented in 2 dimensions, which contradicts Theorem 5.6 [Is]. Therefore X must be non-abelian. By the local structure of G, X is a 2-group and since it must admit an automorphism of order 3, X ∼ = Q8 . Recall that ∼ ∼ C normalizes X. Since Aut(X) = S4 , we obtain that C/CC (X) = Z3 . We also know that |CG (X)| = 2. Since 3 | (r a + ε) (where ε ∈ {±1}), (r a + ε)/3 = 2. Thus a = 1 and r ∈ {5, 7}. The result now follows immediately from the 2-local structure of G. Lemma 2.16. Suppose that G = PSp2 (r a ) with a ≥ 1, p is a prime distinct from r, and t and X are such that (G, t, X) is of type (H ). If X = 1, then a = 1, r ∈ {5, 7}, p = 2 and X ∼ = Z2 × Z2 . Proof. If p is odd, then because of the coprimeness, we must have the same thing happening in G as in 2G. So, by Lemma 2.15, X = 1. Therefore p = 2 and r is odd. Since X must admit an automorphism of order 3, the 2-local structure of G implies that X ∼ = S4 , we have = Z2 × Z2 . Recall that C normalizes X. Since Aut(X) ∼ that C/CC (X) ∼ = Z3 . We also know that |CC (X)| = 2. Since 3 | (r a + ε) (where ε ∈ {±1}), (r a + ε)/3 = 2. Thus a = 1 and r ∈ {5, 7}. The result now follows. Lemma 2.17. Suppose that G = C2 (r a ) with a ≥ 1, p is a prime distinct from r, and t and X are such that (G, t, X) is of type (H ). Then a = 1, r ∈ {5, 7}, p = 2 and X ∼ = Z3 × Z2 × A1 (r) and = Q8 . Moreover, t ∈ G is an element with CG (t) ∼ ∼ A1 (r) = Sp2 (r). Proof. Assume first that p is odd. Because of the coprimeness, we may suppose that G = Sp4 (r a ). If X were non-abelian, we would need at least p dimensions to represent it faithfully. Since p ≥ 5 and G has a 4-dimensional module, X must be abelian. Since [X, t] = X, X t is a Frobenius group. Let V be a natural module for G. Then V |X t = V1 ⊕ V2 where V1 is 1-dimensional and V2 is of dimension 3. Reducing the situation to the 3-dimensional case and applying Lemma 2.9, we obtain X = 1.

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Therefore p = 2 and r is odd. If a ≥ 2, then Proposition 1.1 together with Corollary 1.10 imply that X is non-abelian. Therefore we may suppose that G = Ga ∼ = PSp4 (r a ). As usual, there exists an involution z ∈ Z(X) such that [z, t] = 1 and [z, C] = 1. Therefore X t ≤ O r (CG (z)). By Theorem 4.5.1 [GLS3], O r (CG (z)) is isomorphic either to A1 (r 2a ), or to A1 (r a ) A1 (r a ). In the former situation we are reduced to the case G ∼ = A1 (r 2a ) and by Lemma 2.15, X = 1. In the latter case, a ∼ C = Z3 ×Z2 ×A1 (r ). Therefore Lemma 2.15 implies that X ∼ = Q8 with r a = 6±1. The result follows immediately. We are now ready to prove the following result. Proposition 2.18. Let G = Cl (r a ) with a ≥ 1, l ≥ 3, p a prime distinct from r and t and X are such that (G, t, X) is of type (H ). Then a = 1, r ∈ {5, 7}, p = 2 and X∼ = Z3 × Z2 × Cl−1 (r). = Q8 . Moreover, t ∈ G with C ∼ ¯ σ ) be a σ -setup of G satisfying the conclusion of Theorem 2.3.14 Proof. Let (G, ¯ which in[GLS3]. By Lemma 4.1.1 [GLS3], there exists a unique element t¯ ∈ Aut(G) duces the automorphism t on G and satisfies Proposition 1.1. Now Theorem 4.2.2 (a,e) [GLS3] together with Proposition 1.1 give us that L = L1 · · · Ls where Li ∈ Lie(r), s ∈ {1, 2} and one of the following conditions hold: a a 1. s = 1 and L = L1 is isomorphic either to A± l−1 (r ), or to Cl−1 (r ); or a a ∼ 2. s = 2 and L = L1 L2 with L1 ∼ = A± n (r ) and L2 = Cl−n−1 (r ). ∼ In both cases C contains a subgroup Z = Zr ×Zr . By Corollary 1.10, p = 2 and X is a non-abelian 2-group. As usual, by minimality, there exists an involution z ∈ Z(X) such that [z, t] = 1 and [z, C] = 1. Therefore XC ≤ CG (z). Suppose that L ∼ = Cl−1 (r a ). Combining this with Theorem 4.5.1 [GLS3], we obtain that (C, CG (z)) a a ± a a a is one of the following pairs: (A± l−1 (r ), Cl−1 (r )), (An (r ) Cl−n−1 (r ), Cn (r )

a Cl−n−1 (r )). Therefore we are reduced to the smaller dimensional case, and by induction, X = 1. Finally suppose that L ∼ = Cl−1 (r a ). Then by Theorem 4.5.1 [GLS3], we obtain that (C, CG (z)) is (Cl−1 (r a ), A1 (r a ) Cl−1 (r a )). Since t acts trivially on the second component of CG (z) and [X, t] = X, X must project on it trivially. So by Lemma 2.17, X ∼ = Q8 and the result follows.

3. Alternating groups Proposition 3.1. Let G = An with n ≥ 5, t ∈ Aut(G) and X ≤ G be such that (G, t, X) is of type (H ). Then either X = 1, or p = 2 and one of the following holds: ∼ E4 ; ∼ A5 and X = (1) G = (2) G ∼ = E4 and t is an element of G with CG (t) ∼ = E9 ; or = A7 , X ∼ ∼ A8 , X ∼ (3) G = = E24 and t is an element of G with CG (t) ∼ = Z3 Z2 .

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Proof. The proof of this proposition is an induction on n. To start the induction, we shall have to go up to n = 14 “by hand”. By direct calculations, we obtain that for 5 ≤ n ≤ 14, either X = 1, or one of the following conditions hold: ∼ E4 ; ∼ A5 and X = (1) G = (2) G ∼ = A7 , X (3) G ∼ = A8 , X

∼ = E4 and t is an element of G with CG (t) ∼ = E9 ; ∼ = E 4 and t is an element of G with CG (t) ∼ = Z3 Z2 . 2

Suppose now that n ≥ 15. Since (| Out(G)|, 3) = 1, t ∈ Inn(G) and without loss of generality we may suppose that t = (123) . . . is the product of l 3-cycles for some l ≥ 1. Consider Z ∼ = Z3 × Z3 such that Z = (123), (456). Since Z ≤ CG (t), Z must act on X and because of coprimeness, X = Xu = CX (u) | u ∈ Z # . If u ∈ Z is a 3-cycle, then CG (u) ∼ = Z3 × An−3 , otherwise CG (u) ∼ = (Z3 × Z3 × An−6 ).Z2 . Since X = [X, t], in both cases considering t|CG (u) , we are restricted to the situation when [Xu , t] t is contained in an alternating group of degree m < n. Since m ≥ 9, induction finishes the proof.

4. Sporadic groups Finally, let us deal with the “small” and “medium” sporadic groups. Proposition 4.1. Let G be a sporadic simple K-group with e(G) ≤ 3. Suppose that t ∈ Aut(G) and X ≤ G are such that (G, t, X) is of type (H ). Then either X = 1, or one of the following conditions holds: (1) G ∼ = Z3 × A4 and X ∼ = E4 ; = M22 , t ∈ 3A with CG (t) ∼ (2) G ∼ = M23 , t ∈ 3A with CG (t) ∼ = Z3 × A5 and X ≤ E16 ; ∼ (3) G = M24

(a) t ∈ 3A with C(3A) ∼ = 3A6 and X ∼ = 26 ; (b) t ∈ 3B with C(3B) ∼ = 26 ; = Z3 × L3 (2) and X ∼ ∼ Z3 × A4 and ∼ J2 , t ∈ 3B with C(3B) = (4) G = XCG (t) ≤ N (J (S2 )) ∼ = 22+4 .(Z3 × S3 );

(5) G ∼ = J4 , t ∈ 3A with C(3A) ∼ = 6M22 and XCG (t) ≤ Q68 (3M22 .2); (6) G ∼ = H e, t ∈ 3B with C(3B) ∼ = Z3 × L3 (2). and X ∼ = E4 .

Proof. Throughout this proof, we shall assume the results of section 5.3 [GLS3]. First of all recall, that e(G) = max{m2,p (G) | p odd}. Thus G is isomorphic to one of the following groups: M11 , M12 , M22 , M23 , M24 , J1 , J2 , J3 , J4 , HS, Mc, Co3 , Co2 , Suz, He, Ly, Ru, O’N, F5 , F3 .

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Since (| Out(G)|, 3) = 1, t ∈ Inn(G). If m3 (G) = 1, then G ∼ = J1 , |G| = 23 · 3 · 5 · 7 · 11 · 19 and CG (t) ∼ = Z3 × D10 . If Pp ∈ Sylp (G) for some odd prime p, then NG (Pp ) does not contain a subgroup isomorphic to CG (t). Therefore p = 2, and since S2 ∼ = E8 while X = [X, t], X∼ E . Since X does not admit an automorphism of order 5, X ≤ CG (P5 ) for some = 4 P5 ∈ Syl5 (G). But m2 (P5 ) = 1, hence X = 1. Suppose that m3 (G) ≥ 2. Then there exists s ∈ CG (t) such that U = s, t ∼ = E9 and Xs = CX (s) = 1. Moreover, we may suppose that Xs = [Xs , t]. Thus we are reduced into the centralizer of some 3-element, and if “nothing happens” for all possible choices of s, we may conclude that X = 1. On the other hand, if something does happen, i.e., Xs = 1 for some choice of s, it determines the possible value of p. Thus we must first look inside the centralizers of the 3-elements of the groups listed above. In order to do so, we will use Tables 5.3a–5.3z [GLS3], p. 262–287. ∼ ∼ M11 : Since G contains a unique conjugacy class of 3-elements, CG (s) = G= Z3 × S3 . Hence X = 1. G∼ = M12 : Looking at the list of all maximal subgroups of G (p. 263 [GLS3]), we obtain X = 1. ∼ G = M22 : Since G contains a unique conjugacy class of 3-elements, CG (s) ∼ = CG (t) ∼ = Z3 × A4 . If X = 1, then p = 2. Looking at the list of all maximal subgroups (p. 264 [GLS3]), we get X ∼ = E4 . ∼ G = M23 : Since G contains a unique conjugacy class of 3-elements, CG (s) ∼ = CG (t) ∼ = Z3 × A5 . If X = 1, then p = 2. Looking at the list of all maximal subgroups (p. 265 [GLS3]), we get X ≤ E16 . ∼ G = M24 : G has two conjugacy classes of 3-elements: 3A whose centralizer is isomorphic to 3A6 , and 3B whose centralizer is isomorphic to Z3 × L3 (2). Hence if X = 1, p ∈ {2, 7}. But P7 ∼ = = Z7 and NG (P7 ) ∼ Z7 .Z3 ×S3 . Hence p = 2. Looking at the list of all maximal subgroups (p. 266 [GLS3]), we get that X ∼ = 26 and both cases are possible, i.e., t ∈ 3B and t ∈ 3B happen. ∼ G = J2 : G contains two conjugacy classes of 3-elements: 3A with C(3A) ∼ = 3A6 and 3B with C(3B) ∼ = Z3 × A4 . If X = 1, then p = 2. Looking at the list of all maximal subgroups (p. 268 [GLS3]), we get t ∈ 32 and XCG (t) ≤ N (J (P2 )) ∼ = 22+4 .(Z3 × S3 ). ∼ ∼ J3 : G contains two conjugacy classes of 3-elements: 3A with C(3A) = G= ∼ E9 31+2 .2. If X = 1, then p = 2. But Z3 × A6 and 3B with C(3B) = no 2-local of G has a 3-part of its order bigger than 9. Hence X = 1. G∼ = 6M22 . = J4 : G has a unique conjugacy class of 3-elements 3A with C(3A) ∼ If X = 1, then from the M22 -case, p = 2. Since C(2A) ≥ NG (3A), XCG (t) ≤ Q68 (3M22 .2). G∼ = Z3 ×S5 . = HS: Since G has a unique conjugacy class of 3-elements, CG (s) ∼ But Xs = 1, and so X = 1.

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G∼ = Mc: G contains two conjugacy classes of 3-elements, and whether s ∈ 3A, or s ∈ 3B, F ∗ (CG (s)) = O3 (CG (s)). If Xs = 1, then Xs t must act faithfully on a 3-group, which contradicts Lemma 1.8. Hence Xs is trivial, and so is X. G∼ = = Co3 : G contains three conjugacy classes of 3-elements: 3A with C(3A) ∼ E .(Z × S ) and 3C with 31+4 .((4 SL2 (9)).2), 3B with C(3B) ∼ = 35 2 5 C(3C) ∼ = Z3 × L2 (8).Z3 . Hence, if X = 1, then p = 2, and since e(G) = m2,3 (G) = 3, t ∈ 3C. Moreover, by Lemma 1.8, s ∈ 3C. Since Xs = 1, t acts as an outer automorphism on Ls ∼ = L2 (8) and Xs ∼ = S3 does not normalize X. Hence Xs = = E4 . But CLs (t) ∼ X = 1. G∼ = = Co2 : G contains two conjugacy classes of 3-elements: 3A with C(3A) ∼ 31+4 .((D8 Q8 )S5 ) and 3B with C(3B) ∼ = Z3 × Aut(PSp4 (3)). Applying Lemma 1.8 if s ∈ 3A, and Proposition 2.1 if s ∈ 3B, we obtain that Xs = 1. Hence X = 1. G∼ = He: G contains two conjugacy classes of 3-elements: 3A with C(3A) ∼ = 3A7 .2 and 3B with C(3B) ∼ = Z3 × L3 (2). Hence, p ∈ {2, 7}. If p = 7, there exists X ∼ = Z7 with NG (X) ∼ = Z7 .Z3 × L3 (2). But the 3-element which acts faithfully on X is of type 3A. Hence, p = 2. Then X ∼ = S4 × L3 (2) contains CG (t) for t ∈ 3B. = E4 and NG (X) ∼ G∼ = = Ru: Since G has the unique conjugacy class of 3-elements, NG (t) ∼ NG (s) ∼ = 3 Aut(A6 ). So, if X = 1, then p = 2. Since no involution of G has a centralizer which contains a subgroup isomorphic to CG (t), CX (CG (t)) = 1. Studying the 2-local structure of G, we get X = 1. G ∈ {Ly, O’N, Suz, F5 , F3 }: No matter what t is, m3 (CG (t)) ≥ 4. mp,3 (G) ≤ 3, X = 1.

Since

The result now follows. Combining Propositions 2.1, 2.2, 2.4, 2.5, 2.18, 3.1 and 4.1, we obtain the results of Theorems 0.2 and 0.3. Acknowledgements. I would like to thank the referee for his valuable and patient comments. I would like to thank Ronald Solomon, under whose kind and thorough direction most of this work was done as a part of my Ph.D. thesis, and Richard Lyons for his invaluable discussions, corrections and support, which made it possible to complete this paper. I would also like to thank Peter Sin for his valuable comments and corrections.

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References [AS]

M. Aschbacher, G. Seitz, Involutions in Chevalley Groups Over Fields of Even Order, Nagoya Math. J. 63 (1976), 1–91.

[Be1]

H. Bender, Über den grössten p -Normalteiler in p-auflösbaren Gruppen, Arch. Math. 18 (1967), 15–16.

[BuGrLy1]

N. Burgoyne, R. L. Griess, R. Lyons, Maximal subgroups and automorphisms of Chevalley groups, Pacific J. Math. 71 (1977), 365–403.

[G]

D. Gorenstein. Finite Groups, Second Edition, Chelsea, New York 1980.

[GLS]

D. Gorenstein, R. Lyons, R. Solomon, The Classification of the Finite Simple Groups, Math. Surveys Monogr. 40.1, Amer. Math. Soc., Providence, RI, 1994.

[GLS2]

D. Gorenstein, R. Lyons, R. Solomon, The Classification of the Finite Simple Groups, Number 2, Math. Surveys Monogr. 40.2, Amer. Math. Soc., Providence, RI, 1996.

[GLS3]

D. Gorenstein, R. Lyons, R. Solomon, The Classification of the Finite Simple Groups, Number 3, Math. Surveys Monogr. 40.3, Amer. Math. Soc., Providence, RI, 1998.

[Is]

I. Martin Isaacs, Character Theory of Finite Simple Groups, Academic Press, New York 1976.

Inna Korchagina, Department of Mathematics, Rutgers University, New Brunswick, NJ 08903, U.S.A. E-mail: [email protected]

Classifying character degree graphs with 5 vertices Mark L. Lewis

Abstract. We study the degree graphs of solvable groups that have 5 vertices. Using known necessary conditions on one hand, and making use of straightforward constructions on the other hand, one can decide if a graph with 5 vertices occurs as a character degree graph for a solvable group– up to four exceptions. In this paper, we study these four exceptions. In particular, we show that two graphs that satisfy the previously known necessary conditions with 5 vertices cannot occur as degree graphs for solvable groups. We give an example showing that a third graph with 5 vertices does occur as the degree graph of a solvable group. Finally, we leave undecided whether one graph with 5 vertices is the character degree graph for a solvable group. 2000 Mathematics Subject Classification: 20c15

1. Introduction Throughout this paper, G will be a finite solvable group, and cd(G) will be the set of (irreducible) character degrees of G. Let ρ(G) be the set of primes that divide degrees in cd(G). The degree graph of G, written (G), is the graph whose vertex set is ρ(G). Two vertices p and q in ρ(G) are adjacent if there is some degree a ∈ cd(G) where pq divides a. These graphs have been studied in several places. The best sources for general information about these graphs are Theorem 14 of [3], Chapter 30 of [4], and Sections 18 and 19 of [12]. In this paper, we wish to consider which graphs with five vertices occur when G is solvable. The following theorem of Pálfy puts a restriction on which graphs can occur as degree graphs of solvable groups. Theorem 1.1 (Pálfy, [13]). Let G be a solvable group, and let π be a subset of ρ(G). If |π| ≥ 3, then there exist primes p, q ∈ π and a degree a ∈ cd(G) so that pq divides a. In particular, any three primes in ρ(G) must have an edge in (G) that is incident to two of those primes. We will say that a graph satisfies Pálfy’s condition if it has the property that every three vertices have an edge incident to two of those vertices. It is easy to see that a disconnected graph that satisfies Pálfy’s condition will have two connected components, and each connected component will be a complete graph. Therefore,

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there are only two possible disconnected graphs with five vertices that meet Pálfy’s condition. Both of these graphs occur as (G) for some solvable group G. We will present examples of groups with these graphs in Section 2. We now turn to the case where (G) is connected. In this case, we look at the diameter of the graph. As usual, the distance between two vertices is the number of edges in the shortest path between the vertices. The diameter of a connected graph is the maximum distance between points in the graph. For disconnected graphs, the diameter of the graph is the diameter of the largest connected component. It is not difficult to see that graphs satisfying Pálfy’s condition have diameter at most 3. It had been conjectured for solvable groups that in fact the diameter of (G) was at most 2, but in [10] we constructed a solvable group G where (G) had diameter 3. The graph we constructed had 6 vertices, and in [11], we proved the following theorem: Theorem 1.2 ([11]). If G has a solvable group and (G) has 5 vertices, then the diameter of (G) is at most 2. As mentioned in [11], this eliminates from consideration two graphs with 5 vertices that satisfy Pálfy’s condition. In Section 2, we will show how to use direct products to construct groups with various degree graphs having 5 vertices by using groups whose degree graphs have 4 or fewer vertices. When these constructions are completed there will remain four graphs with 5 vertices that satisfy Pálfy’s condition. These graphs are shown in Figure 1. The purpose of this paper is to show that (1) and (2) in Figure 1 cannot occur as degree graphs, and to construct a group whose degree graph is (3). Unfortunately, we have not been able to determine whether or not (4) is the degree graph of a solvable group. We codify this in the following theorem: Main Theorem. The graphs with 5 vertices that arise as (G) for some solvable group G are precisely the graphs with diameter at most 2 that satisfy Pálfy’s condition except for graphs (1), (2), and possibly (4) of Figure 1. .

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Figure 1. Graphs under consideration

2. Constructions We first present two groups G that have the two possible disconnected graphs with 5 vertices as (G). Both of these groups are affine semi-linear groups. For the first example, take the Galois field of order 210 acted on by its full multiplication group and

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then its Galois group. It is not difficult to see that this group has character degree set {1, 2, 5, 10, 3 · 11 · 31}. This gives a graph with two connected components: {2, 5} and {3, 11, 31} which are both complete graphs. For the second example, take the Galois field of order 216 acted on by its full multiplication group and then its Galois group. In this case, the character degree set is {1, 2, 4, 8, 16, 3 · 5 · 17 · 257}, and the degree graph has connected components: {2} and {3, 5, 17, 257} where the second component is a complete graph. These examples are in no way unique. Similar constructions will yield examples of both graphs, and it is not difficult to find examples where the groups have odd orders. One way to build groups from smaller groups is by taking direct products. It is not difficult to see that if H and K are groups, then ρ(H × K) = ρ(H ) ∪ ρ(K). Furthermore, there is an edge between p and q in ρ(H × K) if one of the following occurs: (1) p, q ∈ ρ(H ) and there is an edge between p and q in (H ), (2) p, q ∈ ρ(K) and there is an edge between p and q in (K), (3) p ∈ ρ(H ) and q ∈ ρ(K), or (4) p ∈ ρ(K) and q ∈ ρ(H ). This follows from Theorem 4.21 of [5]. All graphs with 3 or fewer vertices that satisfy Pálfy’s condition are known to arise as (G) for some solvable group G. In fact, the only graph with 3 or fewer vertices that does not satisfy Pálfy’s condition is the graph with 3 isolated vertices. Graphs with 4 vertices that satisfy Pálfy’s condition do occur as degree graphs for some solvable group. The two graphs with 4 vertices that satisfy Pálfy’s condition and do not occur as degree graphs are shown in Figure 2. For Graph (1) of Figure 2, this is proved in [14], and for Graph (2), this is the content of [15]. So we may refer to this fact later, we state the following theorem. Theorem 2.1 ([14] and [15]). If G is a solvable group, then (G) is neither of the graphs shown in Figure 2. ·

·

· (1)

·

·

·

·

·

(2)

Figure 2. 4-vertex graphs that do not occur

Notice that Graph (2) of Figure 2 is the only graph with four or fewer vertices that has diameter 3. Therefore, combining Theorem 1.2 with Theorem 2.1, we see that if G is solvable group and |ρ(G)| ≤ 5, then the diameter of (G) is at most 2. On page 25 of [3], Huppert listed all the graphs with at most 4 vertices that can arise as (G) for some solvable group G. For the record, there are 14 graphs with 5 vertices that satisfy Pálfy’s condition. We have seen that two of these graphs are disconnected and do occur as degree graphs. Two of the graphs have diameter 3, and by Theorem 1.2, they do not occur as degree graphs. This leaves ten graphs to consider, and of these six graphs can be obtained by looking at groups of the form H × K where ρ(H ) and ρ(K) are disjoint, and |ρ(H ) ∪ ρ(K)| = 5. The five graphs with 4 vertices give five graphs arising in the form H × K where |ρ(H )| = 4 and |ρ(K)| = 1. Since

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there are two graphs with 2 vertices and three graphs with 3 vertices, it would seem that we should get six graphs arising from H × K where |ρ(H )| = 3 and ρ(K) = 2, but it turns out that one graph arises in two different fashions, so we get five graphs of this form. We should mention that it is necessary to show for each of the pairs of graphs above that there exist groups H and K with the given graphs where ρ(H ) and ρ(K) are disjoint. Since for every prime p there is a group K with ρ(K) = {p}, this is fine when |ρ(H )| = 4 and |ρ(K)| = 1. When |ρ(H )| = 3 and |ρ(K)| = 2, it is less trivial, but still not difficult to show that such H and K will exist. The two lists have four graphs in common, so we get six graphs in this fashion, and we are left with the four graphs found in Figure 1.

3. Graph (1) of Figure 1 Throughout this paper, we will make use of the following results without further reference. By Itô’s theorem (Corollary 12.34 of [5]), we know that a solvable group G has a normal abelian Sylow p-subgroup if and only if p ∈ ρ(G). Suppose N is a normal subgroup of G, and the character θ ∈ Irr(N ) extends to G. Using Gallagher’s theorem (Corollary 6.17 of [5]), we can show that cd(G|θ ) = {θ (1)a | a ∈ cd(G/N )}. If (|G : N |, |N |) = 1, then we know via Corollary 6.28 of [5] that every character in Irr(N ) will extend to its stabilizer in G. Let F be the Fitting subgroup of G. In Lemma 18.1 of [12], it is shown that if G ⊆ F , then |G : F | ∈ cd(G). Let F2 /F be the Fitting subgroup of G/F . On page 254 of [12], a character in Irr(G) is produced that is divisible by all the prime divisors of |F2 : F |. We will also make use of the detailed information collected in [7] regarding a solvable group G where (G) is disconnected. It is actually quite easy to show that (1) from Figure 1 does not occur. In fact, we prove a couple of much more general results which are still quite easy. The hypotheses of these lemmas will arise in an inductive argument. Lemma 3.1. Let G be a solvable group, and suppose p is a prime in ρ(G). For every proper normal subgroup H of G, suppose that (H ) is a proper subgraph of (G). Assume that no subgraph of (G) obtained by removing the vertex p or by removing one or more of the edges incident to p is (H ) for any normal subgroup H of G. Then G = O p (G). Proof. Let M = O p (G). If M = G, then we are done, so we assume that M < G. By hypothesis, (M) is a proper subgraph of (G). Since |G : M| is a power of p, the only vertex of (G) that could be missing is p, and any edges that are missing must be incident to p. Thus, (M) is a subgraph of (G) obtained by deleting p or by deleting one or more of the edges incident to p. Since no such graph can be the degree graph of a normal subgroup of G, this is a contradiction, so we must have G = M.

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Lemma 3.2. Let be a graph with n vertices. Suppose that no proper subgraph of with either n vertices or n − 1 vertices occurs as (G) for some solvable group G. Then is not the degree graph (G) for some solvable group G. Proof. Let G be a counterexample with |G| minimal. Since is not empty, G > 1, and since G is solvable, there must be a prime p so that O p (G) < G. On the other hand, the minimality of |G| will imply that (H ) is a proper subgraph of for every proper normal subgroup H . Furthermore, any subgraph of obtained by removing p or edges incident to p will have n or n − 1 vertices, and so any graph obtained in this fashion cannot be the graph of a (normal) subgroup of G. Therefore, we obtain a contradiction via Lemma 3.1 which said O p (G) = G. We can now apply Lemma 3.2 to eliminate Graph (1) of Figure 1. Corollary 3.3. The pentagon (Graph (1) of Figure 1) is not the degree graph of a solvable group. Proof. We claim that no proper subgraph of Graph (1) of Figure 1 is the degree graph of a solvable group. The only proper subgraphs with 4 or 5 vertices that are not excluded by Pálfy’s condition are the two graphs in Figure 2, and these graphs are excluded by Theorem 2.1. It follows that the pentagon satisfies the hypotheses of Lemma 3.2, and applying that lemma, the pentagon is not the degree graph of any solvable group.

4. A nonabelian normal Sylow subgroup In the next three sections, we work to prove that Graph (2) of Figure 1 cannot occur. The next two sections will consist of lemmas that can be applied to a more general setting than just the graph under consideration. In this section, we concentrate on groups having a normal nonabelian Sylow subgroup. With this in mind, we make the following hypothesis. Hypothesis 4.1. Let G be a solvable group with a normal nonabelian Sylow psubgroup P for some prime p. Let π be the set of primes in ρ(G) that are adjacent to p in (G), and let ρ be the set of primes in ρ(G) that are not adjacent to p. Thus, ρ(G) is the disjoint union: {p} ∪ π ∪ ρ. The next lemma deals with the case where the p-complement is acting faithfully on P and the Fitting subgroup of the p-complement is a π-group. We show that the primes involved in the degrees of the p-complement lie in ρ. Lemma 4.2. Assume Hypothesis 4.1. Suppose that for every prime q ∈ π there is a prime r ∈ ρ so that q and r are not adjacent in (G). Let H be a p-complement for G. Assume that H acts faithfully on P , that ρ ⊆ ρ(H ), and that the Fitting subgroup of H is a π-group. Then ρ(H ) = ρ.

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Proof. If π is empty, then the result is immediate, so we may assume that π is not empty. Also, we can find a nonlinear character θ ∈ Irr(P ). We know that p divides every degree in cd(G|θ), and so no prime in ρ divides any degree in cd(G|θ ). By Theorem 12.9 of [12], G/P has an abelian Hall ρ-subgroup. Let F be the Fitting subgroup of H , and label the primes in π by π = {q1 , q2 , . . . , qm }. For each i, let Fi be the Sylow qi -subgroup of F , and let Ci = CH ([P , Fi ]P /P ). Since F = F1 × F2 × · · · × Fn , we have the following product and intersection: [F, P ] = [F1 , P ][F2 , P ] . . . [Fn , P ] and m i=1 Ci = CH ([P , F ]P /P ). Because F acts faithfully on P , the action of F on P /P will be faithful. Applying Fitting’s Lemma, F will act faithfully on [F, P ]P /P . On the other hand, if CH ([P , F ]P /P ) > 1, then CH ([P , F ]P /P ) will contain a minimal normal subgroup of H which would be contained in F . As CF ([P , F ]P /P ) = 1, m we conclude that CH ([P , F ]P /P ) = 1 and i=1 Ci = 1. For each i = 1, . . . , m, we fix a nonprincipal character λ ∈ Irr([P , Fi ]P /P ). Let ri be the prime in ρ that is not adjacent in (G) to qi . Since |H : CH (λ)| divides every degree in Irr(G|λ) and qi divides |H : CH (λ)|, it follows that ri does not divide |H : CH (λ)|. We see that CH (λ) contains a Sylow ri -subgroup of H . Observe that λ extends to P CH (λ), and using Gallagher’s theorem, ri ∈ ρ(CH (λ)). As usual, CH (λ) has a normal abelian Sylow ri -subgroup. Note that Ci ⊆ CH (λ), so if Ci contains a Sylow ri -subgroup of H , then H would have a normal abelian Sylow ri -subgroup which is a contradiction since ri ∈ ρ ⊆ ρ(H ). We conclude that a Sylow ri -subgroup of H acts nontrivially on [P , Fi ]P /P . By Lemma 1 of [8], either H /Ci ∼ = SL2 (3) or GL2 (3) and |[P , Fi ]P /P | = 9 or H /Ci has a normal abelian subgroup that acts irreducibly on [P , Fi ]P /P . Since (|P |, |H |) = 1, it must be that H /Ci has an abelian normal subgroup that acts irreducibly on [P , Fi ]P /P . From Lemma 2.2 of [12], H Ci /Ci is nilpotent. Because m i=1 Ci = 1, this implies that H is nilpotent. In particular, H ⊆ F , and thus, |H : F | ∈ cd(H ) = cd(G/P ). Since ρ ⊆ ρ(H ) and no prime in ρ lies in ρ(F ), we conclude that |H : F | is divisible by all the primes in ρ. Since every prime in π is not adjacent to some prime in ρ, it follows that |H : F | is not divisible by any prime in π , and F is the Hall π-subgroup of H . Our next goal is to prove that F is abelian. This will imply that H has an abelian normal Hall π -subgroup, and thus, ρ(H ) ⊆ ρ. It is sufficient to prove that Fi is abelian for each i, and so, we assume Fi is not abelian for some i, and we obtain a contradiction. Let Ri be a Sylow ri -subgroup of H . We claim that P Fi Ri is a normal subgroup of G. Since H /F is abelian, we see that F Ri is normal in H . Let Ei and Hi be Hall qi -complements of F and H , so F = Fi × Ei . There is a character ν ∈ Irr(Fi ) that is not linear. We see that ν extends to Fi CHi (ν). Since qi divides every degree in cd(H |ν) ⊆ cd(G), it follows that ri does not divide |Hi : CHi (ν)| and ri divides no degree in cd(CHi (ν)). As usual, we may use Gallagher’s theorem and Itô’s theorem to see that CHi (ν) contains a unique Sylow ri -subgroup of H . By conjugating, we may assume that Ri ⊆ CHi (ν).

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For any character µ ∈ Irr(Ei ), the stabilizer CHi (ν × µ) = CHi (ν) ∩ CHi (µ) contains some Sylow ri -subgroup for H . Since CHi (ν × µ) ⊆ CHi (ν), the uniqueness of Ri forces Ri ⊆ CH (ν × µ) ⊆ CH (µ), and hence, Ri stabilizes every character in Irr(Ei ). This implies that Ri centralizes Ei (see Lemma 3.2 (b) of [6]). It follows that Fi Ri is a normal subgroup of F Ri . Since Fi Ri is a Hall subgroup of F Ri , it is characteristic, and we use the fact that F Ri is normal in H to see that Fi Ri is normal in H . Therefore, P Fi Ri is a normal subgroup of G. It is not difficult to see that (P Fi Ri ) has two connected components: {ri } and {qi , p}. Looking at the classification in [7], we see that the only solvable groups K where (K) has two connected components and the Fitting subgroup of K/F (K) is not abelian has ρ(K) = {2, 3}. As |ρ(P Fi Ri )| = 3 > 2, this is a contradiction, and Fi is abelian. The next lemma deals with the situation where the primes dividing the degrees of the p-complement of G are precisely the primes in ρ. The lemma finds a subgroup whose degree graph has two connected components. In this lemma, it will be useful to have some additional notation. Let N be a normal subgroup of G. Then Irr(G|N) = {χ ∈ Irr(G) | N ⊆ ker(χ)}, and we define cd(G|N) = {χ (1) | χ ∈ Irr(G|N)}. Given a character θ ∈ Irr(N), we write cd(G|θ ) = {χ (1) | χ ∈ Irr(G|θ )}. If n is a positive integer, we define π(n) to be the set of primes dividing n. Lemma 4.3. Assume Hypothesis 4.1. Let H be a p-complement for G. Suppose that P is minimal normal in G, and ρ(H ) = ρ is not empty. Then there is a nonempty subset π ∗ of π so that (P H ) has two connected components: ρ and π ∗ . Furthermore, if |ρ| = n, then |π ∗ | ≥ 2n − 1. (In fact, P H satisfies the hypotheses of Example 2.4 of [7].) Proof. Since P is minimal normal in G, it follows that P ⊆ Z(P ). For any nonprincipal character α ∈ Irr(P ), we see that P CH (α) is the stabilizer of α in G. Using the Glauberman correspondence (Theorem 13.1 of [5]), we can find a character θ ∈ Irr(P |α) so that θ is CH (α)-invariant. On the other hand, θP = θ (1)α, and CH (θ ) = CH (α). The stabilizer of θ in G is P CH (α), and θ extends to H CH (α). It follows that p|H : CH (α)| divides every degree in cd(G|θ ), and so the prime divisors of |H : CH (α)| must lie in π. Also, we may use Gallagher’s theorem to see that ρ(CH (α)) ⊆ π . It follows that CH (α) contains a full Hall ρ-subgroup of H as a normal abelian subgroup. Now, α extends to P CH (α), its stabilizer in P H , and the prime divisors of degrees in cd(P H |α) must lie in π . Letting π ∗ be the set of prime divisors of degrees in cd(P H |P ), we conclude that π ∗ ⊆ π . Because ρ(H ) = ρ is nonempty, we use Itô’s theorem to see that H does not have a normal abelian Hall ρ-subgroup. Taking α as before, CH (α) < H , and π ∗ is not empty. We have cd(P H ) = cd(P H /P ) ∪ cd(P H |P ). We know that ρ(P H /P ) = ρ(H ) = ρ. In the previous paragraph, we showed that the degrees in cd(P H |P ) were divisible only by primes in π ∗ which is a subset of π. It follows that (P H ) has two connected components: ρ and π ∗ . Observe that P is a noncentral, abelian, normal Sylow p-subgroup of P H , and using Lemmas 3.1–3.6 of [7], the solvable

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groups with two connected components having such a subgroup satisfy the hypotheses of Example 2.4 of [7]. Let F and E/F be the Fitting subgroups of P H and P H /F . We know that the two connected components of (P H ) are π(|P H : E|) and π(|E : F |). Since P ⊆ F , we have that π(|P H : E|) = ρ(P H /F ) ⊆ ρ(H ) = ρ. This yields by the proof of Theorem 3 of [14] that if |ρ| = n, then |π ∗ | ≥ 2n − 1. The remaining lemmas in this section deal with groups where the neighbors of p are partitioned in two sets with no adjacencies between the sets. To make this more clear, we state the following hypothesis. Hypothesis 4.4. Assume Hypothesis 4.1. Suppose that π is the disjoint union of nonempty sets π1 and π2 , and assume that no prime in π1 is adjacent in (G) to any prime in π2 . This next lemma shows that G/P must have a normal abelian Hall π -complement. Lemma 4.5. Suppose Hypothesis 4.4 holds. Then π(G/P ) ⊆ π which is equivalent to saying G/P has a normal abelian Hall π -complement. Proof. Note that the equivalence of the conditions in the conclusion is due to Itô’s theorem. Let H be a Hall p-complement of G. Suppose that there exist nonlinear characters θ1 , θ2 ∈ Irr(P ) so that p1 ∈ π1 divides |H : CH (θ1 )| and p2 ∈ π2 divides |H : CH (θ2 )|. It follows that ppi divides every degree in cd(G|θi ), and thus, |H : CH (θi )| is divisible only by primes in πi . In particular, CH (θi ) contains a Hall πi -complement of H . Furthermore, θi extends to P CH (θi ), and using Gallagher’s theorem ρ(CH (θi )) ⊆ πi . By Itô’s theorem, CH (θi ) has a normal abelian Hall πi complement. Now, both CH (θ1 ) and CH (θ2 ) will normalize a Hall π -complement of H . Conjugating if necessary, we may assume that they normalize the same Hall π-complement of H . Since |H : CH (θ1 )| and |H : CH (θ2 )| are relatively prime, we conclude H = CH (θ1 )CH (θ2 ), and H has a normal abelian Hall π-complement, and the lemma is proved in this case. We may assume that we do not have a pair a characters like θ1 and θ2 in the previous paragraph. Because π1 and π2 are nonempty, there exist characters χ1 , χ2 ∈ Irr(G) so that ppi divides χi (1) for i = 1, 2 and primes pi ∈ πi . For each i, let γi be an irreducible constituent of (χi )P . We see that p divides γi (1), so γi is not linear. Using the previous paragraph, we cannot have both CH (γ1 ) < H and CH (γ2 ) < H . This implies that H = CH (γi ) for either i = 1 or i = 2. We fix the value i so that H = CH (γi ), and we note that γi extends to G. Applying Gallagher’s theorem, ρ(G/P ) = ρ(H ) ⊆ π . In addition to assuming Hypothesis 4.4, we now assume that the only primes dividing degrees in cd(G/P ) all lie in π1 . We show that the product of a Hall π2 subgroup of G with P will be normal in G and have an abelian quotient. Lemma 4.6. Suppose Hypothesis 4.4 holds. Assume that ρ(G/P ) ⊆ π1 , and let P2 be a Hall π2 subgroup of G. Then G ⊆ P P2 .

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Proof. Since no prime in π2 lies in ρ(G/P ), we use Itô’s theorem to see that G/P has a normal abelian Hall π2 -subgroup, and so P P2 is normal in G. Choose the character ψ ∈ Irr(G) so that ψ(1) is divisible by pp2 for some prime p2 ∈ π2 . Note that ψ(1) will be divisible only by primes in {p} ∪ π2 , and |G : P P2 | is divisible by no primes in {p} ∪ π2 . It follows that ψ(1) is relatively prime to |G : P P2 |, and so, ψP P2 is irreducible by Corollary 11.29 of [5]. By Gallagher’s theorem, ψ(1)a ∈ cd(G) for every degree a ∈ cd(G/P P2 ). Since ρ(G/P P2 ) contains no primes in {p} ∪ π2 , we see that 1 is the only degree in cd(G/P P2 ), and so, G/P P2 is abelian. This remaining lemma looks at the case where both π1 and π2 have primes dividing degrees in cd(G/P ). Lemma 4.7. Suppose Hypothesis 4.4 holds. Assume that ρ(G/P ) intersects both π1 and π2 nontrivially. Then one of the following holds: (1) G/P has a central Hall ρ-subgroup. (2) There is a prime r ∈ ρ so that G/P has a noncentral Sylow r-subgroup, and every other prime q ∈ ρ is adjacent in (G) to every prime in ρ(G/P ). Proof. By Lemma 4.5, we know that ρ(G/P ) ⊆ π . Since (G/P ) is a subgraph of (G), we see that (G/P ) has two connected components: ρ(G/P ) ∩ π1 and ρ(G/P ) ∩ π2 . If G/P has a central Hall ρ-subgroup, then the result holds; so we assume that G/P does not have a central Hall ρ-subgroup. By Theorem 5.5 of [7], G/P has at most one noncentral normal Sylow subgroup, and we will let r be the prime so that G/P has a noncentral Sylow r-subgroup. Since G/P has a normal abelian Hall ρ-subgroup (by Lemma 4.5), we see that if r is not in ρ, then G/P would have a central Hall ρ-subgroup, and we assumed this did not happen. Thus, we have r ∈ ρ. If ρ = {r}, then we are done, so we assume that ρ contains primes other than r. Looking at Lemmas 3.1–3.6 of [7], we see that if K is a solvable group where (K) is disconnected and K has a noncentral Hall ρ(K)-complement, then K satisfies the hypotheses of Example 2.4 of [7]. Therefore, G/P satisfies the hypotheses of Example 2.4 of [7]. Let H be a p-complement for G, and note that H ∼ = G/P . The Fitting subgroup of H has the form V × Z where V is a minimal normal r-subgroup of H and Z = Z(H ) (see Lemma 3.4 of [7]). Anything centralizing V will centralize V Z, and as V Z is the Fitting subgroup of H , we conclude that CH (V Z) = V Z. We now consider a prime q ∈ ρ with q = r. We need to show that q is adjacent to every prime in ρ(G/P ). For the purpose of finding a contradiction we assume that q is not adjacent to the prime s ∈ ρ(G/P ). Since q ∈ ρ ⊆ ρ(G), we see that the Sylow q-subgroup Q of H is not central in G. Since Q is central in H , we see that Q must act nontrivially on P , and this implies that Q acts nontrivially on P /P . Let C = CH ([P , Q]), and observe that C = CH ([P , Q]P /P ). Consider a nonprincipal character λ ∈ Irr([P , Q]P /P ). As q divides |H : CH (λ)|, it follows that s does not divide |H : CH (λ)|. Also, λ must extend to P CH (λ), so by Gallagher’s theorem s ∈ ρ(CH (λ)). Using Itô’s theorem, CH (λ) has a normal abelian Sylow s-subgroup of H . Now, C ⊆ CH (λ).

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If C contains a Sylow s-subgroup of H , then H would have a normal abelian Sylow s-subgroup which is a contradiction since s ∈ ρ(H ). Thus, a Sylow s-subgroup of H acts nontrivially on [P , Q]P /P . We have shown that H satisfies the hypotheses of Lemma 1 of [8] with respect to the prime s. By that result, either H /C ∼ = SL2 (3) or GL2 (3) with |[P , Q]P : P | = 9 or H /C has a normal abelian subgroup that acts irreducibly on [P , Q]P /P . Since |P | and |H | are coprime, we do not have H /C ∼ = SL2 (3) or GL2 (3). Since H /C has an abelian normal subgroup that acts irreducibly on [P , Q]/P , the Fitting height of H /C is at most 2 (see Theorem 19.3 of [12]). If V ⊆ C, then V C = V × C since V is minimal normal in H . We obtain C ⊆ Z. It follows that H /C has Fitting height 3, and this is a contradiction, so V ⊆ C ⊆ CH (λ). Let S be the Sylow s-subgroup of H contained in CH (λ), and recall that S is a normal subgroup of CH (λ). Because V is also a normal subgroup of CH (λ), we obtain V S = V × S, and S ⊆ CH (V ) = V × Z. We conclude that S is central in H . This contradicts the fact that s ∈ ρ(H ), and the lemma is proved.

5. Disconnected degree graphs In this section, we further the study of disconnected degree graphs. We begin with a general lemma about groups where the connected components of the degree graph have specified sizes. This next lemma should be viewed as a refinement of Theorem 5.4 of [7]. Lemma 5.1. Let G be a solvable group, and suppose that (G) has two connected components π and ρ where |π| = n > 1 and |ρ| = 2n − 1. Let F and E/F be the Fitting subgroups of G and G/F , respectively, and let Z be the center of G. Then G satisfies the hypotheses of Example 2.4 of [7], π = π(|G : E|), ρ = π(|E : F |), and |G : E| is square-free. Furthermore, there exists a prime q so that F /Z is an elementary abelian q-group of order q |G:E| and E/F is cyclic of order (q |G:E| − 1)/(q − 1). Finally, if m is a divisor of |G : E|, then the number of prime divisors of (q m − 1)/(q − 1) equals the number of nontrivial divisors of m which equals 2|π(m)| − 1. Proof. Reading the proof of Theorem 5.4 in [7], we see that G satisfies the hypotheses of Example 2.4 of [7], and from that example, we see that G has Fitting height 3. In addition from Lemma 3.4 of [7], we have (|G : E|, |E : F |) = 1, π = π(|G : E|) and ρ = π(|E : F |), and |F : Z| = q |G:E| for some prime power q and (q |G:E| −1)/(q−1) divides |E : F | which divides q |G:E| − 1. We wish to apply the Zsigmondy prime theorem to |E : F |. To do this, we need to see that we do not have any of the exceptional cases of the Zsigmondy prime theorem. Suppose 2 divides |G : E| and q is odd. It follows that 2 divides q + 1 = (q 2 −1)/(q −1). On the other hand, (q 2 −1)/(q −1) divides (q |G:E| −1)/(q −1) which

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divides |E : F |. Thus, 2 divides |G : E| and |E : F |, and this is a contradiction. Either |G : E| is odd, or q is even. Also, if q = 2 and 6 divides |G : E|, then 3 = (22 − 1)/(2 − 1) is a divisor of (2|G:E| − 1)/(2 − 1), which divides |E : F |. This is also a contradiction. We have shown that the exceptions to the Zsigmondy prime theorem cannot occur in our situation. We may apply the Zsigmondy prime theorem to see that the number of prime divisors of (q |G:E| − 1)/(q − 1) that do not divide q − 1 is at least the number of nontrivial divisors of |G : E|. Since |E : F | has 2n − 1 prime divisors, none of whom divide |G : E|, we see that |G : E| has at most 2n − 1 nontrivial divisors. Because n primes divide |G : E|, |G : E| has at least 2n − 1 nontrivial divisors. If some prime divides |G : E| with multiplicity more than 1, then |G : E| would have more divisors, so |G : E| is square-free. Furthermore, we see that the Zsigmondy primes account for all of the prime divisors of |E : F |, and so each prime dividing |E : F | must be a Zsigmondy prime for (q, m) where m is a nontrivial divisor of |G : E|. Further, this implies that the number of prime divisors of (q m − 1)/(q − 1) equals the number of nontrivial divisors of m which is 2|π(m)| − 1. We conclude that no prime dividing q − 1 divides |E : F |, and thus, |E : F | = (q |G:E| − 1)/(q − 1). If q is not prime, then the Zsigmondy prime theorem could be applied to the divisors of q, to see that (q m − 1)/(q − 1) would have more prime divisors than the number of nontrivial divisors of m, and so, q must be prime. In the next lemma, we consider adjacencies in (G) when G has a normal subgroup M so that (G/M) is similar to the degree graph studied in Lemma 5.1. Lemma 5.2. Let G be a group, and suppose M is a normal subgroup of G so that G/M is a solvable group that satisfies the hypotheses of Example 2.4 of [7]. Let F /M and E/F be the Fitting subgroups of G/M and G/F , respectively, and let r be the prime power so that |F /Z| = r |G:E| , where Z/M = Z(G/M). Suppose that there exist primes p ∈ π and q ∈ ρ and a degree a ∈ cd(G) so that pq divides a. If s ∈ π is not adjacent to q in (G), then (r |G:E| − 1)/(r |G:E|/s − 1) divides a. Proof. Let S/F be a Sylow s-subgroup of G/F . From [7], we see that we may view F /Z as the additive group of the field F of order r |G:E| , E/F as a subgroup of the multiplicative group of F, and G/E as the Galois group of F. Also by [7], the order of E/F is divisible by (r |G:E| − 1)/(r − 1), and |E/F | divides (r |G:E| − 1). We know that the action of S/F on E/F is the action coming from the Galois automorphism of order s. In F, the automorphism of order s has a fixed field of order r |G:E|/s , and looking at the associated elements in the multiplicative group of F, we obtain |E : C| = (r |G:E| − 1)/(r |G:E|/s − 1), where C/F = CE/F (S/F ). On the other hand, there exists a character χ ∈ Irr(G) with χ (1) = a. Let λ be an irreducible constituent of χF . Write T for the stabilizer of λ in G. Since all the Sylow subgroups of G/F are cyclic, it follows that all the Sylow subgroup of T /F are cyclic and λ extends to λˆ ∈ Irr(T ). There exists a character τ ∈ Irr(T /F ) so that

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ˆ G = χ . Since G/F has a normal abelian Hall ρ-subgroup, q ∈ ρ(T /F ), and (τ λ) thus, q does not divide τ (1). We deduce that q divides |G : T | or q divides λ(1). Because s does not divide χ(1), it will not divide |G : T |. Thus, T /F contains a Sylow s-subgroup of G/F . By conjugating, we may assume that S ⊆ T . Now, q divides every degree in cd(G|λ) and qs divides no degree in cd(G). We conclude that s divides no degree in cd(T /F ). Hence, T /F has a normal abelian Sylow s-subgroup, and so S/F is normal in T /F , and in particular, E ∩ T ⊆ C. This implies that |E : C| divides |G : T |. Since |G : T | divides a and |E : C| = (r |G:E| − 1)/(r |G:E|/s − 1), we obtain the desired conclusion.

6. Graph (2) of Figure 1 We now prove that Graph (2) of Figure 1 does not arise as the degree graph of a solvable group. Theorem 6.1. There is no solvable group G where (G) is the graph found in (2) of Figure 1. p1 p3 HH p2

p4

p5

Figure 3. Labeled Graph of Theorem 6.1

Proof. Suppose that G is a counterexample with |G| minimal. Label the primes in ρ(G) as in Figure 3. Our first goal is to show that G does not have any normal nonabelian Sylow subgroups. If P is a normal Sylow p-subgroup for some prime p, we have shown in Lemma 3 of [11] that cd(G/P ) = cd(G) \ {p}. From Theorem 2.1, we know that the graphs in Figure 2 cannot be the degree graphs for subgroups of G, and the only disconnected graph with 4 vertices that satisfies Pálfy’s condition has an isolated vertex and the remaining three vertices form a complete subgraph. We conclude that none of the subgraphs of (G) with vertex sets of {p1 , p2 , p3 , p5 } or {p1 , p2 , p4 , p5 } can arise as (H ) for a solvable group H . If G had a normal Sylow p3 -subgroup or normal Sylow p4 -subgroup P , then (G/P ) would be a subgraph of (G) with one of these two sets as its vertex set. Thus, G cannot have a normal Sylow p3 -subgroup nor a normal Sylow p4 -subgroup. Claim 1. Suppose H is a group where ρ(H ) = ρ(G), (H ) is a subgraph of (G), and |H | < |G|. Then (H ) has two connected components: {p1 , p2 } and {p3 , p4 , p5 }. In particular, H has no nonabelian normal Sylow subgroups. Proof. By minimality in the choice of G, we know (H ) = (G). Applying Pálfy’s condition to p1 , p2 , and p5 in (H ), we see that p1 and p2 must be adjacent in (H ).

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If (H ) is disconnected, we have already seen that Pálfy’s condition forces each component to be a complete graph, and so the two connected components must be {p1 , p2 } and {p3 , p4 , p5 }. In Lemma 5.1, we saw that H must satisfy the hypotheses of Example 2.4 of [7], and so, every normal Sylow subgroup of H is abelian. We assume (H ) is connected, and we work for a contradiction. If p3 and p4 are not adjacent, then (H ) will be a subgraph of Graph (1) of Figure 1. The only connected proper subgraph of Graph (1) of Figure 1 has diameter 4 which contradicts Pálfy’s condition, and so, (H ) will be Graph (1) of Figure 1 which was disallowed by Corollary 3.3. Therefore, p3 and p4 must be adjacent in (H ). Applying Pálfy’s condition to the sets {p1 , p4 , p5 } and {p2 , p3 , p5 }, we see that there must be an edge between p3 and p5 and an edge between p4 and p5 in (H ). Finally, since (H ) is connected, it must have an edge between p1 and p3 or between p2 and p4 . In fact, (H ) must contain both of these edges, or else it has diameter 3 which is not allowed by Theorem 1.2. We conclude (H ) = (G) which is a contradiction. We now work to prove G = O p3 (G) = O p4 (G). We show that this is true for p3 , and a similar argument will work to show that it is true for p4 . By the minimality of G, if H is a proper normal subgroup of G, then (H ) is a proper subgraph of (G). If p3 is removed from (G), we get Graph (2) from Figure 2 which does not occur by Theorem 2.1. On the other hand, removing one or more edges incident to p3 from (G), we obtain a graph that violates Pálfy’s condition, has diameter 3, or is Graph (1) of Figure 1, and we know that none of these graphs occur by Theorem 1.1, Theorem 1.2, and Corollary 3.3. We then use Lemma 3.1 to see that O p3 (G) = G. We show that G does not have a normal Sylow p1 -subgroup. A similar argument shows that G does not have a normal Sylow p2 -subgroup. We assume P is a normal Sylow p1 -subgroup of G, and we look for a contradiction. By Lemma 4.5, we see that ρ(G/P ) ⊆ {p2 , p3 }. If G/P is abelian, then O p4 (G) < G which is a contradiction. If |ρ(G/P )| = 1, then by Lemma 4.6, we see that G ⊆ P Q where Q is either a Sylow p2 -subgroup or Sylow p3 -subgroup of G. In either case, we again get O p4 (G) < G which is our contradiction. The final possibility is that ρ(G/P ) = {p2 , p3 }. In this case, we can apply Lemma 4.7 where ρ = {p4 , p5 }. If G/P has a central Sylow p4 -subgroup, then O p4 (G) < G, a contradiction. Thus, we have conclusion (2) of Lemma 4.7 with r = p4 . This implies that p5 must be adjacent to p2 which is a contradiction. Thus, G does not have a normal Sylow p1 -subgroup. We have shown that if G has a nonabelian Sylow p-subgroup for some prime p, then p must be p5 . We now eliminate this possibility. Claim 2. G does not have a normal Sylow p5 -subgroup. Proof. Suppose that G has a normal Sylow p5 -subgroup P . Let H be a p5 -complement for G, and let C = CH (P ). Note that C is normal in G and P C = P × C. Thus, any prime in ρ(C) must be adjacent to p5 in (G). It follows that ρ(C) ⊆ {p3 , p4 }, and hence, p1 p2 divides |H : C|. Since H /C acts faithfully on P , the primes in π(|H : C|) must lie in ρ(G/C), and so, {p1 , p2 } ⊆ ρ(G/C). Looking at the subgraphs of (G)

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that are allowed to be the degree graphs of solvable groups, we conclude that either ρ(G/C) = {p1 , p2 , p5 } or ρ(G/C) = ρ(G). Suppose ρ(G/C) = {p1 , p2 , p5 }. As we stated in the last paragraph, p3 and p4 will not divide |H : C|. Fix a nonlinear character θ ∈ Irr(P ). Then θ extends to θ × 1C in P × C. Given a character χ ∈ Irr(G|θ × 1C ), we see that p5 divides χ (1) and χ (1)/θ(1) divides |G : P C| = |H : C|. Since p1 and p2 do not divide χ (1) and the only primes that possibly divide |H : C| are p1 and p2 , we conclude that χP C = θ × 1C . This implies that χP = θ. Using Gallagher’s theorem, we have χ (1)a ∈ Irr(G) for every degree a ∈ Irr(H ) = Irr(G/P ), and so, ρ(H ) ⊆ {p3 , p4 }. Then H has a normal abelian Hall {p1 , p2 }-subgroup, and either O p3 (G) < G or O p4 (G) < G. This is a contradiction, so ρ(G/C) = ρ(G). On the other hand, G/C has a normal nonabelian Sylow subgroup P C/C. In view of Claim 1, we conclude that C = 1. Therefore, H acts faithfully on P , and P is the Fitting subgroup of G. Consider a nonlinear character θ ∈ Irr(P ) and any character χ ∈ Irr(G|θ ). We know that p5 divides both θ(1) and χ(1). It follows that p1 and p2 do not divide χ (1) nor χ (1)/θ (1). As this is true for all characters χ ∈ Irr(G|θ ), we use Theorem 12.9 of [12] to see that G/P has an abelian Hall {p1 , p2 }-subgroup. Let F be the Fitting subgroup of H , and observe that F P /P = F (G/P ). There exists a degree in cd(G) that is divisible by all the prime divisors of |F |. Suppose that p1 divides |F |. We know that p4 does not divide |F |. If p3 does not divide |F |, then F is a {p1 , p2 }-subgroup. Since H has an abelian Hall {p1 , p2 }-subgroup, it follows that F is the Hall {p1 , p2 }-subgroup of H . We have either O p3 (G) < G or O p4 (G) < G, and this is a contradiction. Thus, p3 must divide |F |, so F is a {p1 , p3 }-subgroup. We can find a character γ ∈ Irr(P F ) so that p1 p3 divides γ (1). For any character ψ ∈ Irr(G|γ ), we see that p1 p3 divides ψ(1), so p2 and p4 do not divide ψ(1) or ψ(1)/γ (1). From Theorem 12.9 of [12], H /F has an abelian Hall {p2 , p4 }-subgroup. Let M = O{p1 ,p3 } (H ), and note that F ⊆ M. Set E/M to be the Fitting subgroup of H /M. Since H /M has an abelian Hall {p2 , p4 }-subgroup, E/M is the Hall {p2 , p4 }subgroup of H /M. It is not difficult to see that ρ(P E) = ρ(G), and by Claim 1, G = P E. Because E/M is abelian, O p4 (G) < G, and this is a contradiction. We conclude that p1 does not divide |F |, and a similar argument shows that p2 will not divide |F |. This implies that F is a {p3 , p4 }-group and p1 , p2 ∈ ρ(H ). We now apply Lemma 4.2 with π = {p3 , p4 } and ρ = {p1 , p2 } to conclude that ρ(H ) = {p1 , p2 }. Suppose that X is a normal subgroup of G that is contained in P so that P /X is a chief factor for G. We know that ρ(G/X) ⊇ ρ(G/P ) = ρ(G)\{p5 } by Lemma 3 of [11], and p5 ∈ ρ(G/X) since G/X has a nonabelian Sylow p5 -subgroup. It follows that ρ(G) = ρ(G/X). By Claim 1, |G/X| = |G|, and X = 1. This implies that P is minimal normal in G. Finally, Lemma 4.3 yields a contradiction since it implies that |π| ≥ 22 − 1 = 3, and we know |π| = 2. We have shown that G has no normal nonabelian Sylow subgroups. Let F be the Fitting subgroup of G. We see that ρ(G) = π(|G : F |), and thus, ρ(G) = ρ(G/(G)) where (G) is the Frattini subgroup of G. Our next goal is to show that (G) = 1.

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We make a refinement of Claim 1. We prove that if M is normal in G with ρ(G/M) = ρ(G), then M = 1. We suppose that M > 1 for the purpose of finding a contradiction. By Claim 1, (G/M) has two connected components: π = {p1 , p2 } and ρ = {p3 , p4 , p5 }. In light of Lemma 5.1, G/M satisfies the hypotheses of Example 2.4 of [7]. Letting F /M and E/F be the Fitting subgroups of G/M and G/F , respectively, we know that |G : E| = p1 p2 and there is a prime q so that |E : F | = (q p1 p2 − 1)/(q − 1). Also, by Lemma 5.1, we see that (q p1 − 1)/(q − 1) is a prime power, so (q p1 p2 − 1)/(q p1 − 1) is divisible by at least 2 primes. Notice that p1 is adjacent to p3 , and p2 is not adjacent to p3 , so we may use Lemma 5.2 to see that p1 must be adjacent to all the primes that divide (q p1 p2 − 1)/(q p1 − 1) which is a contradiction since p1 is adjacent to only p3 in ρ and at least 2 primes in ρ divide (q p1 p2 − 1)/(q p1 − 1). Since ρ(G/(G)) = ρ(G), this implies that (G) = 1. By Hilfssatz III.4.4 of [2], there is a subgroup H so that G = H F and H ∩ F = 1. We write E for the Fitting subgroup of H . Claim 3. F is a minimal normal subgroup of G. Proof. Suppose that there is a normal subgroup N of G so that 1 < N < F . By a result of Gaschütz (Satz III.4.5 of [2]), there is a normal subgroup M of G so that F = N ×M. Since N > 1 and M > 1, we have ρ(G/N) < ρ(G) and ρ(G/M) < ρ(G). For any prime p ∈ ρ(G) \ ρ(G/N), we know that G/N has a normal abelian Sylow p-subgroup. The class of finite groups with an abelian and normal Sylow p-subgroup is a formation, so p must lie in ρ(G/M). (For information regarding formations, we suggest consulting [1].) We conclude that ρ(G) = ρ(G/N) ∪ ρ(G/M). If p ∈ ρ(G) \ ρ(G/N), then p is not in ρ(G/F ) = ρ(H ); so E must then contain the Sylow p-subgroup of H . Since p ∈ ρ(G), it follows that p divides |H |, and so, p divides |E|. Similarly, any prime in ρ(G) \ ρ(G/M) will divide |E|. Recall that cd(G) contains a degree divisible by all the prime divisors of |EF : F | = |E|. We conclude that ρ(G) \ (ρ(G/M) ∩ ρ(G/N)) lies in a complete subgraph of (G). Therefore, ρ(G) \ (ρ(G/M) ∩ ρ(G/N)) lies in one the subsets: (1) {p1 , p2 }, (2) {p3 , p4 , p5 }, (3) {p1 , p3 }, or (4) {p2 , p4 }. We begin by supposing that (1) occurs. This implies that E contains a Hall {p1 , p2 }subgroup of H . Since cd(G) has a degree divisible by all the primes dividing |E|, we see that |E| is divisible by no other primes, and E is the Hall {p1 , p2 }-subgroup of H . We can find a character χ ∈ Irr(G) with p1 p2 dividing χ (1). Let θ be an irreducible constituent of χF E . Now, χ(1)/θ(1) divides |G : F E| and χ (1) is relatively prime to |G : F E|. We determine that χF E = θ. Since p1 and p2 divide θ (1) and the only possible prime divisors of a ∈ cd(G/F E) are p3 , p4 , or p5 , we conclude via Gallagher’s theorem that cd(G/F E) = {1} and G/F E is abelian. We now have that O p3 (G) < G, a contradiction. Thus, (1) cannot occur. Suppose (2) occurs. This implies that ρ(G/N) ∩ ρ(G/M) ⊇ {p1 , p2 }. Since ρ(G/N ) and ρ(G/M) are not ρ(G), {p1 , p2 , p3 , p5 }, nor {p1 , p2 , p4 , p5 }, we may assume without loss of generality that ρ(G/N) = {p1 , p2 , p5 } and ρ(G/M) =

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{p1 , p2 , p3 , p4 }. Now, (G/N) has two connected components: {p1 , p2 } and {p5 }. From Theorem 5.5 of [7], G/N has either a central Sylow p3 -subgroup or a central p4 -subgroup. This would imply O p3 (G) < G or O p4 (G) < G which is a contradiction. Thus, (2) cannot occur. Suppose (3) occurs. This implies that ρ(G/N) ∩ ρ(G/M) ⊇ {p2 , p4 , p5 }. We know that one of ρ(G/N) or ρ(G/M) must contain p1 , say ρ(G/N). Since ρ(G/N ) < ρ(G), this implies that ρ(G/N) = {p1 , p2 , p4 , p5 }. We have previously seen that this set cannot be the degree set for a subgroup of G, so this is a contradiction. Thus, (3) cannot occur. A similar proof shows that (4) cannot occur, and the claim is proved. Let p be a prime divisor of |E|. We know via Lemma 2.10 of [9] that every degree in cd(G|F ) is divisible by all the prime divisors of |E|. There is a prime q ∈ ρ(G) that is not adjacent to p in (G). Consider a nonprincipal character λ ∈ Irr(F ). We know that every degree in cd(G|λ) is divisible by p, so CH (λ) contains a Sylow qsubgroup of H as a normal subgroup. Using Lemma 1 of [8], either (1) H ∼ = SL2 (3) or GL2 (3) or (2) H as an normal abelian subgroup that acts irreducibly on F . We know that ρ(G) = π(|H |). If (1) occurs, then π(|H |) = {2, 3} which is not allowed since |ρ(G)| = 5. Thus, (2) must occur. In case (2), H /E is abelian, and |H : E| ∈ cd(G). There is a degree in cd(G) divisible by all the prime divisors of |E|. Any prime in π(|H : E|) ∩ π(|E|) would be adjacent in (G) to all the primes in ρ(G). Since (G) has no such vertex, |H : E| and |E| are relatively prime. If p3 divides |H : E|, then O p3 (H ) < H and O p3 (G) < G. Since this was excluded, p3 does not divide |H : E| and p3 divides |E|. We have that p2 divides |H : E|. We then see that p5 divides |E|. This yields π(|H : E|) = {p1 , p2 } and π(|E|) = {p3 , p4 , p5 }. For i = 3, 4, 5, write Ei for the Sylow pi -subgroup of E. Let A be a Hall p3 complement for H , so H = E3 A and E3 ∩ A = 1. We can find a character χ ∈ Irr(G) with p1 p3 dividing χ(1). We know that p4 does not divide χ (1), so χ ∈ Irr(G|F ). It follows that χ ∈ Irr(G/F ) and χH is irreducible. Let θ be an irreducible constituent of χE3 . Now, p3 divides θ(1), so E3 is not abelian. The stabilizer of θ in H is E3 CA (θ ). Observe that E4 ⊆ CA (θ). Using the usual arguments, p2 ∈ ρ(CA (θ )), and CA (θ ) contains a Sylow p2 -subgroup P2 of H as a normal abelian subgroup. We see that P2 centralizes E4 , and P2 E = E3 E5 P2 × E4 . Since H /E is abelian, P2 E is normal in H . Let K = E3 E5 P2 , and note that F K is normal in G. Furthermore, (F K) has two connected components: {p2 } and {p3 , p5 }. Thus, F K is one of the examples in [7], but this is a contradiction as the only examples where EF /F is not abelian are Examples 2.2 and 2.3 of [7] and in both of those cases ρ(G) = {2, 3}. This is the final contradiction, and the theorem is proved.

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7. An example of Graph (3) of Figure 1 Example 7.1. We now construct a solvable group G so that (G) is the graph in Figure 1 (3). The group we construct will be similar to those groups constructed in [6]. We begin with a prime p1 with the property that 2p1 − 1 = p2 , a Mersenne prime, and (2p1 + 1)/3 is a power of a third prime p3 where the primes p1 , p2 , and p3 are different from 2 and 3. For example, we can take p1 = 5, p2 = 31, and p3 = 11. We take F to be the field of size 2p1 and E to be the field extension of F whose size p is 22p1 . Observe that E has the field automorphism σ defined by σ (α) = α 2 1 for all 2 α ∈ E. We know that F is the fixed field under σ and σ = 1. We take R to be the skew polynomial ring with coefficients in E and indeterminate X. We define Xα = α σ X to be the relation for multiplication involving an element α ∈ E and the indeterminate X. We observe that X3 R = RX3 is an ideal of R, and we have the quotient R = R/RX 3 . We let x denote the image of X in R, and we note that the Jacobson radical of R is Rx. We know that Q = 1 + Rx = {1 + αx + βx 2 | α, β ∈ E} is a group of order 4p 2 1 . It is not difficult to see that commutator [1 + α1 x + β1 x 2 , 1 + α2 x + β2 x 2 ] = 1 + (α1 α2σ + α1σ α2 )x 2 for αi , βi ∈ E. Since (α1 α2σ + α1σ α2 )σ = α1σ α2 + α1 α2σ , it is not difficult to show that Q = {1 + δx 2 | δ ∈ F}. Let C be the multiplicative group of E, so C is cyclic of order 22p1 − 1. Take G to be the Galois group of E over its base field, so G is a cyclic group of 2p1 . One can p show that C acts on Q via (1 + αx + βx 2 )c = 1 + αcx + βc2 1 +1 x 2 and G acts on R via (1 + αx + βx 2 )τ = 1 + α τ x + β τ x 2 for τ ∈ G. Also, G acts on C in the natural manner. Let C1 be the subgroup of C of order 2p1 + 1 and C2 the subgroup of order 2p1 − 1 = p2 . We see that the action of C2 on Q is a Frobenius action. On the other hand, let Z be the centralizer in Q of C1 , and we see that Z = {1 + βx 2 | β ∈ E}. By Fitting’s lemma, Q/Q = [Q, C1 ]Q /Q × Z/Q . We take P = [Q, C1 ]Q . Observe that |Q : P | = |Z : Q | = 2p1 , so P has order 23p1 . It is not difficult to show that Z(P ) = P = Q and that C and G both normalize P . We define G to be the semi-direct product P CG. An easy computation shows cd(CG) = {1, 2, p1 , 2p1 }. Note that C acts transitively on the nonprincipal characters in Irr(P /P ). If δ is a nonprincipal character in Irr(P /P ), then δ lies in an orbit of size 22p1 − 1. We may assume that the stabilizer of δ in G is P G. Since G is cyclic, we see that Irr(P G|δ) consists of extensions of δ. We conclude that cd(G|δ) = {22p1 − 1}, and thus, cd(G/P ) = {1, 2, p1 , 2p1 , 22p1 − 1}. Suppose λ is a nonprincipal character in Irr(P ). We see that C2 acts transitively on the nonprincipal characters of Irr(P ), so λ lies in an orbit of size 2p1 −1 = p2 . Without loss of generality, we may assume that the stabilizer of λ in G is P C1 G. Observe that P /P is a chief factor in P C1 . Using Problem 6.12 of [5], we see that λ must be fully-ramified with respect to P /P . Let λˆ be the unique irreducible constituent of λP . It follows that the stabilizer of λˆ in G is P C1 G. Since all the Sylow subgroups of C1 G are cyclic, λˆ will extend to P C1 G. One can show that cd(C1 G) = {1, 2, 2p1 }. By

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ˆ = Gallagher’s theorem (Corollary 6.17 of [5]), we obtain cd(P C1 G|λ) = cd(P C1 G|λ) p p +1 p +1 p p +1 p +1 1 1 1 1 1 1 ,2 p1 }, and thus, cd(G|λ) = {2 p2 , 2 p2 , 2 p1 p2 }. Since all {2 , 2 of the nonprincipal characters of Irr(P ) are conjugate, we obtain cd(G) = {1, 2, p1 , 2p1 , 22p1 − 1, 2p1 p2 , 2p1 +1 p2 , 2p1 +1 p1 p2 }. Since 22p1 − 1 = p2 (2p1 + 1) and the only prime divisors of 2p1 + 1 are 3 and p3 , we obtain the graph in Figure 4. 2

p1

HH

p2

HH

p3

3

Figure 4. Graph of Example

References [1]

K. Doerk and T. Hawkes, Finite Soluble Groups, de Gruyter Exp. Math. 4, Walter de Gruyter, Berlin 1992.

[2]

B. Huppert, Endliche Gruppen I, Springer-Verlag, Berlin 1983.

[3]

B. Huppert, Research in representation theory at Mainz (1984–1990), in Representation Theory of Finite Groups and Finite-Dimensional Algebras, ed. by G. O. Michler and C. M. Ringel (Bielefeld, May 15–17, 1991), Progr. Math. 95, Basel 1991, 17–36.

[4]

B. Huppert, Character Theory of Finite Groups, de Gruyter Exp. Math. 25, Walter de Gruyter, Berlin 1998.

[5]

I. M. Isaacs, Character Theory of Finite Groups, Academic Press, New York 1976.

[6]

I. M. Isaacs, Coprime group actions fixing all nonlinear irreducible characters, Canad. J. Math. 41 (1989), 68–82.

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M. L. Lewis, Solvable groups whose degree graphs have two connected components, J. Group Theory 4 (2001), 255–275.

[8]

M. L. Lewis, Bounding Fitting heights of character degree graphs, J. Algebra 242 (2001), 810–818.

[9]

M. L. Lewis, Derived lengths of solvable groups having five irreducible character degrees I, Algebr. Represent. Theory 4 (2001), 469–489.

[10] M. L. Lewis, A solvable group whose character degree graph has diameter 3, Proc. Amer. Math. Soc. 130 (2002), 625–630. [11] M. L. Lewis, Solvable groups with character degree graphs having 5 vertices and diameter 3, Comm. Algebra 30 (2002), 5485–5503.

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[12] O. Manz and T. R. Wolf, Representations of Solvable Groups, Cambridge University Press, Cambridge 1993. [13] P. P. Pálfy, On the character degree graph of solvable groups I: three primes, Period. Math. Hungar. 36 (1998), 61–65. [14] P. P. Pálfy, On the character degree graph of solvable groups, II: disconnected graphs, Studia Sci. Math. Hungar. 38 (2001), 339–355. [15] J. Zhang, On a problem by Huppert,Acta Scientiarum Naturalium Universitatis Pekinensis 34 (1998), 143–150. Mark L. Lewis, Department of Mathematical Sciences, Kent State University, Kent, OH 44242, U.S.A. E-mail: [email protected]

Heights of characters and defect groups Alexander Moretó

1. Introduction An important result in ordinary character theory is the Ito–Michler theorem, which asserts that a prime p does not divide the degree of any irreducible character of a finite group G if and only if G has a normal abelian Sylow p-subgroup. The famous Brauer’s height zero conjecture can be thought as the block version of this result. Given a block B with defect d the height of a character χ ∈ Irr(B) is the integer h such that χ (1)p = pa−d+h , where |G|p = pa . The height zero conjecture asserts that the height of any character χ ∈ Irr(B) is zero if and only if the defect group is abelian. The goal of this note is to discuss some possible extensions of the Ito–Michler theorem and its “block version”, the height zero conjecture. P. Fong [3] proved that all characters in a block with abelian defect group have height zero in a p-solvable group. The converse was proved by D. Gluck and T. Wolf in [4]. (However, both parts of the height zero conjecture remain open for arbitrary finite groups.) In their work in [4] Gluck and Wolf prove in fact a stronger result, namely that if e is the largest height of the characters in B, then the derived length of the defect group D cannot exceed 2e + 1 (we will keep this notation throughout this note). This generalized a result of Isaacs [8] which asserts that the derived length of a Sylow p-subgroup of a p-solvable group does not exceed 2f + 1, where pf is the largest p-part of the degrees of the irreducible characters of G (we will also maintain this notation in the remaining of this paper). The following conjecture asserts that these results should hold for arbitrary finite groups. Conjecture A. Let G be a finite group. Then the derived length of a Sylow p-subgroup is bounded in terms of f , where p f is the largest p-part of the degrees of the irreducible characters of G. More precisely, if B is a p-block of G with defect group D, then the derived length of D is bounded in terms of the largest height of the irreducible characters in B. It is clear that the first statement of Conjecture A follows from the second statement for the principal block. Isaacs bound was improved to a logarithmic bound in the case

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of solvable groups in [15] and it seems reasonable to hope that a logarithmic bound should hold in both statements for arbitrary groups. The work in [15] began as an attempt to relate p-parts of character degrees not with the derived length of a Sylow p-subgroup P but with the largest degree of the irreducible characters of P . It was conjectured in [12] that if we write b(P ) to denote the largest degree of the irreducible characters of P then logp b(P ) is bounded by a function of f , where pf is the largest p-part of the degrees of the irreducible characters of G. Now, we restate this conjecture and present its block version. Conjecture B. Let P be a Sylow p-subgroup of a finite group G and b(P ) = pb . Then b is bounded in terms of f . More precisely, if B is a p-block of G with defect group D then logp b(D) is bounded in terms of e. Again, the first statement of the conjecture follows from the second one for the principal block. It is tempting to conjecture that the bound logp b(D) ≤ 2e holds. For p ≤ 3, there are examples due to Isaacs in [8] and [12] that show that this bound would be best possible. It is clear that Conjecture B implies Conjecture A and that this bound would imply the desired logarithmic bound in Conjecture A (using Theorem D of [13], for instance). Our final conjecture relates the derived length of a Sylow p-subgroup (or of a defect group) and the number of p-parts of character degrees (or the number of heights of the characters in the block). Conjecture C. The derived length of a Sylow p-subgroup of a finite group is bounded in terms of the number of different p-parts of the degrees of the irreducible characters of the group. More precisely, the derived length of the defect group of a block B is bounded in terms of the number of different heights of the characters in B. As before, the second statement for the principal block implies the first statement and it is clear that this conjecture implies Conjecture A. It also seems tempting to conjecture that a logarithmic bound should hold but, among other things, one would need to solve the hard problem that says that the derived length of a p-group is bounded by a logarithmic function of the number of character degrees. In this note we discuss these conjectures for several important classes of finite groups: in Section 2 we consider p-solvable groups, in Section 3 the general linear groups in the defining characteristic, in Section 4 the symmetric groups and in Section 5 the sporadic groups. We prove that most of them hold for these groups. Before proceeding to do this, we consider the possible existence of reversed inequalities. There are p-groups of derived length 2 with arbitrarily many character degrees, so there is no hope that we can find any class of reversed bound in the situation of Conjectures A and C. Also, as Frobenius groups whose complement is an abelian p-group show, there is no possible reversed inequality for the first part of Conjecture B. Hence, it is perhaps surprising that something can be said about the second part of Conjecture B. It has recently been proved in [14] that if G is p-solvable, then the

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height of any character in a p-block with defect group D does not exceed 2 logp b(D) if p is odd and 5 logp b(D) if p = 2. However, as shown by an example due to G. Malle (see Section 4 of [14]) there is no hope to obtain any bound for arbitrary groups.

2. p-solvable groups As commented in the introduction, Conjecture A was proved for p-solvable groups in [8] and [4]. A logarithmic bound for the first statement for solvable groups was obtained in [15] For solvable groups, the first statement of Conjecture B was recently proved in Corollary B of [15] and that of Conjecture C in Theorem A of [13]. In the remainder of this section we discuss the block forms of these conjectures. Our first result reduces the p-solvable case of the modular form of Conjecture B to a problem in ordinary character theory. Theorem 2.1. The second statement of Conjecture B for p-solvable groups holds if the following is true: if Z is a cyclic central p -subgroup of a p-solvable group G, P ∈ Sylp (G), λ ∈ Irr(Z) and for any χ ∈ Irr(G|λ), χ (1)p ≤ p n , then logp b(P ) is bounded in terms of n. At first sight this statement looks very similar to that of the first part of Conjecture B, which was proved for solvable groups in [15]. However, it doesn’t seem possible to prove it using the methods of [15] Proof of Theorem 2.1. We begin working toward a proof of the second statement of Conjecture B for p-solvable groups. Using an argument due to Fong (see Theorems 9.14 and 10.20 of [16] or the first paragraph of the proof of Theorem A of [14]) we may assume that the defect group of B is a Sylow p-subgroup of G and that Irr(B) = Irr(G|θ) where θ ∈ Irr(Op (G)) is G-invariant. Put N = Op (G). Let (G∗ , N ∗ , θ ∗ ) be a character triple isomorphic to (G, N, θ) (see Definition 11.23 of [10]). By Theorem 5.2 of [9] and Theorem 11.28 of [10], we may assume that N ∗ ≤ Z(G∗ ) is a cyclic p -group. Furthermore, by Lemma 11.24 of [10] we know that the sets of p-parts of the degrees of the characters of G∗ that lie over θ ∗ and the set of p-parts of the degrees of G that lie over θ coincide. Also, by the definition of character triple, the Sylow p-subgroups of G/N and G∗ /N ∗ are isomorphic, and the result follows. We remark that the same proof would allow to obtain the corresponding restatement of the block form of Conjecture C for p-solvable groups. Using the results of [15] it is possible to reduce the block form of Conjecture B to a somewhat more restricted situation, but we do not think that it is worth including it here.

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3. General linear groups The goal of this section is to prove Conjectures A, B and C for GL(n, q) where q = pu . We prove the following result. Lemma 3.1. Suppose that G = GL(n, q), where q = p u and p is a prime. Then the set of p-heights of the ordinary irreducible characters of G lying in any of the p-blocks of defect bigger than 0 of G contains {ut (t − 1)/2 | t = 1, . . . , n − 1}. Proof. By [2], G has q − 1 blocks of defect zero and q − 1 blocks of full defect. Let B be a block of G of full defect whose characters lie over a character ρ ∈ Irr(Z(G)). By Proposition 3.1 of [18], the number of characters in B whose p-part of the degree is p ui is bigger than 0 whenever there is a partition µ of n such that n (µ) = i, where if µ = (a1l1 , . . . , aδlδ ) then

n (µ) =

δ aj

j =1

2

lj .

Now, it suffices to consider the partitions µt = (t, 1n−t ) for t = 1, . . . , n − 1. The result follows. Now, we can prove Conjectures A, B and C for the general linear group. Note that the bounds we obtain are good. By the structure of the blocks of GL(n, q) we may assume that B has full defect. Theorem 3.2. Let G = GL(n, q), where q is a power of p. Suppose that B is a block of full defect of G. Then the derived length of a Sylow p-subgroup P of G is bounded logarithmically in terms of the number of heights in B and also in terms of the largest height of the characters in B. Furthermore, logp b(P ) ≤ 2e. Proof. By Satz III.16.3 of [6], the derived length of P is of the order of log n. We have just proved that the number of heights is at least n − 1 and that the largest height is at least (n − 1)(n − 2)/2, so the first claim follows. In order to prove the second claim it suffices to use, for instance, the information on the character degrees of P that appears in [7].

4. Symmetric groups In this section we will find the set of heights of the characters in the blocks of the symmetric groups when p ≥ 5. As a consequence, we will see that Conjectures A, B and C hold for these groups. We need to recall some terminology. It is well-known that the irreducible characters of the symmetric group Sn are labelled by the partitions of n. If λ is a partition of n and [λ] is the Young diagram associated to λ a p-hook of

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[λ] is the part of the diagram associated to a hook of length p. The diagram (or the partition) obtained by successively removing all the rim p-hooks successively is called the p-core of λ (it is known that this does not depend on the order). The p-weight of a partition λ is the number of rim p-hooks that have to be removed before obtaining the p-core. By Nakayama’s conjecture (which was proved independently by Brauer [1] and de Robinson [19]), we know that two characters of Sn belong to the same block if and only if the associated partitions have the same p-core. Hence, it makes sense to define the weight of a block as the weight of the partition associated to any character of the block. For more details on all these concepts see [11]. Next, we need to recall some notation from [17]. We write c(p, n) to denote the number of p-core partitions of n and Fp (x) is the formal power series Fp (x) =

∞

c(p, n)x n .

n=0

We also put Fp (x)s =

∞

Cp (s, n)x n .

n=0

Now, we can prove the key result of this section. Theorem 4.1. If p ≥ 5, then the set of heights of the characters in any p-block B of Sn is the set of integers {0, 1, 2, . . . , (w − a0 − a1 − · · · − ar )/(p − 1)}, where w is the weight of B and w = a0 + a1 p + · · · + ar p r is the p-adic decomposition of w. Proof. It was proved in Corollary 3.8 of [17] that the maximal possible height of characters in B is (w − a0 − a1 − · · · − ar )/(p − 1) and that the number of characters of such height is Cp (p, w). It was proved in [5] that for every integer n there exists a p-core partition on n when p ≥ 5. Hence c(p, n) > 0. This means that for any s all the coefficients of Fp (x)s are non-zero positive integers. Hence it follows from Proposition 3.5 of [17] that the set of heights is the one asserted in the statement of the theorem (here we are using the fact that the set Ea (p, w) defined in [17] is not empty for all a with 0 ≤ a ≤ (w − a0 − a1 − · · · − ar )/(p − 1)). Corollary 4.2. Let B be a p-block of Sn for p ≥ 5 and D the defect group of B. Then logp b(D) is the maximum height of the characters in B. Proof. Write b(D) = p b . We have to show that b = (w − a0 − a1 − · · · − ar )/(p − 1), where w is the weight of B and w = a0 +a1 p+· · ·+ar p r is the p-adic decomposition of w. It is well-known that D is isomorphic to a Sylow p-subgroup of Spw . We have that pw = a0 p + a1 p2 + · · · + ar p r+1 , so D is isomorphic to the direct product of a0 copies of a Sylow p-subgroup of Sp , a1 copies of a Sylow p-subgroup of Sp2 ,…, ar copies of a Sylow p-subgroup of Spr+1 . It was proved in [7] that the largest character

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degree of a Sylow p-subgroup of Spn (for n ≥ 2) is p1+p+···+p

n−2

. We deduce that

b = a1 + a2 (1 + p) + a3 (1 + p + p 2 ) + · · · + ar (1 + p + · · · + p r−1 ) =

r

ai (p i − 1)/(p − 1) = (w − a0 − a1 − · · · − ar )/(p − 1),

i=1

as desired. Hence, we have proved that Conjecture B holds for symmetric groups when p ≥ 5. It is also clear from Theorem 4.1 that Conjectures A and C also hold in this case. Similarly, one could prove that all these conjectures also hold when p ≤ 3 for symmetric groups, but this would require some more tedious calculations using the structure of the 2-cores and 3-cores and the results of [17].

5. Sporadic groups Using the information provided by the GAP character table library, we have checked that the bound logp b(D) ≤ 2e holds for the sporadic groups. Actually, in order to check this bound we do not need to know the structure of the defect groups; it suffices to know their order. Acknowledgements. I thank G. Navarro for helpful conversations. This research was partially supported by the Spanish Ministerio de Ciencia y Tecnología, grant BFM2001-0180, the FEDER and the Basque Government.

References [1]

R. Brauer, On a conjecture by Nakayama, Trans. Roy. Soc. Canada Sect III 41 (1947), 11–19.

[2]

S. W. Dagger, On the blocks of the Chevalley groups, J. London Math. Soc. 3 (1971), 21–29.

[3]

P. Fong, On the characters of p-solvable groups, Trans. Amer. Math. Soc. 98 (1961), 263–284.

[4]

D. Gluck, T. R. Wolf, Brauer’s height conjecture for p-solvable groups, Trans. Amer. Math. Soc. 282 (1984), 137–152.

[5]

A. Granville, K. Ono, Defect zero p-blocks for finite simple groups, Trans. Amer Math. Soc. 348 (1996), 331–347.

[6]

B. Huppert, Endliche Gruppen I, Springer-Verlag, Berlin 1967.

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[7]

B. Huppert, A remark on the character-degrees of some p-groups, Arch. Math. 59 (1992), 313–318.

[8]

I. M. Isaacs, The p-parts of character degrees in p-solvable groups, Pacific J. Math 36 (1971), 677–691.

[9]

I. M. Isaacs, Partial characters of π -separable groups, in Representation Theory of Finite Groups and Finite-Dimensional Algebras, ed. by G. O. Michler and C. M. Ringel (Bielefeld, May 15–17, 1991), Progr. Math. 95, Birkhäuser, Basel 1991, 273–287.

[10] I. M. Isaacs, Character Theory of Finite Groups, Academic Press, New York 1976; reprinted by Dover, New York 1994. [11] G. James, A. Kerber, The Representation Theory of the Symmetric Group, Encyclopedia Math. Appl. 16, Addison-Wesley, Reading, MA 1981. [12] A. Moretó, Characters of p-groups and Sylow p-subgroups, in Groups St Andrews 2001 in Oxford, vol. II, London Math. Soc. Lecture Note Ser. 305, Cambridge Univ. Press, Cambridge 2003, 412–421. [13] A. Moretó, Derived length and character degrees of solvable groups, Proc. Amer. Math. Soc. 132 (2004), 1599–1604. [14] A. Moretó, G. Navarro, Heights of characters in blocks of p-solvable groups, to appear in Bull. London Math. Soc. [15] A. Moretó, T. R. Wolf, Orbit sizes, character degrees and Sylow subgroups, Adv. Math. 184 (2004), 18–36; Erratum ibid. p. 409. [16] G. Navarro, Characters and Blocks of Finite Groups, London Math. Soc. Lecture Note Ser. 250, Cambridge University Press, Cambridge 1998. [17] J. Olsson, McKay numbers and heights of characters, Math. Scand. 38 (1976), 25–42. [18] J. Olsson, K. Uno, Dade’s conjecture for general linear groups in the defining characteristic, Proc. London Math. Soc. 72 (1996), 359–384. [19] G. de B. Robinson, On a conjecture by Nakayama, Trans. Roy. Soc. Canada Sect III 41 (1947), 20–25. Alexander Moretó, Departament d’Àlgebra, Universitat de València, 46100 Burjassot, València, Spain E-mail: [email protected]

Problems on characters and Sylow subgroups Gabriel Navarro∗

I think that this is a very good opportunity to publicize some problems on characters and Sylow subgroups which, in my opinion, deserve attention. It is annoying that even for p-solvable groups, I am unable to solve most of these. Of course, it is very well possible that some of them have negative answers, but even if this is the case, that information will still be very valuable. There are hardly any new results on this note, but many questions. In fact, it all comes back to Problem 12 in the famous paper of R. Brauer [1]. If G is a finite group and P is a Sylow p-subgroup of G, Brauer asked how much the character table ct(G) of G determines the structure of P . In particular, he was interested in whether or not ct(G) determines if P is abelian. As this is answered in [7], a very natural question is the following. Question 1. Does ct(G) determine |P : P |? The invariant |P : P | is the p-part of the size of the important section NG (P )/P of G. Of course, the McKay conjecture asserts that there is a bijection Irrp (G) → Irr(NG (P )/P ), where Irrp (G) is the set of complex irreducible characters χ ∈ Irr(G) of degree not divisible by p. But this does not give a great deal of information unless, of course, the group NG (P )/P is abelian. We have a strong form of the McKay conjecture ([5], [12]) which is somehow relevant for the purpose of this talk. Conjecture 2. Let G be a finite group of order n and let p be a prime. Let e be a nonnegative integer and let σ ∈ Gal(Qn /Q) be any Galois automorphism sending e every p -root of unity ξ to ξ p . Then σ fixes the same number of characters in Irrp (G) as in Irrp (NG (P )). Using Conjecture 2, we were able to compute the exponent of P /P from ct(G) in [5]. As the reader interested in p-solvable groups will easily notice, Problem 1 reduces (in that case) to the following: Suppose that G has a normal elementary abelian p-subgroup N . Does ct(G) determine |N : N ∩ P |? ∗ Research partially supported by the Ministerio de Ciencia y Tecnología, Grant BFM2001-1667-C03-02.

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Of course, there is another variation to Brauer’s question. Question 3. Does ct(G) determine |Z(P )|? Next, we show how to find the exponent of Z(P ) from the character table of G. This is elementary. If G is a finite group and g ∈ G, let us denote by Fg = Q(χ (g)|χ ∈ Irr(G)). Also, Qn = Q(ξ ) is the cyclotomic field, where ξ ∈ C has order n. Notice, then, that Fg ⊆ Qo(g) . Surely the following must be well-known. Theorem 4. Suppose that G is a finite group and let g ∈ G. Then |NG ( g ) : CG (g)| = |Qo(g) : Fg |. Proof. Write o(g) = n and let ξ be a primitive n-th root of unity. Given an integer k coprime with n, there is a unique αk ∈ Gal(Qn ) sending ξ → ξ k and, in fact, Gal(Qn ) = {αk | (k, n) = 1}. Also, if ψ is any character of G, then ψ(g)αk = ψ(g k ). Write H = NG ( g )/CG (g). Given h ∈ H , there is an integer k coprime to n, uniquely determined modulo nZ, such that g h = g k . Now, the map f : H → Gal(Qn /Q) given by f (h) = αk is a one to one group homomorphism. Hence, it suffices to show that f (H ) = Gal(Qn /Fg ). Now, given h ∈ H , we have that g h = g k . Therefore, if χ ∈ Irr(G), we have that χ (g) = χ (g h ) = χ (g k ) = χ (g)αk , so f (H ) ⊆ Gal(Qn /Fg ). Conversely, if αt ∈ Gal(Qn /Fg ), then we have that αt fixes all χ (g) for χ ∈ Irr(G). Therefore, g and g t are G-conjugate. Hence, there exists x ∈ G such that g t = g x . Then x ∈ NG ( g ) and f (xCG (g)) = αt . The following was pointed out to me by M. Isaacs. Corollary 5. Suppose that 1 = g is a p-element of G such that |CG (g)|p = |G|p . If Fg = Q, then o(g) = p. Suppose that Fg = Q, and let e ≥ 1 be the integer such that Fg ⊆ Qpe but Fg is not contained in Qpe−1 . Then o(g) = pe . Proof. Suppose that o(g) = pf . By hypothesis and Theorem 4, we know that |Qpf : Fg | is not divisible by p. Hence, if Fg = Q, then necessarily f = 1. Suppose now that Fg = Q. We have that Fg ⊆ Qpf so 1 ≤ e ≤ f . If e < f , then p = |Qpf : Qpf −1 | divides |Qpf : Fg |, and this is not possible. Notice that if a Sylow p-subgroup of G is abelian, then Corollary 5 proves that we can find out from ct(G) the orders of the p-elements of G. Corollary 6. Suppose that G is a finite group and let P ∈ Sylp (G). Then ct(G) determines the exponent of Z(P ). Proof. We have that ct(G) determines the primes dividing the order of the elements of G (by Higman’s Theorem (8.21) of [4]) and the sizes of their centralizers. Now, notice that if g is a p-element of G, then |CG (g)|p = |G|p if and only if g ∈ Z(Q) for some Sylow p-subgroup Q of G. Now, Corollary 5 applies.

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We have already mentioned the groups P /P and NG (P )/P . So a natural problem is: Question 7. Does ct(G) determine |NG (P )|? It is a consequence of Conjecture 2, that ct(G) determines if NG (P ) = P ([12]). For solvable groups, we showed in [6], that ct(G) determines the set of primes dividing |NG (P )|, but of course, this is far from what we want. It is particulary disturbing in this problem that we do not know how to deal with what seems to be an easy case. Suppose that G has a normal p-complement K. Does ct(G) determine |CK (P )|? In this case, by using the Glauberman correspondence, it is very easy to determine the number of irreducible characters of CK (P ), but we cannot go much further than that. Perhaps, the problem is that we are very short of examples. Given a p-group P acting faithfully on p -groups K1 and K2 of the same size, when is it true that K1 P and K2 P have the same character table? Being unable to prove much, I decided to try Problem 7 in the case when we have the easiest possible Sylow subgroup: when P is cyclic. It follows from [7] that ct(G) determines if P is cyclic. But there is an elementary argument. Theorem 8. Suppose that P ∈ Sylp (G) has order p a , where a ≥ 2. Then P is cyclic if and only if there is a p-element g ∈ G such that Fg ⊆ Qpa but Fg is not contained in Qpa−1 . Proof. Suppose that P = g . Then Fg ⊆ Qpa . If Fg ⊆ Qpa−1 , then p = |Qpa : Qpa−1 | divides |Qpa : Fg |, and this is impossible by Theorem 4. Conversely, if g is a p-element such that Fg ⊆ Qpa but Fg is not contained in Qpa−1 , then necessarily g has order greater than or equal to pa . Now, using Theorems 4 and 8 and Corollary 5, we see that ct(G) determines |NG (P )| if P is cyclic. Since we know (if we believe Conjecture 2) that ct(G) determines when NG (P )/P is trivial, it is natural to ask the following. Question 9. Does ct(G) determine if NG (P )/P is abelian? About Problem 9 we can only say that we gave a positive answer in [11] for p-solvable groups. Unfortunately, our methods did not work for general groups. Question 10. Does ct(G) determine the character (1P )G ? This problem immediately leads us to study characters of groups which vanish off its p-elements. Could it be true that these characters are exactly those induced from characters of their Sylow subgroups? That would be quite an advantage toward a solution to 10. And, in fact, this is the case for p-solvable groups. (This is a not so well-known consequence of work by M. Isaacs.) Unfortunately, this is not true in full generality. If G = A6 and p = 3, then the character ψ = 2χ1 + χ2 + 3χ3 + 2χ6 + 4χ7

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vanishes off the 3-elements of G (where we are using the ATLAS notation), and ψ is not induced from any character of a Sylow 3-subgroup of G. We remark that in the case where P is cyclic, we have that the character (1P )G evaluated on the p-element x ∈ G is |NG ( x ) : P |. Therefore ct(G) determines (1P )G by Theorems 4 and 8 and Corollary 5. Notice that, using Frobenius reciprocity, Problem 10 would be solved if the following has a positive answer. Question 11. If K is a conjugacy class of p-elements of G, does ct(G) determine |K ∩ P |? We showed in [9] that this is the case, again, for p-solvable groups. Finally, we leave the character table of G, and we concentrate on the complex group algebra CG. Question 12. Does CG determine if P G? This sounds as a very elementary problem, but it seems hard to solve. At least, work in [2] and [3] show that the answer to 12 is positive when |G| is divisible by two primes only. Now, what kind of information do we have on the degrees of the irreducible characters of a group G with a normal Sylow p-subgroup P ? Given a nonnegative integer e, the following numbers seem relevant: χ (1)2 . se (G) = χ∈Irr(G),χ(1)p =p e

Lemma 13. Suppose that G/M is a p -group. Then se (G) = |G : M|se (M). Proof. Temporarily, let us write Irrpe (G) for the set of those χ ∈ Irr(G) such that χ (1)p = pe . Now, let be a complete set of representatives of the G-orbits of the action of G on Irrpe (M). It is then clear that Irrpe (G) = Irr(G|θ ) θ∈

is a disjoint union (where Irr(G|θ) is the set of irreducible characters of G lying over θ ). Now, if Tθ is the stabilizer of θ in G, by the Clifford Correspondence we have that |G : Tθ |2 ψ(1)2 se (G) = θ∈

=

ψ∈Irr(Tθ |θ )

|G : Tθ | |Tθ : M|θ (1)2

θ∈

= |G : M|

2

θ∈

|G : Tθ |θ (1)2

= |G : M|se (M), as desired.

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We see that if G has a normal Sylow p-subgroup P , then |G|p divides se (G) for all e. Although this is a very strong condition, M. Isaacs has found, with some effort, an example of a group satisfying this and not having a normal Sylow p-subgroup. On the other hand, notice that by Lemma 13, we have that s0 (G)p = |G|p if G has a normal Sylow p-subgroup. We do not know of any example where this holds and G has not a normal Sylow p-subgroup. (By work in [2], it is not possible to find counterexamples in groups of order divisible by two primes only.) However, it is very much possible that the condition s0 (G)p = |G|p is a rare numerical accident with no further significance. Having talked about s0 (G)p , we cannot resist mentioning a few facts on s0 (G)p , which come back to Problem 1. First, by using the fact that χ (1)2 , |G| = χ∈Irr(G)

notice that |G| ≡ s0 (G)

mod p2 .

In particular, we have that p divides s0 (G) if p divides |G|. (Also, s0 (G)p = p if |G|p = p.) Lemma 14. Suppose that Op pp (G) = G. Then s0 (G)p = |P : P |. Proof. By Lemma 13, we may assume that G has a normal p-complement K. Let Irr P (K) be the set of P -invariant irreducible characters of K. Then, it is easy to check that ˆ | λ ∈ Irr(G/G K)}, {θλ Irrp (G) = θ∈Irr P (K)

is a disjoint union, where θˆ ∈ Irr(G) is an extension of θ to G. Hence s0 (G) = |P : P | θ (1)2 . θ∈Irr P (K)

Now, if is a P -orbit on Irr(K) − Irr P (K), notice that p divides Hence, θ(1)2 mod p, |K| ≡ and we deduce that

2 ψ∈ ψ(1) .

θ∈Irr P (K) θ∈Irr P (K) θ(1)

2

is not divisible by p.

Unfortunately, there are many examples where the conclusion of Lemma 14 fails for general groups. Even more, A. Moretó has found an example where s0 (G)p = |G|p

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and P is not abelian (this is not so easy to find, though). However, we have the following. Theorem 15. Suppose that P ∈ Sylp (G) is abelian of order pa , and assume Brauer’s height zero conjecture. Then s0 (G) ≡ |NG (P )|

mod pa+1 .

In particular, s0 (G)p = |P |. The proof of Theorem 15, for which we require the height zero conjecture, follows from a fact on dimension of blocks (which was known to B. Külshammer). If B is a Brauer p-block of G, recall that the dimension of B is dim(B) = χ (1)2 . χ∈Irr(B)

Theorem 16. Suppose that B is a block of a finite group G with defect group D, and let b be the Brauer first main correspondent of B. Then dim(B)p ≡ (|G : NG (D)|2 )p dim(b)p

mod p.

Proof. This follows by using the formula in the third paragraph of Theorem (2.1) in [8]. Proof of Theorem 15. Suppose that b1 , . . . , bk are all the blocks of NG (P ). Hence, k

|NG (P )| =

dim(bj ).

j =1

Since P NG (P ), we have that all these blocks have defect group P . Now, let Bi = (bi )G , and by Brauer’s first main theorem, notice that {B1 , . . . , Bk } are exactly the blocks of maximal defect of G. By using the height zero conjecture, we have that Irrp (G) =

k

Irr(Bj )

j =1

is a disjoint union. Hence, s0 (G) =

k

dim(Bj ).

j =1

Now, by Theorem 16, we have that dim(Bj )p ≡ dim(bj )p

mod p

since |G : NG (P )| ≡ 1 mod p. Also, by Brauer’s Theorem (3.28) of [10], we have that dim(Bj )p = pa = dim(bj )p ,

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281

and therefore dim(Bj ) ≡ dim(bj )

mod pa+1 .

Now the proof of Theorem 15 easily follows. Acknowledgements. I thank Thomas Breuer, Marty Isaacs and Alexander Moretó for useful conversations on this paper.

References [1]

R. Brauer, Representations of Finite Groups, in Lectures on Modern Mathematics (ed. by T. L. Saaty), Vol. I, J. Wiley, New York 1963, 133–175; also in Richard Brauer, Collected Papers, Vol. II, MIT Press, Cambridge, MA 1980, 183–225.

[2]

J. Cossey, T. Hawkes, Computing the order of the nilpotent residual of a finite group from knowledge of its group algebra. Arch. Math. (Basel) 60 (1993), 115–120.

[3]

I. M. Isaacs, Recovering information about a group from its complex group algebra, Arch. Math. (Basel) 47 (1986), 293–295.

[4]

I. M. Isaacs, Character Theory of Finite Groups, Academic Press, New York 1976; reprinted by Dover, New York 1994.

[5]

I. M. Isaacs, G. Navarro, New refinements of the McKay conjecture for arbitrary finite groups, Ann. of Math. 156 (2002), 333–344.

[6]

I. M. Isaacs, G. Navarro, Character tables and Sylow normalizers, Arch. Math. (Basel) 788 (2002), 430–434.

[7]

W. Kimmerle, R. Sandling, Group theoretic and group ring theoretic determination of certain Sylow and Hall subgroups and the resolution of a question of R. Brauer, J. Algebra 171 (1995), 329–346.

[8]

G. Michler, Trace and defect of a block idempotent, J. Algebra 131 (1990), 496–501.

[9]

G. Navarro, Fusion in the character table, Proc. Amer. Math. Soc. 126 (1998), 165–166.

[10] G. Navarro, Characters of Blocks of Finite Groups, London Math. Soc. Lecture Note Ser. 250, Cambridge University Press, Cambridge 1998. [11] G. Navarro, Sylow normalizers and character tables II, Israel J. Math. 132 (2002), 277–283. [12] G. Navarro, The McKay conjecture and Galois automorphisms, to appear in Ann. of Math. Gabriel Navarro, Departament d’Àlgebra, Facultat de Matemàtiques, Universitat de València, 46100 Burjassot, València, Spain E-mail: [email protected]

Ovals in finite projective planes via the representation theory of the symmetric group Alan R. Prince

Abstract. We discuss the method, introduced by John Thompson, of studying ovals in finite projective planes via the representation theory of the symmetric group.

1. Introduction An oval configuration of involutions in the symmetric group Sn , where n is even, is a set of fixed-point-free involutions of Sn with the property that, given any product of two disjoint transpositions (a, b)(c, d) ∈ Sn , there is a unique involution in the set which contains (a, b)(c, d) in its expression as a product of disjoint transpositions. A counting argument shows that an oval configuration in Sn must contain (n − 1)(n − 3) involutions. An oval (sometimes called a hyperoval) in a projective plane of even order m is a set of m + 2 points, no three of which are collinear. If O is an oval in a projective plane of even order m, then by labelling the points of O by 1, 2, . . . , n, where n = m + 2, every point P ∈ O determines a fixed-point-free involution in Sn as follows: each line through P meeting O meets it in precisely two points i, j and the product of the disjoint transpositions (i, j ) corresponding to the lines through P meeting O is the required involution of Sn . The set of these involutions, as P ranges over the m2 − 1 points ∈ O, forms an oval configuration in Sn . Any oval configuration of involutions in Sn gives a start to constructing a projective plane of order m = n − 2 with an oval O but the lines not meeting O still need to be constructed and this may not be possible. All the known examples, except one, of oval configurations of involutions arise, via the construction described, from ovals in projective planes of even order. The exception is an oval configuration in S10 , discovered by Mathon [5]. It is easy to show that there are no oval configurations in S8 . Lam et al showed, by a long exhaustive computer search, that there are no oval configurations in S12 [3] and this formed part of the proof, by computer, of the non-existence of a projective plane of order 10 [4]. Since the known projective planes of even order all have order 2t , the known oval configurations of involutions in Sn occur only when n has the form 2t + 2 (there is

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always at least one example, if n is of this form, since the Desarguesian plane of order 2t contains an oval). The existence problem for n not of this form is open, for n ≥ 14. The author has shown that there are no oval configurations of involutions in S14 , which are invariant under conjugation by a Frobenius subgroup of order 39 [6]. The study of ovals in finite projective planes, via combinatorial properties of sets of involutions in the symmetric group, originated with Buekenhout [1] and the method can be applied also to ovals in projective planes of odd order n (which contain n + 1 points). For a survey of results on ovals in finite projective planes, see [2]. The idea of tackling the problem of the existence of oval configurations of involutions in Sn , n even, by using the representation theory of Sn was introduced by Thompson [8]. In Thompson’s theory, a certain upper-triangular matrix and its inverse play a key role and the main result of this paper is the observation that this matrix can be described precisely: a criterion is given for precisely when an entry in the matrix is nonzero and, moreover, it is shown that the nonzero entries are ±1 (Theorem 3.1). This result makes the computation of the inverse more feasible. We give the result, in Section 4, of some computations in S8 , as an illustration of the general theory. Of course, the aim is to be able to apply the theory in a case such as S14 to construct an oval configuration of involutions (and possibly a projective plane of order 12)!

2. Thompson’s theory In this section, we summarise some of the theory contained in [8]. Let Ck denote the set of all products of k disjoint transpositions in Sn , n = 2m, and let Mk be the free Zmodule with basis consisting of the elements of Ck . Denote the norm element x∈C k x y, of Mk by ν(Mk ). Let M = Mm and N = M2 . For x ∈ Cm , define λ(x) = where the sum is over all y ∈ C2 with y involved in x. Extending by linearity gives a linear map λ : M → N.(π)Let χ be the character of the action of Sn on Cm . In [8], it is shown that χ = χ , where π ranges over P , the set of all partitions of n with even parts. If T is a standard tableau associated to a partition π ∈ P , let u(T ) ∈ Cm denote the involution interchanging the entries in cells (i, 2j − 1) and (i, 2j ) throughout T . If H is the subgroup of the column group of T , consisting of all permutations fixing elements in the odd columns of T , set sg(h)u(T )h γ (T ) = h∈H

A key result of [8] is that the elements γ (T ) form a Z-basis for M. We also associate with the standard tableau T another element δ(T ) ∈ M, defined to be the sum of all the elements of Cm in the row group of T . We view M as a lattice with orthonormal basis Cm , so that we have an inner product, denoted · , · , defined on M. Using the usual ordering of partitions and the lexicographic ordering of standard tableaux for

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285

each partition, the tableaux are linearly ordered and the matrix C of inner products γ (Ti ), δ(Tj ) is upper triangular with 1’s on the diagonal. A different ordering of the standard tableau is used in [8] but the conclusion remains the same, with the lexicographic ordering, since Ti > Tj implies that there exist i, j in the same column of Ti and the same row of Tj . The coordinates, with respect to the δ(T )-basis, of the dual basis to the γ (T )-basis are given by the rows of (C −1 )t . The element ν(M) satisfies λ(ν(M)) = cν(N), where c = (n−5)(n−7) · · · 5·3·1. Let Q be the set of partitions [n], [n − 2, 2], [n − 4, 4], [n − 4, 2, 2]. The elements γ (T ), where T is a standard tableau associated with a partition of P − Q, form a Z-basis of the kernel K of λ. If ν(M) = a(T )γ (T ), where the sum is over all standard tableaux, then the row vector of coefficients a(T ) is (. . . , ι(δ(T )), . . . )C −1 , where ι(δ(T )) = ν(M), δ(T ) . For the tableaux T corresponding to π ∈ Q, the coefficients a(T ) are divisible by c. Thus, a(T )γ (T )} ν0 (M) = c−1 {ν(M) − π ∈P −Q

satisfies λ(ν0 (M)) = ν(N), ν0 (M) ∈ M, showing that ν(N) ∈ im(λ). If O is any set of fixed-point-free involutions of Sn , then the condition for O to be an oval configuration is just λ( x∈O x) = ν(N). Thus, the problem of finding an oval configuration is equivalent to that of finding an element m ∈ M, with coefficients 0, 1 when expressed in terms of the Cm -basis or equivalently of norm (n − 1)(n − 3), such that λ(m) = ν(N).

3. The upper triangular matrix C In this section, our main result is proved, giving a precise description of the upper triangular matrix C. First, we introduce some notation and terminology. If standard tableaux T and T have the property that no two elements in the same row of T are in the same column of T we write T ∼ T . If T ∼ T , then we can construct a rectangular tableau S as follows: place k in the (i, j )-cell of S if k is in the ith row of T and the j th column of T . Since T ∼ T , each cell in S is either empty or is assigned a single entry. We may assume that the cells of S form a rectangular array with the number of rows being the number of rows of T , while the number of columns is the number of columns of T . If T and T have the same shape and we ignore empty cells, then S is an ordinary tableau, not necessarily standard, of the same shape (see, for example, [7]). In the present context, however, we are also interested in the construction if T and T do not have the same shape, when the non-empty cells need not form the shape of a Ferrers diagram of a partition. The following are two examples of the construction of S.

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Alan R. Prince

1 4 5

2 6 7

3

8

1 4 1 4

2 6 2 6

3 7 3

5 8 8

7

5

1 4 6

2 5 7

3

8

1 4 1 4

2 6 2

3 7 3

6

7

5 8 8 5

If T ∼ T and S is the associated rectangular tableau, constructed as described, then there is a unique column permutation σ of T which when applied to T reorders the elements in each column so that they are in the order they appear in the columns of S. Let θ (T , T ) = sg(σ ). If S is a rectangular tableau, then a domino of S is a pair of cells in positions (i, 2j − 1), (i, 2j ) for some i, j (i.e. consecutive cells in the ith row in positions 1,2 or 3,4 or …). We say that S satisfies the domino condition if the cells in any domino are both non-empty or both empty. In the above examples, the rectangular tableau S satisfies the domino condition in the first case but not the second. With this notation and terminology in place, we can now state our main theorem. Theorem 3.1. If T ∼ T and the associated rectangular tableau S satisfies the domino condition, then γ (T ), δ(T ) = θ (T , T ). Otherwise, γ (T ), δ(T ) = 0. Proof. Suppose that T ∼ T and the associated rectangular tableau S satisfies the domino condition. Let u(S) ∈ Cm be the involution interchanging the entries in each domino. Then, u(S) is the unique involution involved in both γ (T ) and δ(T ) and appears in γ (T ) with coefficient θ(T , T ). Thus, γ (T ), δ(T ) = θ (T , T ). Suppose that T ∼ T and the associated rectangular tableau S does not satisfy the domino condition. Then, no involution is involved in both γ (T ) and δ(T ) and hence γ (T ), δ(T ) = 0. Finally, if T ∼ T , then there exist elements i, j occurring in the same row of T and the same column of T . Thus, the transposition (i, j ) is in the row group of T and the column group of T and it follows that γ (T ), δ(T ) = 0, since involutions involved in both γ (T ) and δ(T ) can be paired off, with contributions to the inner product being +1 and −1 for each pair (see also [8], where this argument is used in proving C is upper triangular). The previous theorem gives a precise description of the upper triangular matrix C. It is a sparse matrix with 1’s down the diagonal and nonzero entries ±1, making the computation of its inverse feasible, at least for n relatively small. We have seen (Section 2) that, if ν(M) = a(T )γ (T ), then the row vector of coefficients is (. . . , ι(δ(T )), . . . )C −1 , where ι(δ(T )) = ν(M), δ(T ) . Thus, these coefficients can be calculated once we know C −1 . For a tableau T corresponding to π ∈ Q, the coefficient a(T ) is divisible by c, while for a tableau T corresponding to π ∈ P − Q, γ (T ) ∈ K, the kernel of λ. Replacing

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each of the coefficients a(T ), for the tableaux T corresponding to π ∈ P − Q, by any integer multiple a(T )∗ of c, leaving the other coefficients a(T ) unchanged, and then dividing by c, we obtain an element m0 ∈ M m0 =

1 a(T )∗ γ (T ) c

which maps to ν(N). The element ν0 (M) of Section 2 is just the case where we set a ∗ (T ) = 0, for each tableau T corresponding to π ∈ P − Q. Alternatively, we could take a(T )∗ to be the multiple of c closest to a(T ), or some other close multiple. It may be possible, computationally, to exploit this idea to find elements m0 ∈ M of norm close to (n − 1)(n − 3), which map to ν(N). We illustrate this idea in the following section where some computations are given for the case S8 .

4. The case S8 It is easy to show that there are no oval configurations in S8 , as we may assume that the involutions containing (7, 8) are: (1, 2)(3, 4)(5, 6)(7, 8) (1, 3)(2, 5)(4, 6)(7, 8) (1, 4)(2, 6)(3, 5)(7, 8) (1, 5)(2, 4)(3, 6)(7, 8) (1, 6)(2, 3)(4, 5)(7, 8) and then attempts to extend this lead quickly to a contradiction, essentially because there are only 6 1-factorisations of the complete graph K6 . However, it is still instructive to consider the general theory in this particular case and we give the results of some calculations relating to S8 in this section. The partitions of 8 with all parts even are 8, 6 + 2, 4 + 4, 4 + 2 + 2, 2 + 2 + 2 + 2 and the corresponding number of standard tableaux is 1, 20, 14, 56, 14 with respective ι(δ(T ) being 105, 15, 9, 3, 1. The 14 elements γ (T ) corresponding to the partition [24 ] are a basis for K. The upper triangular matrix C is given by Theorem 3.1. Writing C as I +N, N has 266 nonzero entries and is nilpotent of index 7. From its inverse I − N + N 2 − N 3 + · · · + N 6 , the row vector of coefficients for ν(M), expressed as a linear combination of the basis vectors γ (T ), can be computed. These coefficients (ordered as described in the general theory) are:

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Alan R. Prince

105 −90 −90

15 15

15 −75

24 −36

−6

54 −27 18 3 −30 15 78 −9 40

15 45

9 −36

15

15

15

15

15

84

66

24

−12

−6

−15

−6 −6 48 0

3 3 −51 −48

−6 −78 −48 −30

−8 −48

22

−8

24 −12 24 39 −66 78 78 −30

88 −20

15 −90 −45 75

3 0 39 24 39 −162 3 −162 45 −36 −33 12 102 −9 75 63 −92

58 −57

0

15

15

15

39

−78

9

−12 −6 9 −678 −15 0 183 93

3 3 −3 −9

−102

210 −117

In this case, c = 3. Note that all the coefficients, except those in the final group, corresponding to the kernel K, are divisible by 3, as the general theory predicts. If we denote the 14 standard tableaux corresponding to [24 ] by T1 , T2 , . . . , T14 (in lexicographic order), then subtracting γ (T1 ) + γ (T2 ) + γ (T3 ) + γ (T4 ) + γ (T6 ) + γ (T7 ) + γ (T8 ) + γ (T9 ) from ν(M) gives a vector in which the coefficients have been changed to the nearest multiple of 3 and results, after dividing the coefficients by 3, and expressing the resulting vector in terms of the original basis of C4 ’s, in a vector of norm 65. However, replacing the coefficient of γ (T1 ) by 42 (instead of 39) and the coefficient of γ (T2 ) by 90 (instead of 87), results in a (0, 1, −1) solution of norm 45 in M which maps to ν(N ) and this appears to be a solution of minimum possible norm. We list this solution in full: 12 · 34 · 56 · 78 + 12 · 34 · 57 · 68 − 12 · 34 · 58 · 67 + 12 · 35 · 48 · 67 + 12 · 36 · 47 · 58 + 12 · 37 · 46 · 58 + 12 · 38 · 45 · 67 + 13 · 24 · 58 · 67 + 13 · 25 · 48 · 67 + 13 · 26 · 45 · 78 + 13 · 27 · 45 · 68 − 13 · 28 · 45 · 67 + 13 · 28 · 46 · 57 + 13 · 28 · 47 · 56 + 14 · 23 · 58 · 67 + 14 · 25 · 36 · 78 + 14 · 25 · 37 · 68 − 14 · 25 · 38 · 67 + 14 · 26 · 38 · 57 + 14 · 27 · 38 · 56 + 14 · 28 · 35 · 67 + 15 · 23 · 46 · 78 + 15 · 23 · 47 · 68 − 15 · 23 · 48 · 67 + 15 · 24 · 38 · 67 + 15 · 26 · 37 · 48 + 15 · 27 · 36 · 48 + 15 · 28 · 34 · 67 + 16 · 23 · 48 · 57 + 16 · 24 · 35 · 78 + 16 · 25 · 38 · 47 + 16 · 27 · 34 · 58 + 16 · 28 · 37 · 45 + 17 · 23 · 48 · 56 + 17 · 24 · 35 · 68 + 17 · 25 · 38 · 46 + 17 · 26 · 34 · 58 + 17 · 28 · 36 · 45 + 18 · 23 · 45 · 67 − 18 · 24 · 35 · 67 + 18 · 24 · 36 · 57 + 18 · 24 · 37 · 56 + 18 · 25 · 34 · 67 + 18 · 26 · 35 · 47 + 18 · 27 · 35 · 46

There is a nice pattern here: the involutions that occur with coefficient −1 are: (1, 2)(3, 4)(5, 8)(6, 7) (1, 3)(2, 8)(4, 5)(6, 7) (1, 4)(2, 5)(3, 8)(6, 7) (1, 5)(2, 3)(4, 8)(6, 7) (1, 8)(2, 4)(3, 5)(6, 7)

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and each gives rise to the involutions with coefficient +1 in the appropriate group above, in the following manner (illustrated for the (1, 2) group only): pair (1, 2) with each of (3, 4), (5, 8) and (6, 7) and take the two C4 ’s that involve each pair of transpositions other than (1, 2)(3, 4)(5, 8)(6, 7). In the same manner, (3, 4)(5, 8) gives rise to the involutions (1, 6)(2, 7)(3, 4)(5, 8) and (1, 7)(2, 8)(3, 4)(5, 8), which occur in the (1, 6) and (1, 7) sets above. Thus, we could write down other similar solutions. For example, we could start with the 5 involutions involving (7, 8), listed at the start of this section, taking each with coefficient −1, and constructing the involutions with coefficients +1 from them, as described. We conclude with an observation about the lattice K. The inner product matrix γ (Ti ), γ (Tj ) for the basis T1 , T2 , . . . , T14 of K is: 24 6 6 4 6 6 4 4 6 4 6 4 6 4 6 24 4 6 −6 4 6 6 4 −4 4 6 −6 −4 6 4 24 6 −6 4 6 6 4 −4 −6 −4 4 6 4 6 6 24 6 6 4 4 6 4 −4 −6 −4 −6 6 −6 −6 6 24 4 −4 −4 4 6 4 −4 −6 6 6 4 4 6 4 24 6 6 4 6 −6 −4 −6 −4 4 6 6 4 −4 6 24 4 6 −6 −4 −6 6 4 4 6 6 4 −4 6 4 24 6 −6 6 4 −4 −6 6 4 4 6 4 4 6 6 24 6 4 6 4 6 4 −4 −4 4 6 6 −6 −6 6 24 −4 4 6 −6 6 4 −6 −4 4 −6 −4 6 4 −4 24 6 −6 6 4 6 −4 −6 −4 −4 −6 4 6 4 6 24 6 −6 6 −6 4 −4 −6 −6 6 −4 4 6 −6 6 24 6 4 −4 6 −6 6 −4 4 −6 6 −6 6 −6 6 24 (Thompson [8] shows that the non-existence of an oval configuration in S8 follows from the fact that K is a doubly even lattice.) The coefficients of ν(M), not divisible by 3, are precisely the coefficients of γ (T1 ), γ (T2 ), γ (T3 ), γ (T4 ), γ (T6 ), γ (T7 ), γ (T8 ), γ (T9 ). It is interesting that the inner product of any two of these vectors is positive.

References [1] F. Buekenhout, Etude intrinseque des ovales, Rend. Mat. Appl. (5) 125 (1986), 525–534. [2] G. Korchmaros, Old and new results on ovals in finite projective planes, in Surveys in combinatorics, 1991 (ed. by A. D. Keedwell), London Math. Soc. Lecture Note Ser. 166, Cambridge Univ. Press, Cambridge 1991, 41–72. [3] C. W. H. Lam, L. Thiel, S. Swiercz and J. McKay, The non-existence of ovals in a projective plane of order 10, Discrete Math. 45 (1983), 319–321.

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[4] C. W. H. Lam, L. Thiel, S. Swiercz, The non-existence of finite projective planes of order 10, Canad. J. Math. 41 (1989), 1117–1123. [5] R. Mathon, The partial geometries pg(5,7,3), Congr. Numer. 31 (1981), 129–139. [6] A. R. Prince, Oval configurations of involutions in symmetric groups, Discrete Math. 174 (1997), 277–282. [7] D. E. Rutherford, Substitutional Analysis, Edinburgh University Publications 1, University Press, Edinburgh 1948; Hafner Publishing, New York 1968. [8] J. G. Thompson, Fixed point free involutions and finite projective planes, in Finite Simple Groups II (ed. by M. J. Collins), Academic Press, London 1980, 321–337. Alan R. Prince, School of Mathematical and Computer Sciences, Heriot-Watt University, Edinburgh EH14 4AS, United Kingdom E-mail: [email protected]

Rank invariance and automorphisms of generalized Kac–Moody superalgebras Urmie Ray

Abstract. We show that all infinite dimensional generalized Kac–Moody superalgebras are characterized by a unique Cartan matrix (up to equivalence) and describe their automorphisms. 2000 Mathematics Subject Classification: 17B

1. Introduction Automorphisms of finite dimensional simple Lie algebras are products of inner automorphisms and diagram automorphisms [10]. Unlike the finite dimensional case, for a Kac–Moody algebra, given a Cartan decomposition, the Borel subalgebra generated by the positive root spaces and the one generated by the negative root spaces are not in general conjugate under the action of inner automorphisms (or in other words of the corresponding Kac–Moody group) ([17], [15], 5.9). Hence the Chevalley involution has to be considered. The aim of this paper is to give the decomposition of an arbitrary automorphism of an infinite dimensional generalized Kac–Moody superalgebra. In order to do so, we show that Cartan subalgebras of generalized Kac–Moody superalgebras are conjugate under the action of inner automorphisms. This well-known result of Kac–Peterson [17] for Kac–Moody algebras therefore remains valid in the more general context. Furthermore, we show that all infinite dimensional generalized Kac–Moody superalgebras G are characterized by a unique Cartan matrix. This matrix is of generalized Kac–Moody type, i.e. it satisfies the usual conditions (a) − (d) (see §1 below). In other words, there are two conjugacy classes of Borel subalgebras, one corresponding respectively to positive and negative root spaces of a giving Cartan decomposition. This also generalizes a classical result of Kac–Peterson for Kac–Moody algebras [17] and implies that not only the rank of a Cartan subalgebra of an infinite dimensional generalized Kac–Moody superalgebra, but also the rank of the Kac–Moody sub-superalgebra generated by all the simple finite type (real) root spaces, are well defined.

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These results suggest that the structure of infinite dimensional generalized Kac– Moody superalgebras is closer to that of Kac–Moody algebras than to finite dimensional Lie superalgebras. Finite dimensional simple Lie superalgebras were extensively studied by many. In particular, their automorphisms were described by Serganova [22] and it is well known that in general, there are more than two conjugacy classes of Borel sub-superalgebras. Hence for finite dimensional simple Lie superalgebras, the situation differs much from the Lie algebra one. However, we show that if the finite dimensional classical Lie superalgebra [13] G is a generalized Kac–Moody superalgebra, then except when the derived Lie superalgebra G = [GG] is isomorphic to the 8 dimensional classical Lie superalgebra A(1, 0), it has a unique Cartan matrix of generalized Kac–Moody type associated to it. In other words, except in that one case, there are at most two conjugacy classes of Borel subalgebras corresponding to a Cartan decomposition of generalized Kac–Moody type, as in the infinite dimensional case. The uniqueness of the Cartan matrix also has a consequence in the Moonshine context. It follows that the moonshine module constructed by Frenkel, Lepowsky and Meurman [9] cannot be constructed as a lattice vertex algebra. That the construction of the Monster vertex algebra could not be as simple as that of the fake monster vertex algebra is known but we give a different proof of this fact.

2. Preliminary results We first need to fix some notations and to remind the reader of a few basic facts about generalized Kac–Moody superalgebras which will be needed later on. For details, see ([19], §1, 2). As usual G0¯ and G1¯ will denote the even and odd part of the Lie superalgebra G. The base field F will be either R or C. Let I be a finite set {1, . . . , n} or a countable one identified with Z+ and S be a subset of I . The set S indexes the generators contained in G1¯ . Let H be a F-vector space with a non-degenerate symmetric bilinear form ( · , ·) containing non-trivial vectors hi indexed by the set I . Set aij := (hi , hj ) and assume that (a) aij = aj i ; (b) aij ≤ 0 if i = j ; (c) (d)

2aij aii ∈ Z if aii > 0; aij aii ∈ Z if aii > 0 and

i ∈ S.

˜ = G(A, ˜ The Lie superalgebra G S, H ) is generated by the vector space H considered as an even abelian subalgebra and elements ei , fi , i ∈ I , satisfying the defining relations:

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(i) [ei , fj ] = δij hi ; (ii) [h, ej ] = (h, hj )ej , h ∈ H ; (iii) [h, fj ] = −(h, hj )fj , h ∈ H ; (iv) deg ei = deg fi = 0¯ if i ∈ S; (v) deg ei = deg fi = 1¯ if i ∈ S. Following Moody’s definition [16], we define the generalized Kac–Moody superalgebra G = G(A, S, H ) by generators and relations. Definition 2.1. The generalized Kac–Moody superalgebra G = G(A, S, H ) with generalized symmetric Cartan matrix A = (aij )i,j ∈I and Cartan subalgebra H is the Lie superalgebra generated by H , ei , and fi , i ∈ I and with defining relations (i)-(v) and with the following extra ones: (vi) 1 (ad ei ) j ∈ I;

−2aij aii

+1

ej = 0 = (ad fi )

−2aij aii

+1

fj for all i ∈ I such that aii > 0 and

(vii) [ei , ej ] = 0 = [fi , fj ] if aij = 0. The cardinality of the indexing set I is called the rank of the Lie superalgebra G. We will show that this concept is well defined. We first need to make some remarks about Cartan matrices. Remarks 2.2. If the matrix A is indecomposable and D is a diagonal matrix with positive entries, then the generalized Kac–Moody superalgebras G(A, S, H ) and G(DA, S, H ) are isomorphic. The matrix DA need not be symmetric and satisfies conditions (b) − (d) and if it is not symmetric, condition (a) has to be replaced by aij = 0 if aj i = 0. Note that one can also take a matrix D with negative entries. To include this possibility, the definition of a Cartan matrix A needs to be extended and modified in an obvious manner. Multiplying by −1 does not change anything: as long as all non-diagonal entries are of the same sign, whether they are taken to be non-positive or non-negative is only a matter of convention. Hence we need the concept of equivalent matrices. Definition 2.3. Two matrices B and C of the same size are said to be equivalent, if there exist non-singular matrices P and D with D diagonal, such that C = DP BP −1 . The generalized Kac–Moody superalgebras corresponding to equivalent Cartan matrices are isomorphic, given the same Cartan subalgebra H and indexing set S. Let G be the generalized Kac Moody superalgebra. Let Q be the root lattice form given by the generalized Cartan matrix A: with basis αi , i ∈ I and bilinear (αi , αj ) = aij . Let Q+ = { i∈I ki αi | ki ∈ Z+ }. For α ∈ Q+ (resp. −α ∈ Q+ ), let Gα be the subspace of G generated by all elements [. . . [ei3 [ei2 , ei1 ]] . . . ] (resp. [. . . [fi3 [fi2 , fi1 ]] . . . ]), where α1 + α2 + α3 + · · · = α (resp. −α). let denote 1 Note that there is a mistake in the statement of condition (vi) in ([20], §5.1).

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the set of roots, i.e. the non-zero elements α ∈ Q for which Gα = 0, + the set of positive roots, 0 the set of even roots and + 0 the set of even positive roots. The Cartan decomposition holds for the generalized Kac–Moody superalgebra G ([15], §1): G = (⊕−α∈ + Gα ) ⊕ H ⊕ (⊕α∈ + Gα ) Let N + := ⊕α∈ + Gα and N0+ = N + ∩ G0¯ . We give a simpler definition for infinite type roots that the one given in ([19], Def. 2.3) for the earlier one excluded roots of norm 0. Definition 2.4. A root α ∈ is said to be of finite type if for any root β ∈ , nα + β is a root for only finitely many integers n. Otherwise it is said to be of infinite type. So we can reformulate ([19], Lemma 2.4 and Prop. 2.6) as follows: Proposition 2.5. A root α ∈ + is of infinite type if for any root β ∈ {α, 21 α, 2α} for which (α, β) < 0, nα + β are roots for all positive integers n, unless α and β are both positive or both negative, β is of finite type and norm 0 and α − β is a root. In the case of a generalized Kac–Moody algebra of rank at least two, the former are the real roots and the latter the imaginary ones. All roots of positive norm are of finite type as in the algebra case. Remark 2.6. By (vii) the support of a root α is always connected. Let αi be a simple root of positive norm, then ri will denote the corresponding reflection when αi is even, and the reflection corresponding to 2αi , when αi is odd. Set W1 to be the group generated by these reflections. Roots of negative norm might be of finite type. The full Weyl group W is the group generated by all reflections corresponding to non-zero norm roots of finite type. Note that when the derived Lie superalgebra [G, G] has infinite dimension, or equivalently when the set of roots is infinite, the group W1 is the full Weyl group W for then G does not contain roots of finite type having negative norm ([19], Cor. 2.5). When G contains roots of the latter type, W1 is a proper subgroup of W . Lemma 2.7. For any root α ∈ + , if α has positive norm, then there exits w ∈ W1 , such that w(α) or 21 w(α) is a simple root. If α is an odd root of norm 0 and finite type, then exits w ∈ W1 such that w(α) is a simple root. For the proof of this result, see ([19], Lemma 2.2). There is an omission in the statement in [19]: as W1 contains the reflection ri corresponding to 2αi when the simple root αi is an odd root of positive norm, a positive root α of positive norm could be conjugate to 2αi rather than αi . Since the generalized Kac–Moody algebra G has no non-trivial ideal intersecting the Cartan subalgebra H trivially ([19], Lemma 2.1), the following two results are immediate.

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Corollary 2.8. The centre of the generalized Kac–Moody superalgebra G is contained in the Cartan subalgebra H . Corollary 2.9. The even part G0¯ of the generalized Kac–Moody superalgebra G is a generalized Kac–Moody algebra. Proof. The invariant, consistent, super-symmetric bilinear form induced on G by the bilinear form on H ([15], 2.2) is non-degenerate since its non-degenerate on H and any ideal intersecting H trivially is trivial. The result then follows by ([19], Theorem 3.3). We fix some more notation before proceeding to the main part of the paper. Let J = I when I = 1 and a11 = 0 and J = {i ∈ I | aii > 0} otherwise. So J is the set indexing the simple roots of finite type with non-zero norm. Let GJ be the Kac–Moody sub-superalgebra of G generated by the elements ei and fi for i ∈ J . We will refer to the cardinal of the set J as the Kac–Moody rank of G. We will show that this is a well defined notion. Let G0 be the Lie subalgebra of G generated by the Cartan subalgebra H , the elements ei , fi belonging to the even part G0¯ of the Lie superalgebra G and by the elements [ei , ei ], [fi , fi ], where ei , fi belong to the odd part G1¯ and the simple root αi is of finite type with non-zero norm. This is clearly a generalized Kac–Moody (even) subalgebra of GJ . Define U + (resp. U − ) as the Lie sub-superalgebra generated by the root spaces Gα where there exits some index i in the support of the root α ∈ + (resp. −α ∈ + ) such that the simple αi is either of infinite type or of finite type with norm 0.

3. Conjugacy of Cartan subalgebras of the Lie superalgebra G We first state an obvious fact: Lemma 3.1. G = U + ⊕ (GJ + H ) ⊕ U − as a direct sum of vector spaces. Proposition 3.2. If the generalized symmetric Cartan matrix A is indecomposable and the derived BKM superalgebra [G, G] is infinite dimensional, then (1) every element of G which act ad-locally finitely on G is contained in the Lie sub-superalgebra GJ + H ; (2) all Cartan subalgebras of G are contained in the Lie sub-superalgebra GJ + H .

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Proof. Suppose that K is a Cartan subalgebra of G not contained in GJ + H . By definition, K is an even subalgebra of G. Since K is a Cartan subalgebra, it acts on G in a locally finite way, i.e. for any element x ∈ K and v ∈ G, there is a finite dimensional subspace of G containing (ad x)n v for all n ∈ Z+ . Let x be an element of K not contained in GJ + H . Write x = u+ + v + u− , where u+ ∈ U + , u− ∈ U − and v ∈ GJ + H . We first assume that u+ = 0. So there exists positive roots β1 , . . . , βr ∈

containing either infinite type simple roots or finite type simple roots with zero norm in their support and root vectors ui ∈ Gβi such that u = ri=1 ui . Suppose that there exists a simple root αi such that for some j = 1, . . . , r, (βj , αi ) < 0. Since K is an even subalgebra, all roots βj are even. So by Corollary 2.9, the roots βj , 1 ≤ j ≤ r are of infinite type as they have non-positive norm and are even and by assumption, [G, G] being is infinite dimensional so that there are no roots of negative norm and finite type ([19], Corollary 2.5). This implies that (ad uj )n ei = 0 for all n ∈ Z+ ([19], Lemma 2.4). We claim that there are infinitely many vectors (ad x)n (ei ), n ∈ Z+ which are linearly independent. Let k ∈ {1, . . . , r} be such that βk is of maximum height with the property that βk + αi is a root. Then for each positive integer n, (ad u+ )n (ei ) is the sum of root vectors, exactly one of which corresponds to the root nβk + αi . Hence there are infinitely many linearly independent vectors (ad u+ )n (ei ), n ∈ Z+ . Indeed for all n ∈ Z+ , the component of (ad u+ )n (ei ) in the nβk + αi root space cannot be a linear combination of components of (ad u+ )r (ei ), r ≤ n − 1 since all the components of u− belong to negative root spaces and those of v belong to root spaces generated by simple finite type root spaces and to the Cartan subalgebra H , it follows that the claim holds. This contradicts the fact that K acts in a locally finite way. So (βj , αi ) ≥ 0

for all j = 1, . . . , r, i ∈ I.

(1)

Let αi be a simple root either of infinite type or of finite type and norm 0 in the support of βj . Then condition (1) forces (βj , αi ) = 0 for all j = 1, . . . , r. Hence the simple root αi has norm 0 and is orthogonal to all the simple roots in the support of each root βj , j = 1, . . . , r. Since the support of a root is connected and βj is an even root, we can deduce that βj = αi is a simple root of infinite type. As βj is simple, (βj , αk ) ≤ 0 for any simple root αk of the generalized Kac–Moody superalgebra G. Hence by (1), the simple root βj is orthogonal to all the simple roots of G. As the Cartan matrix is assumed to be indecomposable, it follows that j = 1 and I = {β1 }. So G is a Heisenberg algebra of rank 1. Therefore the Cartan subalgebra H is unique and in particular K = H , contradicting assumptions. So u+ = 0.

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If u− = 0, then similar arguments lead to a contradiction. It follows that x ∈ GJ + H and this forces the Cartan subalgebra K to be contained in GJ + H , contradicting assumptions and proving the result. Proposition 2.2 implies the first main result of this paper. We remind the reader that the derived Lie group of the Lie algebra G0 consists of the inner automorphisms. Theorem 3.3. All Cartan subalgebras of the generalized Kac–Moody superalgebra G are conjugate under the action of the derived Lie group of G0 . Proof. We first suppose that the derived Lie superalgebra [G, G] is infinite dimensional. Let H1 be a Cartan subalgebra of G. Then H1 ≤ H + GJ by Proposition 2.2. Let K be a finite subset of the indexing set J and HK = H1 ∩ (H + GK ). Write Z for the centre of the Lie superalgebra H + GK . By Corollary 2.8 applied to the generalized Kac–Moody superalgebra H + GK , Z ≤ HK ∩ H . For i ∈ K, let Vi+ (resp. Vi− ) be the HK -submodule of H + GK generated by the vector ei (resp. fi ) and φi+ (resp. φi− ) be the Lie algebra homomorphism from HK to gl(Vi+ ) (resp. gl(Vi− )). Then Z = ( i∈K ker φi+ )∩( i∈K ker φi− ). As the action of HK on G is locally finite, the modules Vi+ and Vi− are finite dimensional. Hence the subspaces HK /(ker φi+ ) and HK /(ker φi− ) are finite dimensional. So the subspace Z has finite co-dimension in HK since the set K is finite. Let L be a complement of Z in H1 . As L is finite dimensional, with obvious modifications for the superalgebra case, the Conjugacy Theorems of Peterson–Kac for Kac–Moody algebras [17] can directly be extended to L + GK . Thus, the subalgebra L of H + GJ is conjugate to a subalgebra of H under the action of some element g of the derived Lie group of G0 . Since Z is central in H + GK and contained in H , it follows that HK is conjugate to a subalgebra of H under the action of this element g. A symmetric argument applied to H shows that HK is conjugate to H under the action of G0 . Similarly H ∩ (H1 + GK ) is conjugate to H1 under the action of the derived Lie group of G0 . Hence H and H1 are conjugate under this action, proving the result. When the derived Lie superalgebra [G, G] is finite dimensional, the argument above shows that we may assume G to be finite dimensional. The result is then a well known result in the Lie algebra case; and has been proved in [22] in the context of Lie superalgebras. The next result is a also a basic one needed for the uniqueness of the Cartan matrix associated to the generalized Kac–Moody superalgebra G. Theorem 3.4. If the set of roots is infinite, then all bases of are conjugate to the base or − under the action of the Weyl group W . Proof. Note that as there are infinitely many roots, the group W1 is the full Weyl group W . Let be the base of defined in section 1 and 2 be another set of simple roots in Q with respect to which the symmetric Cartan matrix is B = (bij ). We may assume that the Cartan matrix A (with respect to ) is indecomposable. It

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is then clear that the Cartan matrix B with respect to 2 is also indecomposable. If the odd root α is of infinite type or has positive norm, then 2α is an even root. By Corollary 2.9, the Lie algebra G0¯ is a generalized Kac–Moody algebra and the set ˜ of 0 . ( ∩ 0 ) ∪ {2α | α ∈ − 0 , (α, α) = 0} can be completed to get a base ˜ Similarly the base 2 gives a base 2 of 0 . By ([15], §5.9, 11.10), we may assume ˜ 2 = . ˜ that We show that 2 ≤ + or ≤ − . If the root α ∈ 2 is not odd of norm 0, then ˜ 2 , and so α ∈ + . α (if α is even) or 2α (if α is odd) is in So suppose that the root α ∈ 2 is odd of norm 0 and that −α ∈ + . As it belongs to a base 2 , it is of finite type as there are no non-trivial ideals intersecting the Cartan subalgebra H trivially ([19], Lemma 2.1). If the Lie superalgebra G has finite growth, then it is an affine Lie superalgebra [14] since there are infinitely many roots. Affine generalized Kac–Moody Lie superalgebras have no odd root of finite type having norm 0 ([19], §2) and so there is nothing to show. Hence we may assume that the growth of the Lie superalgebra G is not finite. Then by ([19], Thm 3.3), either the matrix B or −B satisfies conditions (a) − (d). Furthermore, by ([19], Lemma 3.17), there are roots of infinite type and non-zero norm or else the indexing set I is infinite and all roots are of finite type. Suppose first that there is a positive (with respect to 2 ) root β of infinite type with non-zero norm. So (β, β) < 0. It follows that the matrix B satisfies conditions (a) − (d) for otherwise by ([19], Lemma 2.2), infinite type roots would have nonnegative norm. Also, depending on whether the root β is even or odd, β or 2β ∈ ˜ and thus β ∈ + . As the matrix B is indecomposable, we may assume that ˜ 2 = , (α, β) = 0, and so (α, β) < 0 by condition (b) satisfied by B. On the other hand, as −α ∈ + , by ([19], Lemma 2.2), there exists w ∈ W1 such that w(−α) ∈ , and so (β, −α) = (w(β), w(−α)) ≤ 0 since w(β) ∈ + . This contradiction implies that α ∈ + . Suppose next that the indexing set I is infinite and all roots are of finite type. Then, there are roots of positive norm and finite type in 2 , and so the matrix B again satisfies conditions (a) − (d). Since there are no roots of negative norm, and positive roots of positive norm with respect to 2 are sums of roots of positive norm belonging 2 different from α), to the base 2 ([19], Lemma 2.2) (and so are sums of roots in n 2 = {α = β0 , β1 , . . . , }, where for all integers n ≥ 0, i=0 βi is an odd root of norm 0, and βn is a root of positive norm and so is in + . Hence there is a minimal j integer n > 0 such that ni=0 βi ∈ + and − i=0 βi ∈ + for all j < n. Let r be the reflection with respect to the root βn . However these are not both positive roots, contradicting ([19], Lemma 2.2). This again shows that α ∈ + . Therefore, the base 2 ≤ + and so 2 = , proving the result. We are now ready to state the result on Borel sub-superalgebras and on the invariance of the Cartan matrix as it is an immediate consequence. Theorem 3.5. Suppose that the derived Lie superalgebra [G, G] is infinite dimensional. Then

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(1) there is a unique (up to equivalence) Cartan matrix associated to the generalized Kac–Moody superalgebra G; (2) all Borel subalgebras are conjugate to B + or B − under the action of the Weyl group W ; (3) the rank and Kac–Moody rank are well defined. Remark 3.6. Suppose that the Lie superalgebra [G, G] has finite dimension. It is well known that in general a simple finite dimensional classical Lie superalgebra has Cartan decompositions giving non-equivalent symmetric Cartan matrices [13]. For example, both 0 −1 0 2 −1 0 −1 2 −1 and −1 0 1 0 −1 0 0 1 −2 are Cartan matrices of A(1, 1). However, from the list of finite dimensional generalized Kac–Moody superalgebras in ([19], §2) and the list of all their simple root systems in ([13], §2.5.4), it follows that G is characterized by a unique Cartan matrix satisfying conditions (a) − (d) except when [G, G] is isomorphic to A(1, 0) when, up to equivalence, there are two such matrices associated to G: 0 −1 −1 0 (with S = I ) and

0 −1 −1 2

(with S = {1}). Furthermore, the Borel sub-superalgebras B + and B − are not W conjugate unless G is a Lie algebra or [G, G] is isomorphic to B(0, n) or to B(1, 1). Though in the latter case, there still are two conjugacy classes of Borel subalgebras (one with Cartan matrix A and the other −A). If [G, G] is isomorphic to A(1, 0), then there are three conjugacy classes of Borel subalgebras. We next give two applications of Theorems 3.4 and 3.5.

3.1. The Monster vertex algebra The crucial first step towards the proof of the Moonshine Theorem was the construction of a Z-graded moonshine module V for the Monster simple group [9]. It has a rich algebraic structure. One of its most important characteristics is that it is a vertex algebra [9], [1]. It would be useful to have a different construction of the Monster vertex algebra for its present one is very complex. In particular this may help to find an adequate integral form on V for modular moonshine questions [21], [6], [4], [5].

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Unlike the Monster vertex algebra, the Fake Monster vertex algebra has a nice simple construction. It is the lattice vertex algebra associated to the even unimodular non-degenerate Lorentzian lattice II 25,1 of rank 26 [1], [2]. Now, an important property of the moonshine module V is that it is a vertex algebra on which the Virasoro algebra acts with central charge 24, and an important aspect of a lattice vertex algebra constructed from an even non-degenerate lattice is that the Virasoro algebra acts on it with central charge equal to the rank of the lattice. Therefore, as the Monster vertex algebra is closely connected to the Leech lattice (i.e. the unique, up to isomorphism, positive definite even lattice of rank 24 not containing vectors of norm 2), it is natural to ask whether it can be constructed as a lattice vertex algebra VL , where L is a Niemeier lattice [8] (i.e. a positive definite even lattice of rank 24) as the Virasoro algebra acts on the VL with central charge 24. We remind the reader that the Virasoro algebra is the unique (up to isomorphism) central extension by a 1 dimensional centre of the Lie algebra of all derivations of the algebra of Laurent polynomials C[t, t −1 ]. It is generated by a central element c and elements Ln , n ∈ Z, satisfying 1 (m3 − m)δm+n,0 c. 12 Furthermore, there is a natural non-degenerate bilinear form on the vertex algebra VL such that the adjoint of the operator Ln is L−n for n ∈ Z. The monster Lie algebra G is constructed as follows: consider the tensor product of the moonshine module V with the lattice vertex algebra VII 1,1 . Tensor products of vertex algebras are vertex algebras. The quotient of the subalgebra V1 = {v ∈ V ⊗ VII 1,1 | L0 v = v; Ln v = 0, n ≥ 1} of V ⊗ VII 1,1 by the radical of the restriction of the bilinear form on V ⊗VII 1,1 to V1 is the monster Lie algebra. This is a generalized Kac–Moody algebra [3]. The Fake Monster Lie algebra is the generalized Kac–Moody algebra constructed in a similar way from the lattice vertex algebra VII 25,1 . Theorem 3.5 gives a different proof of the only known result that the Moonshine module does not as simple a construction as a lattice vertex algebra VL . [Lm , Ln ] = (m − n)Lm+n +

Corollary 3.7. If L is a Niemeier lattice, then VL is not isomorphic to the Moonshine module V . Proof. Suppose that the moonshine module is isomorphic to the lattice vertex algebra VL , where L is a Niemeier lattice. The direct sum L ⊕ II 1,1 is an even unimodular Lorentzian lattice of rank 26. Up to isomorphism there is a unique lattice II 25,1 with this property. It is the root lattice of the Fake Monster Lie algebra [2]. Therefore the tensor product vertex algebra V = VL ⊗ VII 1,1 is isomorphic to the Fake Monster vertex algebra. It therefore follows that the monster Lie algebra G is isomorphic to the Fake Monster Lie algebra. Hence by Theorem 3.5, they both have the same Cartan matrix (up to reordering). As this is false, it shows that these generalized Kac–Moody algebras are not isomorphic and proves the result.

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3.2. Automorphisms of generalized Kac–Moody superalgebras We finally give the decomposition of an arbitrary automorphism of the generalized Kac–Moody superalgebra G. Let ω be the Chevalley automorphism of period 4 of G [14]: i.e. fi if i ∈ S ω(ei ) = −fi otherwise, ω(fi ) = −ei for all i ∈ I , and ω(h) = −h for all h ∈ H . It is the Chevalley involution if G is a generalized Kac–Moody superalgebra. We remind the reader that a diagram automorphism θ of G satisfies θ ei = eθ˙ (i) and θfi = fθ˙ (i) for all i ∈ I , where θ˙ is a bijection on the indexing set I keeping the Cartan matrix invariant, i.e. aθi, ˙ θj ˙ = ai,j and such that θ˙ (S) = S. Consider the natural map from the root lattice Q to the Cartan subalgebra H : αi → hi , i ∈ I . In general this map is not injective: Suppose hi = hj , i = j ∈ I . Then aki = akj for all k ∈ I , i.e. the i-th and j -th columns of the Cartan matrix are equal. In particular this implies that the simple roots αi and αj are of infinite type or are both odd roots of norm 0 and finite type. When this is the case, it may be that hi = hj . However as i = j , αi = αj . Note that this map is always injective when G is a Kac–Moody superalgebra since then all the simple roots have positive norm. Set Ghi = {x ∈ G : [h, x] = (hi , h)x},

G−hi = {x ∈ G | [h, x] = −(hi , h)x}

and Ii = {j ∈ I | aki = akj for all k ∈ I }. Gαj and G−hi = j ∈Ii G−αj can have dimension greater than 1

Then Ghi = j ∈Ii when i ∈ I − J . Note that the Weyl group W clearly acts on the subspace of H generated by the elements hi , i ∈ I . We choose a set of representatives from each class of indices Ii : Iˆ = {i ∈ I | i ≤ j, for all j ∈ Ii }. The vectors ej (resp. fj ), j ∈ Ii form a basis of Ghi (resp. G−hi ). For simplicity of notation later, we choose an ordering of Ii = {j1 , . . . , jni }, where (s) (s) ni = dim Ghi and write ei = ejs and fi = fjs . For each i ∈ Iˆ, for each bijective linear map µi of the vector space Ghi ⊕ G−hi satisfying µi (Ghi ) = Ghi , µi (G−hi ) = G−hi , and [µi (ej ), µi (fj )] = µi ([ej , fj ]) for all j ∈ Ii , let µˆi be the automorphism of G given by µi (ej ), if j ∈ Ii µi (fj ), if j ∈ Ii µˆi (fj ) = µˆi (ej ) = otherwise; otherwise. ej fj

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First, a preliminary result. We write Aut(G) for the group of automorphisms of the generalized Kac–Moody algebra G, and In(G) for the derived Lie group of G0 . Lemma 3.8. (i) For fixed i ∈ Iˆ, the set Ai = {µˆ i | µi ∈ GL(Ghi )} is a group isomorphic to the general linear group GL(ni ), where ni = dim Ghi . (ii) The set D of diagram automorphisms is a group. (iii) The group of inner automorphisms In(G) is normal in G. (iv) For any i, j ∈ Iˆ, [Ai , In(G)] = 1 = [Ai , Aj ], and Ai is normal in Ai .D. (v) The Chevalley automorphism ω commutes with every element in Ai , D and In(G). Proof. (i) is obvious and (ii) and (iii) are standard results (the proof of the semisimple finite dimensional case applies). (v) and the second equality of (iv) are easy to see. (s) (s) With the above notation, for j ∈ Iˆ, for any k ∈ J , µˆ i ([ek , ej ]) = [ek , µˆ i (ej )] since dim Ghk = 1. The same holds with ek replaced by fk . Therefore µˆ i commutes with every inner automorphism. Since θ −1 µˆ i θ = µˆ θ˙ −1 i , Ai is normal in Ai .D. We are now ready to give the decomposition of an arbitrary automorphism of a generalized Kac–Moody superalgebra. Theorem 3.9. Suppose that the set of roots is infinite. Let φ be an automorphism of the generalized Kac–Moody superalgebra G. Then there is a diagram automorphism φ1 , an inner automorphism φ2 of G, and for each i ∈ Iˆ − J , automorphisms µi of G such that φ= µi ωi φ1 φ2 for some integer 0 ≤ i ≤ 3. i∈=I ˆ −J

This decomposition (in the given order) is unique. Proof. Let φ be an automorphism of the generalized Kac–Moody superalgebra G. Then φ(H ) is a Cartan subalgebra of G. So by Theorem 3.3, there exists an inner automorphism (i.e. an element of the derived Lie group G0 ) τ1 such that ψ1 = φτ1−1 fixes H . Then for all elements h ∈ H , [ψ1 (h), ψ1 (ei )] = ψ1 ([h, ei ]) = (h, h1 )ψ1 (ei )

for all i ∈ I ;

Thus ψ1 (ei ) is a root vector. Furthermore ψ1 ([G, G]) = [G, G] as its a Lie algebra homomorphism. Hence ψ1 fixes the subspace of H generated by the elements hi as < hi : i ∈ I >= H ∩ [G, G]. Clearly the ψ1 (ei ), ψ1 (fi ) and H are generators of the Lie superalgebra G satisfying relations (i)–(vii) of Definition 1.1. Also ψ1 induces a linear bijection of the root lattice Q keeping the set of roots invariant and mapping the base to another base of . Thus Theorem 3.4 implies that this isomorphism of the root lattice is given by an element w ∈ W . Now, there is an inner automorphism τ2 of the generalized Kac–Moody superalgebra G inducing the Weyl group element w (the argument remains the same as in the case of finite dimensional semisimple Lie

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algebras: see [10]). Hence ψ2 = ψ1 τ2−1 is an automorphism of G mapping the Cartan subalgebra to itself and ψ2 (ei ) ∈ Ceψˆ 2 (αi ) ,

ψ2 (fi ) ∈ Ceψˆ 2 (−αi ) ,

ψ2 (hi ) = hψˆ 2 (αi )

for all i ∈ I.

Set φ1 = τ2 τ1 . Since ω(h) = −h for all h ∈ H , the above implies that there exists some 0 ≤ l ≤ 3, and a diagram automorphism φ2 of G such that φφ1−1 ωl φ2−1 (Gα ) = Gα

for all α ∈ .

The decomposition of φ follows. Since we only need In(G)is normal in Aut(G), to prove uniqueness,

to check that (ω Ai ).D ∩ In(G) = 1. Suppose that φ = ωl µθ, where µ ∈ Ai and θ ∈ D is an inner automorphism. As φ keeps the Cartan subalgebra H invariant, there is an element w ∈ W corresponding to φ. Either w() = or −. For any root α, the height of w(α) or of −w(α) is the same as that of α. Suppose first that there are roots of infinite type. By Corollary 2.9, the even part ˜ ⊇ . By ([15], 11.13.3), G0¯ is a generalized Kac–Moody algebra and has a base + any element α ∈ Q with connected support containing an even number of indices in S satisfying (α, αi ) ≤ 0 is a positive even root of infinite type provided that it is not a ˜ with (β, β) ≤ 0. So for all α ∈ K, w(α) > 0, and multiple mβ for m ≥ 2 and β ∈ so w() = . Thus (w(α), w(αi )) ≤ 0 for all i ∈ I implies that (w(α), αj ) ≤ 0 for all j ∈ I . Hence w(α) = α by ([15], 3.12.b). This implies that w = 1. Suppose next that all roots are of finite type and that the indexing set I is countably infinite. Let w = ri1 . . . , rim be a reduced expression for w. Then as any finite subset of the indexing set I generates a finite subset of , there is a simple root αj ∈ such that (αj , αik ) = 0 for all 1 ≤ k ≤ m. Thus w(αj ) = αj and so w() = . However, w(αim ) < 0 by ([15], 3.11.b). This contradiction forces w = 1. Hence in all cases, φ acts as the identity on G. From this it easily follows that l = 0 and µ = 1 = θ. Remarks 3.10. 1. In the above definition of diagram automorphisms, we could have included the maps µi by considering the entire indexing set I rather than Iˆ. However, as the following example of the Monster Lie algebra shows the above method is more convenient. Indeed in practice, the roots are more naturally considered as elements of the Cartan subalgebra H than of an abstract root lattice Q, where the roots are linearly independent. 2. When i ∈ J , i.e. the corresponding simple root is of finite type, we do not have to include the automorphism µi . Indeed, the map given by ei → cei for non-zero constants c is accounted for by inner automorphisms. As an immediate consequence of Lemma 3.8 and Theorem 3.9, we can deduce that: Theorem 3.11. When dim([G, G]) = ∞, the automorphism group of the generalized Kac–Moody superalgebra G is the direct product of a central subgroup and of a

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semidirect product:

Aut(G) = ω × D Ai × In(G) .

Example 3.12. To construct the Monster Lie algebra 0 M, we take the Cartan subalgebra −1 ˆ H = C2 , with bilinear form given by the matrix −1 0 . The set I (as defined above) is countably infinite (S = ∅). Let h0 = (1, −1), hi = (1, i) for any integer i > 0. Then, the element (m, n) of Z 2 is a root of M with multiplicity c(mn), where for each n ∈ Z, c(n) is the coefficient of q nin the q-expansion of the modular invariant j (q) = q −1 + c(1)q + c(2)q 2 + · · · = n c(n)q n . The vectors (1, n) are the simple roots and have multiplicity c(n). Let us denote the bilinear form on H by u.v, for u, v ∈ H . As (1, n).(1, n) = −2n, the only simple root of finite type is (1, −1). Therefore the Weyl group W is cyclic of order 2 and GJ is isomorphic to sl2 . Hence the group Inn(M) of inner automorphisms of M is isomorphic to SL2 . The group of diagram automorphisms is trivial. Indeed (1, n).(1, n) = −2n. Hence the identity map is the only bijection of the set Iˆ keeping the bilinear form invariant, since under such a map the norm of the image of (1, n) must be the same as the norm of (1, n). For i > 0, the groups Ai are isomorphic to the general linear group glc(i) and the Chevalley involution has order 2. It follows that the group of automorphisms of G is isomorphic to: GLc(i) × SL2 . Aut(M) ∼ = Z2 × i>0

are representations of the Monster group. Thus the Monster The root spaces G(1,n) group is a subgroup of i>0 Ai . Remark 3.13. The finite dimensional case is classical when G is a Lie algebra and has been treated in [22] when G is a simple finite dimensional Lie superalgebra. Hence it need not be included in this paper. Acknowledgements. The author is very grateful to Richard Borcherds for useful discussions, and thanks the Centre de Recerca Matèmatica, the Institut des Hautes Etudes Scientifiques, the Max-Planck Institute and the University of California, Santa Cruz, for their financial support and hospitality.

References [1]

R. E. Borcherds, Vertex algebras, Kac–Moody algebras, and the Monster, Proc. Natl. Acad. Sci. USA 83 (1986), 3068–3071.

[2]

R. E. Borcherds, The Monster Lie algebra, Adv. Math. 83 (1990), 30–47.

[3]

R. E. Borcherds, Monstrous Moonshine and monstrous Lie superalgebras, Invent. Math. 109 (1992), 405–444.

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[4]

R. E. Borcherds, Modular Moonshine III, Duke Math. J. 93 (1998), 129–154.

[5]

R. E. Borcherds, Problems in Moonshine, in First International Congress of Chinese Mathematicians (Beijing, 1998), ed. by Lo Yang and S.-T. Yau, AMS/IP Stud. Adv. Math. 20, Amer. Math. Soc., Providence, RI 2001, 3–10.

[6]

R. E. Borcherds, A. J. E. Ryba, Modular moonshine II, Duke Math. J. 83 (1996), 435–459.

[7]

N. Bourbaki, Lie groups and Lie algebras, Part 1, Hermann, Paris 1975.

[8]

J. H. Conway, N. J. A. Sloane, Sphere Packings, Lattices and Groups, 3rd ed., Grundlehren Math. Wiss. 290, Springer-Verlag, New York, Berlin 1999.

[9]

I. B. Frenkel, J. Lepowsky, J. Meurman, Vertex operator algebras and the Monster, Pure Appl. Math. 134, Academic Press, Boston 1988.

[10] W. Fulton, J. Harris, Representation Theory, A First Course, Grad. Texts in Math. 129, Springer-Verlag, New York 1991. [11] O. Gabber, V. G. Kac, On defining relations of certain infinite-dimensional Lie algebras, Bull. Amer. Math. Soc. 5 (1981), 185–189. [12] V. G. Kac, Simple Irreducible graded Lie algebras of finite growth, Math. USSR-Izv. 2 (1968), 1271–1311. [13] V. G. Kac, Lie Superalgebras, Adv. Math. 26 (1977), 8–96. [14] V. G. Kac, Infinite-Dimensional Algebras, Dedekind’s η-Function, Classical Möbius Function and the Very Strange Formula, Adv. Math. 30 (1978), 85–136. [15] V. G. Kac, Infinite dimensional Lie algebras, 3rd ed., Cambridge University Press, Cambridge 1990. [16] R. V. Moody, A new class of Lie algebras, J. Algebra 10 (1968), 211–230. [17] V. G. Kac, D. H. Peterson, Infinite flag varieties and conjugacy theorems, Proc. Natl. Acad. Sci. USA 80 (1983), 1778–1782. [18] V. G. Kac, S. P. Wang, On automorphisms of Kac–Moody algebras and groups, Adv. Math. 92 (1992), 129–129. [19] U. Ray, A Characterization Theorem for a certain class of graded Lie superalgebras, J. Algebra 229 (2000), 405–434. [20] U. Ray, Generalized Kac–Moody algebras and some related topics, Bull. Amer. Math. Soc. 38 (2001), 1–42. [21] A. J. E. Ryba, Modular Moonshine?, in Moonshine, the Monster, and Related Topics (South Hadley, 1994), ed. by Chongying Dong and Geoffrey Mason, Contemp. Math. 193, American Mathematical Society, Providence, RI 1996, 307–336. [22] V. V. Serganova, Automorphisms of simple Lie superalgebras, Izv. Akad. Nauk. SSSR Ser. Mat. 48 (3) (1984), 585–598; Math. USSR-Izv. 24 (1985), 539–551. Urmie Ray, CRM, Apartat 50, 08193 Bellaterra, Spain E-mail: [email protected]

Bounding numbers and heights of characters in p-constrained groups Geoffrey R. Robinson

Abstract. Positive confirmation of Brauer’s k(B)-problem (which is to prove that the number of ordinary irreducible characters in a p-block B is bounded above by the order of a defect group for B ) has recently been achieved in the case that B is a block of a p-solvable group, which represents the culmination of the work of many authors, beginning with Nagao [6] and Knörr [4]. By contrast, the k(B) problem for p-constrained groups remains open at present. In this article, we present a new character-theoretic conjecture for p-constrained groups, which is, for p-solvable groups, strictly stronger than the k(B)-problem. The new conjecture seems neither to imply, nor to be implied by, the k(B)-problem for the general p-constrained group. In some ways, though, it is more precise for such a group. A key observation is Theorem 1.3, which may be of independent interest, and which clears the way for the charactertheoretic methods successfully employed in the k(GV )-problem to be used in more general situations. In particular, we prove that the new conjecture holds in fairly general circumstances, to be described more precisely later. The generalized characters constructed in Theorem 3.1 may also be of independent interest. 2000 Mathematics Subject Classification: 20C20

1. The conjecture and proof in special cases We consider a finite group G which has a non-trivial Abelian normal (p)-subgroup V such that CG (V ) is a p-group. Let S be a Sylow p-subgroup of G, and set P = S/V . For each v ∈ CV (S), set H (v) = CG (v)/V and let H (v)p denote the set of p-regular elements of H (v). We abbreviate H (1) to H . We let kd (G) denote the number of irreducible characters, χ , of G which satisfy p d χ (1)p = |G|p . We make the following: Conjecture 1.1. Under the hypotheses above, we have ∞ |H (v)p | kd (G) : v ∈ CV (S) . ≤ max p 2d |V ||H (v)| d=0

In particular,

∞ kd (G) d=0

p2d

with equality only possible if V ∈ Sylp (G).

≤

1 , |V |

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Remark 1.2. If it were the case that |H (v)p | |Hp | max : v ∈ CV (S) = , |H (v)| |H | this would allow a cleaner statement of the conjecture, but A. Maróti has recently furnished us with examples where this equality fails. We will henceforth refer to the second version of the inequality as the “weak version” of the conjecture. Note that it asserts that a necessary condition for equality is that V ∈ Sylp (G), but does not assert that this is sufficient to yield equality. Another interpretation of the conjecture (in the weak form, for ease of exposition), is that |G|p2 . χ(1)p2 ≤ |V | χ∈Irr(G)

For purposes of comparison, the reader might like to note that (after reconciling notation), it is a consequence of Brauer–Feit [1] that ∞ kd (G) d=0

p 2d

≤ 1,

and (implicitly) that ∞ kd (G) d=0

p 2d

≤

|Gp | . |G|

While our conjectured bound may at first glance look only marginally sharper, the difference is significant. For example, if we additionally assume that G/V has a normal p-complement, then (the stronger version of) our conjectured bound (suitably re-written) translates to µ(1)p2 ≤ |G|p . µ∈Irr(G)

This serves to illustrate that our conjectured bound is sometimes sharper than the bound predicted in Brauer’s k(B)-problem, which in the last example would reduce to k(G) ≤ |G|p . It also shows that the new conjecture (even in its weak form) implies the k(GV )-conjecture, and hence Brauer’s k(B) problem for p-solvable groups. Another consequence of the weak form of our conjecture would be that G/V should have at most |V | p-blocks of defect 0, or, more precisely, that G should have at most |V | irreducible characters which are V -projective (an irreducible character χ is V -projective under the current hypotheses if and only if its degree is divisible by [G : V ]p ). Notice that the weaker version of our conjecture would be sure to hold for G if we knew that k(G) ≤ |V |, since the degree of any irreducible character of G divides [G : V ] by Ito’s Theorem. However, it is not generally true that k(G) ≤ |V | under our hypotheses.

Bounding numbers and heights of characters in p-constrained groups

309

We also note that the inequality ∞ kd (G) d=0

≤

p 2d

1 |G|p

does not hold in general when G is p-solvable with Op (G) = 1, as the example G = GL(2, 3) already illustrates (for p = 2). We finally remark that the case of p = 3 and G the Hessian group of order 34 23 (recall that G is the semi-direct product on an extra-special group of order 27 and exponent 3 with SL(2, 3), where the action is the one obtained naturally from the fact that SL(2, 3) = Sp(2, 3)) shows that the inequality ∞ kd (G) d=0

p2d

≤

1 |Op (G)|

can fail even for p-solvable G (with Op (G) = 1), when Op (G) is non-Abelian. Hence the assumption that V is Abelian is crucial. Notation. If a finite group X acts on a p-group U, we will say that the Brauer character of X afforded by U is that afforded by the action of X on the direct sum of the chief factors of XU contained within U, viewed as GF(p)X-modules. We will prove that the conjecture holds in the case that the Brauer character afforded by V is real-valued, or more generally, in the case that there is an element v ∈ CV (S) such that ResG CG (v) (V ) has a submodule U on which no non-identity p-regular element acts trivially, and which affords a real-valued Brauer character. These are similar to the circumstances under which the k(GV )-conjecture was shown to hold in [9] (where S = V ), though the proof is in some ways more delicate in this case. We also recall some general facts which are useful, and will be freely used in what follows: i) If X is a finite group with a Sylow p-subgroup Q, and φ is a Brauer character (for the prime p) of X, then the class function of X which vanishes on all p-singular elements, and agrees with |Q|φ on all p-regular elements, is a generalized character. ii) X is as above, and X has a normal p-subgroup W, then there is a natural bijection between generalized characters of X which vanish on all p-singular elements and generalized characters of X/W which vanish on all p-singular elements. In this bijection, the generalized character θ of X corresponds to the generalized character θ ∗ of X/W which satisfies θ ∗ (xW ) =

θ (x) |CW (x)|

for all p-regular x ∈ X. The first assertion above is well-known. The second is frequently used in [9], and is due to W. F. Reynolds. The next auxiliary result is true in complete generality.

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Theorem 1.3. Let X be a finite group, A be an Abelian normal subgroup of X, and let µ be a complex irreducible character of X. Then for every y ∈ X, [X : CX (y)]µ(y) [A : CA (y)]µ(1) is an algebraic integer. i Proof. Let ω = exp( 2π |X| ) and let R = Z[ω]ρ be the localization at some prime ideal ρ of Z[ω]. Since ρ is arbitrary, it suffices to prove that

[X : CX (y)]µ(y) ∈ R. [A : CA (y)]µ(1) Let λ be a linear constituent of ResX A (µ), and let H = IX (λ). We may write X µ = IndH (ζ ) for some irreducible character ζ of H which lies over λ. Let S + denote the sum of the elements of S in RX, whenever S is a subset of X. A routine calculation shows that µ((y X )+ ) ζ ((y X ∩ H )+ ) = , µ(1) ζ (1) so it suffices by induction to consider the case that H = X, since y X ∩ H is a union of conjugacy classes of H, and since [A : CA (y )] = [A : CA (y)] whenever y is an X-conjugate of y. Hence, we may, and do, suppose that λ is X-stable. Let V be an RX-module affording character µ, and let σ : X → EndR (V ) be the associated representation. We view E = EndR (V ) as an interior X-algebra in the usual fashion, employing σ . We note that A acts trivially on E, since elements of A act as scalars on V by the stability of λ. Hence we see that X Tr X CX (y) (yσ ) = [A : CA (y)] Tr ACX (y) (yσ ),

while Tr X ACX (y) (yσ ) is a scalar matrix with entries in R. From this, we conclude that [X : CX (y)]µ(y) ∈ R, [A : CA (y)]µ(1) as required. Now let us return to our particular situation. If p is an odd prime, we let ε be the sign for which p ≡ ε (mod 4), and we let R √ be the ring of algebraic integers of Q[ εp]. If p = 2, we let R = Z. A key result for the cases of the conjecture we can prove, which is related to Knörr’s notion in [4], of “containing a square” is: Lemma 1.4. Let K be a finite group which acts on a p-group U in such a way that U affords a real-valued Brauer character of K. Then there is an R-combination of Brauer characters, θ, of K such that:

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311

i) |θ (x)|2 = [U : CU (x)] for each p-regular x ∈ K. √ ii) θ is integer valued if p = 2, and for p odd, θ (x) has the form ε(x)( εp)m(x) for some sign ε(x) and some integer m(x) (for each x ∈ K). Proof. It suffices to consider the case that U is elementary Abelian and completely reducible as GF(p)K-module. In that case, by the results of [9], which rely on the results of Gow [3], we can find an integer-valued Brauer character ψ of K such that either: i) ψ(x)2 = [U : CU (x)] for all p-regular x ∈ K, or ii) ψ(x)2 = [U : CU (x)] for all p-regular x ∈ K which act with determinant 1 on U , and ψ(x)2 = [U :CpU (x)] for all p-regular x ∈ K which act with determinant −1 on U . In the second case (which only occurs when p is odd), let L be the normal subgroup of K consisting of those elements which acts with determinant 1 on U . Let√χ be the √ 1+ εp class function which takes the value 1 on L and εp on K\L. Then χ = 2 1K + √ 1− εp 2 λ,

where λ is the character afforded by the determinant on U . Hence χ is an R-combination of characters of K, so we may set θ (x) = χ (x)ψ(x) for all p-regular x ∈ K to obtain the desired conclusion. Now let us turn to the proof of the conjecture in some special cases. Suppose first that CG (V ) is a p-group, and that V contains a normal subgroup U of G such that CG (U ) is a p-group and such that U affords a real-valued Brauer character. We let R be as above. Let P be a Sylow p-subgroup of G/V . We have an Rcombination of Brauer characters of G, say θ, such that |θ (x)|2 = [U : CU (x)] for all p-regular x ∈ G. We may, and do, regard θ as an R-combination of Brauer characters of G/V . Then |P |θ is an R-combination of characters of projective indecomposable modules of G/V , and the class function,ψ, of G defined by ψ(x) = |CV (x)||P |θ (x) for all p-regular x ∈ G is an R-combination of characters of projective indecomposable modules of G. Let us compute ψ, µ for an irreducible character µ of G. We know from (x)]µ(x) is an algebraic integer multiple of [V : CV (x)] for Theorem 1.3 that [G:CGµ(1) −1

) is an algebraic integer multiple of each p-regular x ∈ G. Hence [G:CG (x)]µ(x)ψ(x µ(1) −1 |P ||V |θ (x ) for each p-regular x ∈ G. Now θ (1) = ±1, while θ (x −1 ) is divisible √ by εp (or by 2 in the case p = 2) for each non-identity p-regular element x of G. Since |G|p = |P ||V |, we deduce that ψ, µ is divisible (in R) by µ(1)p but not by √ εpµ(1)p (or by 2µ(1)2 in case p = 2). In the following calculation, the reader may find it helpful to recall that there is a bijection between the classes of p-regular elements of G and of G/V , and that if x is G (x)] a p-regular element of G, then the number of conjugates of xV in H is [G:C [V :CV (x)] . We have |G| ψ, ψ = [G : CG (x)][U : CU (x)]|CV (x)|2 |P |2 ≤ x∈Gp /G

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Geoffrey R. Robinson

≤

[G : CG (x)]|V ||CV (x)||P |2 ,

x∈Gp /G

so that |H | ψ, ψ ≤

x∈Gp /G

[G : CG (x)]|V ||P |2 [V : CV (x)]

and hence

ψ, ψ ≤ Since

ψ, ψ ≥

|V ||P |2 |Hp | . |H | µ∈Irr(G)

µ(1)p2 ,

dividing both sides by |P |2 |V |2 gives the result. To be more precise, when p is odd, we see that

ψ, ψ = ψ σ , ψ σ where σ is a Galois automorphism fixing p -roots of unity and sending negative), and we note that

√ εp to its

|α σ |2 + |α|2 ≥ 2 for any non-zero α ∈ R. We have now dealt with the case that V affords real-valued Brauer character, or more generally, when it has a subgroup U which affords real-valued Brauer character and for which CG (U ) is a p-group. Suppose now that there is some non-identity element v ∈ CV (S) such that ResG CG (v) (V ) has a subgroup U with the same property with respect to CG (v). We may define a class function δ analogously for CG (v) so that |δ(x)|2 = [U : CU (x)] for all p-regular x ∈ CG (v). We may, and do, regard δ as an R-combination of Brauer characters of H (v) = CG (v)/V . Then |P |δ is an R-combination of characters of projective indecomposable modules of H (v), and the class function,ψ, of CG (v) defined by ψ(x) = |CV (x)||P |δ(x) for all p-regular x ∈ CG (v) is an R-combination of characters of projective indecomposable modules of CG (v). Now (letting ζ denote a primitive complex |v|-th root of unity), we define the Z[ζ ]combination of characters β of CG (v) via β(vx) = ψ(x) if x is p-regular, β(x) = 0 otherwise. We then consider the induced class function ψ = IndG CG (v) (β), which vanishes outside the p-section of v in G, and which has the same norm as β does. Again, we know from Theorem 1.3 that whenever µ is an irreducible charac(xv)]µ(xv) is an algebraic integer multiple of [V : CV (xv)] for each ter of G, [G:CGµ(1)

Bounding numbers and heights of characters in p-constrained groups

313

−1

) is an algebraic integer multiple p-regular x ∈ CG (v). Hence [G:CG (xv)]µ(xv)ψ((xv) µ(1) −1 of |P ||V |δ(x ) for each p-regular x ∈ CG (v). Again, δ(1) = ±1, while δ(x −1 ) is √ divisible by εp (or by 2 in the case p = 2) for each non-identity p-regular element x (v)]µ(v) of G. Furthermore, [G:CGµ(1) lies in Z[ζ ]\(1 − ζ )Z[ζ ], since µ is in the principal (and only) p-block of G, and v ∈ Z(S). We deduce that ψ, µ is divisible (in Z[ζ ]) by µ(1)p but not by (1 − ζ )µ(1)p . The computation showing that

ψ, ψ ≤

|V ||P |2 |H (v)p | |H (v)|

follows as before, using the fact that

ψ, ψ = β, β. We now need to invoke a standard argument using algebraic conjugation (essentially dating back to Burnside, and exploiting the arithmetic- geometric mean inequality in the case of totally positive algebraic integers). For every τ ∈ Gal(Q[ζ ]/Q), we have

ψ τ , ψ τ ≤

|V ||P |2 |H (v)p | , |H (v)|

and (for µ as above), we know that

ψ τ , µ ∈ Z[ζ ]\(1 − ζ )Z[ζ ]. µ(1)p Hence

τ ∈Gal(Q[ζ ]/Q)

| ψ τ , µ|2 ≥ | Gal(Q[ζ ]/Q)|µ(1)p2 .

Hence we may still conclude that µ∈Irr(G)

µ(1)p2 ≤

|V ||P |2 |H (v)p | , |H (v)|

which completes the proof that the strong form of the conjecture holds under the present hypotheses. Remark 1.5. The interested reader may care to note that it follows from the proof of the conjecture in the cases above, that if H is a subgroup of Sn and G = Z2 H, then we have χ(1)22 ≤ 23n−2 . χ∈Irr(G)

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Geoffrey R. Robinson

This already follows from the weak form of the conjecture, using the fact that a Sylow 2-subgroup of Sn has order at most 2n−1 . Notice in particular, that this yields χ(1)22 ≤ 23n−2 . χ∈Irr(H )

2. Clifford-theoretic reduction and residual obstacles The Clifford-theoretic reductions of the paper [9] and [8] carry over without any real difficulty to the present situation. We have seen that a sufficient condition for the conjecture to hold is the existence of an element v ∈ V with [G : CG (v)] coprime to p such that ResG CG (v) (V ) has a subgroup U which affords a real Brauer character for CG (v) and no non-trivial p-regular element of CG (v) acts trivially on U . In the situation of [9], the condition that [G : CG (v)] is coprime to p is automatically satisfied, but in this new setting, we need to keep track of the condition. However, this is routine. This means that one critical configuration which remains to consider after Cliffordtheoretic reductions have been performed is the following: We have a finite group L which acts faithfully and absolutely irreducibly on a GF(p)L-module U . L has a Sylow p-subgroup T , and there is no u ∈ U T for which ResL CL (u) (U ) has a faithful submodule W which affords a real-valued Brauer character. Furthermore, we may suppose that either: i) F ∗ (L) = Z(L)Oq (L), where q = p is a prime, and Oq (L) has all characteristic Abelian subgroups central. Furthermore, Q also acts absolutely irreducibly on U, or: ii) F ∗ (L) = Z(L)E, where E is a quasi-simple subgroup acting absolutely irreducibly on U . However, if E is a group of Lie type in characteristic p, for example, it is extremely likely that there will indeed be no such element u, since U T is 1-dimensional, so it seems that a different approach will be needed if such sections are involved in our group G. Our next result is relevant to this “natural characteristic” situation, and is really concerned with the case when U as above is the natural module, which may be the most difficult case. Returning to our group G, let be a set of representatives for the orbits of Irr(V ) under the action of G. For a non-trivial λ ∈ , let Sλ = IS (λ), and assume (as we may) that Sλ ∈ Sylp (IG (λ)). Let Pλ = Sλ /V . Suppose that IG (λ)/V is still p-constrained (in the strong sense that its largest normal p-subgroup contains its own centralizer). Let us count the sum of the squares of the p-parts of the degrees of those irreducible characters of G which lie over (G-conjugates of) λ. This is [P : Pλ ]2 times the corresponding sum for IG (λ). Let Vλ be the kernel of λ. Then all irreducible characters of Hλ which lie over λ contain Vλ in their kernels and V /Vλ is cyclic and central in IG (λ)/Vλ . Furthermore, λ may be identified with an IG (λ)/Vλ -stable linear character of V /Vλ .

Bounding numbers and heights of characters in p-constrained groups

We wish to evaluate

χ∈Irr(IG (λ),λ)

315

χ (1)p2 .

We may replace IG (λ) by Hλ = IG (λ)/Vλ and set Zλ = V /Vλ . By hypothesis, Hλ has only one p-block. From [7], we know that χ∈Irr(Hλ )

χ(1)p2 ≤

|Hλ |p2 |Zλ |

.

We wish to count those characters which lie over a linear character of maximal order of Zλ . We note that this contribution is independent of the algebraic conjugate chosen, and that there are at most (p − 1)|Zλ |/p such algebraic conjugates. Hence χ∈Irr(IG (λ),λ)

χ(1)p2 ≤

p|Pλ |2 . p−1

We have thus proved: Theorem 2.1. Let G, V be as above and assume further that IG (λ)/V is p-constrained for each non-trivial linear character λ of V . Then χ∈Irr(G):V ≤ker(χ)

χ(1)p2 ≤

p|P |2 (|| − 1) , p−1

where is a set of representatives for the orbits of G on Irr(V ). Remark 2.2. We remind the reader that the weak form of the conjecture asserts in this notation that we should have χ(1)p2 ≤ |P |2 |V |, χ∈Irr(G)

so that under the hypotheses of the theorem above, we are some considerable way to achieving this conjectured inequality. In fact, if G/V is itself p-constrained, the inequality holds, using the Brauer–Feit bound. More generally, the inequality holds if G/V has at most |V |− p(||−1) p−1 p-blocks. In the case that G/V is a quasi-simple group of Lie type in characteristic p and V is its natural module, there are 1 + |Z(G/V )| 1 p-blocks of G/V , and we certainly have |Z(G/V )| < (|V | − 1) 2 , so the required inequality holds comfortably.

3. Some generalized characters In this section, we note the existence of some generalized characters which are relevant to the questions considered earlier, but also have some independent interest. The relevance of these generalized characters lies in the fact that if the finite group G acts

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faithfully on a GF(p)G-module U, we are interested in constructing a virtual Brauer character θ such that |θ(y)|2 ≤ [U : CU (y)] for all p-regular y ∈ G, θ (y) ∈ π for all p-regular y ∈ G# , and θ(1) ∈ π (where π is a suitable prime ideal containing the rational prime p). Theorem 3.1. Let X be a finite group, p be a prime, n be a positive integer relatively prime to p. Let U be a GF(p)X-module, and for each g ∈ X, let mg (n, U ) denote the number of primitive n-th roots of unity (counting multiplicities) among the eigenvalues of g on U ⊗GF(p) GF(p). Define a class function ψn,U of X via ψn,U (g) = pmg (n,U ) for all g ∈ X. Then ψn,U is a generalized character of X. Proof. We note that ψn,U is constant on p -sections, so it suffices to check that ResX E (ψn,U ) is a generalized character for each Brauer elementary p -subgroup, E, of X. Hence we may, and do, assume that X is a p -group. We note that the result is visibly true when n = 1, since ψ1,U is the permutation character afforded by the action of X on the vectors of U . Let χ be the (Brauer) character of X afforded by U . Suppose that n > 1, and that prime divisors of n. Let J = {1, 2, . . . , m}. For each p1 , p2 , . . . , pm are the distinct subset I ⊆ J, set pI = i∈I pi , where the empty product is defined as 1. It is easy to check that for every g ∈ X, we have mg (n, U ) = (−1)|I | ResX pn (χ ), 1,

g

I ⊆J

so that

|I |even |CU (g

ψn,U (g) =

|I |odd |CU (g

I

n pI

)|

n pI

)|

.

Using so-called Adams operations, we know that for each integer r, the class function for which g → |CU (g r )| for all g ∈ X is a generalized character. Hence ψn,U is an integer-valued ratio of generalized characters whose values are coprime to |X|, so is a generalized character. This last fact is a result of G. Mason [5], but we include a proof of a slightly stronger version of Mason’s result for the convenience of the reader: Lemma 3.2. Let X be a finite group, γ be an algebraic integer-valued class function of X with the property that γ (g n ) = γ (g)σn whenever n is an integer coprime to |X| and σn is a Galois automorphism sending |X|-th roots of unity to their n-th powers. Suppose that there is a generalized character β of X such that β.γ is a generalized character of X and such that g∈X:γ (g)=0 β(g) is coprime to |X| (this quantity is an integer under the current hypotheses). Then γ is a generalized character. Proof. We may multiply β by various algebraic conjugates if necessary, and assume that β is integer-valued. We may find a positive integer r so that β(g)r ≡ 1 (mod |X|)

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whenever γ (g) = 0. Hence we may replace β by β r if necessary, and assume that (g) is an β(g) ≡ 1 (mod |X|) whenever γ (g) = 0. We now have that γ (g)β(g)−γ |X| algebraic integer for all g ∈ X. It follows that β.γ − γ , χ is an algebraic integer whenever χ is an irreducible character of X. Since our hypotheses together imply that β.γ , χ ∈ Z and that γ , χ ∈ Q, we deduce that γ , χ ∈ Z. Hence γ is a generalized character, as χ was arbitrary. Remark 3.3. We note that n>1 ψn,U takes the value [U : CU (x)] for all p-regular x ∈ X. We also remark that it is not really essential that U should be a module over the prime field to construct the generalized characters above, but that this is no real loss of generality in any case (first, it is no loss of generality to assume that U is irreducible. In that case, the argument works for any finite field of characteristic p and size q, as long as we replace pmg (n,U ) by q mg (n,U ) . However, the same generalized character can clearly be constructed from an irreducible GF(p)X-module).

References [1] R. Brauer and W. Feit, On the number of irreducible characters of finite groups in a given block, Proc. Nat. Acad. Sci. U.S.A. 45 (1959), 361–365. [2] D. Gluck and T. Wolf, Brauer’s height conjecture for p-solvable groups, Trans. Amer. Math. Soc. 282 (1984), 137–152. [3] R. Gow, On the number of characters in a block and the k(GV )-problem for self-dual V , J. London Math. Soc. (2), 48 (199), 441–451. [4] R. Knörr, On the number of characters in a block of a p-solvable group, Illinois J. Math. 28 (1984), 181-210. [5] G. Mason, Some applications of quasi-invertible characters, J. London Math. Soc. (2) 33 (1986), 40–48. [6] H. Nagao, On a conjecture of Brauer for p-solvable groups, J. Math. Osaka City University 13 (1962), 35–38. [7] G. R. Robinson, On the number of characters in a block, J. Algebra 138 (1991), 515–521. [8] G. R. Robinson, Further reductions for the k(GV )-problem, J. Algebra 195 (1997), 141–150. [9] G. R. Robinson and J. G. Thompson, On Brauer’s k(B) Problem, J. Algebra 184 (1996), 1143–1160. Geoffrey R. Robinson, School of Mathematics and Statistics, University of Birmingham, Birmingham B15 2TT, England E-mail: [email protected]

Classification of buildings Mark A. Ronan∗

Introduction This paper discusses the classification of buildings of all types: spherical, affine, etc. The question of constructing these buildings, which goes hand in hand with their classification, can be dealt with in several ways. The spherical buildings of classical type are constructed using classical forms on vector spaces, and other spherical buildings can be constructed using algebraic groups. More generally buildings can be constructed from groups having a BN-pair, and this is how the existence of affine buildings is established. There is also a direct construction [RT1] that I shall explain in Sect. 5. It produces the groups as by-products. Some types of buildings do not admit a classification because wild examples can be constructed. I shall deal with this too. The structure of the paper is as follows. The first section sets the scene, introducing apartments and buildings. Then in Sections 2 and 3 we deal with spherical and affine buildings – spherical ones arise largely from algebraic groups, and affine ones arise when the base field for the group has a specified valuation into the integers. Section 4 deals with twin buildings. These arise largely from Kac–Moody groups in the same way that spherical buildings arise from algebraic groups. They are like spherical buildings, but with infinite Weyl groups. Section 5 then explains some constructions of buildings, mainly using blueprints, and this leads to a short Section 6 mentioning some questions and conjectures. The assumption throughout this paper is that the reader has some familiarity with a few basic ideas in the theory of buildings, particularly regarding spherical buildings, which are the best known ones. For more details I suggest Tits’ original book [T3] and my book [R2]. The purpose of the present article is as a guide to known classification results, and some open questions. This paper originates from my lecture at the Florida conference celebrating John Thompson’s 70th birthday. I would like to thank the organizers for inviting me and thank my audience for their enthusiastic response. ∗ Partially supported by the NSA

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1. Buildings and apartments An apartment is a Coxeter complex for some Coxeter group W . This is a group generated by involutions s1 , . . . , sn along with defining relations (si sj )mij = 1 (when mij is infinity si and sj generate an infinite dihedral group). The rank of W is n. All Coxeter groups are non-trivial [T2], and moreover if J is a subset of {1, . . . , n} then the group WJ generated by the set {sj }j ∈J (along with the relations for pairs of generators) embeds as a subgroup of W . If for each j in J and k in K, each sj commutes with each sk , then the direct product WJ × WK embeds as a subgroup of W . A Coxeter group having no direct factors WJ is called irreducible. A Coxeter complex for W is a tiling of some space, such as a sphere, Euclidean space, or hyperbolic space in which the maximal dimensional tiles, called chambers, are mutually isometric. A chamber has n faces of codimension 1, called panels (n being the rank of W ), and the group acts simple-transitively on the set of chambers. Each conjugate of an si is called a reflection, and its fixed point set is called a wall. The halves of the Coxeter complex on either side of a wall are called roots. Here are some examples. Spherical apartments: These are tilings of an (n − 1)-sphere, and each chamber is an (n − 1)-dimensional simplex. The walls are reflecting hyperplane sections and the roots are hemispheres. Two chambers in an apartment are said to be opposite if they are antipodal on the sphere. Affine apartments: These are tilings of Euclidean space, and when W is irreducible each chamber is an (n − 1)-simplex. Otherwise the tiling is a direct product of lower dimensional tilings, the simplest example being a tiling of the Euclidean plane by rectangles, for which W is the direct product of two infinite dihedral groups. The walls are reflecting hyperplanes and the roots are half-spaces. Hyperbolic apartments: These are tilings of hyperbolic space, and there are many of them. Even when one restricts to the hyperbolic plane there are infinitely many. For example any triangle having dihedral angles that divide 180◦ and add up to less than 180◦ can be used to tile the hyperbolic plane, and there is a rank 3 Coxeter group in each case. Another example is provided by tiling the hyperbolic plane with regular hyperbolic n-gons (n ≥ 5) having dihedral angles of 90◦ . The chambers are n-gons, and the Coxeter group has rank n; it is generated by involutions si and relations (si si+1 )2 = 1 as i ranges over the integers modulo n. Buildings. A building of type W is a union of apartments of that type. Any two chambers lie in a common apartment, and any two apartments intersect in a convex subset (see [R2] for a formal definition). A building is called spherical, affine, etc. if its apartments are. As with Coxeter complexes, the codimension 1 faces of a chamber

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are called panels, and the number of panels on a chamber is the rank of the building. When the chambers are simplexes, the rank is one greater than the dimension. This is the case for all spherical and all irreducible affine buildings. Buildings of rank zero are vacuous, those of rank 1 are discrete sets of points, and those of rank 2 are bipartite graphs. A spherical building of rank 2 is called a generalized polygon, and a non-spherical building of rank 2 is a tree. Buildings, like their apartments, can be reducible or irreducible. Every building is a direct product of irreducibles in a unique way (see e.g. [R2, 3.10]), so the classification of buildings reduces immediately to the irreducible case. We shall first consider spherical buildings, which is where the whole subject started.

2. Spherical buildings and algebraic groups Buildings were invented by J. Tits who, back in the early 1950s, wanted to generalize classical projective geometry to the exceptional simple Lie groups (those of types E6 , E7 , E8 , F4 and G2 ), and thereby construct such groups over arbitrary fields. The group E6 had been dealt with by Dickson [D] using a cubic form in 27 dimensions, but Tits wanted to deal with all the exceptional groups in a uniform manner. In 1955, Claude Chevalley published his famous paper [Ch] on “Chevalley groups” and this was soon followed by variations on his methods by Chevalley himself, Steinberg [St] (Variations on a theme of Chevalley), Tits and others. These “twisted” Chevalley groups are now embraced by the theory of algebraic groups, where they appear as groups of rational points over fields, such as finite fields, that are not algebraically closed. While the theory of algebraic groups was moving forward, Tits developed the theory of buildings. He created axioms that could be given combinatorially, and other axioms that could be given group-theoretically in terms of a BN-pair. This was presented in his book [T3] where he goes on to classify spherical buildings. As mentioned earlier it suffices to consider the irreducible ones (those having a connected diagram), and in rank at least 3 this means one of the types An (n ≥ 3), Bn = Cn (n ≥ 3), Dn (n ≥ 4), E6 , E7 , E8 , F4 . The other types of finite Coxeter groups W , namely H3 and H4 , were eliminated by Tits [T4] using the non-existence of Moufang generalized 5-gons (see 2.4). Tits’ approach was to show that an irreducible spherical building of rank ≥ 3 must be determined by its local data, and must admit automorphisms taking one local datum to another. The chambers on a codimension 2 face of a spherical building form a rank 2 building, referred to as a rank 2 residue of , and the term local means the union of the rank 2 residues on a given chamber. To state the relevant theorem, let c be a chamber, and let E1 (c), respectively E2 (c), denote the subcomplex formed by chambers having a face of codimension 1, respectively 2, in common with c. A building is thick if each panel lies on at least three chambers.

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Theorem 2.1 (Tits [T3, 4.1.1]). Let and be thick irreducible spherical buildings of rank at least 3. If (c, d) is a pair of opposite chambers in , then any isomorphism from to is uniquely determined by its action on E1 (c) ∪ {d}. Theorem 2.2 (Tits [T3, 4.16]). Let (c, d) and (c , d ) be pairs of opposite chambers in thick spherical buildings and that are irreducible and of rank at least 3. Then any isomorphism from E2 (c) ∪ {d} to E2 (c ) ∪ {d } extends to a unique isomorphism from to . In particular the local structure determines the global structure. When and are the same building, these theorems lead directly to the existence of root groups, showing that must be Moufang. Moreover the rank 2 residues must be Moufang (the term “Moufang” was taken by Tits from its use in the special case of projective planes, which are buildings of type A2 ). In particular all buildings belonging to A2 sub-diagrams are Moufang planes, and this is the starting point for the classification. The classification in rank at least 3. In all cases except B3 and F4 there are subdiagrams of type A3 , and this implies (by a classical theorem of projective geometry – see [T3, 6.7]) that the planes must be Desarguesian, rather than just Moufang. Desarguesian means they are coordinatized by a field or skew-field. The classification of all spherical buildings of rank ≥ 3 is as follows. Type An (n ≥ 3): These all arise from projective space of dimension n over a field (possibly non-commutative), and the result is a classical theorem of projective geometry. The finite case: same as above except that finite fields are commutative. Types Dn (n ≥ 4), E6 , E7 , E8 : All these diagrams have single bonds. The existence of A3 subdiagrams implies that the planes are coordinatized by a field or skew-field, and the D4 subdiagram implies it cannot be a skew-field (see [T3, 6.12]). Thus there is one building of each of these types for each commutative field. The finite case: no change. Type Bn = Cn (n ≥ 3): These are called polar spaces by Tits. When n ≥ 4, there is an A3 sub-diagram, so all the projective planes are coordinatized by a field or skewfield K. Tits then showed that the polar spaces are embeddable in a vector space over K, possibly infinite dimensional, and arise from algebraic forms (quadratic forms, sesquilinear forms, pseudo-quadratic forms). This is hard work! When n = 3 there is no A3 subdiagram and the projective planes could be nonDesarguesian, though they must be Moufang. A Moufang non-Desarguesian plane is coordinatized by an alternative division algebra, and its automorphism group arises from a rational form of E6 (see [T1]). A B3 building having such planes arises from a rational form of E7 . These, along with the embeddable polar spaces give a complete classification of all buildings of type Bn (n ≥ 3).

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The finite case: Rank 3 is no longer a special case. All polar spaces are embeddable in a vector space whose dimension lies between 2w and 2w + 2 where w is the Witt index of the form. Type F4 : There are five types. One arises from a split group of type F4 , and exists over any field. A second one, known as 2E6 , arises from a quasi-split form of E6 . A third involves quaternionic planes and is a form of E7 . A fourth has a B3 residue with non-Desarguesian planes (arising from a rational form of E7 – see the paragraph above), and comes from a rational form of E8 . The fifth arises from a “mixed group” of type F4 over two non-perfect fields of characteristic 2. For details see [T3, Ch.10] and cf. [R2, Ch.8]. The finite case: Only the first two occur. Theorem 2.3. The above list gives a classification of all irreducible spherical buildings of rank at least 3. Rank 2 spherical buildings. For buildings of rank 2, known as generalized polygons, there is no classification unless one assumes the existence of root group automorphisms, in which case they are called Moufang (see above). Proofs published in 1979 by Tits [T5] and Weiss [W] showed that for Moufang m-gons, m must be 3, 4, 6 or 8, assuming there are at least three chambers per panel. More than twenty years later Tits and Weiss [TW] completed a classification of Moufang buildings of rank 2. Theorem 2.4 (see [TW, Ch.17]). For a Moufang m-gon, m = 3, 4, 6 or 8, and there is a complete list in each case. The finite case: When a generalized m-gon is finite and irreducible, and has at least three edges per vertex, W. Feit and G. Higman [FH] proved that m = 3, 4, 6 or 8. This remarkable result has nothing to do with the Moufang condition; there is no assumption on the existence of isomorphisms. In the cases m = 6 or 8 the only known finite examples are those arising from algebraic groups, so there may well be a classification theorem in these cases, but little progress has been made. For non-classical finite generalized m-gons, see [VM].

3. Affine buildings and discrete valuations Affine buildings first emerged from work of Iwahori and Matsumoto [IM] for Chevalley groups over p-adic fields. The general theory was then developed by Bruhat and Tits [BT], and a classification was finally given by Tits [T7]. Affine buildings arise from semisimple groups over fields endowed with integer valuations; fields such as the rational numbers Q with the p-adic valuation

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vp (p n a/b) = n, where p is a prime not dividing a or b. The completion of Q with respect to this valuation is the field of p-adic numbers Qp . A quite different example is obtained from a field k(t) of rational functions over a coefficient field k. There are many different integer valuations of k(t), one of which is given by v( fg ) = deg g − deg f , where f and g are polynomials. The completion of k(t) with respect to this valuation is the field k((t)) of Laurent series. Each field with a discrete valuation has a residue field, obtained from the ring comprising those elements having valuation ≥ 0, and factoring out by the maximal ideal of elements having valuation ≥ 1. Completion with respect to the valuation has no effect on the residue field. For example Q (with valuation vp ) and Qp have residue field Fp (the field of p elements), and k(t) and k((t)) have residue field k. Given a field K having an integer valuation with residue field k, a semisimple group over K acts on an affine building with local structure over the field k. For example if the group is SLn (K) and if x is a vertex of the affine building, then St(x) (the residue of x formed from the chambers having x as a vertex) is the spherical building for SLn (k). One immediate consequence of this is that two affine buildings, for example those for SLn (Qp ) and SLn (Fp ((t))), can be locally isomorphic but not globally isomorphic. The fields Qp and Fp ((t)) have residue field Fp , as does any totally ramified extension of Qp , so there are infinitely many isomorphism classes of affine buildings of dimension n having the same local structure. In particular Theorem 2.2 on spherical buildings fails for affine buildings. The simplest example of an affine building is the tree for SL2 (K) where K has an integer valuation with residue field k (see [S] for details). The star of each vertex in this tree is a set of edges in bijective correspondence with the rational points of the projective line over k. For example if k is Fp there are p + 1 edges per vertex. Notice that the trees for SL2 (Qp ) and SL2 (Fp ((t))) are isomorphic because if all vertices of a tree have the same valency, then this valency determines the tree up to isomorphism. However as soon as we move to SL3 the affine buildings for Qp and Fp ((t)) are not isomorphic, and in fact they differ already at the second neighborhood of a vertex. The building at infinity. Since the local-to-global theorem (Theorem 2.2) for spherical buildings fails for affine buildings, a classification cannot be given by local data. However, an affine building of dimension n has a spherical building of dimension n−1 at infinity, and for n ≥ 3 it is possible to use the classification of spherical buildings. The simplest example of an affine building is a tree; its building at infinity is its set of ends. The building at infinity arose in the work of Bruhat and Tits [BT], and was then used by Tits [T7] in the following theorems. Theorem 3.1 ([T7, Sect. 14]). Every irreducible affine building of dimension at least 3 has at infinity a spherical building of dimension at least 2 whose base field is complete with respect to an integer valuation. This valuation uniquely determines the affine building.

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Corollary 3.2 ([T7, Sect. 15]). The locally finite affine buildings that are irreducible and of dimension d ≥ 3 are the affine buildings of simple algebraic groups of rank d over locally compact fields that are non-discrete and totally discontinuous. Affine buildings of dimension 2 (rank 3) do not admit a classification because of a free construction [R1] – see Sect. 5 for more details. Non-complete fields. In the theorem above the field is complete with respect to the discrete valuation, and it is reasonable to ask what happens in the case of fields K, such as Q and k(t), that are not complete. The difference occurs with the system of apartments. Each affine apartment has a spherical apartment at infinity, and all apartments of the spherical building at infinity arise from affine apartments in this way. In the complete case all affine apartments are used, but in the non-complete case a certain subset of apartments, called a system of apartments is used. Going to infinity only in these apartments gives the spherical building for the group over K rather than the complete spherical building for the group over the complete field K. For example the SL2 affine building is a tree. Any two ends of this tree span an apartment, and the complete building at infinity is the set of all ends – it is in bijective . For the lesser field correspondence with the projective line over the complete field K K one has the same tree but a lesser set of ends, in bijective correspondence to the points of the projective line over K. Any two of these ends span an apartment in the system of apartments for SL2 (K). Tits’ work [T7] deals extensively with systems of apartments, and the appropriate definitions can be found in Section 1 of that paper.

4. Twin buildings and Kac–Moody groups Kac–Moody Lie algebras generalize semisimple Lie algebras, allowing infinite Weyl groups W , and they lead to Kac–Moody groups in the same way that semi-simple Lie algebras lead to Chevalley groups. When W is a finite group (the spherical case) the algebra is finite dimensional (and semisimple); when W is infinite the algebra is infinite dimensional. The buildings have the same rank as W and are finite dimensional. The first method of attaching groups to these algebras was given by Moody and Teo [MT]. Years later, Tits wrote a detailed study of Kac–Moody groups [T8] generating them using root groups Uα where α ranges over the set of roots of the Coxeter complex for W (recall that a root is half of a Coxeter complex, on either side of a reflecting hyperplane). These root groups, each isomorphic to the additive group of the base field k, generate a group G. If N denotes the subgroup of G normalizing the set of root groups, and T denotes the subgroup normalizing each individual root group, then as in the case of Chevalley groups, N/T is isomorphic to the Coxeter group W . The subgroup T , called a torus, is isomorphic to a direct product of copies of the multiplicative group of k.

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If c denotes a chamber of the Coxeter complex, the positive roots are those containing c, and the negative roots are those not containing c. The positive and negative root groups generate groups U+ and U− , which along with the torus generate subgroups B+ and B− respectively. The pairs (B+ , N) and (B− , N) are BN-pairs for G. A very important difference between the spherical case and the non-spherical case is that B+ and B− are conjugate in the spherical case, but not otherwise. The reason is simple. The element conjugating B+ to B− in a Chevalley group is the longest word in W , but when W is infinite there is no longest word. The BN -pairs (B+ , N) and (B− , N) generate two buildings + and − whose rank is the rank of W . There is a W -valued “codistance” between chambers of + and − , providing (+ , − ) with the structure of a twin building – see [T9] for the twin building axioms. When W is finite (the spherical case) then + = − , and the codistance is the usual W -valued distance pre-multiplied by the longest word in W . A spherical building is twinned with itself. Two chambers, one in one building, one in the other, are said to be opposite one another when the codistance between them is the identity element of W (and two vertices are opposite if they belong to opposite chambers and have the same type). Each twin building (+ , − ) contains twin apartments, which are pairs (A+ , A− ) of apartments one in each building, such that each chamber of A+ is opposite a unique chamber of A− . In the non-spherical case a twin apartment can be thought of as having its two components on opposite sides of a sphere, so that a chamber in one component of a twin apartment is opposite its antipode in the other component. Example. The group SLn over a ring of Laurent polynomials k[t, t −1 ] is a Kac– Moody group of affine type and acts naturally on a twin building (+ , − ). Both + and − are buildings for SLn (k(t)) but using different valuations v+ and v− , where v+ (t) = 1 and v− (t −1 ) = 1. The stabilizers of opposite vertices x+ and x− are isomorphic to SLn (k[t]) and SLn (k[t −1 ]) respectively, the stabilizer of the pair (x+ , x− ) being SLn (k). When k is a finite field, this implies that SLn (k[t]) acts discretely on − , and similarly with SLn (k[t −1 ]) acting on + . This makes affine twin buildings useful in the study of arithmetic groups over a ring of Laurent polynomials (see [A]). 2-spherical twin buildings. When all rank 2 residues have bounded diameter, or in other words are spherical (this eliminates twin tree residues), the twin building is called 2-spherical. In this case Tits’ methods for spherical buildings can be used to prove that the rank 2 residues must be Moufang. A complete proof is given in [R3, Theorem 3]. Theorem 4.1 ([R3]). Let be a thick 2-spherical twin building with no direct factors of rank 2. Then every residue of type {i, j } for which mij ≥ 3 is Moufang. In particular mij is 2, 3, 4, 6 or 8 for all {i, j }.

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Moreover one can show that the local structure usually determines the global structure, as in 2.2. Recall that E1 (c), respectively E2 (c), means the union of the rank 1, respectively rank 2, residues containing the chamber c. The following theorems are generalizations of 2.1 and 2.2 for spherical buildings. An isometry is a map preserving distances and codistances, and a building is thick if each panel is on at least three chambers. Theorem 4.2 ([R3]). Let and be 2-spherical twin buildings of rank at least 3 that are thick (at least 3 chambers per panel). If (c, d) is a pair of opposite chambers in , then any isomorphism from to is uniquely determined by its action on E1 (c) ∪ {d}. Theorem 4.3 ([MR]). Let (c, d) and (c , d ) be pairs of opposite chambers in thick 2spherical twin buildings and respectively, and assume has no residues of type Sp4 (2), G2 (2), G2 (3) or 2F4 (2). Then any isometry from E2 (c) ∪ {d} to E2 (c ) ∪ {d } extends to a unique isometry from to . In particular the local structure of determines its global structure. The reason for excluding certain residues over the fields of 2 or 3 elements is that in these cases the set of chambers opposite a given chamber is not connected. In all other cases the set of chambers opposite a given chamber is connected, and the building is said to satisfy condition (co). Using this local-to-global theorem the classification of 2-spherical twin buildings has been achieved in the following way (see [M1] and [M2] for further details). Theorem 4.4 ([M2]). Let be an irreducible, 2-spherical twin building satisfying (co). Then one of the following holds: (1) is associated with a classical group over a division ring that is infinite dimensional over its center. In this case is of spherical or of affine type. (2) can be obtained from the class of twin buildings associated with Kac–Moody groups over algebraically closed fields by a sequence of constructions (CS), (FP), (L), (FP), (FP). (CS) refers to Convex Sub-buildings of the same type, obtained by descending to a suitable subfield. (L) refers to Limits, such as moving from finite dimensional to infinite dimensional forms of finite Witt index. (FP) refers to Fixed Points. All three procedures occur in the spherical case when descending from a building over an algebraically closed field, but are only needed in the sequence (CS), (FP), (L). In the non-spherical case there are added complications because the diagram can contain circuits, and the fixed point procedure can involve descent from a building of infinite rank to one of finite rank (see [M2] for more details).

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When a twin building fails to be 2-spherical it has rank 2 residues that are trees. These residues and their opposites form twin trees. A single tree is a simple structure, but twin trees are far more complicated. A single tree can be twinned with an isomorphic copy of itself in infinitely many different ways, as I shall now point out. Twin trees. The definition of a twin tree can be given in terms of an integer codistance between vertices (see [RT2]), rather than using a W -valued codistance between chambers as in the higher rank case [T9]. When this codistance is zero the vertices are said to be opposite, and two such vertices necessarily have the same valency. When the trees are thick (meaning that every vertex has valency at least three), then vertices in the same tree at even distance from one another, or vertices in opposite trees at even codistance from one another, have the same valency. Thus the two trees are isomorphic and each tree is semi-homogenous. Example. In the twin tree for the group SL2 over a ring of Laurent polynomials Fq [t, t −1 ], where Fq is the field having q elements, each vertex has valency q + 1. Moufang twin trees and Kac–Moody groups. Rank 2 Kac–Moody algebras Lie are constructed using a field k along with a generalized Cartan matrix b2 a2 , where a and b are negative integers. When ab = 1, 2 or 3 the algebras are finite dimensional, giving rise to Chevalley groups of types A2 , B2 = C2 , and G2 , respectively. Their buildings are generalized m-gons for m = 3, 4, and 6, respectively. When ab ≥ 4 the algebra is infinite dimensional and the buildings are twin trees. If the field k is finite with q elements, then the twin trees have valency q + 1, and the example above is the special case where a = b = −2. These “Kac–Moody” twin trees are Moufang in the sense that they admit root groups (see [RT2] for the definition of root groups for twin trees), but they are by no means the only Moufang examples. In the case of valency 3 (three edges per vertex) Tits [T10] has produced uncountably many Moufang twin trees. Constructing twin trees. In the Kac–Moody examples over a finite field, the number of edges per vertex is q + 1 where q is a prime power or an infinite cardinal number. An obvious question is whether examples can exist for any q. The answer is that there are uncountably many, as the next theorem shows. Theorem 4.5 ([RT3, 8.2]). A thick semi-homogeneous tree whose set of vertices has cardinality α admits 2α isomorphism classes of twinnings, and among these 2α have trivial automorphism group. The method of proof is to examine a partial twinning between a tree and a ball of radius 2 around a given vertex. This partial twinning is called a uniform 2-ball. In [RT3] we show two main things: (1) that a uniform 2-ball can have a wild structure, and (2) that the uniform 2-balls at different vertices of a tree twinned with T can be

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independent of one another. The point is that one can amalgamate uniform 2-balls in a random way to obtain a twin tree. We now return to single buildings, and find that wildness can occur there too. There is a difference, however. Wildness does not need the absence of rank 2 spherical residues, as it does with twin buildings, but the absence of rank 3 spherical residues.

5. Construction and existence of buildings of all types In [R1] I showed that a classification of affine buildings of rank 3 (dimension 2) was not possible, even in the locally finite case. The method used a free construction and applies to any building whose rank 3 residues are non-spherical. It works by starting at one chamber and working outwards. The rank 2 residues can be chosen independently, the only restriction being the obvious one that if two rank 2 residues have a rank 1 residue in common then they confer the same valency on this rank 1 residue. Assuming valency at least 3 this implies that two rank 1 residues of the same type have the same valency. In order to state a theorem, let us say that a building has parameters (qi ), i ∈ I , if every rank 1 residue of type i has qi + 1 chambers; we allow the possibility that qi may be infinite. This means that every rank 2 residue of type {i, j } is a generalized mij -gon with parameters (qi , qj ); if mij is infinity this means it is a tree. The fact that every rank 3 residue is non-spherical means that m1ij + m1j k + m1ki ≤ 1 for every set of three types i, j , k ∈ I . Define (qi ) to be an admissible set of parameters if for each i, j there exists a generalized mij -gon having parameters (qi , qj ). For example if mij = 3 then qi = qj and in all known finite examples qi is a prime power. Theorem 5.1 ([R1]). Let (qi ) be an admissible set of parameters, and W a Coxeter group having no rank 3 spherical subgroups. Then there exists a building of type W with these parameters, whose rank 2 residues of type {i, j } range through any desired set of generalized mij -gons having parameters (qi , qj ). This theorem, of course does not apply to irreducible, spherical buildings of rank at least 3, and for that we need a more sophisticated approach. This approach, developed by Tits and the author uses the concept of a “blueprint”. Blueprint constructions. If a building contains irreducible spherical rank 3 residues (namely of types A3 or B3 ) then things are very different. For example in an A3 or B3 residue two rank 2 sub-residues intersecting in a rank 1 residue must give compatible structures on that intersection. For example if this rank 1 residue is a projective line it must receive the same projective structure from all rank 2 residues containing it. This compatibility problem led the author and J. Tits [RT1] to a new method of construction

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that, although not as free as the earlier one in [R1], applies to buildings of all types. Here is how it works. Let I be the indexing set for the fundamental roots, or equivalently for the different types of panels in the building. Attach to each i in I a set Ui . The idea is to use the sets Ui , along with the Coxeter group, to parametrize the building, starting with one chamber b and working outwards. A gallery from b to another chamber x is a sequence of chambers b = x0 , x1 , . . . , xd = x with each one adjacent to the next; we say it has type (i1 , . . . , id ) if xj −1 is ij -adjacent to xj . Using the sets Ui we then specify x by a sequence, where uj ∈ Uj , and (i1 , . . . , id ) is the type of a minimal gallery from b to x. If b is joined to x by minimal galleries of different types then it is necessary to know when sequences of two different types yield the same chamber. Fortunately, if we fix the end chambers b and x, one minimal gallery can be transformed into another by a sequence of alterations (called elementary homotopies) in rank 2 residues (see [T6, (3.4)]). The datum consisting of the sets Ui along with these rank 2 homotopies, for each type of rank 2 residue, is called a blueprint. A building that can be parametrized in accordance with a blueprint is said to conform to that blueprint, and a blueprint is called realisable if there exists a building conforming to it. The secret in producing realisable blueprints is to let the Ui be root groups for the fundamental roots – see [RT1] for details. Clearly any realisable blueprint must be realisable in all rank 3 spherical residues, and these are the residues that formed an obstruction to the previous construction. What is surprising is that this is the only obstruction. Theorem 5.2 ([RT1, Theorem 1]). A blueprint is realisable if its restriction to each rank 3 spherical subdiagram is realisable. In this case there is a unique building which conforms to it. [Note: the statement in [RT1] inadvertently omits the word spherical]. The reason the rank 3 spherical restriction is sufficient can be explained in terms of galleries as follows. A sequence of rank 2 homotopies may yield a gallery having the same type as the initial gallery. But the initial sequence (ui1 , . . . , uid ) and the final sequence (vi1 , . . . , vid ) must yield the same end chamber, so uij = vij . To ensure this happens, it suffices to check such self-homotopies in rank 3 spherical residues because of a theorem for Coxeter complexes due to Tits [T6, Prop.4]. Foundations. In order to avoid specifying an explicit blueprint, the authors [RT1] define a foundation to be an amalgam of rank 2 residues containing a given chamber. It is said to support a building if there exists a building and a chamber c in that building such that E2 (c) – the union of the rank 2 residues containing c – is isomorphic to the foundation concerned. The point of using foundations is the following powerful existence theorem.

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Theorem 5.3 ([RT1, Theorem 2]). Let E be a foundation. If each rank 3 residue of type A3 or B3 supports a building, then E supports a building. The theorem is proved by producing a realisable blueprint, and hence a building conforming to it. It is now time to ask which buildings these are that conform to blueprints. In the spherical case all buildings conform to a blueprint. In particular the theorems above prove the existence of spherical buildings, such as those of type E8 , that were previously only known from the existence of the Chevalley groups. In the affine case the buildings conforming to a blueprint are those for groups over a power series field, rather than a p-adic field. The p-adic affine buildings do not conform to blueprints, apart from an exceptional case for SL3 (Q2 ). The affine buildings over power series fields are precisely those that arise from Kac–Moody groups. For example in the last section the Kac–Moody group SLn (k[t, t −1 ]) acts on an affine twin building, its two components being the affine building for SLn (k((t))).

6. Questions and conjectures The last paragraph brought out a connection between twin buildings, and buildings conforming to a blueprint. This leads to the following questions. 1. Do the buildings of a twin conform to a blueprint? The answer is yes in the 2-spherical case when condition (co) is satisfied because Theorem 4.3 shows that a twin building is Moufang, and this yields root groups that give a natural blueprint for the building. 2. Are buildings that conform to a blueprint twinnable? The immediate answer is no because there are examples conforming to some odd-ball blueprints that are not twinnable. For example there are A˜ 2 blueprints involving non-Moufang planes, but by Theorem 4.1 there are no twin buildings of such type. Of course this example does not involve rank 3 spherical residues, and begs the question of what happens when such residues exist. Here is a conjecture. Conjecture 6.1. If a building conforms to a blueprint and all its rank 3 residues are spherical, then it admits a twinning. Conforming to a blueprint is a strong condition, and this conjecture seems quite likely to be true. On the other hand we can forget about blueprints and just ask what happens if all rank 3 residues are spherical. This is true of every affine building having rank ≥ 4, and we already know such buildings that are not twinnable (for example the p-adic ones). But what about the non-affine case? Hyperbolic buildings of rank ≥ 4 in which all rank 3 residues are spherical are known only from Kac–Moody groups. Are there others, comparable to the p-adic affine buildings? No-one knows of suitable groups, so it is worth asking the following question.

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Question 6.2. Can one prove that a non-affine building whose rank 3 residues are spherical is necessarily twinnable? If this were true there would be a classification of such buildings, by using twin buildings, as in 4.4, at least when condition (co) is satisfied. To attack this question one might proceed in two steps: (i) twin the building with a single chamber, and (ii) extend outwards to neighboring chambers, obtaining a twinning with another building. The first step seems to be impossible in the p-adic affine case, and is probably where the obstruction lies, if there is one.

References [A]

P. Abramenko, Twin Buildings and Applications to S-Arithmetic Groups, Lecture Notes in Math. 1641, Springer-Verlag, Berlin 1996.

[BT]

F. Bruhat and J. Tits, Groupes réductifs sur un corps local, I. Données radicielles valuées, Inst. Hautes Études Sci. Publ. Math. 41 (1972), 5–252.

[Ch]

C. Chevalley, Sur certains groupes simples, Tohoku Math. J. (2) 7 (1955), 14–66.

[D]

L. Dickson, Linear Groups, originally published 1900, reprinted by Dover, New York 1958.

[FH]

W. Feit and G. Higman, The nonexistence of certain generalized polygons, J. Algebra 1 (1964), 114–131.

[IM]

N. Iwahori and H. Matsumoto, On some Bruhat decomposition and the structure of the Hecke ring of a p-adic Chevalley group, Inst. Hautes Études Sci. Publ. Math. 25 (1965), 5–48.

[MT]

R. Moody and K. Teo, Tits systems with crystallographic Weyl groups, J. Algebra 21 (1972), 178–190.

[M1]

B. Mühlherr, Locally split and locally finite twin buildings of 2-spherical type, J. Reine Angew. Math. 511 (1999), 119–143

[M2]

B. Mühlherr, Twin Buildings, in Tits Buildings and the Model Theory of Groups, ed. by K. Tent, London Math. Soc. Lecture Note Ser. 291, Cambridge University Press, Cambridge 2002, 103–117.

[MR]

B. Mühlherr and M. Ronan, Local to Global Structure in Twin Buildings, Invent. Math. 122 (1995), 71–81.

[R1]

M. Ronan, A Construction of Buildings with no Rank 3 Residues of Spherical Type, in Buildings and the Geometry of Diagrams (Como 1984), ed. by L. A. Rosati, Lecture Notes in Math. 1181, Springer-Verlag, Berlin 1986, 242–248.

[R2]

M. Ronan, Lectures on Buildings, Perspect. Math. 7, Academic Press, Boston 1989.

[R3]

M. Ronan, Local Isometries of Twin Buildings, Math. Z. 234 (2000), 435–455.

[RT1] M. Ronan and J. Tits, Building Buildings, Math. Ann. 278 (1987), 291–306.

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[RT2] M. Ronan and J. Tits, Twin Trees I, Invent. Math. 116 (1994), 463–479. [RT3] M. Ronan and J. Tits, Twin Trees II, Israel J. Math. 109 (1999), 349–377. [S]

J.-P. Serre, Arbres, Amalgames, SL2 ; Astérisque 46, 2nd corr. ed., Soc. Mathém. de France, Paris 1979; English translation: Trees, Springer-Verlag, Berlin 1980.

[St]

R. Steinberg, Variations on a theme of Chevalley, Pacific J. Math. 9 (1959), 875–891.

[T1]

J. Tits, Le plan projectif des octaves et les groupes de Lie exceptionelles, Acad. Roy. Belgique. Bull. Cl. Sci. (5) 39 (1953), 309–329.

[T2]

J. Tits, Le problème de mots dans les groupes de Coxeter. Ist. Naz. Alta Mat., Symposia Math. 1 (1968), 175–185.

[T3]

J. Tits, Buildings of Spherical Type and Finite BN-Pairs, Lecture Notes in Math. 386, Springer-Verlag, Berlin 1974.

[T4]

J. Tits, Endliche Spiegelungsgruppen, die als Weylgruppen auftreten, Invent. Math. 45 (1977), 283–295.

[T5]

J. Tits, Non-existence de certains polygones généralisés, I, II, Invent. Math. 36 (1976), 275–284; ibid. 51 (1979), 267–269.

[T6]

J. Tits, A Local Approach to Buildings, in The Geometric Vein – The Coxeter Festschrift, ed. by C. Davis, B. Grünbaum and F. A. Sherk, Springer-Verlag, New York 1982, 519–547.

[T7]

J. Tits, Immeubles de type affine, in: Buildings and the Geometry of Diagrams (Como 1984), ed. by L. A. Rosati, Lecture Notes in Math. 1181, Springer-Verlag, Berlin 1986, 159–190.

[T8]

J. Tits, Uniqueness and Presentation of Kac–Moody Groups over Fields, J. Algebra 105 (1987), 542–573.

[T9]

J. Tits, Twin Buildings and Groups of Kac–Moody Type, in Groups, Combinatorics and Geometry (Durham 1990), ed. by M. W. Liebeck and J. Saxl, , London Math. Soc. Lecture Note Ser. 165, Cambridge University Press, Cambridge 1992, 249–286.

[T10]

J. Tits, private communication

[TW]

J. Tits and R. M. Weiss, Moufang Polygons, Springer Monogr. Math., Springer-Verlag, Berlin 2002.

[VM]

H. Van Maldeghem, Generalized Polygons, Monogr. Math. 93, Birkhäuser Verlag, Basel 1998.

[W]

R. Weiss, The Nonexistence of certain Moufang Polygons, Invent. Math. 51 (1979), 261–266.

Mark A. Ronan, Mathematics Department, University of Illinois at Chicago, 851 S. Morgan, mc 249, Chicago, IL 60607, U.S.A. E-mail: [email protected]

Characters of algebra groups and unitriangular groups Josu Sangroniz∗

Abstract. In this work we study the irreducible complex characters of finite algebra groups, with a special interest in the groups of unitriangular matrices over finite fields. By an algebra group we mean a group of the form 1 + J , where J is the Jacobson radical of a finite associative algebra over a commutative ring R and the additive group of J is a p-group. We start with an exposition of Kirillov’s method adapted to our setting, which yields a description of the irreducible characters of the algebra group 1 + J when J p = 0. Although this method can be applied immediately to the unitriagular groups when the size of the matrices does not exceed the characteristic of the field, it fails in general to describe all the irreducible characters. However, we shall show that the method is still useful to describe some of the irreducible characters, namely, those whose degree is small or large enough.

1. Introduction This work is a contribution to the study of the (complex, irreducible) characters of the unitriangular groups (i. e., the groups formed by the n × n upper-triangular matrices with entries in a finite field and all of whose diagonal elements are 1). Isaacs found it useful in [6] to consider unitriangular groups only as part of a larger family of groups: the algebra groups. By an algebra group he means a group of the type G = 1 + J , where J is the Jacobson radical of a finite dimensional algebra over a finite field with identity element 1. This setting lends itself to inductive arguments which do not always work when confined to unitriangular groups. In the abovementioned work, Isaacs proved for instance that the degrees of the irreducible characters of algebra groups are always powers of q, the size of the underlying field, a result originally conjectured for unitriangular groups. Our goal here is twofold. First we explain Kirillov’s method of coadjoint orbits (see [11]) in the context of algebra groups. Our presentation, inspired by [10], also reflects some of the ideas of Isaacs’ in [7]. For an alternative approach, see [2]. We also change slightly the concept of algebra group by allowing the algebras to be ∗ This research has been supported by the Feder, the Spanish Ministerio de Ciencia y Tecnología, grant BFM2001-0180 and the University of the Basque Country, grant 1/UPV00127.310-E-13817/2001.

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defined over a commutative ring rather than a field. As we shall see, this seemingly unimportant modification has some noteworthy consequences. In the second part of this paper we focus our attention on unitriangular groups. As it was proved by Isaacs and Karagueuzian in [8], there are serious limitations in Kirillov’s method when we try to apply it to unitriangular groups if the size of the matrices exceeds the characteristic of the field. However, we shall show that even in this case the method still works to describe the irreducible characters provided the degree is small or large enough. These results can be seen as a generalization of [13] (where the irreducible characters of degree 1, q and q 2 are considered) and [1] (which includes a description of the characters of maximal degree). I would like to thank M. Isaacs and A. Jaikin for their helpful comments. Part of this work was done while I was visiting the Department of Mathematics of the University of Wisconsin-Madison in the fall 2002. I would like to express my gratitude for their kind hospitality.

2. Kirillov’s method and algebra groups In this section R will denote a commutative ring with unity and J will be a multiplicatively closed submodule inside an associative R-algebra with identity 1. We shall also assume that J n = 0 for some positive integer n and that the additive group of J is a finite p-group. Under these hypotheses, the set G = 1 + J is a finite p-group. A group like G will be called an algebra group. Some preliminary work is necessary before stating the key result of Kirillov’s method. We start by introducing a convenient notion of ‘multiplicative bilinearity’. A multiplicative bilinear alternating form on J will be for us a map B : J × J → C∗ satisfying the following conditions: (i) B(u + v, w) = B(u, w)B(v, w) and B(u, v + w) = B(u, v)B(u, w) for all u, v, w ∈ J . (ii) B(ru, u) = 1 for all u ∈ J and r ∈ R. Expanding the equality B(r(u + v), u + v) = 1 and using (ii), we obtain B(ru, v)B(rv, u) = 1. For r = 1 this yields B(u, v) = B(v, u)−1 for all u, v ∈ J , so B(ru, v) = B(rv, u)−1 = B(u, rv). The radical of B is the submodule defined by: rad B = {u ∈ J | B(u, v) = 1 for all v ∈ J } = {u ∈ J | B(v, u) = 1 for all v ∈ J }. If rad B happens to be trivial we shall say that B is non-degenerate. Suppose for a moment that this is the case. Then there is a canonical isomorphism φ between J and Jˆ, the set of irreducible (linear) characters of J , mapping u ∈ J to Bu , which sends v ∈ J to B(u, v) ∈ C∗ . For a submodule (or subgroup) H ≤ J we define H ⊥ to be the set of elements of Jˆ whose kernel contains H . It is clearly isomorphic, as

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a group, to J /H, so in particular |H ⊥ | = |J : H |. Now suppose that H is isotropic with respect to B (that is B(u, v) = 1 for all u, v ∈ H). Then φ maps H into H ⊥ , whence |H | ≤ |H ⊥ | = |J : H| and |H|2 ≤ |J |. We claim that the equality holds precisely when H is a maximal isotropic submodule of J . Indeed, if H is maximal isotropic and we are given ζ ∈ H ⊥ , we can represent ζ as ζ = Bu for some u ∈ J and the fact that ζ is in H ⊥ translates into the condition B(u, v) = 1 for all v ∈ H . Then K = Ru + H, the submodule generated by u and H , is also isotropic, whence K = H and u ∈ H. This means that φ is an isomorphism between H and H ⊥ , which is simply another way to formulate our claim. The degenerate case is reduced to the non-degenerate one by the usual procedure of considering B as a form on J / rad B. It follows that all maximal isotropic submodules of J have the same order, namely (|J || rad B|)1/2 . The algebra group G = 1 + J acts by conjugation on the additive group J (it is actually the action of a group on an R-module), so it also acts by duality on Jˆ. Given ζ ∈ Jˆ, we denote by ζ the orbit of ζ under this action (these are the coadjoint orbits in Kirillov’s terminology). There is a close relation between ζ and the multiplicative bilinear form Bζ defined by Bζ (u, v) = ζ ([u, v]), where [ , ] denotes the standard Lie bracket in J . It can be checked routinely that the correspondence u → 1 + u maps the radical of Bζ onto the stabilizer of ζ , S. In particular, |ζ | = |G : S| =

|J | = |H / rad Bζ |2 , | rad Bζ |

(2.1)

where H ≤ J is a maximal isotropic submodule of J . This shows that the lengths of the orbits are perfect squares. If R = Fq is a finite field we can identify Jˆ with the dual vector space J ∗ : simply fix a non-principal linear character ψ : Fq → C∗ of the additive group of the field and notice that the map λ → ψ λ gives a bijection from J ∗ onto Jˆ. If ζ = ψ λ, we can also identify Bζ with the ordinary bilinear map Bλ given by Bλ (u, v) = λ([u, v]) and both the radical of Bζ and the usual radical of Bλ coincide. The action of G on Jˆ translates into an action of G on J ∗ and, for λ ∈ J ∗ , the size of the orbit λ of λ is q rk Bλ . The next step is to prove that the submodule H in (2.1) can be chosen to be multiplicatively closed. The key observation here is that if B is a multiplicative bilinear form on J and U ≤ J is an isotropic submodule, then any maximal isotropic submodule W inside U⊥ = {v ∈ J | B(u, v) = 1 for all u ∈ U} is also maximal isotropic when viewed in the whole J (the reason for this is that any isotropic submodule of J containing W is actually contained in U⊥ ). Proposition 1. Let ζ ∈ Jˆ and B = Bζ . Then there exists a maximal isotropic submodule which is multiplicatively closed. In addition, this submodule can be taken to contain any particular isotropic ideal of J .

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Proof. We argue by induction on the order of J . The result is obvious if B is trivial. Otherwise, J = 0 and, by refining the series {J k }, we obtain a chain of ideals J = J1 ⊃ J2 ⊃ · · · ⊃ Jr+1 = 0 such that all the factor modules Ji /Ji+1 are cyclic. We can also suppose that this series contains any particular isotropic ideal A. Let Ji , i ≥ 1, be the largest non-isotropic ideal in the series. Since Ji /Ji+1 is cyclic, Ji+1 cannot be contained in the radical of ⊥ is properly contained in J and it is not difficult B, so its orthogonal submodule Ji+1 to see that is multiplicatively closed (the key is that the bilinear form B = Bζ has the ⊥ . By property B(uv, w) = B(u, vw)B(v, wu)). Notice also that A ⊆ Ji+1 ⊆ Ji+1 ⊥ has a maximal isotropic submodule multiplicatively closed containing induction Ji+1 A which, by the previous remark, gives the desired submodule. If ζ ∈ Jˆ and B = Bζ , we shall denote by Hζ the submodule whose existence we have just proved in the last proposition and we set Hζ = 1 + Hζ , which is an algebra subgroup of G = 1 + J . We insist that in the sequel this notation will be used without any further explanation. Since Hζ is isotropic, the subgroup Hζ fixes the restriction of ζ to Hζ and the function ϕζ defined by ϕζ (1 + u) = ζ (u) is a class function of Hζ . We compute now the induced function ϕζG . Theorem 2. Let ζ ∈ Jˆ and ζ = ϕζG . Then, for all u ∈ J , µ(u). ζ (1 + u) = |ζ |−1/2 µ∈ζ

Proof. By definition of ζ , ζ (1 + u) =

1 ◦ 1 ◦ g ϕ ζ (1 + ug ) = ζ (u ), |Hζ | |Hζ | g∈G

◦

g∈G

◦

where ϕ ζ (resp. ζ ) is the extension of ϕζ (resp. ζ ) which vanishes outside Hζ (resp. Hζ ). By the orthogonality relations on the group J /Hζ we have that |J : Hζ |, if v ∈ Hζ ; µ(v) = 0, otherwise, µ∈J /Hζ

hence, ζ (1 + u) =

1 1 ζ (ug )µ(ug ) = ν(ug ), |J | g∈G |J | g∈G µ∈J /Hζ

(2.2)

ν∈E

where E is the set of linear characters of J which agree with ζ on Hζ . Note that the subgroup Hζ acts on E (because Hζ fixes ζ|Hζ ) and contains the stabilizer S of ζ in

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G (because of the maximality of Hζ ), thus the length of the orbit of ζ under Hζ is /Hζ | = |J |/|Hζ |, so bearing |Hζ : S| = |Hζ |/| rad Bζ |. On the other hand, |E | = |J in mind the formula for the order of Hζ we conclude that |E | = |Hζ : S|, which means that Hζ acts transitively on E . Then if ν ∈ E , there exists h ∈ Hζ such that ν = ζ h and ν(ug ) = ζ h (ug ) = ζ (ug ). g∈G

g∈G

g∈G

Putting this into (2.2) we obtain ζ (1 + u) =

|E | g 1 g ζ (u) = ζ (u) |J | |Hζ | g∈G

g∈G

|Gζ | = µ(u) = |ζ |−1/2 µ(u). |Hζ | µ∈ζ

µ∈ζ

Corollary 3. Let G = 1 + J and, in addition to the usual assumptions, suppose that J p = 0. Then any irreducible character of G is induced by a linear character of an algebra subgroup of G, that is, of the form 1 + H for H a multiplicatively closed submodule of J . Moreover, if χ is an irreducible character of G, then there exists ζ ∈ Jˆ, unique up to the action by G, such that µ(u). χ(exp u) = |ζ |−1/2 µ∈ζ

Proof. The exponential is well-defined because of the condition J p = 0 and maps bijectively J onto G. For ζ ∈ Jˆ we set χζ and αζ to be, respectively, the functions on G and Hζ defined by χζ (exp u) = ζ (1 + u), u ∈ J ; αζ (exp u) = ϕζ (1 + u) = ζ (u), u ∈ Hζ . By the Hausdorff formula (and the isotropy of Hζ ) it is clear that αζ is a linear character of the subgroup Hζ and, by the last theorem, χζ = αζG , so χζ is a genuine character of G. Now, (χζ , χζ ) = 1 or 0 depending on whether ζ and ζ are in the same orbit by G or not (this an easy consequence of the orthogonality relations for the linear characters of J ), so we have produced as many irreducible characters of G as coadjoint orbits. Finally, we know, by Brauer’s permutation lemma, that the number of coadjoint orbits is the same as the number of orbits of the action of G on J , which is of course the number of classes of G, thus the characters χζ account for all the irreducible characters of G. Under the hypotheses of the last corollary, the technique we have used also allows us to describe explicitly the irreducible characters of the subgroups of 1+J associated to Lie subalgebras of J (we insist that J p = 0). To be more precise about the relation between Lie subalgebras and subgroups, note that if L is a Lie subalgebra of J then,

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by the Hausdorff formula, the set L = exp L = {exp u | u ∈ L} is a subgroup of G = 1 + J . This subgroup acts by conjugation on L since for u, v ∈ L, uexp v = u +

[u, v] [u, v, v] . . . , v] [u, v, p−2 + + ··· + 1! 2! (p − 2)!

(2.3)

is an element lying on L. The formula (exp u)g = exp(ug ) tells us that this action is permutation equivalent to the action of L on itself by conjugation, so the number of classes of L coincides with the number of orbits of L acting on L which, by Brauer’s ˆ For permutation lemma, is also the number of orbits of the dual action of L on L. ˆ ζ ∈ L we define the bilinear form Bζ as before. An element exp v ∈ L fixes ζ if and only if . . . , v] [u, v, p−2 [u, v] [u, v, v] + + ··· + ∈ Ker ζ 1! 2! (p − 2)!

(2.4)

for all u ∈ L, a condition that is certainly fulfilled if [v, L] ⊆ Ker ζ . Conversely, if . . . , v] (2.4) holds for all u ∈ L, by substituting u successively by [u, v], . . . , [u, v, p−3 we realize that [u, v] ∈ Ker ζ (for any u ∈ L), whence [v, L] ⊆ Ker ζ . Thus the exponential maps the radical of Bζ onto the stabilizer of ζ . In particular, ζ is fixed by L if and only if [L, L] ⊆ Ker ζ . ˆ Then Proposition 4. Let L be a finite nilpotent Lie algebra over a ring R and ζ ∈ L. there exists a Lie subalgebra Hζ such that Hζ is a maximal isotropic submodule with respect to Bζ . Proof. Mimic the proof of Proposition 1 (obviously, changing the ideals Ji by Lie ideals). Note also that if I is a Lie ideal of L, then the orthogonal submodule I ⊥ is a Lie subalgebra. The computations in the proof of Theorem 2 can be carried out almost word for word to the context of Lie algebras, so that the following analogous result to Corollary 3 can be obtained. Theorem 5. Let L = exp L, where L is a Lie subalgebra of J and J satisfies the conditions of Corollary 3. Then any irreducible character of L can be induced by a linear character of a subgroup exp H for some Lie subalgebra H ⊆ L. In addition, ˆ unique up to the action of L, if χ is an irreducible character of L, there exists ζ ∈ L, such that for all u ∈ L, χ(exp u) = |ζ |−1/2 µ(u). µ∈ζ

It is clear that under the hypotheses of the last theorem the nilpotency class of the group L is at most p − 1. It should be noted that actually any p-group of class less than p can be realized in this way. Indeed, suppose G is such a group and let pe be its exponent. We consider the group algebra RG, where R is the ring of integers

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modulo p e and as usual, denote by the augmentation ideal of RG. Then, by a e result of Moran [14], G ∩ (1 + p ) = Gp γp (G) = 1, so G can be embedded in the algebra group associated to J = /p , an algebra that satisfies our requirement J p = 0. Finally, the exponential induces a bijection between the Lie subrings (in this case there is no need to distinguish between subrings and subalgebras) of J and the subgroups of 1 + J (because of the Hausdorff formula and its inverses), so our group G is isomorphic to a group exp L for a Lie subalgebra L of J . Notice that L is the Lie ring corresponding to G by the Lazard correspondence. For some purposes it may be more useful to change the point of view and start with an abstract Lie ring L of class less than p whose underlying additive group is a p-group. Then we can pass to its group counterpart G and realize G as exp L for some Lie subalgebra L inside an associative algebra J with J p = 0. Since L is also the Lazard correspondent to G, it follows that L and L are isomorphic as Lie rings. We can use this fact for instance to obtain the degrees of the irreducible characters of G performing all computations in L instead of G: by Theorem 5 this ˆ If L is actually an set consists of the numbers |L : rad Bζ |1/2 for ζ running on L. 1 algebra over Fp , these numbers can be rewritten as p 2 rk Bλ with λ ∈ L∗ . We give as an application another construction of a p-group (for odd p) with prescribed character degrees, different from the ones proposed by Isaacs in [4] and Isaacs and Slattery in [9]. Theorem 6. Let p be an odd prime and X any set of powers of p containing 1. Then X is the set of degrees of the irreducible characters of some p-group of class 2. Proof. By the preceding discussion the statement can be reformulated in the following terms: for any set Y of non-negative integers containing 0, there exists a Lie algebra L over Fp of class 2 such that the set of numbers 21 rk Bλ | λ ∈ L∗ equals Y . Suppose then that the positive numbers in Y are n1 > · · · > nr and take a vector space L over Fp of dimension 2n1 + r with a basis x1 , . . . , x2n1 , y1 , . . . , yr . Set [xi , xj ] = yk and [xj , xi ] = −yk for i + j = 2n1 − k + 1, 1 ≤ i ≤ nk , 1 ≤ k ≤ r and define the rest of the Lie products among the basis elements to be zero. This clearly endows L with a Lie algebra structure of class 2. If λ is a linear functional on L, the rank of Bλ is the rank of the antisymmetric matrix (λ([xi , xj ])), and this number is twice the largest nk for which λ(yk ) = 0 (or 0 if λ annihilates all the yk ). Note that if X = {1, pn1 , . . . , pnr }, n1 > · · · > nr , the order of the group we have constructed is p 2n1 +r , compared to p2n1 +n2 +···+nr +r in Isaacs’ example.

3. Unitriangular groups For the rest of this work we shall be concerned mainly with unitriangular groups, so all the algebras will be supposed to be defined over (finite) fields. Recall that in this case Jˆ can be identified with the dual space J ∗ by means of a fixed non-principal

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character ψ of the additive group of the field. If the characteristic is p and J p = 0 the exponential map is no longer defined but a natural substitute for it is the truncated

p−1 exponential map Exp : J → G defined by Exp u = i=0 ui /i!. It is easy to see that Exp is a bijection between J and G (actually any polynomial map with constant term 1 is a bijection from J to G). Notice that we reserve the notation exp for the ordinary exponential map defined when J p = 0. There is also a ‘Hausdorff formula’ for Exp: Lemma 7. Let u, v ∈ J . Then there exists γ ∈ J p + [J, J ] such that Exp u Exp v = Exp(u + v + γ ). Proof. The result is clear if J p = 0 by the Hausdorff formula, so we can suppose J p = 0. Let n > p be such that J n−1 = 0 but J n = 0. Arguing inductively the formula can be assumed to be true for the algebra J /J n−1 . The algebra group associated to this factor algebra can be identified naturally with the factor group (1 + J )/(1 + J n−1 ), thus the relation for the truncated exponential on J /J n−1 gives rise to the equality in 1 + J : Exp u Exp v = Exp(u + v + γ )(1 + w), for some γ ∈ J p + [J, J ] and w ∈ J n−1 . Since wJ = J w = 0, it is clear that Exp(z + w) = (Exp z)(1 + w) for any z, thus Exp u Exp v = Exp(u + v + γ + w) and γ + w ∈ J p + [J, J ] (remember that J n−1 ⊆ J p ). If λ ∈ J ∗ we take a subalgebra Hλ which is a maximal isotropic subspace with respect to Bλ and define the class function αλ on Hλ = 1 + Hλ = Exp Hλ by αλ (Exp u) = ψ(λ(u)). For u ∈ J we also set χλ (Exp u) = αλG (Exp u) (notice that this notation is consistent with the notation in Section 2). Then, by Theorem 2, ψ(µ(u)). (3.1) χλ (Exp u) = |λ |−1/2 µ∈λ

By the definition, it is clear that χλ is a character of G whenever αλ is a character of Hλ . Moreover, in this case χλ is an irreducible character (because (χλ , χλ ) = 1). This p happens for instance when Hλ ⊆ Ker λ (by the last lemma). It should be noted at this point that there is no hope to describe all the irreducible characters of the unitriangular groups by means of a formula like (3.1). The reason is that these formulas always yield functions that take values in the field Q(ε) for ε a primitive pth root of unity, in particular, they are rational valued if p = 2. However, there exist matrices over the field with two elements which are not conjugate to their inverses (see examples in [8]), so these groups do have characters which are not rational valued (actually, for any prime p there exist unitriangular groups over Fp with characters whose values are not in Q(ε), see [15]). As an illustration of the ideas above, we determine the irreducible characters of the unitriangular groups of ‘small degree’. For matrices of size n × n over the finite

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field Fq this group will be denoted by Un (q) and un (q) will be the associated nilpotent algebra. Theorem 8. Let 0 ≤ k ≤ log2 p − 1. Then the correspondence λ → χλ induces a bijection between the irreducible characters of Un (q) of degree q k and the coadjoint orbits of size q 2k . Proof. Suppose first that λ is a linear functional on J = un (q) whose orbit has size q 2k for some k in the specified range. We know that the codimension of Hλ is precisely k. Then, there exists an ascending series of subalgebras Hi , 0 ≤ i ≤ k, beginning with Hλ and ending in J such that each one is an ideal of the next. We have that 2 ⊆ H , thus J 2k ⊆ H and, since p ≥ 2k+1 , Hi+1 i λ Jp ⊆ J2 p

k+1

k

k

= [J 2 , J 2 ] ⊆ [Hλ , Hλ ] ⊆ Ker λ, k+1

k

k

so Hλ ⊆ Ker λ (note that the equality J 2 = [J 2 , J 2 ] is a property of the algebra p un (q) and not a general one). We conclude that Hλ + [Hλ , Hλ ] ⊆ Ker λ. This implies by the previous lemma that αλ is a linear character of Hλ = 1 + Hλ and, as remarked before, this is all we need to ensure that χλ is a character of G = Un (q). Conversely suppose now that χ in an irreducible character of G of degree q k and 0 ≤ k ≤ log2 p − 1. We need to find λ ∈ J ∗ such that χ = χλ . By [5, Theorem 5.12] we have that G(k+1) ≤ Ker χ (it is crucial here to use the fact that the degrees of the irreducible characters of G are powers of q) and it is well-known k+1 that G(k+1) = 1 + J 2 , hence χ can be viewed as a character χ of the factor group k+1 k+1 G/(1 + J 2 ), which is isomorphic to the group associated to J = J /J 2 . Again, p as p ≥ 2k+1 , J = 0, thus χ is determined by a linear functional on J , which lifts to a linear functional λ on J . Now, the formula for χ , when viewed for G, is exactly the equality χ = χλ that we wanted to prove. Theorem 2 yields a description of the irreducible characters of Un (q) for n ≤ p. However this result can be pushed further if we make use of some specific properties of the algebra un (q). If r and s are two positive integers with r + s = n, then un (q) = Kr,s ⊕ Ar,s , where u 0 0 a Kr,s = | u ∈ ur (q), v ∈ us (q) and Ar,s = | a ∈ Mr×s (q) . 0 v 0 0 (3.2) Notice that K = Kr,s is an algebra and A = Ar,s is an ideal with A2 = 0. For algebras J admitting such a decomposition, the hypothesis J p = 0 in the previous section can be relaxed a bit. This will suffice to extend the result on the characters of Un (q) to the case n < 2p and, when no condition is imposed on n, to describe the irreducible characters with large degree. We begin work toward our goal with two rather technical lemmas. Lemma 9. Suppose J = K ⊕ A, where K is a subalgebra, A an abelian ideal and K i Ap−i = 0 for all 0 ≤ i ≤ p (for i = 0 and i = p, read this condition as Ap = 0

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and K p = 0, respectively). Then, if u ∈ K and a ∈ A, there exists an element γ ∈ [J, J ] such that exp u exp a = Exp(u + a + γ ). Proof. Note first that there is no harm in replacing K by the subalgebra generated by u inside J , so without loss of generality, we can suppose that K is a commutative algebra. We claim that [K, A], the subspace spanned by all the Lie products [u, a] for u ∈ K and a ∈ A, is an ideal of J . Indeed, if u, u ∈ K and a, a ∈ A, a [u, a] = a ua − a au = a ua − aa u = [a u, a] = 0 ∈ [K, A], where we have made use both of the commutativity of A and the fact that it is an ideal. Also u [u, a] = u ua − u au = uu a − u au = [u, u a] ∈ [K, A]. Something similar happens when multiplying on the right. Thus [K, A] is an ideal and, in the quotient algebra J = J /[K, A], the elements of K and A commute, so p

J = (K + A)p =

p

i

K A

p−i

= 0,

i=0

that is J p ⊆ [K, A]. In particular J p ⊆ [J, J ] and the result follows directly from Lemma 7. Lemma 10. Let J be as in Lemma 9 and suppose that λ ∈ J ∗ is invariant by the algebra group G = 1 + J (or equivalently, [J, J ] ⊆ Ker λ). Then, the function αλ is a linear character of G. Proof. We observe first that the group G is the semidirect product of 1 + K and 1 + A and, because K p = Ap = 0, the usual exponential is defined between K and A and these subgroups. Hence, any element g ∈ G can be written as g = exp u exp a for some u ∈ K, a ∈ A. Now, if another element g ∈ G is written similarly, we obtain by using the Hausdorff formula on K and the commutativity of A that

gg = exp u exp a exp u exp a = exp u exp u (exp a)exp u exp a

= exp(u + u + γ ) exp(a exp u + a ), for some γ ∈ [K, K]. By using the previous lemma we finally obtain

gg = Exp(u + u + a exp u + a + γ ), where γ lies in [J, J ]. Applying the definition of αλ and bearing in mind the invariance of λ, we conclude that

αλ (gg ) = ψ(λ(u) + λ(u ) + λ(a exp u ) + λ(a )) = ψ(λ(u) + λ(u ) + λ(a) + λ(a )).

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On the other hand, again by the last lemma, we have g = exp u exp a = Exp(u + a + γ1 ), with γ1 ∈ [J, J ], so αλ (g) = ψ(λ(u + a)) and similarly for g . It is now clear that αλ (gg ) = αλ (g)αλ (g ). Theorem 11. Let J be as in Lemma 9. Then the functions χλ are the irreducible characters of G = 1 + J . Proof. All we have to see is that for any λ ∈ J ∗ , the function αλ is a character of Hλ . We can suppose that A ⊆ Hλ , so that Hλ = (Hλ ∩K)⊕A and the hypotheses on J are inherited by Hλ . The result follows now from Lemma 10, because [Hλ , Hλ ] ⊆ Ker λ by the very definition of Hλ . Corollary 12. Suppose n < 2p and let λ1 , . . . , λr be the different coadjoint orbits of the unitriangular group G = Un (q). Then χλ1 , . . . , χλr are the different irreducible characters of G. Proof. Choose positive integers r and s with r < p, s ≤ p and n = r + s. Then if we define K = Kr,s and A = Ar,s as in (3.2), it is clear that K p = 0 = K i Ap−i for 0 ≤ i ≤ p − 2 (remember that A2 = 0) and it is immediate to check that also K p−1 A = 0, hence the last theorem can be applied. We turn now to the problem of determining the irreducible characters of Un (q) of large degree. ‘Large’ will mean here one of the [p(p − 1)/4] + 1 largest possible values. Some explicit computations must be carried out in the algebra un (q). For the sake of notation, K and A will be as in (3.2), which sometimes will be convenient to identify with ur (q) ⊕ us (q) and Mr×s (q), respectively. Given δ ∈ A∗ , there is a unique matrix m ∈ Ms×r (q) such that for all a ∈ Mr×s (q), δ(a) is the trace of am, δ(a) = tr(am). Now, if g = x0 y0 is an element in 1 + K, then the corresponding matrix for δ g is y −1 mx. By an adequate choice of x and y it is possible to obtain in this way a matrix, which we shall call quasi-monomial, in which at most one single element in every row and column is non-zero. Actually, one can prove that the set of quasi-monomial matrices is a set of representatives for the action of Ur (q) × Us (q) on Ms×r (q) given by m(x,y) = y −1 mx. Thus, there is no loss of generality if we assume from the outset that the matrix corresponding to δ is quasi-monomial. Let us calculate now the orthogonal subspace of A with respect of Bλ if λ ∈ J ∗ is an extension of δ. Since A2 = 0, A is isotropic, so A ⊆ A⊥ and A⊥ = K0 ⊕ A, where K0 = K ∩ A⊥ . The Lie product of an element (u, v) ∈ K and a ∈ A is the element ua − av in A, hence (u, v) ∈ K0 if and only if tr(ua − av)m = 0 for all a ∈ Mr×s (q) or, equivalently, tr(mu)a = tr a(vm). By specializing this condition for the matrices a = ei,j , whose (i, j ) entry is 1 and is zero elsewhere, the equality mu = vm follows. Hence K0 = {(u, v) ∈ ur (q) ⊕ us (q) | mu = vm}. The following definition is taken from [8].

(3.3)

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Definition. We say that a subalgebra of un (q) is a pattern algebra if it has a basis consisting entirely of matrices of the type ei,j . Of course, any pattern algebra determines and is determined by a set of pairs P ⊆ {(i, j ) | 1 ≤ i < j ≤ n} whose cardinality is the dimension of the algebra. Also, given an arbitrary set of pairs P , the condition for the subspace spanned by the matrices ei,j with (i, j ) ∈ P to be a pattern algebra is that whenever two pairs (i, j ) and (j, k) are in P , so is (i, k). Although it is not always the case that K0 is a pattern algebra, its structure is closely related to them, as shown by the next result. Recall that a subspace of a direct sum is called a subdirect sum if both projections are surjective. Lemma 13. Let m ∈ Ms×r (q) be a quasi-monomial matrix. Then the algebra K0 defined by (3.3) is a subdirect sum of two pattern algebras. Proof. Since m is quasi-monomial, if an element mi,j is not zero, then there is no k other than j with mi,k = 0. This defines an injective correspondence σ from {1, . . . , s} into {1, . . . , r}, but not necessarily defined on the whole domain {1, . . . , s}. Then, an element (u, v) ∈ ur (q) ⊕ us (q) is in K0 if and only if for all 1 ≤ i ≤ s and 1 ≤ j ≤ r, mi,σ (i) uσ (i),j = vi,σ −1 (j ) mσ −1 (j ),j ,

(3.4)

where we make the convention that, whenever σ (i) or σ −1 (j ) is not defined, the corresponding side of (3.4) is read as zero. Set P1 to be the set of pairs (k, j ) with 1 ≤ k < j ≤ r such that σ −1 (k) is not defined or both σ −1 (k) and σ −1 (j ) are defined and σ −1 (k) < σ −1 (j ). There is a pattern algebra K1 associated to P1 as one can easily check. We claim now that if (u, v) ∈ K0 , then u ∈ K1 . For, otherwise, the matrix u would have a nonzero element uk,j with (k, j ) ∈ P1 , which means that σ −1 (k) is defined and either σ −1 (j ) is not defined or else, σ −1 (k) ≥ σ −1 (j ). In any of the two cases (3.4) implies uk,j = 0, a contradiction. Similarly, one obtains that v ∈ K2 , where K2 is the pattern algebra given by the entries (i, l) with 1 ≤ i < l ≤ s for which σ (l) is not defined or both σ (i) and σ (l) are defined and σ (i) < σ (l). Thus, K0 is a subalgebra of K1 ⊕ K2 . It only remains to be proved that the projections on K1 and K2 are onto. We consider only the first one, the other being similar. Thus, it suffices to exhibit for each (k, j ) ∈ P1 a matrix v such that (ek,j , v) ∈ K0 . Take v = 0 if σ −1 (k) is not defined and v = m−1 m −1 e −1 , otherwise. −1 σ −1 (j ),j σ (k),k σ (k),σ (j ) This lemma has the following useful consequence for our purposes. Lemma 14. Let K0 be as in the previous lemma. If K0l = 0, then the dimension of K0 is at least l(l + 1)/2. Proof. Note first that it suffices to prove the lemma for pattern algebras. Indeed, K0 is a subdirect sum of pattern algebras and, since K0l = 0, the l-th power of one of these algebras cannot be zero. On the other hand their dimensions are at most that

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of K0 , so all we have to prove is that if K is a pattern algebra and K l = 0, then dim K ≥ l(l + 1)/2. Now, the hypothesis K l = 0 implies that there exist matrices ei1 ,j1 , . . . , eil ,jl in K whose product is not zero. This forces j1 = i2 , . . . , jl−1 = il . Put il+1 = jl . Then for any 1 ≤ r ≤ s ≤ l, eir ,is+1 = eir ,ir+1 . . . eis ,is+1 ∈ K and in particular, dim K ≥ l(l + 1)/2. We need one more result before giving our last theorem. Proposition 15. Let A be an abelian ideal of J with Ap = 0 and let χ be an irreducible character of G = 1 + J . Take an irreducible constituent τ of χ1+A and denote by δ the linear functional on A defined by τ (exp a) = ψ(δ(a)) for all a ∈ A. Suppose that the conclusion of Theorem 11 holds for the algebra group 1 + A , where A = {u ∈ J | [u, A] ⊆ Ker δ}. Then there exists λ ∈ J ∗ such that χ = χλ . Proof. The inertia group of τ in G is T = 1 + A , the stabilizer of δ, thus by Clifford’s correspondence, there exists an irreducible character of T η such that χ = ηG and η1+A = η(1)τ . By the hypothesis on the characters of T , η is given by Kirillov’s formula for some linear functional δ of A . This can also be restated in the following way: if H is a subalgebra which is a maximal isotropic subspace of A with respect to Bδ , then η = α T , where α is the class function defined on 1 + H by α(Exp h) = ψ(δ (h)). We can suppose that H contains A. Then α is an extension of τ , whence δ is an extension of δ. Now we take λ ∈ J ∗ , an extension of δ to J . Clearly A is A⊥ , the orthogonal subspace of A with respect to Bλ and A is isotropic, so H is also a maximal isotropic subspace in J with respect to Bλ (recall the remark just before Proposition 1). Hence, χλ = α G = (α T )G = ηG = χ and we are done. The set of character degrees of Un (q) is well-known (this set is determined in [3] and [12] under the additional assumption that only q-powers belong to it and, in fact, this hypothesis can be dropped by Isaacs’ result in [6]): it consists of all the powers q i with 0 ≤ i ≤ f (n), where n(n−2) if n is even; 4 f (n) = n−1 2 if n is odd. 2 Moreover, q f (n) is the index of the abelian subgroup 1 + Ar,s for r = s = n/2, if n is even and r = (n − 1)/2, s = (n + 1)/2, if n is odd. In the sequel, we simply write K and A instead of Kr,s and Ar,s for these particular values of r and s. On the other hand, the size of a coadjoint orbit cannot exceed q 2f (n) . This is so because, as we know, the number of elements in the orbit of λ is q 2m , where 2m is the rank of Bλ . The number m is also the codimension of a maximal isotropic subspace and A, which has codimension f (n), is isotropic, whence m ≤ f (n).

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Theorem 16. Let G = Un (q) and 0 ≤ k ≤ p(p − 1)/4. Then the map λ → χλ gives a bijection between the irreducible characters of G of degree q f (n)−k and the coadjoint orbits of size q 2(f (n)−k) . Proof. Suppose first that λ is a coadjoint orbit of size q 2(f (n)−k) with k in the specified range. We pick as usual a subalgebra H which is a maximal isotropic subspace of un (q) containing A. There is no loss of generality if we suppose that the restriction of λ to A is represented by a quasi-monomial matrix m so that A⊥ = K0 ⊕ A and K0 is as defined in (3.3). Since H is isotropic, H ⊆ A⊥ , so H = (H ∩ K0 ) ⊕ A. If we p−1 prove that K0 = 0, then of course (H ∩ K0 )p−1 = 0 and we can apply Lemma 10 to conclude that the function αλ is a character of 1 + H . But αλG = χλ , so this last function is also a character and the theorem will follow in one direction. Suppose p−1 then by way of contradiction that K0 = 0. By Lemma 14, dim K0 ≥ p(p − 1)/2. To estimate the dimension of H ∩ K0 note that this is a maximal isotropic subspace of K0 (otherwise, we could find an isotropic subspace containing H properly), so its dimension is at least half the dimension of K0 and actually, this inequality is strict because equality occurs only when the alternating form is non-degenerate, which is not our case since the center of K0 is inside the radical of the restriction of Bλ to K0 . In short, 1 p(p − 1) dim K0 ≥ ≥ k. (3.5) 2 4 But the dimension of H ∩ K0 can be computed as follows: write 2m for the rank of Bλ and r for the dimension of its radical. Then, the assumption on the size of λ is that m = f (n) − k, thus dim H ∩ K0 >

dim H ∩ K0 = dim H − dim A = dim H − dim un (q) + f (n) = f (n) − m = k, in contradiction with (3.5). In the reverse direction, suppose that χ is an irreducible character of Un (q) of degree q f (n)−k , again with 0 ≤ k ≤ p(p − 1)/4. We take τ , an irreducible constituent of χ1+A , and δ ∈ A∗ such that τ (1 + a) = ψ(δ(a)) (observe that A2 = 0, so exp a = 1 + a). At the expense of replacing τ by one of its conjugates, we can suppose that δ is represented by a quasi-monomial matrix m. Thus, A = {u ∈ un (q) | [u, A] ⊆ Ker δ} = K0 ⊕ A, where K0 is as in (3.3). Bearing in mind Proposition 15 and Theorem 11, it suffices p−1 to prove that K0 = 0. Suppose this is not true, so that dim K0 ≥ p(p − 1)/2. By Clifford’s correspondence, there exists an irreducible character η of the inertia group of τ , which is T = 1+A , lying over τ such that χ = ηG . But T is the semidirect product of 1 + K0 and 1 + A and the latter is a normal and abelian subgroup, so τ extends to T and, if we take a fixed extension τ , then by Gallagher’s Theorem [5, Corollary 6.17], η = βτ for some irreducible character β of 1 + K0 . Then η(1)2 = β(1)2 < |1 + K0 |. But χ (1) = |G : T |η(1), whence η(1) = |T : 1 + A|/q k = |1 + K0 |/q k and it

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follows that |1 + K0 | < q 2k , that is 2k > dim K0 ≥ p(p − 1)/2, a contradiction. This completes the proof.

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C. A. M. André, Basic sums of coadjoint orbits of the unitriangular group, J. Algebra 176 (1995), 959–1000.

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C. A. M. André, Irreducible characters of finite algebra groups, in Matrices and group representations, Universidade de Coimbra, Coimbra 1999.

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B. Huppert, A remark on the character-degrees of some p-groups, Arch. Math. 59 (1992), 313–318.

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I. M. Isaacs, Sets of p-powers as irreducible character degrees, Proc. Amer. Math. Soc. 96 (1986), 551–552.

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I. M. Isaacs, unpublished notes.

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I. M. Isaacs and M. C. Slattery, Character degree sets that do not bound the class of a p-group, Proc. Amer. Math. Soc. 130 (2002), 2553–2558.

[10] D. Kazhdan, Proof of Springer’s hypothesis, Israel J. Math. 28 (1977), 272–286. [11] A. A. Kirillov, Variations on the triangular theme, in Lie Groups and Lie Algebras: E. B. Dynkin’s Seminar, Amer. Math. Soc. Transl. (2) 169, Amer. Math. Soc., Providence, RI 1995, 43–73. [12] G. I. Lehrer, Discrete series and the unipotent subgroup, Compositio Math. 28 (1974), 9–19. [13] M. Marjoram, Irreducible characters of small degree of the unitriangular group, Irish Math. Soc. Bull. 42 (1999), 21–31. [14] S. Moran, Dimension subgroups modulo n, Proc. Camb. Phil. Soc. 68 (1970), 579–582. [15] A. Vera-López and J. M. Arregi, Computing in unitriangular matrices over finite fields, preprint. Josu Sangroniz, Departamento de Matemáticas, Facultad de Ciencias, Universidad del País Vasco, Apartado 644, 48080 Bilbao, Spain E-mail: [email protected]

The signalizer method Ronald Solomon∗

General remarks The fundamental problem of the simple group classification may be formulated vaguely as follows: Given certain (local) information about a finite group G, determine whether or not G has a proper normal subgroup. More precisely, we may say (at least, a posteriori) that the problem is: Given a finite group whose local structure differs in some way from the local structure of a known simple group, prove that G has a proper normal subgroup. By local subgroup, we mean the normalizer of a non-identity p-subgroup of G. Let π be a set of primes and a set of Sylow p-subgroups of G, one for each p ∈ π . By π-local information we will mean the structure and embedding of the subgroups NG (R) as R ranges over the non-identity subgroups of members of . The work of Frobenius and Burnside from the early 1890s until 1910 achieved significant progress on the following part of the problem: Given a finite group G. Ascertain whether G has a nontrivial abelian quotient. In particular Burnside introduced what we now call the transfer homomorphism, which was further refined in ensuing decades by Grün, Wielandt, Yoshida and others. Thus by 1950, it is more or less fair to say that one could safely assume that G had no nontrivial solvable quotients. The next question was: Does G have a nontrivial solvable normal subgroup? In the context of finite groups, all successful approaches to date have proceeded from the following point of view. The prime graph (G) has vertex set the prime divisors of G with {p, q} an edge if and only if G has an element of order pq. We must begin our p-local study of G at some prime p and, since the Odd Order Paper [FT] of Feit and Thompson, 2 has been the obvious choice. We can then move among the primes p ∈ π2 where π2 is the set of primes which are vertices of the connected component of (G) containing the vertex 2. Other primes may not be accessible. ∗ Partially supported by an NSF research grant.

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The following small example illustrates the difficulty. Let G∗ = V H , where V is elementary of order 112 and H ∼ = SL(2, 5) acting faithfully on the normal subgroup V . Now let G be a finite group with the same 2-local structure as G∗ . Then π2 = {2, 3, 5}. Question: Given a group G with the same π2 -local data as G∗ , can one show that ∼ G = G∗ ? Answer: No. H as the same π2 -local data as G∗ . Weaker question: Given a group G with the same π2 -local data as G∗ , can we show that G has a nontrivial solvable normal subgroup? Answer: Yes. Note that if G ∼ = H , then Z(G) is a nontrivial solvable normal subgroup of G. Thus we may assume that H < G. G is then a member of a class of groups called Frobenius groups and a partial answer is given by the following theorem of Frobenius from 1901. Frobenius’ Theorem ([GLS2, 9.10]). Let G be a finite group with a proper subgroup H such that H ∩ gH g −1 = 1 for all g ∈ G − H . Then G has a proper normal subgroup K with K ∩ H = 1 and G = KH . Indeed K is a solvable (in fact, nilpotent) normal subgroup of G. (This fact was first proved by Thompson in his PhD dissertation in 1958 [GLS2, 9.11 (i)].) Frobenius finds K as the kernel of a nontrivial character of G. This may be regarded as a subtle (complexified) counting argument, made possible by the fact that conjugates of H intersect trivially. The methodology does not seem to extend far beyond the trivial intersection situation, though remarkable tours de force in this vein were achieved in the 1950s and early 1960s by Brauer, Suzuki, Feit and others. In particular Brauer and Suzuki extended the Frobenius-type result to all groups with (generalized) quaternion Sylow 2-subgroups ([GLS2, 15.2]). New methods were needed to deal with finite groups having a rich subgroup lattice with many nontrivial intersections. This is where the story of signalizers begins.

Signalizers We start with a trivial observation: Suppose that G is a finite group with proper subgroups M and N such that G = M, N. If X M and X N , then X G. Normality in both M and N is an unlikely condition to achieve. Much more likely is subnormality. It is always useful to bear in mind the fundamental example: G∗ = V H , where V is an elementary abelian r-group for some odd prime r and H is a nonabelian simple group.

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Assume that G is a finite group with the same 2-local structure as G∗ . We would like to prove that G has a nontrivial normal r-subgroup. Since H is nonabelian simple, the Brauer–Suzuki Theorem implies that H contains a Klein 4-group U ∗ and so CV (u∗ ) = 1 for some involution u∗ ∈ U ∗ . Moreover CV (u∗ ) ∩ N V ∩ N N for every subgroup N of G. So we may hope to prove that there is a nontrivial rsubgroup of CG (u) which is subnormal in every 2-local subgroup of G which contains it. (Here u is the involution of G corresponding to u∗ ∈ G∗ .) Methods to attack such problems were provided by Feit and Thompson in the Odd Order Paper. For a subgroup A of G, they define IG (A) to be the set of all Ainvariant subgroups of G which intersect A trivially, and IG (A; π ) to be the set of all π-subgroups in IG (A), for π a set of primes. Typically A is a Sylow p-subgroup of G or, more generally, a p-subgroup of G which is Sylow in ACG (A). The following fundamental lemma treats the solvable situation. Thompson–Bender Lemma ([GLS2, 23.3]). Let X be a solvable group and let p be an odd prime. Let A be a p-subgroup of X which contains every element of order p in CX (A). Then Op (X) is the unique maximal member of IX (A, p ). Under some circumstances we can say more. Glauberman’s K-Theorem ([GG]). Let X be a solvable group of odd order and let r be a prime with F (X) = Or (X). Let R be any r-subgroup of X containing F (X). Then there exists a nontrivial characteristic subgroup of R, K(R), such that K(R) X. Indeed the hypothesis of solvability can be weakened to a p-stability hypothesis. In particular if A is a p-subgroup of X for some prime p = r, then we may choose R to be a maximal element of IX (A, r). We let I∗X (A, r) denote the set of maximal elements of IX (A, r).

Signalizers: moving around We have yet to confront the fundamental issue – comparing the structures of two different maximal subgroups X and Y of a finite simple group G, where X and Y have a large intersection. Let us suppose then that there is a prime r such that F ∗ (X) and F ∗ (Y ) are both r-groups. Suppose further that p is a prime with p = r and that A is a p-subgroup of X ∩ Y . If Glauberman’s K-Theorem applies to both X and Y , then X = NG (K(R)) and Y = NG (K(S)), where R ∈ I∗X (A, r) and S ∈ I∗Y (A, r). Moreover Or (X) ≤ R and Or (Y ) ≤ S. Thus if F ∗ (X) ∩ F ∗ (Y ) = 1, then likewise R ∩ S = 1. What we need is a Sylow-type theorem relating R and S. If A = 1, then R and S are Sylow r-subgroups of G and the ordinary Sylow Theorem will suffice. In general we need an A-equivariant Sylow Theorem. This is provided in many contexts by the Feit–Thompson Transitivity Theorem.

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Feit–Thompson Transitivity Theorem ([GLS2, 23.12]). Let A be an abelian psubgroup of the finite group G which contains every element of order p in CG (A). Let r be a prime different from p and suppose that every r-local subgroup of G is solvable. If R, S ∈ I∗G (A, r) with R ∩ S = 1, then S is CG (A)-conjugate to R. Moreover if every p-local subgroup of G is solvable and A has p-rank at least 3, then all members of I∗G (A, r) lie in the same CG (A)-orbit. Corollary (A Uniqueness Theorem). Let G be a finite group and let p and r be distinct primes. Let A be a abelian p-subgroup of G with m(A) ≥ 3, such that A contains every element of order p in CG (A). Suppose further that every r-local subgroup H of G is solvable with F ∗ (H )(= F (H )) = Or (H ), and every p-local subgroup of G is solvable. If CG (A) is contained in a maximal r-local subgroup M of G, then M is the unique maximal r-local subgroup of G containing A. (Note: If Or (G) = 1, then M = G.) Proof. Let M be as given and suppose that N is a maximal r-local subgroup of G containing A. Let R ∈ I∗M (A, r) and let S ∈ I∗N (A, r). By Glauberman’s KTheorem, M = NG (K(R)) and N = NG (K(S)). Thus R, S ∈ I∗G (A, r). Now by the Transitivity Theorem, S is CG (A)-conjugate to R. Thus N = M c for some c ∈ CG (A). But CG (A) ≤ M, whence N = M, as claimed. This is an example of a Uniqueness Theorem. Typically one hopes to establish eventually that M is a strongly p-embedded r-local subgroup of G, and then to derive a contradiction. Of course the hypothesis of solvability of r-local and p-local subgroups is unrealistic in the general case, and much of the work of Gorenstein and Walter and somewhat later, of Bender, was directed towards finding the appropriate generalizations. The hypothesis that F ∗ (H ) = Or (H ) for every r-local H is also unjustified in general, but the following theorem of Bender hints at the fact that this is still the most important case. Bender’s Maximal Subgroup Theorem ([GLS2, 19.5]). Let M and N be distinct maximal subgroups of the finite simple group G. Suppose that there exist X ≤ F (M) and Y ≤ F (N ) with NG (X) ≤ N and NG (Y ) ≤ M. Then there exists a prime p such that F ∗ (M) = Op (M) and F ∗ (N) = Op (N). Notice that the hypothesis X ≤ F (M) is equivalent to the assumption that X is a nilpotent subnormal subgroup of M. Typically the Maximal Subgroup Theorem is used in the following way. We have a maximal subgroup M of the finite simple group G and we wish to establish that NG (X) ≤ M for many of the subnormal nilpotent subgroups X of M. Suppose then that NG (X) ≤ N with N maximal in G. Let Y be a non-trivial subgroup of NF (M) (X) with Y M. (For example let Y = Z(F (M)).) Then Y ≤ N and NG (Y ) ≤ M. So, if we can just show that Y ≤ F (N), then the hypotheses of the Maximal Subgroup Theorem hold and we either have the desired conclusion that NG (X) ≤ M or we have that F ∗ (M) and F ∗ (N ) are p-groups for

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some prime p. In the latter case we may try to use Glauberman’s K-Theorem and transitivity arguments to get a contradiction. The main problem then is to force subgroups like Y to lie in F (N). We now impose a hypothesis, which plays a central role in the Signalizer Analysis. Signalizer Hypothesis. There exists a subgroup M and an element t ∈ M of prime order r with CG (t) ≤ M. There exists a t-invariant r -subgroup Z of M, and there exists a second subgroup N containing Z, t. Hence Z ∈ IN (CN (t), r ), i.e. Z is a CN (t)-invariant r -subgroup of N. The general philosophy is that such r-signalizers should “sink” into Op (N ), and indeed, if they are p-groups, into Op (N) ≤ F (N). We subdivide the discussion into two cases. Case 1: [Z, t] = 1. This is the A × B Lemma case. Lemma. If N is solvable, then Z ≤ Or (N). Proof. Let N = N/Or (N). The result follows from the A × B Lemma applied to the action of t × Z on Or (N ). An immediate application is the following result. Balance Theorem of Thompson. Let t1 and t2 be commuting r-elements of G and assume that CG (ti ) is solvable for i = 1, 2. Then Or (CG (ti )) ∩ CG (tj ) = CG (ti ) ∩ Or (CG (tj )). The equation is called the Balance Equation. An easy consideration of commuting semisimple elements in finite groups of Lie type (or of commuting r-elements with large (but distinct) supports in alternating groups) quickly reveals that the Balance Equation routinely fails in the absence of solvable centralizers. Gorenstein’s idea was to restrict attention to a proper subgroup of Or (CG (t)) for which the analogue of Thompson’s Balance Equation will hold. The subgroup {1} is an obvious choice, but this yields no useful information. Again we should bear in mind the fundamental example where G∗ = V H , V a vector space in characteristic p, H a nonabelian simple group. Consider the following specific example: G∗ = V H where V is elementary abelian of order 113 and H = SL(3, 11) acting faithfully on V . If t is an involution of H , then O2 (CG∗ (t)) = Vt Ht , where |Vt | = 11 and |Ht | = 5. Moreover the eigenspaces of t and of Ht on V coincide. Thus if we take t to be an involution of H commuting with t, then O2 (CG∗ (t)) ∩ CG∗ (t ) = Ht O2 (CG∗ (t )).

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Thus Balance fails. But the subgroup we wanted to identify in any case was V . So instead of O2 (CG∗ (t)) we should choose Vt and then of course Vt ∩ CG∗ (t ) = 1 ≤ Vt . The problem though is to name Vt in the hypothetical finite simple group G which has the same 2-local structure as G∗ . If we set A = t, t , then the following solution suggests itself. Define θ1 (a) = [O2 (CG (a)), A] for each a ∈ A# . Now we see that in our example θ1 (a) = CV (a) for each a ∈ A# . For technical reasons, it turns out to be preferable to define θ(a) = [O2 (CG (a)), A]A , where A =

O2 (CG (a)).

a∈A#

Notice that A = 1 in our example and so again θ (a) = CV (a), as desired. The function θ on A# is an example of a signalizer functor and it was introduced by Goldschmidt in [G]. It was later used in the treatment of 2-components of type 2An [S] and, with some refinements, by Aschbacher in his characterization of groups of Lie type in odd characteristic [A]. When extended to arbitrary primes p, it was dubbed the 23 -balanced functor by Gorenstein and Lyons [GLS2, 21.9] and became the first of a family of k + 21 -balanced functors they defined. Earlier, versions of had been used by Gorenstein and Walter to define k-balanced functors for k ≥ 2. But I have gotten ahead of myself and have not defined a signalizer functor. The following definition is due to Goldschmidt, refining an earlier definition by Gorenstein. Here we let P (G) denote the set of all subsets of G. Definition. If A is an elementary abelian r-subgroup of the finite group G, then an A-signalizer functor is a function θ : A# → P (G) satisfying for all a, a ∈ A# : (1) θ (a) is an A-invariant r -subgroup of CG (a); and (2) θ (a) ∩ CG (a ) = CG (a) ∩ θ(a ). The crucial theorem is the Signalizer Functor Theorem. A preliminary version was proved by Gorenstein. Then, for r = 2, the theorem was proved by Goldschmidt. Under the additional hypothesis that θ(a) is a solvable group for all a ∈ A# , the theorem was proved by Glauberman. Finally, under a weak assumption on proper simple sections of G (valid for all (known) simple groups), the full theorem was established by McBride. The Signalizer Functor Theorem ([GLS2, 21.3,21.4]). Let A be an elementary abelian r-subgroup of the finite group G with m(A) ≥ 3. Let θ be an A-signalizer functor on G such that either

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(1) θ (a) is solvable for all a ∈ A# ; or (2) G has the r-McBride property ([GLS2, 2.6]). Then WA = θ (a) : a ∈ A# is an r -subgroup of G. One then hopes to prove that NG (WA ) is a strongly r-embedded r -local subgroup of G; and then that WA G with θ(a) = CWA (a) for all a ∈ A# . If this is accomplished, then we will (in our example) have reconstructed the normal vector space V from the pieces visible inside r-local subgroups of G. As noted in the example above, a naive choice of θ will often lead to the failure of the Balance Equation. If the 23 -balanced functor is used, then often it suffices to establish the following: Xa := [θ(a) ∩ CG (a ), A] ≤ Or (CG (a )). Let C = CG (a )/Or (CG (a )). Then, as in the proof of Thompson’s Balance Theorem, the A × B Lemma shows that [Or (C), Xa ] = 1. It easily follows that an obstruction to balance must be an a, Xa -invariant quasisimple component K of C such that 1 = Xa ≤ [Or (C K,Xa ,a (a)), A]. Unfortunately such “locally unbalanced” configurations are ubiquitous in alternating groups and groups of Lie type in odd characteristic. Fortunately we are permitted to make a judicious choice of A as well. The prototypical good choice is the following: A ≤ Z ∗ (K) = Z ∗ (K1 K2 K3 ), where K/O2 (K) ∼ = SL2 (q) × SL2 (q) × SL2 (q) for some odd q. This is the choice made by Aschbacher in the “generic” case of his characterization of groups of odd Lie type, and it leads to an easy verification of local balance. Good generation of (most) Lie type groups (proved by Seitz and Lyons [GLS3, 7.3.3]) then leads easily to the dichotomy that either the Goldschmidt functor is trivial (as desired) or that G has a strongly embedded subgroup, contradicting the Suzuki-Bender Theorem ([GLS4, Theorem SE]). The triviality of the Goldschmidt functor then leads easily to a weak (but sufficient) version of Thompson’s B-Conjecture, namely that L2 (CG (a)) is semisimple for all a ∈ A# . A similar and almost equally easy argument may be found in [GLS5] to handle the “wide” Lie type groups of characteristic 2. As usual the “narrow” groups cause all the mischief. Case2: [Z, t] = 1. This case plays an important role in Bender’s proof of the 2-Signalizer Functor Theorem and also in the analysis of the Special Odd Case [GLS1, Part 2, 14.2]. In particular, since the Signalizer Functor Theorem requires that m(A) ≥ 3, this case plays a crucial role in the analysis of simple groups G with m2 (G) = 2. Though it has antecedents in the work of Thompson, the shape of this analysis was largely sculpted by Helmut Bender and it is often called the Bender Method. We assume in this case a slightly modified version of the Signalizer Hypothesis.

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Signalizer Hypothesis. N is a finite group and p and r are distinct primes with t ∈ N of order r and Z a CN (t)-invariant p-subgroup of N with [Z, t] = Z. We let N = N/Or (N) and we wish to prove that Z = 1, whence Z is a nilpotent subnormal subgroup of N. In the solvable case (or more generally, when E(N) = 1), we confront a Hall-Higman Configuration. Dropping the bars, we may consider H = QZ t, where Q is a Z t-invariant q-group for some prime q = p and Z = [Z, t] acts faithfully on V := [Q/(Q), Z]. Lemma. If Q, Z and t are as above with p odd (or r not a Fermat prime) and if Z is CH (t)-invariant, then q = r. Proof. A theorem of Hall-Higman type implies that t has a free summand in its action on V unless p = 2 and r is a Fermat prime. In particular, CV (t) = 1. If also r = q, then CV (t) = CQ (t)(Q)/(Q). But Z ∈ IN (CN (t), p), whence [CQ (t), Z] ≤ Q ∩ Z = 1, a contradiction. In particular, if p = 2 and Z, t ≤ N , then Z centralizes O2 (F (N)). Indeed the following is true. Lemma (Thompson [T, 5.17]). Z centralizes F (N) unless O2 (N) has nilpotence class 2. We cannot say much more because of examples such as the following: H = GL2 (3), Q = Q8 , |Z| = 3, |t| = 2; and H = QZ t ≤ L t, L ∼ = U3 (4), |Q| = 26 , |Z| = 5, with t an involutory graph automorphism of L. We conclude with a few remarks about the way this Case 2 analysis is used in the classification of simple groups G of Special Odd Type. We begin with a simple group G and a 2-central involution t of G. The Special Odd Type hypothesis quickly implies that F ∗ (CG (t)) is not a 2-group. If M = CG (t) is a maximal subgroup of G, then Bender’s Maximal Subgroup Theorem is effective and a Case I type analysis gives control of O2 (CG (t)). Suppose then that CG (t) is properly contained in a maximal subgroup M of G. With some exceptions (notably the dihedral Sylow 2-subgroup case), we easily conclude that F ∗ (M) = F (M) has odd order. The heart of the analysis is aimed at proving that NG (X) ≤ M for most t-invariant subgroups X of F (M). (After that, a short argument using the G-fusion of involutions yields a contradiction or leads to the parabolic structure of PSL(3, pn ).)

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Thus assume that CG (t) ≤ M with M a maximal subgroup of G and with F ∗ (M) of odd order. Let X be a non-trivial t-invariant subgroup of F (M) and embed NG (X) in a maximal subgroup N of G. Then half of the hypotheses of Bender’s Maximal Subgroup Theorem apply to the pair (M, N ). It remains to find Y ≤ F (N) with NG (Y ) ≤ M. We proceed by a bootstrap argument, incrementally forcing more and more subgroups Y of F (M) to satisfy NG (Y ) ≤ M. Typically we arrive at the following situation: Set Y = [Op (M) ∩ N, t] with Y = 1. (This must occur for some odd prime p by the A × B Lemma.) Suppose we know that NG (Y0 ) ≤ M for some Y0 ≤ Y . Then it will suffice to show that Y ≤ Op (N). Note that Y = [Y, t] and Y ∈ IN (CN (t), p). So the Case 2 analysis applies to yield that the obstruction to Y ≤ Op (N ) must be either a class 2 2-subgroup of N/Op (N) or a quasisimple component of N/Op (N ). Korchagina has done an exhaustive analysis of the quasisimple obstructions which can arise if t has order 2 or 3. These seem to be the crucial cases for the Classification proof. (In the Special Odd Case, t has order 2.) If Y can be shown to lie in F (N ), then Bender’s Maximal Subgroup Theorem yields that either M = N (the desired conclusion) or that both F ∗ (M) = Op (M) and F ∗ (N ) = Op (N). In the latter case Glauberman’s K-Theorem and transitivity arguments usually suffice to yield a contradiction. The existence of large 2-group and quasisimple obstructions to “pushing down” signalizers makes this method (at least in the naive form outlined here) impractical in large groups. Particularly serious is the possibility of a quasisimple component K of N/Op (N ) with K ∼ = PSL(n, pm ) and containing a maximal parabolic H with H = Y CK (t), i.e. Y projects onto the unipotent radical of a maximal parabolic subgroup H of K and t lies in the center of a Levi complement of H . Fortunately, the Signalizer Functor Method, as outlined in Case I, is quite effective in large groups. Thus the only case where it seems necessary to confront the situation described above is when G has semidihedral or wreathed Sylow 2-subgroups and K ∼ = PSL(3, pm ). Luckily in this situation we are rescued by a modular character argument of Brauer. (There is in fact an unpublished alternate local approach to this configuration, due to Aschbacher.)

Concluding remarks The Signalizer Method pioneered by Feit and Thompson in the Odd Order Paper has proved to be remarkably robust, extending from the original context of solvable locals of odd order to the general context of arbitrary local subgroups (with known composition factors). It has provided the mechanism to prove weak versions of the B-Theorem (whose full-strength version, stated below after a few definitions, is a corollary of the Classification Theorem):

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Definitions. Let H be a finite group and let p be a prime. A p-component L of H is a subnormal subgroup of H minimal subject to covering a quasisimple component L of H = H /Op (H ). The Gorenstein–Walter p-layer of H , Lp (H ) is the product of all p-components of H . Then Bp (H ) is the product of those p-components of H which are not quasisimple, while Ep (H ) is the product of all quasisimple p-components of H . Thus Lp (H ) = Bp (H )Ep (H ) with Bp (H ) ∩ Ep (H ) ≤ Z(Ep (H )). Note that a p-component must by definition have order divisible by p. Thus E2 (H ) = E(H ), but in general Ep (H ) may be a proper characteristic subgroup of E(H ). The B-Theorem. Let G be a finite group, p a prime and H a p-local subgroup of G. Then Bp (H ) ≤ Bp (G). In particular, Op (Bp (H )) ≤ Op (G). This theorem guarantees the semisimplicity of p-layers in p-local subgroups of a finite simple group G. We can then use an amalgam of quasisimple p-components to identify G. The weak version of the B-Theorem establishes the result only for certain p-locals of a K-proper simple group G, but enough p-locals to produce the desired amalgam of quasisimple p-components. Is the good the enemy of the better in this context? Perhaps. We can only hope that some new researchers will bring fresh skeptical eyes to this problem and find a beautiful new approach. We offer a few random comments: In the Killing–Cartan classification of semisimple Lie groups and Lie algebras the analogue of the Signalizer Problem is the Solvable Radical Problem, which is brilliantly resolved by the Killing form and Cartan’s Criterion. Is there an analogue – perhaps some trilinear map on a finite group G which detects the existence of a nontrivial solvable normal subgroup? Lyons has observed a suggestive analogy between the Signalizer Functor Theorem and the Curtis–Tits Theorem (characterizing the finite groups of Lie type from local data). Let A be an elementary abelian r-subgroup of G of rank 3 and let θ be a signalizer functor defined on A. For each hyperplane B of A define θ (b). θ(B) = b∈B #

The subgroups θ(B) play the role of the rank 1 subgroups in the Curtis–Tits set-up, and the subgroups θ(a) play the role of the rank 2 subgroups. The condition θ(a) = θ(B) ∩ θ(a) : B ≤ A, |A : B| = r is an immediate consequence of the Balance Condition. The Signalizer Functor Theorem then asserts that if the rank 1 and rank 2 subgroups are all r -subgroups, then the full group θ(1) := θ(a) : a ∈ A# is also an r -group.

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This suggests the possibility of bypassing the Signalizer Functor Theorem and proceeding directly to a characterization of finite groups generated by a family of rank 1 groups KB , labelled by the hyperplanes B of some fixed elementary abelian r-group A, with K/Or (K) ∈ Chev(p) of BN -rank 1, for some fixed distinct primes p and r, subject to a Balance Condition specifying the nature of the rank 2 amalgams of these rank 1 groups. In any event, Lyons’ remark highlights the crucial role played by the number 3 in both the Signalizer Functor Theorem and the Curtis–Tits Theorem. Knowledge of rank 1 and rank 2 groups is presupposed and only for rank at least 3 is new information obtained. Brauer proved that for any finite group H and any prime p, Op (H ) is the intersection of the kernels of the ordinary irreducible characters in the principal p-block of H . This certainly suggests a role for modular character theory in the analysis of signalizers, a fact which has not escaped the attention of Alperin, Broué, Puig, Geoff Robinson and others. Certain nagging doubts persist, however. For one thing, there is no nice relationship between Op (H ) and Op (G) for H a p-local subgroup of G in general. Even if G is of characteristic p-type, i.e. all p-locals of G are p-constrained (even solvable), problems arise when the p-rank of G is at most 2. The B-Theorem is a general fact, but no one has given a representation theoretic characterization of Bp (G). The fact that general statements about signalizers and strongly p-embedded subgroups only become true for p-rank at least 3 seems to argue against a role for character theory, unless p-modular results specific to the case of p-rank at least 3 are uncovered. Thus far, the contrary seems to be true, i.e. the strongest p-modular results are obtained for p-rank at most 2. Indeed this “miracle” saves the Classification proof. Borovic has extended Signalizer Analysis to the context of infinite groups of finite Morley rank. He discovered that Signalizer Analysis is more elegant in this setting because one can name the vector-space part of the signalizer effectively. This is due to the fact that one can easily distinguish the additive group of an algebraically closed field from the multiplicative group of an algebraically closed field. By contrast there is no way to distinguish the additive group of the field of order 2 from the multiplicative group of the field of order 3. Perhaps then suffering is inevitable in the finite case. While looking forward to new minds with new insights, we must also look back with gratitude for the creativity and perseverance of the heroes of the “first generation” signalizer analysis: Feit and Thompson, Gorenstein and Walter, Suzuki and Bender, Goldschmidt and Glauberman, Alperin and Brauer, Gorenstein and Harada, Aschbacher, Gorenstein and Lyons; but most of all: John G. Thompson.

References [A]

M. Aschbacher, A characterization of Chevalley groups over fields of odd order, Ann. of Math. 106 (1977), 353–468.

[FT]

W. Feit and J. G. Thompson, Solvability of groups of odd order, Pacific J. Math. 13 (1963), 775–1029.

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[G]

D. M. Goldschmidt, Weakly embedded 2-local subgroups of finite groups, J. Algebra 21 (1972), 137–148.

[GG]

G. Glauberman, Prime power factor groups of finite groups, II, Math. Z. 117 (1970) 46–56.

[GLS1] D. Gorenstein, R. Lyons, R. Solomon, The Classification of the Finite Simple Groups, Math. Surveys Monogr. 40.1, Amer. Math. Soc., Providence, RI 1994. [GLS2] D. Gorenstein, R. Lyons and R. Solomon, The Classification of the Finite Simple Groups, Number 2, Math. Surveys Monogr. 40.2, Amer. Math. Soc., Providence, RI 1996. [GLS3] D. Gorenstein, R. Lyons and R. Solomon, The Classification of the Finite Simple Groups, Number 3, Math. Surveys Monogr. 40.3, Amer. Math. Soc., Providence, RI 1998. [GLS4] D. Gorenstein, R. Lyons and R. Solomon, The Classification of the Finite Simple Groups, Number 4, Math. Surveys Monogr. 40.4, Amer. Math. Soc., Providence, RI 1999. [GLS5] D. Gorenstein, R. Lyons and R. Solomon, The Classification of the Finite Simple Groups, Number 5, Math. Surveys Monogr. 40.5, Amer. Math. Soc., Providence, RI 2002. [GLS6] D. Gorenstein, R. Lyons and R. Solomon, The Classification of the Finite Simple Groups, Number 6, Math. Surveys Monogr. 40.6, Amer. Math. Soc., Providence, RI, to appear. [K2]

I. Korchagina, 2-Signalizers in almost simple groups, to appear.

[K3]

I. Korchagina, 3-Signalizers in almost simple groups, in Finite Groups 2003 (C. Y. Ho, P. Sin, P. H. Tiep, A. Turull, eds.), Proceedings of the Gainesville Conference on Finite Groups, March 2003, Walter de Gruyter, Berlin 2004, 229–246. R. Solomon, Finite groups with intrinsic 2-components of type Aˆ n , J. Algebra 33 (1975), 498–522.

[S] [T]

J. G. Thompson, Nonsolvable finite groups all of whose local subgroups are solvable, I, Bull. Amer. Math. Soc. 74 (1968), 383–437.

Ronald Solomon, Department of Mathematics, The Ohio State University, Columbus, Ohio 43210, U.S.A. E-mail: [email protected]

Classical groups in dimension 5 which are Hurwitz M. Chiara Tamburini and Alexandre E. Zalesski

Abstract. In this paper we provide new examples of Hurwitz classical groups. 2000 Mathematics Subject Classification: 20D, 20F

1. Introduction We recall that a group G is said to be (2, 3, k)-generated if it can be generated by an involution and an element of order 3 whose product has order k. A finite (2, 3, 7)generated group is called a Hurwitz group. The relevance of such groups is due to the famous result of Hurwitz, that a finite group of order 84(g − 1) can act faithfully on a compact Riemann surface of genus g if and only if it is (2, 3, 7)-generated. Moreover 7 is the first value of k for which the triangle group T (2, 3, k) is infinite and non-soluble. The smallest Hurwitz group is PSL2 (7), already discovered by Klein. But the first significant examples are due to Macbeath who, in 1969 [12], classified the Hurwitz groups of type PSL2 (q). They are precisely PSL(2, p) if p ≡ 0, ±1 (mod 7), and PSL(2, p3 ) if p ≡ ±2, ±3 (mod 7). This result provided in particular many transitive permutational representations of T (2, 3, 7): they arise from the action of PSL2 (q) on the cosets of its subgroups. These representations, and a clever method of joining them via handles developed by Graham Higman, culminated in the famous theorem of Conder that for n > 167 the alternating groups An are Hurwitz [2]. He also proved the stronger result that for k ≥ 7 almost all of the alternating groups are (2, 3, k)generated, and an even more striking generalization of this was given in [6], where it is shown that any Fuchsian group surjects almost all of the alternating groups. There is now evidence that the property of being a Hurwitz group is quite common among finite simple groups. In fact Conder’s constructive permutational methods were later generalized to the linear context, and led to the discovery that most finite classical groups are Hurwitz (see [10] and [11]). On the other hand, using charactertheoretic methods, Malle determined the precise values of q, m and n for which the exceptional simple groups of Lie type G2 (q), 2 G2 (32m+1 ), 3 D4 (p n ) and 2 F4 (22n+1 ) are Hurwitz (see [13, 14] and, for Ree groups, see also [7] and [16]). Finally, by the contribution of several authors, it is known exactly which of the 26 sporadic simple groups are Hurwitz (see, for example, [3] and [25]). For more information in this very

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wide area we refer to the papers in the references and the literature quoted there, and to the recent survey [22]. Although the above mentioned results on classical groups are satisfactory if considered asymptotically with respect to the ranks of the groups under consideration, the lower bounds for their ranks are certainly much higher than necessary for the existence of Hurwitz generators. But a constructive search of Hurwitz generators for groups of Lie type of small rank cannot be done in a uniform way. The main difficulty consists in the fact that the property of being Hurwitz may depend also on the size of the field and its characteristic, and not only on the rank and type. Evidence of this is given by the above mentioned results of Macbeath and Malle, and by this paper. Further evidence is given in [4], where is made a systematic application of a well-known formula of Scott [17] to show that many classical groups of small rank are not Hurwitz. In this paper we concentrate on the small rank case. Let us first recall that the Hurwitz subgroups of PSL3 (F), where F is an algebraically closed field of positive characteristic, are isomorphic to those discovered by Macbeath, as proved in [1]. And this fact, which also applies to PSL4 (F) (see [22]), may have erroneously discouraged, for a long time, the search for (projective) linear Hurwitz groups. Actually the reason why PSL3 (F) and PSL4 (F) do not contain new Hurwitz groups is due to Scott’s formula which gives a very strong restriction for small dimensional matrix groups. However in dimension 5, we detect new examples. In fact the main result of this paper is the following: Theorem 1.1. Assume that k ≥ 7 is a prime number. If k = p, set q = p. Otherwise, let n be the order of p modulo k, and suppose n ≡ 0 (mod 4). Set q = pn if n is odd, n q = p 2 if n is even. Then the following groups are (2, 3, k)-generated: PSL5 (q), if p ≡ 1 (mod 5); PSU5 (q 2 ), if p ≡ −1 (mod 5); PSU5 (q 4 ), if p ≡ ±2 (mod 5). If k = 7, the only other irreducible Hurwitz subgroups of PSL5 (F) are isomorphic to PSL2 (q), with q as above, except PSL2 (8) and PSL2 (27). We note that our assumptions on n are certainly satisfied if k ≡ 3 (mod 4). For k = 7, in each characteristic p = 0, 5 we obtain one new example of a Hurwitz group. Namely, if p = 7, this is the group PSU5 (74 ). Otherwise, every p = 7 determines n (equal to 1 or 3), hence q, and depending on the residue of p modulo 5, one can pick a Hurwitz group among the three groups in the theorem list. We emphasize that SL5 (q) and SU5 (q 2 ) are not Hurwitz for any q according to [4]. Our proof, which uses the knowledge of the subgroups of SL5 (p m ), is based on results on product of matrices, Scott’s formula, Lang-Steinberg’s theorem and the notion of rigidity. The latter notion is widely explored in the theory of Riemann surfaces and their applications, in particular, to differential equations and Galois theory. Especially a version of the rigidity notion, introduced by John Thompson in [23], plays a prominent role in recent developments of inverse Galois theory.

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365

2. Preliminary results In this paper F denotes an algebraically closed field of characteristic p > 0. Definition 2.1. Let a1 , a2 , a3 ∈ GLn (F). The triple (a1 , a2 , a3 ) is called linearly g g g rigid if a1 a2 a3 = 1 and, for all elements gi in GLn (F) satisfying a1 1 a2 2 a3 3 = 1, gi g gi there exists g ∈ GLn (F) such that ai = ai for i = 1, 2, 3. (Here ai is meant to be gi−1 ai gi ). Given a group H and a representation f : H → GLn (F), let V be the vector space Fn , considered as an H -module, via f . For any subset A of H , define VA as the subspace of fixed points of f (A) and denote by dVA its dimension over F. Define dˆVA in the same way, with respect to the dual representation. Then a special case of Scott’s formula reads Theorem 2.2. Assume that H is generated by x and y. Then y

xy

dVx + dV + dV ≤ n + dVH + dˆVH .

(2.1)

a = According to the notation used in [4], for each a ∈ M = Matn (F), we set dM dim CM (a). Applying formula (2.1) to the action of f (H ) by conjugation on M, we have the following:

Corollary 2.3. Suppose that f : H → GLn (F) is irreducible, and H = x, y. Then y

xy

x + dM + dM ≤ n2 + 2. dM

(2.2)

The following lemma appears as Theorem 2.3 in [21]: Lemma 2.4. Let a, b ∈ GLn (F) and set (ab)−1 = c. Suppose that the group a, b is irreducible and that b c a dM + dM + dM = n2 + 2.

Then the triple (a, b, c) is linearly rigid. In the following, for each power q of p, σ will denote one of the Frobenius automorphisms φq or φq− of GLn (F), defined respectively by: q

(gij ) → (gij )

and

q

(gij ) → (gj i )−1 .

GLn (q) is the subgroup of the fixed points of φq , i.e. GLn (q) = CGLn (F) (φq ). Similarly Un (q 2 ) = CGLn (F) (φq− ). By a well known result of S. Lang and R. Steinberg (see [20]), the map GLn (F) → GLn (F) given by g → g −1 σ (g) is surjective. Corollary 2.5. [19] Let σ be as above and assume that A is a set permuted transitively by GLn (F). Assume further that τ : A → A is a map such that τ (g(a)) = σ (g) (τ (a)) for all g ∈ GLn (F) and all a ∈ A. Then there exists a ∈ A such that τ (a) = a.

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Proof. Let a ∈ A. As GLn (F) acts transitively on A, there exists g ∈ GLn (F) such that g (τ (a)) = a. By Lang-Steinberg’s theorem, there exists x ∈ GLn (F) such that x −1 σ (x) = g. Then x −1 (σ (x) (τ (a))) = a implies σ (x) (τ (a)) = x(a), i.e. τ (x(a)) = x(a). We conclude that a = x(a) is fixed by τ . The following theorem is known and it is used a few times in [15] but it is not stated explicitly. We provide a proof for the reader’s convenience. Theorem 2.6. Let (a1 , a2 , a3 ) be a linearly rigid triple, with ai ∈ GLn (F). For i = 1, 2, 3, let Ci be the conjugacy class of ai and suppose that Ci ∩CGLn (F) (σ ) is nong empty, where σ is as above. Then there exists g ∈ GLn (F) such that ai ∈ CGLn (F) (σ ), for all i = 1, 2, 3. Proof. Set W = GLn (F)3 and make GLn (F) act on W via g g g (w1 , w2 , w3 ) → w1 , w2 , w3 for all (w1 , w2 , w3 ) ∈ W and all g ∈ GLn (F). Define A = {(b1 , b2 , b3 ) ∈ C1 × C2 × C3 | b1 b2 b3 = 1} . Clearly A is invariant under this action and the rigidity of (a1 , a2 , a3 ) is equivalent to the fact that GLn (F) acts transitively on A. Let τ : W → W be the map defined by (w1 , w2 , w3 ) → (σ (w1 ), σ (w2 ), σ (w3 )) . From Ci ∩ CGLn (F) (σ ) = ∅, it follows that σ (Ci ) ⊆ Ci for i ≤ 3. Indeed, if g σ (g) ci ∈ Ci ∩ CGLn (F) (σ ) then σ ci = ci for all g ∈ GLn (F). Thus τ (C1 × C2 × C3 ) ⊆ C1 × C2 × C3 whence also τ (A) ⊆ A. Moreover g g g g g g τ (b1 , b2 , b3 ) = σ (b1 ), σ (b2 ), σ (b3 ) = σ (b1 )σ (g) , σ (b2 )σ (g) , σ (b3 )σ (g) , that is, τ satisfies the assumptions of Corollary 2.5. Hence there is in A a triple fixed by τ . This means that there exist bi ∈ Ci such that b1 b2 b3 = 1 and σ (bi ) = bi . The latter is equivalent to bi ∈ GLn (q) (respectively ∈ Un (q 2 )) for i ≤ 3.

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367

3. Hurwitz groups Lemma 3.1. Consider the matrices x and y in SL5 (F), of respective orders 2 and 3, defined as follows: 1 0 −1 1 −1 0 1 , 1 0 0 0 1 y = x = 0 1 1 0 0 1 0 0 1 0 Moreover, for each prime k ≥ 7, consider z in SL5 (F), of order k, defined as follows. If p = k, 1 0 0 0 0 1 1 0 0 0 z= 0 1 1 0 0 0 0 1 1 0 0 0 0 1 1 If p = k, let ε denote a primitive k-th root of 1 in F and set θj = j + −j for an integer j . For all integers = j such that 1 ≤ , j ≤ k−1 2 define 1 0 −1 1 θ z = z,j = 0 −1 1 θj Assume p = 5 and let 1 = η ∈ F be such that η5 = 1. Then there exist x conjugate to x and y conjugate y such that xy = ηz. Moreover G = x, y is an irreducible subgroup of SL5 (F). Proof. The existence of x and y with the required properties follows from Silva [18, Theorem 2] if p = 7, and from Kurtz [8, Theorem 1.1] if p = 2, 3. As to the irreducibility of G, we note that any abelian epimorphic image of a group generated by two elements whose orders divide 2 and 3 respectively, has order dividing 6. So, assume by contradiction that U is a proper G-invariant subspace of F5 . Then, for each g ∈ G, det(g|U )6 = 1. But ηI5 ∈ G and the restriction of ηI5 to U has determinant of order 5. We conclude that G is irreducible. Remark 3.2. It maybe worth noting that the previous Lemma actually holds also in characteristic 0. Theorem 3.3. Assume p = 5, k ≥ 7 a prime, and let G = x, y be as in Lemma 3.1. If k = p, set q = p. Otherwise, let n be the order of p modulo k, and suppose

368

M. Chiara Tamburini and Alexandre E. Zalesski n

n ≡ 0 (mod 4). Set q = p n if n is odd, q = p 2 if n is even. Then q is the minimum power of p such that: (1) G is conjugate to a subgroup of SL5 (q), if p ≡ 1 (mod 5); (2) G is conjugate to a subgroup of SU5 (q 2 ), if p ≡ −1 (mod 5); (3) G is conjugate to a subgroup of SU5 (q 4 ), if p ≡ ±2 (mod 5). Proof. Recall that G is irreducible. From y

ηz

x dM + dM + dM = 13 + 9 + 5 = 52 + 2 it follows that the triple x, y, (ηz)−1 is linearly rigid by Lemma 2.4. Let us denote by Cx , Cy , Cηz and Cz the conjugacy classes of x, y, ηz and z. From x , y ∈ SL5 (p) it follows that

Cx ∩ SL5 (p m ) = ∅,

Cy ∩ SL5 (p m ) = ∅

for all m ≥ 1.

Moreover, noting that any permutation matrix preserves the hermitian form defined by the identity matrix, and recalling that SL2 (pm ) SU2 (p2m ) preserves a nondegenerate hermitian form, it is easy to see that Cx ∩ SU5 (p2m ) = ∅,

Cy ∩ SU5 (p 2m )

for all m ≥ 1.

From xy = ηz, with o(η) = 5 and o(z) = k, it follows that ηI5 ∈ (xy)k . Thus ηI5 belongs to any conjugate of G. Thus, by Theorem 2.6, G is conjugate to a subgroup of SL5 (p ), for some , precisely when ηI5 ∈ SL5 (p ) and Cz ∩ SL5 (p ) = ∅. Similarly G is conjugate to a subgroup of SU5 (p 2 ), for some , precisely when ηI5 ∈ SU5 (p 2 ) and Cz ∩ SU5 (p 2 ) = ∅. By definition of q, we have GF(q) = GF(p) (θ ) = GF(p) θj . Hence z ∈ SL5 (q). Moreover z ∈ SU5 (q 2 ) ∩ SU5 (q 4 ). This fact, if p = k, is easily seen recalling that SL2 (pm ) SU2 (p2m ) preserves a non-degenerate hermitian form. And, if p = k, one may consider for example the matrix 1 0 0 0 0 −1 1 0 0 0 0 −1 1 0 0 0 −2−1 1 1 0 0 −2−1 1 1 1 conjugate to z, and preserving the hermitian form defined by J = antidiag(1, 1, 1, 1, 1). Our claims follow immediately noting that, as q is supposed to be a power of p with odd exponent, we have q ≡ p (mod 5). Finally q is the minimum power of p for which cases (1), (2) and (3) hold. This is clear if p = k. On the other hand, if p = k, it follows from the facts 2 2 that the characteristic polynomial of z is (t − 1)(t − θ t + 1)(t − θj t + 1) and GF(q) = GF(p) θ + θj , θ θj since we are assuming n ≡ 0 (mod 4).

Classical groups in dimension 5 which are Hurwitz

369

Theorem 3.4. Let k, n and q be defined as in Theorem 3.3. (1) The following groups are (2, 3, k)-generated: PSL5 (q),

if p ≡ 1

(mod 5);

PSU5 (q ), if p ≡ −1

(mod 5);

PSU5 (q ), if p ≡ ±2

(mod 5).

2 4

(2) If k = 7, the only other irreducible Hurwitz subgroups of PSL5 (F) are isomorphic to PSL2 (q) with q as above, except PSL2 (8) and PSL2 (27). Proof. (1) We claim that the group G = x, y of Theorem 3.3 coincides respectively with SL5 (q) in (1), SU5 (q 2 ) in (2), and SU5 (q 4 ) in (3). The centre of G has order 5. In particular G cannot be a simple group and cannot be an orthogonal group. Moreover G is irreducible, as shown in Lemma 3.1. Thus inspection of the subgroups of SL5 (pm ) (see [5] and [24]) gives that G coincides with SL5 (p ) with p ≡ 1 (mod 5), or SU5 (p2 ) with p ≡ −1 (mod 5) or SU5 (p4 ) with p ≡ ±2 (mod 5), for some . Since q is the minimum power of p for which cases (1), (2) and (3) of Theorem 3.3 hold, and q ≡ p (mod 5), we conclude p = q. (2) Let H be the linear preimage of an irreducible (2, 3, 7)-generated subgroup of PSL5 (F). Then, up to conjugation, we may assume H = x, y with x 2 = y 3 = I and either (i) (xy)7 = ηI5 where η ∈ F has order 5, or (ii) (xy)7 = 1.

y

xy

x ≥ 13, d In both cases dM M ≥ 9 and dM ≥ 5. On the other hand, Scott’s formula (2.2) and irreducibility force the condition y

xy

x + dM + dM ≤ 25 + 2. dM

Thus x must be conjugate to x and y must be conjugate to y . Moreover, if p = 7, z = (xy)5 must have five different eigenvalues and determinant 1. Thus z can only be conjugate to one of the matrices z,i , as in Lemma 3.1, with k = 7. And, if p = 7, z must be conjugate to the Jordan block J5 . In case (i), by the rigidity condition, H is conjugate to the group G of Theorem 3.3. In case (ii), by the rigidity condition, the group H can belong to at most 3 conjugacy classes. Let us recall that the groups PSL2 (p m ) have an embedding into SL5 (F), arising from the action on homogeneous polynomials of degree 4 in 2 variables. In this action, the three non-conjugate (2,3,7) generating triples of PSL2 (q), when Hurwitz according to Macbeath’s Theorem, remain non-conjugate in SL5 (F). This embedding is irreducible precisely when p ≥ 5. Thus, in case (ii) with p = 2, 3, H coincides with one of the groups in the statement. Finally, if p = 2, 3, H is finite by 2.6 and H preserves a symmetric bilinear form by [4]. It follows p = 3 as groups in characteristic 2 preserving symmetric bilinear forms are reducible. Thus it could only be an irreducible subgroup of 5 (3m ) PSp4 (3m ), for some m. Moreover, by [4], H cannot be SL5 (3a ) or SU5 (3a ), for any a ≤ m.

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Hence, inspection of the irreducible subgroups of 5 (3m ), H should be isomorphic to 5 (3a ) for some a. But these groups are not even (2, 3)-generated by [9].

References [1]

J. Cohen, On non-Hurwitz groups and non-congruence subgroups of the modular group, Glasgow Math. J. 22 (1981) 1–7.

[2]

M. Conder, Generators for alternating and symmetric groups, J. London Math. Soc. (2) 22 (1980), 75–86.

[3]

M. D. E. Conder, R. A. Wilson and A. J. Woldar, The symmetric genus of sporadic groups, Proc. Amer. Math. Soc. 116 (1992), 653–663.

[4]

L. Di Martino, C. Tamburini and A. Zalesskii, On Hurwitz groups of low rank, Comm. in Algebra 28 (2000), 5383–5404.

[5]

L. Di Martino and A. Wagner, The irreducible subgroups of PSL5 (q), where q is odd, Results Math. 2 (1979), 54–61.

[6]

B. Everitt, Alternating quotients of Fuchsian groups, J. Algebra 223 (2000), 457–476.

[7]

G. A. Jones, Ree groups and Riemann surfaces, J. Algebra 165 (1994), 41–62.

[8]

C. Kurtz, On the product of diagonal conjugacy classes, Comm. Algebra 29 (2001), 769–779.

[9]

M. W. Liebeck and A. Shalev, Classical groups, probabilistic methods, and the (2, 3)generation problem, Ann. of Math. 144 (1996), 77–125.

[10] A. Lucchini and M. C. Tamburini, Classical groups of large rank as Hurwitz groups, J. Algebra 219 (1999), 531–546. [11] A. Lucchini, M. C. Tamburini and J. S. Wilson, Hurwitz groups of large rank, J. London Math. Soc. (2) 61 (2000), 81–92. [12] A. M. Macbeath, Generators of the linear fractional groups, Proc. Symp. Pure Math. 12 (1969), 14–32. [13] G. Malle, Hurwitz groups and G2 (q), Canad. Math. Bull. 33 (1990), 349-357. [14] G. Malle, Small rank exceptional Hurwitz groups, Groups of Lie type and their geometries, in Groups of Lie Type and Their Geometries (Como, 1993), ed. by W. M. Kantor and L. Di Martino, London Math. Soc. Lecture Note Ser. 207, Cambridge University Press, Cambridge 1995, 173–183. [15] G. Malle and B. H. Matzat, Inverse Galois Theory, Springer Monogr. Math., SpringerVerlag, Berlin 1999. [16] C. H. Sah, Groups related to compact Riemann surfaces, Acta Math. 123 (1969), 13–42. [17] L. L. Scott, Matrices and cohomology, Ann. of Math. 105 (1977), 473–492. [18] F. C. Silva, The eigenvalues of the product of matrices with prescribed similarity classes, Linear and Multilinear Algebra 34 (1993), 269–277.

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[19] T. A. Springer and R. Steinberg, Conjugacy classes, in A. Borel, R. Carter, C. W. Curtis, N. Iwahori, Seminar on Algebraic Groups and Related Finite Groups, Lecture Notes in Math. 131, Springer-Verlag, Berlin 1970; corr. 2nd printing 1986. [20] R. Steinberg, Endomorphisms of linear algebraic groups, Mem. Amer. Math. Soc. 80, Amer. Math. Soc., Providence, RI 1968. [21] K. Strambach and H. Völklein, On linearly rigid tuples, J. Reine Angew. Math., 510 (1999), 57–62. [22] C. Tamburini and M. Vsemirnov, Hurwitz groups and Hurwitz generation, Survey article, to appear. [23] J. Thompson, Some finite groups which appear as Gal(L/K), where K ⊆ Q(µn ), J. Algebra 89 (1984), 437–499. [24] A. Wagner, The subgroups of PSL5 (2a ), Results Math. 1 (1978), 207–226. [25] R. A. Wilson, The Monster is a Hurwitz group, J. Group Theory 4 (2001), 367-374. M. C. Tamburini, Dipartimento di Matematica e Fisica, Università Cattolica del Sacro Cuore, Via Musei 41, 25121 Brescia, Italy E-mail: [email protected] A. E. Zalesski, School of Mathematics, University of East Anglia, Norwich NR4 7TJ, UK E-mail: [email protected]

A classification of Weyl groups as finite {3, 4}-transposition groups Franz G. Timmesfeld

1. Introduction A set D of involutions of the finite group G is called a set of {3, 4}-transpositions of G, if the following hold: (1) G = D and D g ⊆ D for all g ∈ G. (2) If d, e ∈ D then o(de) ≤ 4. Finite groups with a set D of {3, 4}-transpositions and containing no solvable normal subgroup have been classified under the additional condition If d, e ∈ D with o(de) = 4, then (de)2 ∈ D.

(+)

(See [2] and [3].) Without this condition a classification has never been tried, since there are many examples (for example the Baby Monster). But using the classification and known properties of involutions in the automorphism groups of the simple groups this should now be no big problem. In this note we go into an other direction, i.e., we look for conditions under which such a {3, 4} transposition group G is a Weyl group, which means we must deal with groups which are not nearly simple. We prove: Theorem 1.1. Let G be a finite group generated by a set D of {3, 4}-transpositions satisfying: (a) If a, b, c ∈ D and X = a, b, c, then X is an epimorphic image of a Weyl group and a, b and c are images of reflections. (b) If ai , x ∈ D, i = 1, . . . , 4 satisfying (i) o(ai x) = 3 for i = 1, . . . , 4 and (ii) o(ai aj ) ≤ 2 for i, j ≤ 4, then X = x, ai | i = 1, . . . , 4 is an epimorphic image of W (D4 ) and x and the ai are images of reflections. Then G is an epimorphic image of a Weyl group W and D is the image of the set of reflections of W .

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Franz G. Timmesfeld

Notice that the Weyl group W of the theorem does not need to be irreducible and that all Weyl groups, which do not have a direct factor isomorphic to W (G2 ), do occur. In fact the group G of the theorem can occur in different disguises as an image of a Weyl group. (For example D8 as W (C2 ), but also as an image of W (C2 ) × W (A1 ), in which case all involutions of D8 are in D.) The main reason for proving the above theorem is, that it will be used to give a presentation type classification of arbitrary ‘Lie-type groups’ (In the sense of [Ti2, II §5]) similar to theorem 2 of [5], in which such a {3, 4}-transposition group plays the role of a Weyl group. (For the proof of theorem 2 of [5] we had to give a classification of W (A ), W (D ) and W (E ) as finite 3-transposition groups satisfying (a) and (b), see [Ti3, (6.7)].) The reader will notice that for most parts of the proof we just rely on condition (a). Hence it should not be difficult to give a classification of {3, 4}-transposition groups G satisfying condition (a) with O2 (G) = 1. (Without condition (b) there exist additional cases in the conclusion of the theorem, for example the symplectic and orthogonal groups over GF(2).) But since for the application I have in mind we must allow O2 (G) = 1, I introduced condition (b).

2. Preliminary results Let in this section D be a set of {3, 4}-transpositions of the finite group G satisfying conditions (a) and (b) of the theorem. We will often use pictorial notation, i.e. we will write for d, e ∈ D: d

e

e

e

d

e

e

e

d

e

e

iff o(de) = 2 iff o(de) = 3 iff o(de) = 4

e

Let F (D) be the graph with vertex set D and edges pairs e, d ∈ D with o(ed) = 3. Lemma 2.1.

(1) There exist no a, b, c ∈ D satisfying

a

b e T e T e c

or

a

b e TT TT e c e

A classification of Weyl groups as finite {3, 4}-transposition groups

(2) If a, b, c ∈ D with

a

c

b

e

375

e

e

then c = a b and a, b, c D8 . (3) If a, b, c ∈ D with

a

e eb JJ

J

e

J c

then a, b, c is isomorphic to W (C3 ) or W (A3 ) ( (4) If a, b, c ∈ D with

a

4 !).

e eb

J J e

c

then a, b, c is isomorphic to

4

or

3.

Proof. Let X = a, b, c. Then by condition (a) of the theorem X is an image of a Weyl group W of rank ≤ 3 and a, b, c are images of reflections. Now there is no rank 3 Weyl group with 3 reflections satisfying (1), so (1) is impossible. In case (2) W is irreducible. Hence it is easy to see that rank W = 3 and then that W W (C2 ). Now (2) follows immediately. In (3) W is also irreducible and thus W W (C3 ), which implies (3). Finally in (4) W W (A3 ) or W (A2 ), since W (C3 ) 4 ×ZZ2 is not generated by 3 reflections satisfying (4). This proves (2.1). Lemma 2.2. Let E ⊆ D be a connectivity component of F (D) with |E| > 1. Then E is a class of 3-transpositions of R = E and E g = E for all g ∈ G. Proof. Suppose false and under all pairs e, f ∈ E with o(ef ) = 4 choose e, f so that e and f are connected by a path of minimal length in F (D) (and whence in F (E)!). Let ei ∈ E, i = o, . . . , n such that e0 = e, en = f and (e0 , e1 , . . . , en ) is such a minimal path from e to f in F (D). Then by (2.1) (1) n > 2. By the minimality of d(e, f ) (d denotes the distance in F (D)) we have o(ei ej ) = 4 for all i, j ≤ n with (i, j ) = (o, n). Moreover by the minimality of the path from e to f we have o(ei ej ) = 3 for all i, j ≤ n with j = i ± 1. Hence o(ei ej ) = 2 for all i, j ≤ n with j = i ± 1 and (i, j ) = (o, n); i.e. we have

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Franz G. Timmesfeld

e2 e1 e

e

e

· e

·

·

·

·

een−2 \\ en−1 e \\ f e

.

Let a be the highest reflection in Y = e1 , . . . , en−1 n . Then o(ea) = 3, since H = e, Y n+1 and thus CH (e) = e × e2 , . . . , en−1 . By the same argument also o(f a) = 3, a contradiction to (2.1) (1). This shows that E is a class of 3-transpositions of R. It remains to show E g ⊆ E for all g ∈ G. Since G = D it suffices to show E d ⊆ E for all d ∈ D. If now d ∈ D centralizes an element e of E then clearly E ∩ E d ⊇ {e} and thus E = E d . So we may assume o(ed) = 4 for all e ∈ E. But then, if e, f ∈ E with o(ef ) = 3, we obtain e e ef JJ

J

e

. J d

Hence by (2.1)(3) o(def ) = 2, a contradiction to ef ∈ E. Corollary 2.3. Let E and R be as in (2.2). Then R is a factor group of W (A ), ≥ 2, W (D ), ≥ 4 or W (E ), 6 ≤ ≤ 8 and E is the image of the reflections. Proof. This is a direct consequence of (2.2) and theorem (6.7) of [5]. Lemma 2.4. Let E and R be as in (2.2) and suppose R W (A ) +1 , ≥ 2. Let d ∈ D − E. Then either d ∈ C(R) or R 4 W (A3 ), R, d is a factor group of W (C3 ) 4 ×ZZ2 and E ∪ d R is the image of the reflections of W (C3 ). Proof. Pick d ∈ D − E. Then, since d normalizes E, d induces even in case = 5 an inner automorphism on R. Hence z = td ∈ C(R) for some t ∈ R. Now o(tt r ) = o(dzd r z) = o(dd r ) ≤ 4 for each r ∈ R. If now d is an isolated vertex of F (D), then o(tt r ) = o(dd r ) ∈ {2, 4} for each r ∈ R. Hence by Baer’stheorem, see [A, (39.6)] t ∈ O2 (R). Hence either t= 1 and d = z ∈ C(R) or R 4 . In the second case obviously R, d = R, z ×Z Z or 4 4 and the conclusion of (2.4) holds. So we may assume that d is not isolated in F (D) and thus we have by (2.2) that o(tt r ) ≤ 3 for each r ∈ R. Hence in case = 5 we obtain t ∈ E (or t = 1 which is to show). But then o(t r d) = o(t r tz) = 6 for some r ∈ R, a contradiction to t r , d ∈ D. Thus we may assume that = 5, t is a conjugate of (12)(34)(56) and E is the class of transpositions of R. But then, if we let e = (12), f = (23) and t r = (14)(25)(36), we obtain

A classification of Weyl groups as finite {3, 4}-transposition groups e

377

e ef JJ

J

e

J dr

and e, f, d r 3 2 , a contradiction to (2.1)(3). Thus in any case t = 1 and d = z ∈ C(R), which proves (2.4). Lemma 2.5. Let E and R be as in (2.2) and suppose R is isomorphic to W (D ) or W (D )/Z(W (D )). Pick d ∈ D − E. Then one of the following hold: (1) d ∈ C(R). (2) d, R W (C ) or W (C )/Z(W (C )) and E is (the image of) the class of short reflections and d R the (image of) the class of long reflections of W (C ). Proof. Suppose first > 4. Then by (2.4) applied to G/O2 (R) we have [R, d] ≤ O2 (R). Hence either (1) holds or there exists an e ∈ E with e = ed ∈ eO2 (R) ∩ E = eO2 (R) . Thus there exists e1 , . . . , ee ∈ E with e−1 e1

e

e−2

e2

e H HH

e …

e

R = e1 , . . . , e and ed = e−1 . e

e

Now by (2.1)(2) we obtain o(dei ) = 2 for i ≤ − 3. Moreover by (2.1)(4) either o(e−2 d) = 2 or 4. In the first case we get e1

e2

e

e …

e−2

e−1

e

e

d

e e

−1 , we and (2) holds. So we may assume o(e−2 d) = 4. But then, setting f = e−2 obtain by (2.1) (3)

e1

e

e2

e …

e−3

e

f

e

e−1

e

d

e

and again (2) holds. Next assume = 4. Then there exist exactly three normal 2-subgroups Ni , i = 1, 2, 3 of R with R/Ni 4 . Thus one of these normal 2-subgroups, say N1 , must be d-invariant. Set G0 = R, d and D0 = D ∩ G0 . Then D0 is a set of {3, 4}-transpositions of G0 satisfying conditions (a) and (b) of the theorem. Hence applying (2.4) to G0 /N1 we obtain [R, d] ≤ N1 . Now arguing in exactly the same manner as in case > 4 we see that either [R, d] = 1 or R, d W (C4 ) or W (C4 )/Z(W (C4 )) and (2) holds.

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Lemma 2.6. Let E and R be as in (2.2) and suppose R is isomorphic to a factor group of W (E ), 6 ≤ ≤ 8. Then D − E ⊆ C(R). Proof. Pick d ∈ D − E. Then d induces an inner automorphism on R. Since E is the only class of 3-transpositions of R satisfying condition (a) of the theorem we obtain either d ∈ C(R) or de ∈ C(R) for some e ∈ E (W (E6 ) U4 (2) contains another class of 3-transpositions. But this class contains 3 involutions satisfying e T e T e which generate a group of order 33 · 2!). But in the second case we obtain as in (2.4) o(deg ) = 6 for some g ∈ R, a contradiction. Lemma 2.7. Let E and R be as in (2.2) and suppose that R W (D ) (resp. W (D )/Z(W (D )). Let d ∈ D − E such that Y = R, d W (C ) or W (C )/Z(W (C ). Then for each x ∈ D − Y one of the following holds: (1) x ∈ C(Y ). (2) = 4, o(xd) = 3 and H = Y, x W (F4 ) or W (F4 )/Z(W (F4 )). Moreover H ∩ D consists of the long and short reflections of W (F4 ). Proof. Suppose first x ∈ C(R). Then o(xd) = 2 or 4 and (2.1)(2) shows o(xd) = 2. Hence (1) holds. So we may by (2.4) assume R, x W (C ) or W (C )/Z(W (C )). Hence if = 4 then d and x centralize O2 (R) and R/O2 (R). But then (2.1)(1) and (2) imply either x ∈ Y or x ∈ C(Y ). Finally if = 4 the same argument shows that d, x permutes the three normal 2-subgroups N1 , N2 and N3 of R with R/Ni 4 as a 3 or x ∈ C(Y ) resp. x ∈ Y . Hence in the first case d, x induces the full group of diagram automorphisms on R W (D4 ) and it is easy to see that Y, x W (F4 ) or W (F4 )/Z(W (F4 )). Since 3 | o(dx), this shows that (2) holds.

3. Proof of the theorem We prove the theorem by induction on |D|. First assume D ∩ Z(G) = ∅. Pick d ∈ D ∩Z(G) and set D1 = D −{d}, G1 = D1 . Then D1 is a set of {3, 4}-transpositions of G1 satisfying the hypothesis of the theorem. So, by induction assumption, there exists a finite Weyl group W1 with set of reflections R1 and a surjective homomorphism ϕ1 : W1 → G1 with ϕ1 (R1 ) = D1 . Hence W1 × t, t ZZ2 , is a Weyl group of type W1 × A1 and it is easy to see that there exists a surjective homomorphism

A classification of Weyl groups as finite {3, 4}-transposition groups

379

ϕ : W1 × t → G with ϕ|W1 = ϕ1 and ϕ(t) = d. (To define ϕ one has to distinguish between the cases d ∈ G1 and d ∈ G1 !) So we may assume Z(G)∩D = ∅. Next assume that there are two isolated vertices a, b of F (D) with o(ab) = 4. Let A = D ∩ a, b and D1 = D − A. If c ∈ D1 , then by (2.1)(1) and (2) c commutes with a and b. Hence G1 = D1 centralizes a, b and moreover, again by induction assumption, we may assume that there exists a finite Weyl group W1 with set of reflection R1 and a surjective homomorphism ϕ1 ·W1 → G1 with ϕ1 (R1 ) = D1 . Thus it is easy to show that G is an image of a Weyl group of type W1 × C2 resp. of type W1 × C2 × A1 with D the image of the reflections. Hence we may assume that L is abelian, where L is the set of isolated vertices of F (D). Let, as in §2, Ei , i = 1, . . . , n be the connectivity components of F (D) of cardinality > 1. We distinguish between the cases: (1) L = ∅. (2) L = ∅. ˙ . . . ∪E ˙ n . If now [Ei , Ej ] = 1 for i = j , then D is a set of In case (1) D = E1 ∪ 3-transpositions of G and it is easy to see that the theorem is a consequence of (2.3). (One just has to look how the Z(Gi ), Gi = Ei are amalgamated!) So assume [E1 , E2 ] = 1. Then by (2.4) - (2.6) H = E1 , e is an image of W (C ), G1 = E1 an image of W (D ) and E1 ∪ eH is the image of the reflections of W (C ) for each e ∈ E2 . Now E2 = eH . Hence by (2.7) we obtain for f ∈ E2 − eH that Y = H, f is an image of W (F4 ) and o(ef ) = 3. Now |Y ∩ E2 | = 12 = |E1 | since G1 W (D4 ). Reversing the roles of E1 and E2 this implies E2 ⊆ Y and thus Y = E1 , E2 and E1 ∪ E2 is the image of the reflections of W (F4 ). We now claim [Ei , Ej ] = 1 for i ≤ 2 and j > 2. Namely, if a ∈ E1 , b ∈ E2 with o(ab) = 4, then by (2.1)(1) and (2) each c ∈ Ej , j > 2 commutes with a and b. Since each Ei is a conjugacy class in Ei and a normal set in G, this proves our claim. So if we set D1 = E1 ∪ E2 , D2 = E3 ∪ · · · ∪ En and Xi = Di for i = 1, 2, then [X1 , X2 ] = 1, X1 is an image of W (F4 ), D1 is the image of the reflections of W (F4 ) and D2 satisfies the hypotheses (a) and (b) of the theorem in X2 . Hence applying the induction assumption to X2 it follows easily that the theorem holds. Finally assume that L = ∅. Since L ∩ Z(G) = ∅ and since L is elementary abelian, there exists for each xi ∈ L a connectivity component Ei of F (D) with [Ei , xi ] = 1. Let Yi = Ei , xi and Li = xiYi . Then by (2.5) Yi is an image of W (Ci ) and Ei ∪ Li is the set of reflections of Yi . By (2.7) D0 ⊆ C(Yi ) for D0 = D − (Ei ∪ Li ), since Li consists of isolated vertices of F (D). Hence applying the induction assumption to D0 and D0 , this shows again that the theorem holds. 2

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References [1] M. Aschbacher, Finite group theory, Cambridge University Press, Cambridge 1986. [2] B. Fischer, Finite groups generated by 3-transpositions, I, Invent. Math. 13 (1971), 232–246. [3] F. G. Timmesfeld, A characterization of Chevalley and Steinberg groups over F2 , Geom. Dedicata 1 (1973), 269–321. [4] F. G. Timmesfeld, Abstract Root Subgroups and Simple Groups of Lie-Type, Monogr. Math. 95, Birkhäuser Verlag, Basel 2001. [5] F. G. Timmesfeld, Presentations for certain Chevalley-groups, Geom. Dedicata 73 (1998), 85–117. F. G. Timmesfeld, Mathematisches Institut, Arndtstraße 2, 35392 Giessen, Germany E-mail: [email protected]

Powerful subgroups of 2-groups Lawrence E. Wilson

Abstract. We improve the bound on the index of a powerful subgroup of a 2-group of rank r. We also prove that certain dimension subgroups are powerful under a natural hypothesis. 2000 Mathematics Subject Classification: 20

1. Introduction When Lubotzky and Mann introduced powerful p-groups in [LM], one immediate goal was to prove results about p-groups in which every subgroup can be generated by at most r elements, called a p-group of rank r. Their primary technique was to prove that such groups have a characteristic powerful subgroup of index bounded by a function of p and r. If we let λ(r) be an integer such that 2λ(r)−1 < r ≤ 2λ(r) , then Theorem 1.14 of [LM] states that a p-group of rank r with p odd has a characteristic powerful subgroup of index at most prλ(r) . In the case of p = 2 the result is weaker. Theorem 4.1.14 of the same paper states that there is a characteristic powerful subgroup of index at most 2r(λ(r)+1) . The primary result of this paper is the following improvement on this bound for 2-groups. Theorem A. Let G be a finite 2-group such that every characteristic subgroup of G can be generated by at most r elements. If λ(r) is defined by 2λ(r)−1 < r ≤ 2λ(r) then λ(r) G has a characteristic powerful subgroup of index at most 2r(λ(r)+1)−(2 −r) . Lubotzky and Mann use their bound to prove (Proposition 4.2.6) that the order of a 2-group of rank r is at most 2r(λ(r)+1)+re where the exponent of the group is 2e . (We note that they use m where we use λ(r) and that there is a typo in the proposition; the m2 should be m.) The same proposition also contains the fact that the derived length of the group is at most λ(e + 1) + λ(r) + 1. Using their proof with our bound does not improve this bound for the derived length; to do so the bound would need to be reduced to at most 2rλ(r) . However, substituting our bound leads to the following improvement in the bound on the order of a 2-group of rank r. Corollary B. Let G and λ(r) be as in T heoremA and let 2e be the exponent of G. λ(r) Then |G| ≤ 2r(λ(r)+1)−(2 −r)+re .

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The proofs of some important results on p-groups involve bounding the rank and deducing the existence of a characteristic powerful subgroup of bounded index. For example, the derived length of a finite p-group is bounded in terms of the order and number of fixed points of a p-automorphism which acts on the group. An early step in the proof is to deduce a bound on the rank of this group and from this that a bounded term of the derived series is contained in a powerful subgroup. The proof then proceeds by considering the special case of powerful p-groups. Again, our improvement is not sufficient to improve this bound on derived length.

2. Notation and tools We will be primarily concerned with the dimension subgroups of a given p-group. The dimension subgroup series is the fastest descending series of G beginning at D1 (G) = G such that [Di (G), Dj (G)] ≤ Di+j (G) and Di (G)p ≤ Dpi (G). Lazard j found a closed form for the dimension subgroups, Dk (G) = ipj ≥k γi (G)p . Let us pause now to review the notation. j For a p-group G and a positive integer j the subgroup Gp is the subgroup generj ated by the elements g p for g ∈ G. Commonly, this subgroup will contain elements j not of the form g p . Another important series in the group G is the lower central series. This is the series defined by γ1 (G) = G and γi+1 (G) = [γi (G) , G]. For subgroups H and K of G we recursively define [H, K; 0] = H and [H, K; + 1] = [[H, K; ], K]. Therefore, γ+1 (G) = [G, G; ]. If p is odd then the p-group G is powerful if γ2 (G) ≤ Gp . A 2-group is powerful if γ2 (G) ≤ G4 . As in Exercise 2.4.i of [DdMS], this condition is equivalent to the condition that γ2 (G) ≤ (G2 )2 . A normal subgroup N of G is said to be powerfully embedded in G if [N, G] ≤ N p for odd primes or if [N, G] ≤ N 4 if p = 2. Again, for p = 2 it suffices to prove that [N, G] ≤ (N 2 )2 . The following lemma was known to Zassenhaus and Jennings. It is a consequence of Lazard’s closed formula for the dimension subgroups and also a consequence of Proposition 11.12 of [DdMS]. We adopt the notation that n∗ = n/p is the least integer such that pn∗ ≥ n. Lemma 2.1. If G is a finite p-group then Dn (G) = γn (G) Dn∗ (G)p . Many results on commutators of dimension subgroups are known. Each is essentially a consequence of the following result of P. Hall, [Ha]. Lemma 2.2. For any group G, elements x and y of G, prime p, and positive integer k the following relations hold: k k k k k−i (1) (xy)p ≡ x p y p modulo γ2 (x, y )p · ki=1 γpi (x, y )p , k k k k−i (2) [x p , y] ≡ [x, y]p modulo γ2 (x, [x, y] )p · ki=1 γpi (x, [x, y] )p .

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We need the following result of [W]. Lemma 2.3. Let G be a finite p-group. Then for positive integers a ≥ b, [Da (G), Db (G)] ≤ γa+b (G) γa ∗ +b (G)p Da+p2 b (G)Dpa+pb (G) Lemmas 2.3 and 2.1 are used to prove the following basic result. In the proof, we first adopt the standard practice of writing Di for Di (G) and i for γi (G). Lemma 2.4. [Di (G), Di (G)] ≤ γ2i (G) γi+i ∗ (G)2 (Di (G)2 )2 for all 2-groups G and all positive integers i. 2 Proof. Lemma 2.3 implies that [Di , Di ] ≤ 2i i+i ∗ D4i . Lemma 2.1 implies that 2 2 D4i ≤ 4i D2i and that D2i ≤ 2i Di . Hence D4i ≤ 2i (Di2 )2 . Substituting this for 2 2 2 D4i in the conclusion from Lemma 2.3, we find that [Di , Di ] ≤ 2i i+i ∗ (Di ) as desired.

A very important tool in this study is Theorem 2.3 (1) of [SS]. Theorem 2.5. If G is a finite p-group then [Dk (G), G; ] =

γi+ (G)p

j

ip j ≥k

Lastly, we include a very basic lemma which will be of great help. Lemma 2.6. Let M and N be normal subgroups of a p-group G and suppose N ≤ N p [N, G]M. Then N ≤ M.

3. Powerful and potent dimension subgroups Chapter 11 of [DdMS] includes a proof of Lazard’s result that if a finitely generated h pro-p group G satisfies γm (G) ≤ Gp with m < p h then G is p-adic analytic. The technique used there is to conclude that Dn (G) = Dn+1 (G) for some n (specifically, n = m if p does not divide m and n = m + 1 otherwise). Lazard’s result is proven by finding that certain subgroups of G are powerful. One way to do this is to use the following theorem of David Riley, Theorem 11.5 of [DdMS]. Theorem 3.1. If G is a finitely generated pro-p group and Dn (G) = Dn+1 (G) then Di (G) is powerful for i ≥ n − n+1 p if p is odd or for i ≥ n for p = 2. We will prove that more dimension subgroups are powerful under the stronger hypothesis of Lazard’s result. Actually, we will be working with the slightly weaker hypothesis that γm (G) ≤ D2h (G). A similar result for odd primes can be found in [W]. We assume that h ≥ 2 as m < 2 produces an assumption met only by the trivial group.

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Theorem 3.2. Let G be a finite 2-group such that γm (G) ≤ D2h (G). (1) If m < 2h−1 + 2h−2 then Di (G) is powerful if i ≥ (2m − 1)/3. (2) If 2h−1 + 2h−2 ≤ m < 2h then Di (G) is powerful if i ≥ 2(m − 2h−1 ). Proof. The proof of this result will be divided into two cases as in the statement of the theorem. Consider first the case that m < 2h−1 + 2h−2 and suppose i ≥ (2m − 1)/3. 2 2 2 Lemma 2.4 implies that [Di , Di ] ≤ 2i i+i ∗ (Di ) . Note that as i ≥ (2m − 1)/3 we have that 2i ≥ (4m − 2)/3 and hence 2i ≥ m. Therefore, 2i ≤ [D2h , G; 2i − m] = 2h +2i−m 22h−1 +2i−m

h

j

22h−j +2i−m

j =2 h

≤

2h +2i−m 22h−1 +2i−m

=

2h +2i−m 22h−1 +2i−m D24h−2 +2i−m

j =2

(Theorem 2.5)

D24h−2 +2j −2 (2i−m)

As m < 2h , Lemma 2.6 implies that 2i ≤ 22h−1 +2i−m D24h−2 +2i−m . We claim that 2h−1 + 2i − m ≥ i + i ∗ . As m ≤ 2h−1 + 2h−2 − 1, we have that 4m ≤ 2h+1 + 2h − 4 = 3 · 2h − 4. Hence 6m − 3 · 2h + 3 ≤ 2m − 1 and thus 2m − 2h + 1 ≤ (2m − 1)/3 ≤ i. Thus 3i + 1 ≤ 2h + 4i − 2m and 2i ∗ ≤ i + 1 so 2i + 2i ∗ ≤ 2h + 4i − 2m and therefore i + i ∗ ≤ 2h−1 + 2i − m. We also claim that 2h−2 + 2i − m ≥ i. As m + 1 ≤ 2h−1 + 2h−2 = 3 · 2h−2 , we have that 3m − 3 · 2h−2 ≤ 2m − 1 ≤ 3i. Therefore i ≥ m − 2h−2 and hence 2h−2 + 2i − m ≥ i. 2 4 2 2 2 We have now proven that 2i ≤ i+i ∗ Di and hence [Di , Di ] ≤ i+i ∗ (Di ) . We ∗ now wish to argue that i + i ≥ m. We take cases on the residue of m mod 3. If m = 3 then i ≥ (6−1)/3 and hence i ≥ 2. Therefore, i ∗ ≥ and i +i ∗ ≥ m. If m = 3 + 1 then i ≥ (6 + 2)/3 and hence i ≥ 2 + 1. Therefore i ∗ ≥ + 1 and so i + i ∗ ≥ 3 + 2 > m. If m = 3 + 2 then i ≥ (6 + 3)/3 = 2 + 1 and so i ∗ ≥ + 1. Therefore i + i ∗ ≥ 3 + 2 = m. Therefore, i+i ∗ ≤ [D2h , G; i + i ∗ − m] = 2h +i+i ∗ −m ≤ 2h +i+i ∗ −m

h j =1 h j =1

j

22h−j +i+i ∗ −m

(Theorem 2.5)

D22h−1 +2j −1 (i+i ∗ −m) = 2h +i+i ∗ −m D22h−1 +i+i ∗ −m

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As m < 2h , Lemma 2.6 implies that i+i ∗ ≤ D22h−1 +i+i ∗ −m . We claim that 2h−1 + i + i ∗ − m ≥ i which is equivalent to i ∗ ≥ m − 2h−1 . As 2i ∗ ≥ i it suffices to argue that i ≥ 2m − 2h . However, we have already shown that i ≥ 2m − 2h + 1. Therefore, we have proven that i+i ∗ ≤ Di2 and therefore [Di , Di ] ≤ (Di2 )2 as desired, completing the proof of (1). Assume now that 2h−1 + 2h−2 ≤ m < 2h and let i ≥ 2(m − 2h−1 ). Lemma 2.4 2 2 2 implies that [Di , Di ] ≤ 2i i+i ∗ (Di ) . We have that 2i ≥ 4m − 2h+1 = m + 3m − 2h+1 ≥ m + 9 · 2h−2 − 2h+1 = m + 2h+1 + 2h−2 . Therefore, 2i ≤ [D2h , G; 2i − m] = 2h +2i−m 22h−1 +2i−m ≤ 2h +2i−m 22h−1 +2i−m =

h

j

22h−j +2i−m

j =2 h j =2

(Theorem 2.5)

D24h−2 +2j −2 (2i−m)

2h +2i−m 22h−1 +2i−m D24h−2 +2i−m

As m < 2h , Lemma 2.6 implies that 2i ≤ 22h−1 +2i−m D24h−2 +2i−m . We claim that 2h−1 + 2i − m ≥ i + i ∗ . If i > 2(m − 2h−1 ) then i ≥ 2(m − 2h−1 ) + 1 and hence 2i ≥ 2(m − 2h−1 ) + i + 1. As 2i ∗ ≤ i + 1 we have that 2i ≥ 2(m − 2h−1 ) + 2i ∗ and hence i ≥ m − 2h−1 + i ∗ . Therefore 2h−1 + 2i − m ≥ i + i ∗ . If i = 2(m − 2h−1 ) then i ∗ = m−2h−1 and i +i ∗ = 3(m−2h−1 ) = 4(m−2h−1 )−m+2h−1 = 2h−1 +2i −m. We also claim that 2h−2 + 2i − m ≥ i which is equivalent to i ≥ m − 2h−2 . As m ≥ 2h−1 + 2h−2 we have that m − 2h ≥ −2h−2 . We have assumed that i ≥ 2 4 2(m − 2h−1 ) = m + m − 2h ≥ m − 2h−2 . Therefore, we have that 2i ≤ i+i ∗ Di 2 2 2 and hence [Di , Di ] ≤ i+i ∗ (Di ) . As i ≥ 2(m−2h−1 ) we have that i ∗ ≥ m−2h−1 and hence i +i ∗ ≥ 3m−3·2h−1 = m + 2m − 3 · 2h−1 . As m ≥ 3 · 2h−2 we know that 2m ≥ 3 · 2h−1 and hence i + i ∗ ≥ m. Therefore, i+i ∗ ≤ [D2h , G; i + i ∗ − m] = 2h +i+i ∗ −m ≤ 2h +i+i ∗ −m

h j =1 h j =1

j

22h−j +i+i ∗ −m

(Theorem 2.5)

D22h−1 +2j −1 (i+i ∗ −m) = 2h +i+i ∗ −m D22h−1 +i+i ∗ −m

As m < 2h , Lemma 2.6 implies that i+i ∗ ≤ D22h−1 +i+i ∗ −m . We claim that 2h−1 + i + i ∗ − m ≥ i which is equivalent to i ∗ ≥ m − 2h−1 . As 2i ∗ ≥ i this

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Lawrence E. Wilson

follows from the assumption that i ≥ 2(m − 2h−1 ). Therefore i+i ∗ ≤ Di2 and hence [Di , Di ] ≤ (Di2 )2 as desired. This completes the proof of Theorem 3.2.

4. Proof of the main result We can now prove Theorem A. Recall that we assume that G is a 2-group of rank r and that λ(r) is determined by 2λ(r)−1 < r ≤ 2λ(r) . We wish to prove that G has a λ(r) powerful subgroup of index at most 2r(λ(r)+1)−(2 −r) . Proof of Theorem A. Notice that we make no claim beyond the known results if r = 2λ(r) . We will assume that r < 2λ(r) . Let us write λ = λ(r). As [Di , Di ]Di2 ≤ D2i each Di /D2i is elementary abelian and hence has size at most 2r . Therefore, |G : D2λ | ≤ 2rλ . We will find a characteristic powerful subgroup λ P of G contained in D2λ with |D2λ : P | ≤ 22r−2 and this will complete the proof. We first note that D2λ /D2λ+1 is elementary abelian and hence of size at most 2r . Therefore, [D2λ , G; r] ≤ D2λ+1 . In particular, 2λ +r ≤ D2λ+1 . Theorem 3.2 (2) implies that D2r is powerful. This provides the base case in an inductive proof of the following claim: either λ D2λ contains a powerful subgroup of index at most 22r−2 or D2(r−2i +1) is powerful for 0 ≤ i < λ(r − 2λ−1 + 1). Assume now that this holds for i < λ(r − 2λ−1 + 1) − 1 and we will attempt to prove it for i + 1. λ If D2λ contains a powerful subgroup of index at most 22r−2 then we are done. λ Otherwise, |D2λ : D2(r−2i +1) | > 22r−2 . As |D2λ : D2λ+1 | ≤ 2r we can conclude that λ

λ

|D2(r−2i +1) : D2λ+1 | < 2r−(2r−2 ) = 22 −r . Hence, [D2(r−2i +1) , G; 2λ − r − 1] ≤ D2λ+1 . Also, [D2λ , G; 2(r −2i +1)−2λ ] ≤ D2(r−2i +1) and so [D2λ , G; r −2i+1 +1] ≤ D2λ+1 . In particular, 2λ +r−2i+1 +1 ≤ D2λ+1 . As i + 1 < λ(r − 2λ−1 + 1) we know that i+1 2 < r − 2λ−1 + 1 and hence 2λ−1 < r − 2i+1 + 1 and hence Theorem 3.2 (2) implies that D2(r−2i+1 +1) is powerful. This completes the induction proof. If we let = λ(r − 2λ−1 + 1) then we know that either D2λ contains a powerful λ subgroup of index at most 22r−2 or D2(r−2−1 +1) is powerful. Arguing as above, λ

we find that either D2λ contains a powerful subgroup of index at most 22r−2 or 2λ +r−2 +1 ≤ D2λ+1 . If 2λ + r − 2 + 1 = 2λ + 2λ−1 then Theorem 3.2 (2) implies that D2λ is powerful. If 2λ + r − 2 + 1 < 2λ + 2λ−1 then Theorem 3.2 (1) implies that Di is powerful for i ≥ (2(2λ + r − 2 + 1) − 1)/3. We know that 2 ≥ r − 2λ−1 + 1 and hence 2+1 ≥ 2r − 2λ + 2 > 2r − 2λ + 1. Therefore, 2λ > 2r − 2+1 + 1 and hence 3 · 2λ > 2 · 2λ + 2r − 2+1 + 2 − 1. We conclude that 2λ > (2(2λ + r − 2 + 1) − 1)/3. Therefore, in either case D2λ is powerful.

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Hence, D2λ contains a powerful subgroup of index at most 22r−2 or D2λ is powerful itself. Either way, we can now conclude that there is a powerful subgroup of λ D2λ of index at most 22r−2 and hence a powerful subgroup of G of index at most λ 2r(λ+1)−(2 −r) . It is worth noting that the only place we used the hypothesis in the proof of the theorem was to deduce that |D2λ(r) (G) : D2λ(r)+1 (G)| ≤ 2r . Hence the result could be about the least such that |D2 (G) : D2+1 (G)| < 22 . In this case, we would get a bound on the index of a powerful subgroup of D2 but without the assumption on the rank we could not get a bound on the index of a powerful subgroup in G. We would also like to point out that all of the results of this paper also hold in finitely generated pro-p groups. One needs to take the quotient by successively large terms of the dimension series. One then finds the results up to closure, but all of the groups in this paper are closed by standard results. Acknowledgment. I would like to thank the referee for a suggestion which simplified the proof of Theorem 3.2.

References [DdMS]

J. D. Dixon et. al., Analytic Pro-p groups, 2nd. ed., Cambridge Stud. Adv. Math. 61, Cambridge University Press, Cambridge 1999.

[Ha]

P. Hall, A contribution to the theory of groups of prime-power order, Proc. London Math. Soc. (2) 36 (1933), 29–95.

[LM]

A. Lubotzky and A. Mann, Powerful p-groups. I. Finite Groups, J. Algebra 105 (1987), 484–505.

[SS]

C. Scoppola and A. Shalev, Applications of dimension subgroups to the power structure of p-groups, Israel J. Math. 73 (1991), 45–56.

[W]

L. Wilson, Dimension subgroups and p-th powers in p-groups, to appear in Israel J. Math.

Lawrence E. Wilson, Department of Mathematics, University of Florida, PO Box 118105, 358 Little Hall, Gainesville, FL 32611–8105, U.S.A. E-mail: [email protected]

Regular orbits of induced modules of finite groups Thomas R. Wolf

Abstract. If W is a H -module for a group H and if S is a transitive permutation group of degree n, then the wreath product K of H by S acts naturally on a module V of dimension n times that of W . Here, we give an exact count of the number of regular K-orbits on V . This can be expressed in a formula dependent only on the number of regular orbits of H on W and the permutation action of S on a set of n elements. This situation arises frequently, e.g. if V is an irreducible G-module for a finite group G and V is not quasi-primitive, then G is isomorphic as a linear group to a subgroup of a wreath product K (as above). The number of regular G-orbits is at least |K : G| times the number of regular K-orbits. 2000 Mathematics Subject Classification: 20C40

1. Introduction If W is a H -module for a group H and if S is a transitive permutation group on a set with | | = n, then the wreath product H S acts naturally on a module V , which as a vector space is the direct sum of n copies of W . Here, we give an exact count of the number of regular H S -orbits on V . This can be expressed in a formula dependent only on the number of regular orbits of H on W and the action of S on the partitions of the set . Our counting arguments also give an exact count of the number of orbits of H S on V . Our emphasis here is on the partitions of the set , versus partitions of the integer n that are frequently used in the representation theory of the symmetric group Sn . The difference in emphasis is presumably due to the fact that any two partitions of the set of the same “type” are necessarily conjugate in Sn , but not necessarily conjugate in S. The above situation arises frequently, e.g. if V is an irreducible G-module for a finite group G and V is not quasi-primitive (i.e. VN is not homogeneous for some normal subgroup N of G), then G is isomorphic as a linear group to a subgroup of H S for a group H (isomorphic to J /I for some I < J < G) and a transitive permutation group S (isomorphic to G/K for a normal subgroup K of G). Furthermore, S may be even chosen to be a primitive permutation group. Of course, if H has infinitely many regular orbits on W , then G has infinitely many regular orbits on V . The formula is essentially for when H has finitely many regular orbits. Indeed, the motivation for this is when |W | and hence |H | are finite, where there are applications to many well-known problems, including conjectures of Brauer.

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Recall = {1 , . . . , l } is a partition of length l of the set if is the disjoint union 1 ∪ · · · ∪ l and each i is non-empty. If S acts on , we let NS () = {s ∈ S | s = } and CS () = {s ∈ S | si = i for all i}. We say that is trivially centralized by S if CS () = 1. While the formula for the number of regular orbits of G is exact, it is probably more easily used to give a lower bound. The following is Corollary 2.6 below. Theorem 1.1. Let S = 1 act transitively on a set and let l ( , S ∗ ) denote the set of partitions of length l of that are trivially centralized by S. If the group H has k regular orbits on the H -module W and V is the “natural” module for G = H S, then the number of regular G-orbits on V is 1

P (k, l)|l ( , S ∗ )|. |S| 2≤l≤| |

Note that P (k, l) = k!/(k − l)! (and equals 0 if k < l). In particular (see Corollary 2.8), the number of regular orbits of G when S = Sn is “k choose n”. When S is a solvable primitive permutation group on , a Theorem of Gluck [1] shows that S has a trivially centralized partition of length two, unless | | < 10. Indeed there are often many of these. In all cases, S has a trivially centralized partition of length four (or less). We discuss this more in Section 3.

2. Counting orbits Hypothesis 2.1. Suppose that W is an H -module and that S = 1 is a transitive permutation group on n letters. We let V be the direct sum of n copies of W and let G be the wreath product H S, so that G acts naturally on V . We let m denote the number of orbits of H on W and let k denote the number of regular orbits of H on W . Assuming Hypotheses 2.1, we have that V = W1 ⊕ · · · ⊕ Wn for subspaces Wi transitively permuted by S G/K where K = H1 ⊕ · · · ⊕ Hn is a normal subgroup of G and the Hi (isomorphic to H ) are permuted transitively by S. Also the action of Hi on Wi is isomorphic to that of H on W , while Hi centralizes Wj for i = j . We identify H with H1 , W with W1 , and with {W1 , . . . , Wn }. Observe that V is a faithful G-module if and only if W is a faithful H -module. Likewise V is an irreducible G-module if and only if W is an irreducible H -module. We first note that x and y in W are G-conjugate if and only if they are H -conjugate. This is because x and y are G-conjugate if and only if they are NG (W )-conjugate and NG (W ) = KNS (W ) = KCS (W ) = H CK (W )CS (W ) = H CG (W ). Also the Gconjugacy class C of x is the disjoint union ∪i (C ∩ Wi ) and each (C ∩ Wi ) is an Hi -orbit on Wi (and is as well a K-orbit on V ). Let O1 , . . . , Om be the distinct H -orbits on W . Then every element of W1 ∪· · ·∪Wn is conjugate to the elements of exactly one Oj . For v = v1 +· · ·+vn ∈ W1 +· · ·+Wn ,

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we let j (v) = {Wi | vi ∈ OjG }. In particular, for each v in V , we have that {W1 , . . . , Wn } is the disjoint union 1 (v) ∪ · · · ∪ m (v). The non-empty i (v) form a partition of = {W1 , . . . , Wn }, which we call the partition of v. It is easy to see that u = u1 + · · · + un and v = v1 + · · · + vn in V are conjugate in K = H1 × · · · × Hn if and only if each pair ui and vi are Hi -conjugate; or equivalently if and only if j (u) = j (v) for all j , 1 ≤ j ≤ m. In particular, [j (v)]k = j (v) for all k ∈ K, all v ∈ V , and for all j , 1 ≤ j ≤ m. We claim that u and v in V are G-conjugate if and only if there is some s ∈ S such that (j (u))s = j (v) for all j , 1 ≤ j ≤ m. If such an s exists, then j (us ) = j (v) for all j and so us and v are K-conjugate by the last paragraph, whence u and v are G-conjugate. On the other hand, if u and v are G-conjugate; there exist k ∈ K and s ∈ S such that v = uks . By the last paragraph, (j (u))k = j (u) for all j . Thus j (v) = (j (u))ks = (j (u))s for all j , 1 ≤ j ≤ m, establishing the claim. For u and v to be G-conjugate, we must have the partitions determined by u and v are G-conjugate, but that is not sufficient. If the partitions of u and v are equal, we still have that u and v are G-conjugate if and only if there is some s ∈ S such that (j (u))s = j (v) for all j for all non-empty j (u). For a given partition of {W1 , . . . , Wn }, the set of vectors v in V that have partition is a union of K-conjugacy classes. If the length of is l, then the number of K-conjugacy classes of elements of V with partition is P (m, l) = m(m − 1)(m − 2) . . . (m − l + 1) = m!/(m − l)!. In particular, there are no elements with partition if the length of is larger than m (we use the convention that P (m, l) = 0 when l > m). Assume that is the partition of v ∈ V . If g = ks ∈ G with k ∈ K and s ∈ S, then v g is K-conjugate to v if and only if v s is K-conjugate to v and this occurs if and only if s centralizes . Of course, in this case, v g has partition . But more generally, v g has partition if and only if s ∈ NS () (i.e. g ∈ NG ()). If we let O be the set of those K-conjugacy classes of elements of V whose partition is , then NG () = KNS () acts on O. For each [v] ∈ O, we have that the stabilizer in G of [v] is precisely KCS (). Thus the number of K-conjugacy classes in O that are NG ()-conjugate to [v] is |NS () : CS ()|. Hence the number of NG ()-orbits on O is |O|/|NS () : CS ()| = P (m, ||)/|NS () : CS ()|. This is precisely the number of NG ()-orbits on O ∗ = {v ∈ V | v has partition }. Since two elements of O ∗ are G-conjugate if and only if they are NG ()-conjugate, P (m, ||)|CS ()|/|NS ()| is the number of G-conjugacy classes C of V for which the partitions of the elements of C are the S-conjugates class of . The number of G-conjugacy classes of V is then P (m, ||)|CS ()|/|NS ()|, where the sum is taken over a set of representatives of each S-conjugacy class of partitions of . Of course, the number of S-conjugates of a partition is |S : NS ()|. Summing over all partitions, we then get that the number of G-orbits on V is then

P (m, ||)|CS ()|/|S|. P (m, ||)|CS ()|/|NS ()||S : NS ()| =

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Summarizing we have: Theorem 2.2. Assume Hypothesis 2.1. The number of orbits of G on V is exactly 1

P (m, ||)|CS ()|, |S| ∈( )

where ( ) is the set of partitions of . The theorem is most useful in giving lower bounds for the number of orbits of G and applied to partitions of short length. Note that the partition of an element v in V is determined by the action of H on W and not dependent upon the action of S on . Corollary 2.3. Assume Hypotheses 2.1. Then the number of G-orbits of elements v in V whose partition has length q is 1

P (m, q)|CS ()|, |S| l()=q

where the sum is taken over all partitions of whose length l() is q. Next we count the number of regular orbits. For v = v1 + · · · + vn to be in a regular G-orbit, then v must be in a regular K-orbit. Now v is in a regular K-orbit if and only if each vi is in a regular Hi -orbit, i.e. whenever j (v) is non-empty, then Oj is a regular orbit of H on W . Again the non-empty j (v) form a partition of = {W1 , . . . , Wn }. If k is the number of regular orbits of H on W and is a partition of , we argue as above that the number of G-orbits of K-regular elements of V with partition or an S-conjugate of is P (k, ||)|CS ()|/|NS ()|. Let v ∈ V be in a regular K-orbit and let be the partition of v. We claim that v is in a regular G-orbit if and only if CS () = 1. For g = ks ∈ G with k ∈ K and s ∈ S, we have from above that v g is K-conjugate to v if and only if s ∈ CS (). If 1 = t ∈ CS (), then v t = v x for some x ∈ K and 1 = tx −1 ∈ CG (v), whence v is not in a regular G-orbit. For the converse, assume that v is not in a regular G-orbit and 1 = g = ks centralizes v. Then v g is K-conjugate to v and so s ∈ CS (). Since v is in a regular K-orbit and 1 = g = ks centralizes v, indeed s = 1 and CS () = 1. This proves the claim. Thus v is in a regular G-orbit if and only if v is in a regular K-orbit and the partition of v is in a regular orbit of S. To count regular G-orbits, we apply the same counting argument used above to count all orbits, except we restrict ourselves to regular H -orbits (vs. all H -orbits) and to partitions that are trivially centralized by S (vs. all partitions). Since CS () = 1 for a trivially centralized partitions of , we get: Theorem 2.4. Assume Hypothesis 2.1 and let ( , S ∗ ) denote the set of partitions of that are trivially centralized by S. Then the number of regular G-orbits on V is exactly ∈( ,S ∗ ) P (k, ||)/|S|. Corollary 2.5. Assume Hypothesis 2.1. G has a regular orbit if and only if there is a trivially centralized partition of whose length is at most the number of regular

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orbits of H on W . In particular, for G to have a regular orbit when | | > 1, H must have at least two regular orbits on W and S must act faithfully on . The necessity that H must have two regular orbits on W above is because S = 1 will centralize the unique partition of length one. When S is a solvable and primitive permutation group, S has a trivially centralized partition of length two, except in a few cases where | | < 10. This is Gluck’s Theorem, which we discuss in more detail later. The following is more convenient for counting regular orbits. Corollary 2.6. Assume Hypothesis 2.1 and let l ( , S ∗ ) denote the set of partitions of length l of that are trivially centralized by S. Then the number of regular G-orbits on V is 1

P (k, l)|l ( , S ∗ )|. |S| 2≤l≤| |

We even note that |S| divides each term P (k, l)|l ( , S ∗ )| in the sum above 1 P (k, l)|l ( , S ∗ )| is the number of regular G-orbits of elements of V whose since |S| partition of {W1 , . . . , Wn } has length l. If = 1 ∪ · · · ∪ l is a partition of length l, then NS ()/CS () faithfully permutes the i and so |NS ()/CS ()| divides l!. If is also trivially centralized, then even |NS ()| divides l!. Every conjugate of is k! trivially centralized and so |l ( , S ∗ )| ≥ |S|/ l!. Now P (k, l)/ l! = l!(k−l)! = kl , known commonly as “k choose l”. Note kl = 0 for k < l. Corollary 2.7. If S has at least one trivially centralizedpartition of length l and if H has k regular orbits on W , then G = H S has at least kl regular orbits of elements of V with partition length l. Corollary 2.8. Assume Hypothesis 2.1 and that S is Sn or An . (1) The number of regular orbits of H Sn on V is nk ; and k . (2) The number of regular orbits of H An on V is 2 nk + (n − 1) n−1 Proof. The only partition of {1, . . . , n} that is trivially centralized by Sn is {{1}, . . . , {n}} and it has, of course, length n. By Corollary 2.6 the number of regular 1 k! P (k, n) = (k−n)!n! = nk . The alternating group has exactly orbits of H Sn is n! one other orbit of trivially centralized partitions of {1, . . . , n}, namely the family of partitions of length n − 1, each such partition consisting of a doubleton and n − 2 singletons. By Corollary 2.6, the number of regular orbits of H An is thus n k k 2 P (k, n) + P (k, n − 1) = 2 + (n − 1) . n! 2 n n−1

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3. Trivially centralized partitions of solvable S Here we give a brief accounting of the number of trivially centralized orbits when S is a primitive solvable permutation group. We restrict ourselves to primitive permutation groups since most applications (e.g. V a quasi-primitive G-module) can be reduced to Hypothesis 2.1 with the additional condition that S is a primitive permutation group. Gluck [1] showed that a solvable primitive permutation group on with | | = n necessarily has a regular orbit on the power set of if n > 9. Note that a non-empty ⊂ is in a regular orbit of S if and only if ∪ ( − ) is a partition of (of length two) that is trivially centralized by S. While there are certainly exceptions to Gluck’s lemma on the existence of trivially centralized partitions of length two, an inspection of the proof of Gluck’s lemma shows that the number of trivially centralized partitions is an exponential function of n. Indeed the argument of Proposition 5.2 of [3] shows that S has at least (2n − |S|23/4 )/2 trivially centralized partitions of length two, and |S| ≤ n13/4 /2 via Corollary 3.6 of [3]. But this says nothing for values of n < 83. Indeed, there maybe none when n < 10. Note that n must be a power of a prime as S is a primitive solvable permutation group. Theorem 3.1. Suppose that S is a solvable primitive permutation group of degree n > 1. Then (i) if n > 9, then there are at least 4|S| trivially centralized partitions of length two; (ii) if |S| is odd, then there are at least |S| trivially centralized partitions of length two; (iii) if 5 ≤ n ≤ 9, there are at least |S| trivially centralized partitions of length three; and (iv) there is a trivially centralized partition of length l ≤ 4. There is one of length l ≤ 3 unless n = 4 and S = S4 . Proof. If n > 9, a refinement of Gluck’s lemma by Zhang [4] shows that S has at least 8 regular orbits on the power set P ( ). Thus there exist 8|S| subsets of such that stabS () = 1. For each such , the stabilizer of the complement c of is also 1 and ∪ c forms a partition of length two of that is trivially centralized by S. Thus there are at least 8|S|/2 trivially centralized partitions of length two. Suppose that |S| is odd. Then Gluck’s Lemma even shows that S has a regular orbit on the power set of (even if S is not primitive and even when S is not transitive). Since n > 1, we may choose a proper non-empty subset of in a regular orbit of S on P ( ). Then c is also in a regular orbit of S and ∪ c forms a partition of length two of that is trivially centralized by S. Since |S| is odd, then and c are not S-conjugate and the partition {, c } of has |S| distinct conjugates. Part (ii) follows. For n = 5 or 7, Part (iii) follows from Corollary 3.4 (below). For n = 8 or 9, Part (iii) follows from Proposition 3.5 or 3.6 (below), respectively. Since n must be a prime power,

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the proof of (iii) is complete. That S has a trivially centralized partition of length at most four is also a direct consequence of Gluck’s lemma and a proof is given in of [2]. However this follows from Part (iii). To see this, we may assume via iii that n < 5. Hence the partition of singletons has length at most four and must be trivially centralized by S. We note part (ii) of Theorem 3.1 remains valid if S is imprimitive or even intransitive. The proof of Theorem 3.1 (iii) will be complete upon examination of small permutation groups below. But first, we show how the results could be used “inductively”. Corollary 3.2. Assume Hypothesis 2.1 and that S is a solvable primitive permutation group. Assume that H has k ≥ 2 regular orbits on W . Then (i) If k > 4, then G has at least k regular orbits on V . Indeed G has at least 2k regular orbits on V , except when k = 5 and S = S4 (and G has 5 regular orbits); (ii) If n > 9, then G has at least 2k regular orbits on V ; (iii) If n > 4 and k ≥ 3, then G has at least 2k regular orbits on V Proof. If n > 9, then G has at least 4P (k, 2) = 4k(k − 1) regular orbits by Theorem 3.1, Part (i) and Corollary 2.6. If n > 4 and k ≥ 3, then G has at least P (k, 3) = k!/(k − 3)! regular orbits by Theorem 3.1, Part (iii) and Corollary 2.6. This proves ii and iii. Now S has a trivially centralized partition of length l with either l < 4 or with l = 4 and S = S4 . By Corollary 2.7, G has at least kl regular orbits. For k > 4 and l < 5, note that kl ≥ 2k, except when k = 5 and l = 4. We have exceptional cases when k and n are small. For n < 5, S is either alternating or symmetric and so the number of regular orbits is given by Corollary 2.8. We let S(n, l) be the number of partitions of length l of a set when | | = n. The numbers S(n, l) are called Sterling numbers of the second kind. If α ∈ , the partitions of may be obtained by modifying the partitions of − {α}, and so we have the following recursive relationship: S(n, l) = S(n − 1, l − 1) + l · S(n − 1, l). These can be easily calculated for small n (e.g. by spreadsheet) and these calculations will be used in Corollary 3.4. We adopt the notation S(n, l) = 0 for l > n. Of course S(n, 1) = 1 = S(n, n). For l = 2, we even have that S(n, 2) = 2n−1 − 1. This is because each of the 2n − 2 non-empty proper subsets of is a member of exactly one partition of length two of , namely = ∪ ( − ). Lemma 3.3. Suppose that S is a solvable transitive permutation group on of prime degree p, so that Sα is cyclic and |Sα | divides p − 1. The number of partitions of

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of length l > 1 that have trivial centralizers in S is

S(1 + p−1 S(p, l) − p q , l) q∈π(Gα )

−

distinct

S(1 +

q,r∈π(Gα )

p−1 qr , l) +

distinct

S(1 +

q,r,s∈π(Gα )

p−1 qrs , l) − . . .

where π(Gα ) is the set of prime divisors of |Gα |. The terms in the alternating sum in parentheses decrease. Proof. Since | | is a prime p, S must be primitive and a subgroup of Zp · Zp−1 and has a normal transitive subgroup M of order p. For a partition of of length larger than one, CM () = 1. It follows that CS () must be a subgroup of one the p point stabilizers, each such stabilizer being cyclic of order dividing p − 1. If g ∈ Gα for some α ∈ and g has order s, then g is a product of a one-cycle and p−1 s s-cycles, whence g has 1+ p−1 orbits on . If a subgroup H of S centralizes = 1 ∪· · ·∪l , s then each i must be a union of H -orbits. If H has r orbits on , then the number of partitions of length l centralized by H is S(r, l), understanding S(r, l) = 0 for r < l. Recalling that Gα is cyclic (of order dividing p − 1), a partition of with nontrivial centralizer in Gα must be centralized by a subgroup of prime order. We may use an inclusion-exclusion argument to get an exact count of the number of partitions of length l, 1 < l < p, that have a non-trivial stabilizer in Gα , namely:

q∈π(Gα )

S(1 +

p−1 q , l) −

distinct

S(1 +

q,r∈π(Gα )

p−1 qr , l) +

distinct

S(1 +

q,r,s∈π(Gα )

p−1 qrs , l) − . . .

.

Each pair of the p point stabilizers intersect trivially, whence the formula in the statement of the lemma is exact. Corollary 3.4. Let S be a solvable transitive permutation group on , with | | = p, a prime. (i) If |S| = p, then S has S(p, l) trivially centralized partitions of length l for all l > 1. In particular, there are 2p−1 − 1 such of length two. (ii) If p ≥ 19, then S has at least 2(p−1)/2 |S| trivially centralized partitions of length two. (iii) If p = 11, 13, or 17 (respectively), then S has at least 6|S|, 20|S|, or 225|S| trivially centralized partitions of length two. (iv) If p = 7, then S has at least 6|S| trivially centralized partitions of length three. If a point stabilizer Sα has order 2, 3, or 6; then the number of trivially centralized partitions of length two is |S|, 2|S|, or 0 (respectively), and (v) If p = 5 < |S|, there are exactly 0, 20, 10 and 1 (respectively) trivially centralized partitions of length 2, 3, 4, or 5 (respectively).

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Proof. Since Gα is cyclic of order dividing p − 1, all parts save Part (ii) can be calculated by the formula in Lemma 3.3. The argument of the lemma shows that the number of partitions of length l with trivial centralizer is at least

p − 1 , l) . S(1 + S(p, l) − p q q∈π(Gα )

Now S(k, 2) = 2k−1 − 1. Also the number of prime divisors of |Gα | is less than log2 (p) and 1 + (p − 1)/q ≤ (p + 1)/2 for all q. Then the number of trivially centralized partitions of length two is at least: 2p−1 − 1 − p log2 (p) ∗ (2(p+1)/2) − 1) ≥ 2((p−1)/2) [2(p−1)/2 − 2p log2 (p)]. For p ≥ 19, note that 2(p−1)/2 − 2p log2 (p) > p(p − 1) ≥ |S|. Proposition 3.5. Let S be a solvable primitive permutation group on , | | = 8. Then Sα is cyclic of order 7 or a Frobenius group of order 21. Then (i) The number of trivially centralized partitions of length three is at least (9/2) · |S|; and (ii) The number of trivially centralized partitions of length two is 0 if |Sα | = 21 and is 3|S|/2 if |Sα | = 7. Proof. Since S is a primitive solvable permutation group of degree 8, S has a unique minimal normal subgroup that is elementary abelian of order 8 and S is the semi-direct product of M and Sα , that acts faithfully an irreducibly of M. The actions of Sα on M and are permutation isomorphic. That Sα is cyclic of order 7 or a Frobenius group of order 21 is routine (e.g. Theorem 2.12 of [3]). A subgroup H of S with k orbits centralizes exactly S(k, l) partitions of length l. The only subgroups of A(23 ) with more than three orbits on are the seven subgroups of order two and the 28 subgroups of order three, and all of these 35 subgroups have exactly four orbits. The only subgroups with exactly three orbits are the 28 cyclic groups of order 6, each containing 2 subgroups of prime order. Thus have the number of trivially centralized partitions of length 3 is S(8, 3) − 35 · S(4, 3) + 28 · S(3, 3) ≥ (9/2) · 168 ≥ (9/2) · |S|. We now consider the partitions of length two. If ∪ c is in a trivially centralized partition of , then is in a regular orbit of S on P ( ). If || = t, then is even in a regular orbit of S on Pt ( ), the collection of all subsets of of size t. Unless t = 3, 4, or 5 and |S| = 8 · 7, then |S| > |Pt ( )| = 8t and S has no regular orbit on Pt ( ). Hence there are no trivially centralized partitions of length two if |S| = 8 · 7 · 3. When |S| = 8·7, note the subgroups of S of order 7 have two orbits on , while those of order two have cycle structure 24 with 4 orbits. Thus the number of trivially centralized partition of length two is at least S(8, 2) − 8 · S(2, 2) − 7 · S(4, 2) = 70 > |S| = 56. We note no element of prime order of S fixes a subset of of size three (or five), and thus the 56 partitions of type (3,5) are all trivially centralized by S. Thus there is at least one partition of of type (4,4) that is trivially centralized by S. As NS ()

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has order 1 or 2, there are at least 28 conjugates of , each trivially centralized by S. Indeed, as there are only 35 partitions of type (4,4), exactly 28 partitions of type (4,4) are trivially centralized. Proposition 3.6. Let S be a solvable primitive permutation group on , | | = 9. (i) If Sα is cyclic or quaternion, then S has at least 2 · |S| or 38 · |S| trivially centralized partitions of length two or three (respectively); (ii) If Sα is not cyclic nor quaternion, then S has no trivially centralized partitions of length two; and (iii) S has at least |S| trivially centralized partitions of length three. Proof. Let M be the minimal normal subgroup of S so that S = MSα and M ∩Sα = 1. Then Sα acts irreducibly on M, so that Sα is an irreducible subgroup of GL(2, 3) and |S| divides 9 · 48. In particular, Sα is cyclic, quaternion of order eight, dihedral of order eight, semi-dihedral of order sixteen, or is isomorphic to SL(2, 3) or GL(2, 3). Now Sα has a unique central involution t that inverts M, whence the cycle structure of t on is 24 · 1. In particular, < t > has 5 orbits and belongs to exactly one point stabilizer. Assume that Sα is cyclic or quaternion. Every element of prime order in S = MSα indeed lies in S0 = M < t >. A partition of with non-trivial centralizer in S has a non-trivial centralizer in S0 . Each of the nine involutions of S0 has 5 orbits on , while each of the four subgroups of S0 order 3 has 3 orbits on . A subgroup with r orbits centralizes exactly S(r, l) partitions of length l. The number of trivially centralized partitions of length l > 1 for S (or S0 ) is at least S(9, l) − 4 · S(3, l) − 9 · S(5, l). As |Sα | divides 8 and |S| divides 72, there are more than 38|S| trivially centralized partitions of length three and at least 3|S|/2 of length two. But if = {, c } is a trivially centralized partition of of length two, then NS () = 1 because || = |c | and so |S| divides the number of trivially centralized partitions of length two. This proves i. Part (iii) follows from Part (i) when Sα is cyclic or quaternion. To prove (ii) and (iii), we may assume that Sα is dihedral of order 8 or has order at least 16. If = ∪ c is a trivially centralized partition of and || = t, then is in a regular 9 orbit of S in its action on Pt ( ) = { ⊆ ||| = t}. If |Sα | > 8, then |S| > t = | t | for all t and so S has no trivially centralized partition of of length two. A dihedral group D of order 8 in GL(2, 3) contains all involutions in a semidihedral Sylow-2-subgroup T of GL(2, 3) and consequently MD contains all elements of prime order belonging to MT . Thus MD has no trivially centralized partition of length two since MT has none. Thus S has no trivially centralized partitions of length two. Part (ii) follows. Finally to prove (iii), it is no loss to assume that Sα GL(2, 3). Fix α, β ∈ with α = β. Now the two-point stabilizer Sαβ is isomorphic to S3 and has orbit pattern 6·13 . In particular, Sαβ also fixes a third point γ ∈ {α, β}. Note that S acts transitively on the 36 subsets of with two elements. Thus |NS {α, β}| = |S|/36 = 12 = 2|Sαβ | =

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399

2|CS {α, β}|. Set N = NS {α, β} and note that the orbits of N are {α, β}, {γ }, and − {α, β, γ }. Now N ≤ Sγ and N is the normalizer of a Sylow-3-subgroup of Sγ and is isomorphic to Z2 ⊕ S3 . Subgroups of N of prime order with their orbit pattern on − {α, β, γ } are: • four subgroups of order two with orbit pattern 23 on − {α, β, γ }; • three subgroups of order two with orbit pattern 22 · 12 on − {α, β, γ }; and • one subgroup of order three with orbit pattern 32 on − {α, β, γ }. Thus the number of subsets of − {α, β, γ } with exactly threeelements fixed by some nonidentity element of N is at most 4 · 0 + 3 · 4 + 1 · 2 < 63 . Thus we may choose ⊆ − {α, β, γ } such that || = 3 and NN () = 1. It follows that the partition = {{α, β}, , [ − {α, β} − ]} of is trivially centralized by N. Now {α, β} is the only subset in of size two and so CS () ⊆ NS ({α, β}) = N . Thus is a partition of of length three and type (2,3,4) that is trivially centralized by S, with γ in the subset of size four. Since the subsets of the partition have distinct sizes, NS () = CS () = 1. Thus has |S| conjugates, each trivially centralized by S.

References [1] D. Gluck, Trivial set-stabilizers in finite permutation groups, Canad. J. Math. 35 (1983), 59–67. [2] A. Moretó and T. Wolf, Orbit sizes, character degrees and Sylow subgroups, Adv. Math. 184 (2004), 18–36; Erratum ibid. p. 409. [3] O. Manz and T. Wolf, Representations of Solvable Groups, London Math. Soc. Lecture Note Ser. 185, Cambridge University Press, Cambridge 1993. [4] J. Zhang, Finite groups with few regular orbits on the power set, Algebra Colloq. 4 (1997), 471–480. Thomas R. Wolf, Department of Mathematics, Ohio University, Athens, OH 45701, U.S.A E-mail: [email protected]

Radical subgroups and p-local ranks Jiping Zhang∗

Abstract. In this paper we investigate some problems on radical p-subgroups and p-local ranks, in particular, conditions on the existence of p-blocks with a given maximal Sylow p-intersection as defect group, and the influence of p-local ranks on the group structure will be provided. 2000 Mathematics Subject Classification: 20C20

Let p be a prime integer and let G be a finite group. An important question in modular representation theory is: When is the given p-subgroup D of G a defect group for some p-block of G? By the First Main Theorem of Brauer we see that if D is a defect group for some p-block then D = Op (NG (D)), that is to say, D is radical. Also, as proved by Green[3], defect groups are Sylow p-intersections, i.e. D = P ∩ P x , where P ∈ Sylp (G) and x ∈ G. Note that Sylow p-intersections are not necessarily radical, thus a Sylow p-intersection is not necessarily a defect group for some p-block. However it is easy to verify the following result. Proposition 1. Suppose that G is a finite group with an abelian Sylow p-subgroup P . Then a proper p-subgroup D of P is radical if and only if D is a Sylow p-intersection in G. Let p and q be two different primes such that p > 2 and p | (q − 1). Then the affine semi-linear group A(q p ) has a non-cyclic abelian Sylow p-subgroup with Op (A(q p )) = 1. Note that A(q p ) has no defect-zero p-blocks. Thus even for finite groups with an abelian Sylow p-subgroup a given radical subgroup is not necessarily a defect group, see [12] for more information. For any finite group G set SIp (G) = {P ∩ P x : P ∈ Sylp (G), x ∈ G \ NG (P )} to be the set of all Sylow p-intersections (= P ). A maximal element (under inclusion) in SIp (G) is called a maximal Sylow p-intersection. We see that maximal Sylow p-intersections are radical. It is thus very interesting to study the following Problem I. When is the given maximal Sylow p-intersection of a finite group G a defect group for some p-block of G ? In the case p = 2, G. Robinson [6] first proved some deep result on Problem I. ∗ Supported by Cheung Kong Scholar’s Programme, National 973 project and RFDP.

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Theorem 2 (Robinson, [6]). Let Q be a maximal Sylow 2-intersection of a finite group G. Then either Q is a defect group for some 2-block of G or (i), (ii) and (iii) are all true: (i) Q is a Sylow 2-subgroup of O2 ,2 (NG (Q)), (ii) NG (Q) is 2-constrained , and CG (Q) has a normal 2-complement, (iii) O2 (NG (Q)) = O2 (NG (T )) whenever T is a Sylow 2-subgroup of G containing Q. Let Q be a maximal Sylow p-intersection of a finite group G. Suppose that Q is not a defect group for any p-blocks of G, then every p-block B of NG (Q) is of the highest defect; in other words, B has a Sylow p-subgroup of NG (Q) as its defect group. We call such groups full p-defective groups. Thus full p-defective groups are of great importance in investigating Problem I. Finite full p-defective groups were first introduced by the author, and the following general structure theorem was proved in [10]. Theorem 3 (Zhang, [10]). Let G be an arbitrary finite group, P a Sylow p-subgroup of G and M a p -subgroup of G such that M = Op (CG (P )). Then G is full p-defective if and only if the following two statements hold: (1) Op (G) = ∪x∈G M x ; (2) F ∗ (G/Op (G)) = (Op (G) · Op (G))/Op (G), and for p = 2, we may have besides F ∗ (G/O2 (G)) = (F × O2 (G))/O2 (G), where F is a normal subgroup of G and all components of F are of type M22 or M24 . From this theorem we have the following corollary which solves Problem I for finite groups with an abelian Sylow p-subgroup Corollary 4. Let G be a finite group with an abelian Sylow p-subgroup. Then every maximal Sylow p-intersection of G is a defect group for some p-block. Proof. Let D be a maximal Sylow p-intersection of G. If D is not a defect group for any p-blocks of G, then, as we noted before, the normalizer H := NG (D) of D in G is a full p-defective group. By the structure theorem above we see that F ∗ (H /Op (H )) = (Op (H ) · Op (H ))/Op (H ). Since a Sylow p-subgroup P of G is abelian and contained in H and by the typical property of generalized Fitting subgroups it follows that P = D is normal in H , which is a contradiction as D is a proper subgroup of P . We are done. From a different approach an answer to Problem I can be derived from the following theorem. To this end we need to introduce the concept of strong radical p-subgroups. A p-subgroup D of a finite group G is called strong radical if NG (D)/D contains a strongly p-embedded subgroup. We see that maximal Sylow p-subgroups are strong radical.

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Theorem 5 (Zhang, [11]). Suppose D is a strong radical p-subgroup of a finite group G then D is a defect group for some p-block of G if and only if there exists a p-regular element x in G such that D is a Sylow p-subgroup of the centralizer CG (x) of x in G. For further studies of Problem I more information are needed on the inner structure of finite full p-defective groups. Let G be a finite full p-defective group, and P a Sylow p-subgroup of G with CG (P ) = Z(P ) × M where Z(P ) is the center of P . In Theorem 3, we see from Lemma 9.2 of ([4], p. 118) that if p = 2 then M = Op (G). In general, for finite full p-defective group G with M = Op (G) one can not control the structure of the quotient of G by the normal subgroup Op ,p (G). In fact any finite group can be embedded into such a quotient of some finite full p-defective group G. However, if M = Op (G) then the group structure of K/CK (Op (G)) can be very much restricted where K is the normal closure of the Sylow p-subgroup P of G, i.e., K = P G = P x : x ∈ G. This is indeed the case if G is a finite solvable full p-defective group. Theorem 6 ([8]). Let G be a finite solvable full p-defective group, and P a Sylow ¯ the quotient group G/(G) p-subgroup of G. Suppose that G = P G . Denote by G ¯ = F (G)H ¯ where (G) is the Frattini subgroup of G. We may assume G with ¯ and the Vi ’s are ¯ ∩ H = 1 and F (G) ¯ = W × V1 × · · · × Vs , where W = Op (G) F (G) ¯ ¯ contained in Op (F (G)). Set M := Op (CG¯ (P¯ )). minimal normal subgroups of G ¯ ¯ Then M = Op (G) unless H /CH (Op (F (G))) isomorphic to a subgroup of the direct product G1 × · · · × Gs , where the Gi ’s satisfy the following statements: (1) If H acts trivially on Vi , then Gi = 1. (2) If Vi is not pseudo-primitive as G-module, then p = 3 and there exists a normal subgroup C in Gi such that Gi /C is isomorphic to A(23 ); if Vi = W1 ⊕ · · · ⊕ Wn where Wj are C-invariant spaces, then Gi /C primitively and faithfully permutes {W1 , . . . , Wn } and furthermore n = 8. (3) If Vi is pseudo-primitive as G-module, then p ≥ 3, Gi ≤ (V ) such that O p (Gi ) is a cyclic p -group; Vi , as an irreducible Gi -module, is faithful and p does not divide |Vi |. Corollary 7. Let G be a finite solvable full p-defective group then a Sylow p-subgroup P of G/CG (Op (G)) is abelian unless p = 3 and P is metabelian. We note that such investigation on finite full p-defective groups has an important connection with the Kronecker equivalence of algebraic number fields; see [5] for reference. In order to get a deeper understanding on Alperin’s weight conjecture and the conjectures by Dade, Robinson [7] introduces the concept of the p-local rank of a finite group, which agrees with the rank of the (B, N )-pair for a group with a (B, N )pair of characteristic p. Let G be a finite group and p a prime divisor of |G|. Given a chain of p-subgroups σ : Q0 < Q1 < · · · < Qn

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of G, define the length |σ | = n, the k-th initial sub-chain σk : Q0 < Q1 < · · · < Qk , and the normalizer of the chain Gσ = NG (Q0 ) ∩ NG (Q1 ) ∩ · · · ∩ NG (Qn ). We say that the p-chain σ is radical if Qi = Op (NG (σi )) for each i, i.e., Q0 is a radical p-subgroup of G and Qi is a radical p-subgroup of NG (σi−1 ) for each i = 0. Write R = R(G) for the set of radical p-chains of G and write R(G)/G for a set of orbit representatives under the action of G. Following [7] we define the p-local rank, plr(G), of G to be the length of a longest chain in R(G). We say that a subgroup H of G is a trivial intersection (T.I.) set in G if H g ∩H = 1 for every g ∈ G \ NG (H ). By [7, 7.1] if plr(G) > 0, then plr(G) = 1 if and only if G/Op (G) has T. I. Sylow p-subgroups. Eaton [2] has verified Dade’s projective conjecture (and a related conjecture of G. R. Robinson) for p-blocks of finite groups of p-local rank 1. It generalizes results by H. I. Blau and G. O. Michler on groups with a T. I. Sylow p-subgroup [1]. Eaton’s proof makes use of the classification of finite simple groups. It seems possible to determine to some extend the finite groups with p-local rank 2 and verify the related conjecture for such groups. However this problem is very difficult and little progress has been made. We call a p-subgroup Q of a given finite group G nontrivially radical if Q is radical in G and Op (G) = Q = P where P is a Sylow p-subgroup of G containing Q. Now we are able to prove the following result which characterizes finite groups with p-local rank 2. Theorem 8. Let G be a finite group with p-local rank plr(G) > 1. Then plr(G) = 2 if and only if every nontrivially radical p-subgroup Q of G is a maximal Sylow pintersection in NG (Q). Proof. Suppose first that G is a finite group with plr(G) = 2. Let Q be a nontrivially radical p-subgroup of G and let R be a maximal Sylow p-intersection in NG (Q). If Q is not a maximal Sylow p-intersection in NG (Q) then it is easy to verify that Op (G) < Q < R < L is a radical chain in G, where L is a Sylow p-subgroup of NG (Q). Thus plr(G) > 2, which is a contradiction. So Q is a maximal Sylow pintersection in NG (Q). Now suppose that every nontrivially radical p-subgroup Q of G is a maximal Sylow p-intersection in NG (Q). Let Op (G) < Q < R2 < ··· < Rn be a radical chain of G such that plr(G) = n. Since Q is a maximal Sylow p-intersection in NG (Q), we see that R2 ∈ Sylp (NG (Q)), so R2 = Rn , n = 2, and thus plr(G) = 2. We are done. The following theorem gives some information on the group structure of finite groups with small p-local rank.

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Theorem 9. Let G be a finite group such that every maximal Sylow p-intersection of G is a T. I. set in G. Then the p-local rank is at most 2. Proof. If plr(G) ≤ 1 then we have nothing to prove. So we now suppose that plr(G) > 1. Let Q be a nontrivially radical p-subgroup of G. If Q is a maximal Sylow pintersection of G then it is also a maximal Sylow p-intersection in NG (Q). If Q is not a maximal Sylow p-intersection of G, then Q is properly contained in a maximal Sylow p-intersection R of G. Now R being a T. I. set in G we have that NG (Q) ≤ NG (R), thus Q < R ∩ NG (Q) ≤ Op (NG (Q)), which is a contradiction since Q is radical. It follows from Theorem 8 that plr(G) = 2. We are done. Theorem 10. Let G be a finite group such that every maximal Sylow p-intersection of G is a T. I. set in G. Then the defect group of any p-block of G is either a maximal Sylow p-intersection of G or isomorphic to {1} or a Sylow p-subgroup of G. Proof. Let B be a p-block of G with defect group D. Suppose D is not trivial or a Sylow p-subgroup of G. Since D is radical, we see from the proof of Theorem 9 that D is a maximal Sylow p-intersection of G. We are done. It seems possible to determine the finite groups with every maximal Sylow pintersection a T. I. set. Please note that finite groups containing a Sylow p-subgroup of order at most p 2 satisfy such condition. Now we consider the relationship between the p-local rank and other invariants of finite groups G. The following result is a typical example of such relationship. Proposition 11. Let G = H P with Op (G) = 1 , where H = Op (G) and P is an elementary abelian p-subgroup. Then the p-rank mp (G) = plr(G) . Proof. Let x be a nontrivial element of P such that the centralizer CG (x) has the maximal possible order among the centralizers of nontrivial elements of P . We claim that Op (CG (x)) = x. Let Tx = CH (x) and note that CG (x) = Tx P . Suppose, on the contrary, that x is a proper subgroup of Px := Op (CG (x)). Then Px is not cyclic of order at least p 2 . Note that Px centralizes Tx . Since H = CH (A) : A ≤ Px , |Px : A| ≤ p one can choose y ∈ Px such that y = 1 and CH (y) is not contained in Tx . Since Tx ≤ CH (y) we have CG (x) = Tx P < CH (y)P = CG (y), which is a contradiction to the choice of x. Thus Op (CG (x)) = x, as claimed. Let P = x × P1 . Repeating the above process for the group Tx P1 we obtain a radical chain of p-subgroups 1 < Q1 < Q2 < · · · < Qn where Q1 = x, Qj = x, x2 , . . . , xj for j = 2, 3, . . . , n, and pn = |P |. Thus mp (G) = n = plr(G). We are done. Theorem 12 ([9]). Let G be a finite p-solvable group with Op (G) = 1. Then we have: (1) The p-rank mp (G) ≤ plr(G). (2) The p-length p (G) ≤ plr(G); and if p (G) = plr(G) with p > 2 then p (G) = 0 or 1.

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Acknowledgements. We thank Professors W. Feit and G. Robinson for their help during the preparation of this paper.

References [1]

H. I. Blau and G. O. Michler, Modular representation theory of finite groups with T. I. Sylow p-subgroups, Trans. Amer. Math. Soc. 319 (1990), 417–468.

[2]

Ch. W. Eaton, On finite groups of p-local rank one and conjectures of Dade and Robinson. J. Algebra 238 (2001), 623–642.

[3]

J. A. Green, Some remarks on defect groups. Math. Z. 107 (1968), 133–150.

[4]

O. Manz and T. R. Wolf, Representations of Solvable Groups, London Math. Soc. Lecture Note Ser. 185, Cambridge University Press, Cambridge 1993.

[5]

Ch. E. Praeger, Covering subgroups of groups and Kronecker classes of fields. J. Algebra 118 (1988), 455–463.

[6]

G. R. Robinson, The number of blocks with given defect group, J. Algebra 84 (1983), 493–502.

[7]

G. Robinson, Local structure, vertices and Alperin’s conjecture, J. London Math. Soc. 72 (3) (1996), 312–330.

[8]

B. S. Wang, J. P. Zhang, On finite solvable full p-defective groups, Algebra Colloquium V10 (2003), 405–412.

[9]

B. S. Wang, J. P. Zhang, On the p-local rank of finite groups, Acta Math. Sin. 19 (2003), 29–34.

[10] J. P. Zhang, On finite groups all of whose p-blocks are of the highest defect, J. Algebra 118 (1988), 129–139. [11] J. P. Zhang, Studies on defect groups, J. Algebra 166 (1994), 310–316. [12] J. P. Zhang, p-regular orbits and p-blocks of defect zero. Comm. Algebra 21 (1993), 299–307. Jiping Zhang, Lmam, The School of Mathematical Sciences, Peking University, Beijing, P. R. China E-mail: [email protected]

List of contributors

Michael Aschbacher, Department of Mathematics, California Institute of Technology, Pasadena, CA 91125-0001, U.S.A. E-mail: [email protected] Christopher P. Bendel, Department of Mathematics, Statistics and Computer Science, University of Wisconsin-Stout, Menomonie, WI 54751, U.S.A. E-mail: [email protected] David J. Benson, Department of Mathematics, University of Georgia, Athens, GA 30602, U.S.A. E-mail: [email protected] Stephen Doty, Mathematics and Statistics, Loyola University Chicago, Chicago, IL 60626, U.S.A. E-mail: [email protected] Norberto Gavioli, Dipartimento di Matematica Pura ed Applicata, Università degli Studi di L’Aquila, Via Vetoio, 67010 Coppito (L’Aquila) AQ, Italy E-mail: [email protected] Meinolf Geck, Institut Girard Desargues, Université Claude Bernard-Lyon I, 21 Avenue Claude Bernard, 69622 Villeurbanne Cedex, France E-mail: [email protected] George Glauberman, Department of Mathematics, University of Chicago, 5734 S. University Ave., Chicago, IL 60637, U.S.A. E-mail: [email protected] Ralf Gramlich, TU Darmstadt, FB Mathematik / AG 5, Schloßgartenstraße 7, 64289 Darmstadt, Germany E-mail: [email protected] Robert M. Guralnick, Department of Mathematics, University of Southern California, DRB 155, 1042 W. 36th Place, Los Angeles, CA 90089-1113, U.S.A. E-mail: [email protected] Allen Herman, Department of Mathematics and Statistics, University of Regina, Regina, SK Canada S4S 0A2, Canada E-mail: [email protected] Gerhard Hiss, Lehrstuhl D für Mathematik, RWTH Aachen, Templergraben 64, 52062 Aachen, Germany E-mail: [email protected] Andrei Jaikin-Zapirain, Departamento de Matemáticas, Facultad de Ciencias, Universidad Autónoma de Madrid, Cantoblanco Ciudad Universitaria, 28048 Madrid, Spain E-mail: [email protected]

408

List of contributors

Ben Klopsch, Mathematisches Institut, Heinrich-Heine-Universität, 40225 Düsseldorf, Germany E-mail: [email protected] Inna Korchagina, Department of Mathematics, Rutgers University, New Brunswick, NJ 08903, U.S.A. E-mail: [email protected] Mark Lewis, Department of Mathematics, Kent State University, Kent, OH 44242, U.S.A. E-mail: [email protected] Frank Lübeck, Lehrstuhl D für Mathematik, RWTH Aachen, 52056 Aachen, Germany E-mail: [email protected] Gunter Malle, Fachbereich Mathematik/Informatik, Universität Kassel, HeinrichPlett-Straße 40, D-34132 Kassel, Germany E-mail: [email protected] Barry Monson, Department of Mathematics and Statistics, University of New Brunswick, Fredericton, NB Canada E3B 5A3, Canada E-mail: [email protected] Valerio Monti, Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate, Università degli Studi di Roma ‘La Sapienza’, via Scarpa, 16, 00161 Roma, Italy E-mail: [email protected] Alexander Moretó, Departamento de Matemáticas, Facultad de Ciencias, Universidad del País Vasco, Apartado 644, Bilbao, Spain E-mail: [email protected] Daniel K. Nakano, Department of Mathematics, University of Georgia, Athens, GA 30602, U.S.A. E-mail: [email protected] Gabriel Navarro, University of Valencia, Dr. Moliner 50, 46100 Burjassot, València, Spain E-mail: [email protected] Alan R. Prince, School of Mathematical and Computer Sciences, Heriot-Watt University, Edinburgh EH14 4AS, United Kingdom E-mail: [email protected] Cornelius Pillen, Department of Mathematics and Statistics, ILB 325, University of South Alabama, Mobile, AL 36688, U.S.A. E-mail: [email protected] Urmie Ray, Université de Reims, 34 Rue des Cheneaux, Sceaux 92330, France E-mail: [email protected] Geoffrey R. Robinson, University of Birmingham, School of Mathematics and Statistics, Edgbaston, Birmingham B15 2TT, United Kingdom E-mail: [email protected]

List of contributors

409

Mark Ronan, Department of Mathematics (M/C 249), University of Illinois at Chicago, 851 S. Morgan, Chicago, IL 60607-0745, U.S.A. E-mail: [email protected] Josu Sangroniz, Departamento de Matemáticas, Facultad de Ciencias, Universidad del País Vasco, Apartado 644, Bilbao, Spain E-mail: [email protected] Carlo Scoppola, Dipartimento di Matematica Pura ed Applicata, Università degli Studi di L’Aquila, Via Vetoio, 67010 Coppito (L’Aquila) AQ, Italy E-mail: [email protected] Ronald M. Solomon, Department of Mathematics, Ohio State University, 231 W. 18th St, Columbus, OH 43210-1101, U.S.A. E-mail: [email protected] M. Chiara Tamburini, Dipartimento di Matematica e Fisica, Università Cattolica del Sacro Cuore, Via Musei 41, 25121 Brescia, Italy E-mail: [email protected] Franz G. Timmesfeld, Mathematisches Institut, Arndtstraße 2, 35392 Giessen, Germany E-mail: [email protected] Lawrence E. Wilson, Department of Mathematics, University of Florida, PO Box 118105, 358 Little Hall, Gainesville, FL 32611-8105, U.S.A. E-mail: [email protected] Thomas R. Wolf, Department of Mathematics, Ohio University, Athens, OH 45701, U.S.A. E-mail: [email protected] Alexandre E. Zalesski, School of Mathematics, University of East Anglia, Norwich NR4 7TJ, United Kingdom E-mail: [email protected] Jiping Zhang, Lmam, The School of Mathematical Sciences, Peking University, 100871, Beijing, China E-mail: [email protected]

List of participants Hossein Andikfar, Department of Mathematics (M/C 249), University of Illinois at Chicago, 851 S. Morgan, Chicago, IL 60607-0745, U.S.A. E-mail: [email protected] Michael Aschbacher, Department of Mathematics, California Institute of Technology, Pasadena CA 91125-0001, U.S.A. E-mail: [email protected] Christine Bachoc, Laboratoire d’Algorithmique Arithmétique, Université Bordeaux 1, 33405 Talence, France E-mail: [email protected] Aron Bereczky, Sultan Qaboos University, Department of Mathematics, P.O.B. 36, Al Khodh 123, Sultanate of Oman E-mail: [email protected] Robert Boltje, Department of Mathematics, University of California, Santa Cruz, CA 95064, U.S.A. E-mail: [email protected] Michel Broué, Institut Henri Poincaré, University of Paris VII, 11 rue Pierre et Marie Curie, 75005 Paris, France E-mail: [email protected] Douglas Brozovic, Department of Mathematics, University of North Texas, PO Box 311430, Denton, TX 76203, U.S.A. E-mail: [email protected] Wai Kiu Chan, Department of Mathematics, Wesleyan University, Middletown, CT 06459, U.S.A. E-mail: [email protected] David Chandler, Department of Mathematics, University of Delaware, Newark, DE 19716-2553, U.S.A. E-mail: [email protected] Naoki Chigira, Department of Mathematical Sciences, Muroran Institute of Technology, Hokkaido 050-8585, Japan E-mail: [email protected] Sarah Cunningham, Department of Mathematics, University of California, Santa Cruz, CA 95064, U.S.A. E-mail: [email protected] Ulrich Dempwolff, Universität Kaiserslautern, Fachbereich Mathematik, Postfach 3049, 67653 Kaiserslautern, Germany E-mail: [email protected]

412

List of participants

Eloisa Detomi, Dipartimento di Matematica pura ed aplicata, Università di Padova, Via G. Belzoni, 7, 35131 Padova, Italy E-mail: [email protected] Richard Dipper, Mathematisches Institut B Fakultät Mathematik, Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany E-mail: [email protected] Gary Ebert, Department of Mathematical Sciences, University of Delaware, Newark, DE 19716, U.S.A. E-mail: [email protected] Harald Ellers, Department of Mathematical Sciences, Northern Illinois University, DeKalb, IL 60115-2888, U.S.A. E-mail: [email protected] Jacqueline Espina, Department of Mathematics, University of California, Santa Cruz, CA 95064, U.S.A. E-mail: [email protected] Clara Franchi, Dipartimento di Matematica e Fisica, Università Cattolica del Sacro Cuore, via Trieste 17, 25121 Brescia, Italy E-mail: [email protected] Michael D. Fried, Department of Mathematics, University of California Irvine, Irvine, CA 92697-3875, U.S.A. E-mail: [email protected] Daniel Frohardt, Department of Mathematics, Wayne State University, Detroit, MI 48202, U.S.A. E-mail: [email protected] Meinolf Geck, Institut Girard Desargues, Université Claude Bernard-Lyon I, 21 Avenue Claude Bernard, 69622 Villeurbanne Cedex France E-mail: [email protected] Stephen P. Glasby, Department of Mathematics, Central Washington University, Ellensburg, WA 98926, U.S.A. E-mail: [email protected] George Glauberman, Department of Mathematics, University of Chicago, 5734 S. University Ave., Chicago, IL 60637, U.S.A. E-mail: [email protected] Adam Glesser, Department of Mathematics, University of California, Santa Cruz, CA 95064, U.S.A. E-mail: [email protected] David Gluck, Department of Mathematics, Wayne State University, Detroit, MI 48202, U.S.A. E-mail: [email protected]

List of participants

413

Jon González, Departamento de Matematicas, Facultad de Ciencias, Universidad del País Vasco, Apartado 644, Bilbao, Spain E-mail: [email protected] Roderick Gow, National University of Ireland, Belfield, Dublin 4, Ireland E-mail: [email protected] Ralf Gramlich, TU Darmstadt, FB Mathematik / AG 5, Schlossgartenstrasse 7, 64289 Darmstadt, Germany E-mail: [email protected] Robert M. Guralnick, Department of Mathematics, University of Southern California, DRB 155, 1042 W. 36th Place, Los Angeles, CA 90089-1113, U.S.A. E-mail: [email protected] David Hemmer, Department of Mathematics, University of Georgia, Athens, GA 30602, U.S.A. E-mail: [email protected] Allen Herman, Department of Mathematics and Statistics, University of Regina, Regina, SK, Canada S4S 0A2 E-mail: [email protected] Marcel Herzog, School of Mathematical Sciences, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel-Aviv University, Tel-Aviv, Israel E-mail: [email protected] Gerhard Hiss, Lehrstuhl D für Mathematik, RWTH Aachen, Templergraben 64, 52062 Aachen, Germany E-mail: [email protected] Chat Yin Ho, Department of Mathematics, University of Florida, PO Box 118105, 358 Little Hall, Gainesville, FL 32611-8105, U.S.A. E-mail: [email protected] Corneliu Hoffman, Department of Mathematics and Statistics, Bowling Green State University, Bowling Green, OH 43403, U.S.A. E-mail: [email protected] Alexander Hulpke, Department of Mathematics, Colorado State University, Fort Collins, CO 80523-1874, U.S.A. E-mail: [email protected] Andrei Jaikin-Zapirain, Departamento de Matemáticas, Facultad de Ciencias, Universidad Autónoma de Madrid, Cantoblanco Ciudad Universitaria, 28048 Madrid, Spain E-mail: [email protected] Michael Kallaher, Department of Pure and Applied Mathematics, Washington State University, Pullman, WA 99164-3113, U.S.A. E-mail: [email protected]

414

List of participants

Thomas Keller, Department of Mathematics, Southwest Texas State University, 601 University Drive, San Marcos, TX 78666-4604, U.S.A. E-mail: [email protected] Radha Kessar, Ohio State University, Department of Mathematics, 231 W 18th St., Columbus, OH 43210-1101, U.S.A. E-mail: [email protected] Alexander S. Kleshchev, Department of Mathematics, University of Oregon, Eugene, OR 97403, U.S.A. E-mail: [email protected] Ben Klopsch, Mathematisches Institut, Heinrich-Heine-Universität, 40225 Düsseldorf, Germany E-mail: [email protected] Inna Korchagina, Department of Mathematics, Rutgers University, New Brunswick, NJ 08903, U.S.A. E-mail: [email protected] Paul Lescot, INSSET, Université de Picardie, 48 Rue Raspail, 02100 Saint-Quentin, France E-mail: [email protected] Mark Lewis, Department of Mathematics, Kent State University, Kent, OH 44242, U.S.A. E-mail: [email protected] Klaus Lux, Department of Mathematics, University of Arizona, 617 N Santa Rita, Tucson, AZ 85721, U.S.A. E-mail: [email protected] Frank Lübeck, Lehrstuhl D für Mathematik, RWTH Aachen, 52056 Aachen, Germany E-mail: [email protected] Richard Lyons, Department of Mathematics, Rutgers University, New Brunswick, NJ 08903, U.S.A. E-mail: [email protected] Kay Magaard, Department of Mathematics, Wayne State University, Detroit, MI 48202, U.S.A. E-mail: [email protected] Mario Mainardis, Dipartimento di Matematica ed Informatica, Università di Udine, via delle Scienze 206, 33100 Udine, Italy E-mail: [email protected] Gunter Malle, Fachbereich Mathematik/Informatik, Universität Kassel, HeinrichPlett-Straße 40, 34132 Kassel, Germany E-mail: [email protected] David McNeilly, Department of Mathematical and Statistical Sciences, University of Alberta, 632 Central Academic Building, Edmonton, Alberta T6G 2G1, Canada E-mail: [email protected]

List of participants

415

Alexander Moretó, Departamento de Matematicas, Facultad de Ciencias, Universidad del País Vasco, Apartado 644, Bilbao, Spain E-mail: [email protected] Marta Morigi, Dipartimento di Matematica, Università di Bo

John Thompson in his Little Hall office Photo: Jane Dominguez, Courtesy CLAS News & Publications.

Finite Groups 2003 Proceedings of the Gainesville Conference on Finite Groups March 612, 2003

Editors C. Y. Ho P. Sin P. H. Tiep A. Turull

≥ Walter de Gruyter · Berlin · New York

Editors Chat Yin Ho Department of Mathematics University of Florida PO Box 118105 358 Little Hall Gainesville, FL 32611-8105 USA e-mail: [email protected]

Peter Sin Department of Mathematics University of Florida PO Box 118105 358 Little Hall Gainesville, FL 32611-8105 USA e-mail: [email protected]

Pham Huu Tiep Department of Mathematics University of Florida PO Box 118105 358 Little Hall Gainesville, FL 32611-8105 USA e-mail: [email protected]

Alexandre Turull Department of Mathematics University of Florida PO Box 118105 358 Little Hall Gainesville, FL 32611-8105 USA e-mail: [email protected]

Mathematics Subject Classification 2000: 20-06; 05Bxx, 17Bxx, 20Cxx, 20Dxx, 20Exx, 20Fxx, 20Gxx, 20Jxx Keywords: buildings, classification of finite simple groups, cohomology of groups, finite geometries, finite p-groups, Lie algebras and superalgebras, representation theory of finite and algebraic groups

P Printed on acid-free paper which falls within the guidelines of the E ANSI to ensure permanence and durability.

Library of Congress Cataloging-in-Publication Data Finite Groups 2003 (2003 : Gainesville, Fla.) Finite Groups 2003 : proceedings of the Gainesville conference on finite groups, March 612, 2003 / edited by Chat Yin Ho … [et al.]. p. cm. ISBN 3-11-017447-2 (cloth : alk. paper) 1. Finite groups Congresses. I. Ho, Chat-Yin, 1946 II. Title. QA174.F56 2004 5121.23dc22 2004021279

ISBN 3-11-017447-2 Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at http://dnb.ddb.de. ” Copyright 2004 by Walter de Gruyter GmbH & Co. KG, 10785 Berlin, Germany. All rights reserved, including those of translation into foreign languages. No part of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording or any information storage and retrieval system, without permission in writing from the publisher. Printed in Germany. Cover design: Thomas Bonnie, Hamburg. Typeset using the authors’ TEX files: I. Zimmermann, Freiburg. Printing and binding: Hubert & Co. GmbH & Co. KG, Göttingen.

To John Thompson on the occasion of his 70th birthday

Preface

During the Academic Year 2002–2003, a number of events took place on both sides of the Atlantic to celebrate the 70th Birthday of our illustrious colleague and friend Professor John G. Thompson. A conference was held in Cambridge, England, in Thompson’s honor. A special issue of the Journal of Algebra was edited to mark the occasion. Here in Gainesville, the occasion was marked by the University of Florida mainly with two events. The Thompson Assistant Professorship was launched, and the whole Academic Year 2002–2003 was a Special Year in Algebra. One of the highlights of the Special Year in Algebra was the conference Finite Groups 2003 of which the present volume is the Proceedings. We are grateful to the Mathematics Department, the College of Liberal Arts and Sciences, and the Office of Research and Graduate Programs at the University of Florida, as well as the National Science Foundation, and the National Security Agency for their support of the Special Year in Algebra in general, and the conference Finite Groups 2003 in particular. Finite Groups 2003 took place in Gainesville from March 6, 2003 to March 12, 2003. As befits a conference in honor of John Thompson, talks on many different topics were given. A list of speakers and their topics is included below. Many of the speakers emphasized the relationship of their work to the work of John Thompson. At the conference banquet, Professor Broué presented John Thompson with a copy of the special issue of the Journal of Algebra edited in honor of his 70th Birthday. The present Proceedings is a collection of articles in honor of John Thompson. They do not represent a record of the talks as given, but, instead, they are articles prepared specifically for these Proceedings. Each article has undergone a strict refereeing process, and we are sorry that, because of the scope and the size of the proceedings, not all interesting articles submitted could be accepted. Most of the articles here present original research results with their proofs. A few are survey articles written by leading researchers. It was the intention of all participants at the conference to honor John Thompson. We hope these Proceedings do honor him, for we, as indeed all group theorists do, owe him a great debt of gratitude for helping make our subject as interesting as it is today. Gainesville, August 2004

C. Y. Ho, P. Sin, P. H. Tiep and A. Turull

List of talks M. Aschbacher (California Institute of Technology): A question of Farjoun C. Bachoc (Université Bordeaux): Codes and designs in Grassmannian spaces A. Bereczky (Sultan Qaboos University): Standard bases for unipotent elements in classical groups R. Boltje (University of California, Santa Cruz): Characterizations of Adams operations on certain representation rings M. Broué (Institut Henri Poincaré): Representations of reductive groups in transverse characteristic W. K. Chan (Wesleyan University): On almost strong approximation for algebraic groups defined over number fields U. Dempwolff (Universität Kaiserslautern): On automorphisms of symmetric designs R. Dipper (Universität Stuttgart): The rational Schur algebra G. Ebert (University of Delaware): Odd order flag-transitive affine planes D. Frohardt (Wayne State University): Genus zero actions M. Geck (Université Claude Bernard-Lyon I): On the computation of some Schur indices for cuspidal unipotent characters S. Glasby (Central Washington University): Modules over non-algebraically closed fields induced from a normal subgroup of prime index G. Glauberman (University of Chicago): An extension of Thompson’s replacement theorem by algebraic group methods D. Gluck (Wayne State University): The final stage of the proof of the k(GV )conjecture J. González (Universidad del País Vasco): The power structure of potent p-groups R. Gow (National University of Ireland): The arithmetic of Steinberg lattices of a Chevalley group R. Gramlich (Technische Universität Darmstadt): Phan-type theorems R. Guralnick (University of Southern California): Conjugacy classes and the noncoprime k(GV )-problem D. Hemmer (University of Georgia): Specht module filtrations for representations of the symmetric group M. Herzog (Tel-Aviv University): The existence of normal subgroups in finite groups G. Hiss (RWTH Aachen): Some observations on products of characters of finite classical groups A. Jaikin-Zapirain (Universidad Autónoma de Madrid): The number of finite pgroups with bounded number of generators

x

List of talks

M. Kallaher (Washington State University): On semiovals T. Keller (Southwest Texas State University): On the k(GV )-conjecture R. Kessar (Ohio State University): Shintani descent and perfect isometries for blocks of finite general linear groups A. Kleshchev (University of Oregon): W -algebras B. Klopsch (Universität Düsseldorf): Enumerating highly non-soluble groups I. Korchagina (Rutgers University): Groups of simultaneously even- and p-type P. Lescot (Université de Picardie): Variations on Thompson’s J -subgroup M. Lewis (Kent State University): Character degree graphs with 5 vertices K. Lux (University of Arizona): A database for basic algebras of simple groups R. Lyons (Rutgers University): Second generation proof G. Malle (Universität Kassel): Families of characters for complex reflection groups D. McNeilly (University of Alberta): Realizability of Weil representations A. Moretó (Universidad del País Vasco): Heights of characters and defect groups G. Navarro (Universitat de València): Problems on characters and Sylow subgroups S. Onofrei (Kansas State University): A point-line characterization of the building of type E7 C. Pillen (University of South Alabama): Extensions for finite Chevalley groups, Frobenius kernels, and algebraic groups A. Prince (Heriot-Watt University): Oval configurations of involutions in the symmetric group D. Riley (Western Ontario University): Beyond the restricted Burnside problem G. Robinson (University of Birmingham): Upper bounds for numbers of characters M. Ronan (University of Illinois at Chicago): Classification of buildings J. Sangroniz (Universidad del País Vasco): Characters of algebra groups N. Sastry (Indian Statistical Institute): On the conjugacy classes of finite Suzuki and Ree groups M. Sawabe (Naruto University of Education): On a p-local geometry associated with a p-subgroup complex C. Scoppola (Università degli Studi di L’Aquila): Space and loop Lie algebras L. Scott Jr. (University of Virginia): Some new examples in 1-cohomology G. Seitz (University of Oregon): Unipotent centralizers and saturation J. Shareshian (Washington Universtity): Topology of intervals in subgroup lattices S. Shpectorov (Bowling Green State University): Classification of amalgams

List of talks

xi

S. Smith (University of Illinois at Chicago) : Quasithin groups – in retrospect R. Solomon (Ohio State University): The signalizer method B. Srinivasan (University of Illinois at Chicago): Remarks on Dade’s conjecture for GL(n, q) K. Uno (Osaka University): Some variations of conjectures on character degrees L. Wilson (University of Florida): Large powerful subgroups of p-groups T. Wolf (Ohio University): Regular and large orbits of induced modules A. E. Zalesski (University of East Anglia): Minimal polynomials of group elements in linear representations of finite groups J. Zhang (Peking University): On p-local rank of finite groups

Table of contents

Preface

vii

List of talks

ix

Michael Aschbacher On a question of Farjoun

1

Christopher P. Bendel, Daniel K. Nakano and Cornelius Pillen Extensions for finite groups of Lie type: twisted groups

29

David J. Benson On the classifying space and cohomology of Thompson’s sporadic simple group 47 Stephen Doty New versions of Schur–Weyl duality

59

Norberto Gavioli, Valerio Monti and Carlo M. Scoppola Just infinite periodic Lie algebras

73

Meinolf Geck On the Schur indices of cuspidal unipotent characters

87

George Glauberman An extension of Thompson’s Replacement Theorem by algebraic group methods

105

Ralf Gramlich Simple connectedness of the geometry of nondegenerate subspaces of a symplectic space over arbitrary fields

111

Robert M. Guralnick and Gunter Malle Classification of 2F -modules, II

117

Allen Herman and Barry Monson On the real Schur indices associated with infinite Coxeter groups

185

Gerhard Hiss and Frank Lübeck Some observations on products of characters of finite classical groups

195

Andrei Jaikin-Zapirain The number of finite p-groups with bounded number of generators

209

xiv

Table of contents

Benjamin Klopsch Enumerating highly non-soluble groups

219

Inna Korchagina 3-Signalizers in almost simple groups

229

Mark L. Lewis Classifying character degree graphs with 5 vertices

247

Alexander Moretó Heights of characters and defect groups

267

Gabriel Navarro Problems on characters and Sylow subgroups

275

Alan R. Prince Ovals in finite projective planes via the representation theory of the symmetric group

283

Urmie Ray Rank invariance and automorphisms of generalized Kac–Moody superalgebras 291 Geoffrey R. Robinson Bounding numbers and heights of characters in p-constrained groups

307

Mark A. Ronan Classification of buildings

319

Josu Sangroniz Characters of algebra groups and unitriangular groups

335

Ronald Solomon The signalizer method

351

M. Chiara Tamburini and Alexandre E. Zalesski Classical groups in dimension 5 which are Hurwitz

363

Franz G. Timmesfeld A classification of Weyl-groups as finite {3, 4}-transposition groups

373

Lawrence E. Wilson Powerful subgroups of 2-groups

381

Thomas R. Wolf Regular orbits of induced modules of finite groups

389

Table of contents

xv

Jiping Zhang Radical subgroups and p-local ranks

401

List of contributors

407

List of participants

411

On a question of Farjoun Michael Aschbacher∗

Let G be a group, H a subgroup of G, Hom(H, G) the set of all group homomorphisms from H into G, and End(G) = Hom(G, G). Following E. Farjoun, we say the embedding of H in G is closed if each member of Hom(H, G) extends uniquely to a member of End(G). Example 1. G is closed in itself for each group G. Example 2. Let H and G be simple. Then H is closed in G iff Aut(G) is transitive on subgroups of G isomorphic to H , Aut Aut(G) (H ) = Aut(H ), and CAut(G) (H ) = 1. In particular if G is the alternating group of degree n, and H is the stabilizer of a point, then H is closed in G for each n > 5 with n = 7. As part of his investigation of idempotent augmented functors, Farjoun posed the following question: Question 1. If H is a finite closed nilpotent subgroup of a group G, is G = H ? For technical reasons it is sometimes easier to work with a slightly weaker condition. Define H to be nearly closed in G if each member of End(H ) extends uniquely to a member of End(G). Question 2. If H is a finite nearly closed nilpotent subgroup of a group G, is G = H ? In this paper we introduce some machinery useful in investigating Question 2 and present some partial results which show the Question has a positive answer in various special cases. For example we reduce Question 2 to the case where H is a p-group for some prime p, we show Question 2 has an affirmative answer when G is finite and the nilpotence class of H is at most 3, and we show Question 2 has an affirmative answer when G is finite for other interesting classes of p-groups. Here are some specifics: Let C be a class of groups such that if G ∈ C and H ≤ G then H ∈ C. Define a group P to be C-rigid or rigid with respect to C if whenever P is nearly closed in G ∈ C then P = G. Our first result says: Theorem 1. Let C be a class of groups and P1 and P2 groups which are C-rigid. Then P1 × P2 is C-rigid. ∗ This work was partially supported by NSF-0203417.

2

Michael Aschbacher

As finite nilpotent groups are the direct product of their Sylow groups, Theorem 1 reduces Question 2 to the case where H is a p-group, so in the remainder of this introduction assume G is a group, p is a prime, and P is a finite p-subgroup of G. Moreover in the rest of the introduction we work in the class C of finite groups, so G is assumed to be finite. The next result shows Question 2 has a positive answer when P is large: Theorem 2. If P is nearly closed in a finite group G and P ∈ Sylp (G) then G = P . Theorem 3. If P is metabelian or of nilpotence class at most 3, then P is rigid with respect to the class of finite groups. A. Mann has a slick proof that groups of class 2 are rigid in the class of all groups; indeed he shows that if P is closed in G then Z2 (P ) ≤ Z2 (G). The method used to prove Theorem 3 can be used to treat other classes of p-groups, but there are obstructions to a simple minded extension of the approach. See Section 9 for more discussion and a sketch of a proof that Question 2 has a positive answer for groups P of class at most 5 generated by 3 elements of order p. One tool for investigating Question 2 involves the relationship between “splittings” of P and G, where a splitting of P is a decomposition of P as a split extension. In Section 2 we consider a “splitting category” and define a class of “splitting solvable” groups, which includes many interesting p-groups. Then we prove: Theorem 4. If P is splitting solvable then P is rigid with respect to the class of finite groups. Finally the near closure property does not inherit to subgroups or homomorphic images, so while the condition is very strong, it is not easy to exploit using traditional techniques from finite group theory. On the other hand various properties following from near closure do inherit. Thus in Section 10 we investigate classes of finite groups satisfying such properties. We would like to show the subnormal closure of P in each group G in the class is P -nilpotent (i.e. G = Op (G)P ); such a result would give an affirmative answer to Question 2 when G is solvable, and give very strong information in the nonsolvable case. Unfortunately examples show this result is not true in general for the classes we’ve considered, even when G is solvable. Nevertheless we include an abbreviated discussion of this approach in one case, since that discussion does at least lead to our final theorem, and since in addition the result may be true if other properties are adjoined to obtain smaller classes. Theorem 5. If p is an odd prime, G is a finite solvable group, and P is a p-subgroup of G such that CG (P ) = Z(G), then Op (G) is the largest p -signalizer for P . Recall a p -signalizer for P is a P -invariant p -subgroup of G. The reader is directed to [2] for basic notation, terminology, and results involving finite groups.

On a question of Farjoun

3

The author would like to thank Yoav Segev for drawing his attention to Farjoun’s question, for many helpful conversations about the problem, and for suggesting improvements to the proofs of a number of lemmas in this paper.

1. Monoids and semigroups We will be concerned with the monoid End(G) of a finite group G, so we begin with some discussion of monoids. In this section M is a finite semigroup; that is M is a set with an associative binary operation · . Recall M is a monoid if M has an identity 1. If an identity exists, it is unique. A zero for M is an element 0 ∈ M such that for each x ∈ M, 0 · x = 0 = x · 0. Again if M has a 0 then it is unique. For X ⊆ M, write [X] for the subsemigroup of M generated by X. Thus [X] consists of all products x1 . . . xn with xi ∈ X for all i. In particular for x ∈ X, [x] = {x n : 0 < n ∈ Z}. An idempotent of M is an element x ∈ M such that x 2 = x. An element y ∈ M is said to be split if y n = y for some n > 1. 1.1. Let a ∈ M, k the least positive integer such that a k = a j for some j > k, and q the least positive integer such that a k+q = a k . Set B = {a j : j ≥ k}. Then: (1) B is a cyclic subgroup of M of order q. (2) B is the set of split elements in [a]. (3) Let sq be the first positive multiple of q such that k ≤ sq. Then a sq is the identity of B and B = a t , for each t ≥ k with (q, t) = 1. Proof. As a k = a k+q , by induction on i, a k+i = a k+i+q for each i ≥ 0. Then by induction on r: (i) a m+rq = a m for each m ≥ k and r ≥ 0. Hence: (ii) B = {a j : j ∈ J }, where J = {k + i : 0 ≤ i < q}. From (i) we see that e := a rq is the identity element of B, for each r such that rq ≥ k, and hence, for each b ∈ B, bq = e. It follows that for each b ∈ B, bq−1 is the inverse of b and B is a group. If a j1 = a j2 for some j1 ≤ j2 in J , then multiplying by a (s+1)q−j1 , we get that e = a (s+1)q = a (s+1)q+j2 −j1 . Multiplying by a k we get a k = a k+(s+1)q+j2 −j1 = a k+j2 −j1 , because a (s+1)q is the identity element of B. But the equality a k = a k+j2 −j1 contradicts the minimality of q, unless j1 = j2 . It follows that |B| = q. Let k ≤ t ∈ Z with (t, q) = 1, and set c := a t . As (t, q) = 1: (iii) tJ = J mod q, so B = [c] by (i)–(iii). This establishes (1) and (3). Finally if a r is split then a r = a t for some t > r, so r ≥ k by minimality of k. Thus (2) holds.

4

Michael Aschbacher

For a ∈ M, let sp(a) = {b ∈ [a] : b is split}. By 1.1, sp(a) is a group, so it has a unique identity id(a). 1.2. Let a ∈ M. (1) sp(a) is a cyclic subgroup of M. (2) id(a) is the unique idempotent in [a]. Proof. Part (1) follows from 1.1. Let e = id(a). Then e2 = e so e is an idempotent. Conversely if f ∈ [a] is an idempotent then f is split, so f ∈ sp(a), and hence e = f as groups have a unique idempotent. Define an equivalence relation ∼ on M by a ∼ b if id(a) = id(b). Write a˜ for the equivalence class of a under ∼, and set sp(a) ˜ = {b : a ∼ b and b is split} 1.3. sp(a) ˜ is a subgroup of M with identity id(a). Proof. Let e = id(a) and H the set of x ∈ M such that xe = ex = x and yx = xy = e for some y ∈ M. It is easy to check that H is a subgroup of M. Then since M is finite, e ∈ [h] and h is split for each h ∈ H , so e = id(h) by 1.2.2. Thus H ⊆ sp(a), ˜ and of course sp(a) ˜ ⊆ H , so the lemma holds. Define a relation ≤ on M by b ≤ a if ab = ba = b. For a ∈ M let M(≤ a) = {b ∈ M : b ≤ a}. Write for the set of idempotents in M. 1.4. .

(1) ≤ is antisymmetric.

(2) ≤ is transitive. (3) a ≤ a iff a ∈ . (4) ≤ is a partial order on . (5) If M has 0 or 1 then 0 ≤ a ≤ 1 for all a ∈ M. (6) For each a ∈ , M(≤ a) is a subsemigroup of M with identity a. (7) If a ∈ then M(≤ a) = aMa and sp(a) ˜ ⊆ M(≤ a). ˜ ⊆ M(≤ a). (8) If a, b ∈ and b ≤ a then sp(b) Proof. If b ≤ a and a ≤ b then a = ab = b, so (1) holds. Let c ≤ b ≤ a. Then ac = a(bc) = (ab)c = bc = c,

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and similarly ca = c. Thus (2) holds. Next a ≤ a iff a 2 = a, so (3) holds. Of course (1)–(3) imply (4). Part (5) is easy. If b, c ≤ a then (bc)a = b(ca) = bc and similarly a(bc) = bc, so (6) holds. Let a ∈ . Then aMa ⊆ M(≤ a) and for x ∈ M(≤ a), x = axa ∈ aMa. Then as a is an identity for the group sp(a), ˜ (7) holds. By (2) and ˜ ⊆ M(≤ b) ⊆ M(≤ a), so (8) holds. (7), if a, b ∈ with b ≤ a, then sp(b)

2. Splittings of groups In this section we establish a category to investigate certain questions involving split extensions of groups which arise in the study of Farjoun’s Question. There is some overlap between our treatment and that of Bechtell in [1]. For example our notion of “splitting series” corresponds to Bechtell’s “complementary splitting systems” and our “splitting simple groups” are his “inseperable groups”. We thank Yoav Segev for calling our attention to Bechtell’s work. In this section G is a group. A splitting of G is a pair (K, C) where K is a normal subgroup of G and C is a complement to K in G. Let S be the set of splittings of G and partially order S by (K, C) ≤ (H, B) if C ≤ B and H ≤ K. Call K the kernel of the splitting and C the complement. Write 0 for the splitting (G, 1) and 1 for the splitting (1, G). Thus 0 and 1 are the least and greatest elements of the poset S, respectively. These splitting are the trivial splittings and the remaining splittings are said to be nontrivial. Define G to be splitting simple if G has no nontrivial splittings. That is G is splitting simple if G cannot be realized as a nontrivial split extension; equivalently there are no nontrivial semidirect product decompositions of G. Of course simple groups are splitting simple but the converse is not true: Example 2.1. A finite abelian group G is splitting simple iff G is a cyclic p-group for some prime p. Example 2.2. Quaternion groups are splitting simple. A splitting series for G is a series 0 = S0 ≤ S1 ≤ · · · ≤ Sn = 1 of splittings Si = (Ki , Ci ). Define a splitting series S = (Si : 0 ≤ i ≤ n) to be maximal if for each 0 ≤ i < n, Si < Si+1 and there exists no splitting S with Si < S < Si+1 . For ≤ Aut(G), define a splitting (K, C) of G to be a -splitting if K and C are -invariant. Define G to be -splitting simple (or -simple for short) if there are no nontrivial -splittings of G. Define a splitting (K, C) of G to be simple if K is C-splitting simple.

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2.3. Let S = (Si : 0 ≤ i ≤ n) be a splitting series for G. Then the following are equivalent: (1) S is maximal. (2) For each 0 ≤ i < n, Ki ∩ Ci+1 is Ci -simple. (3) For each 0 ≤ i < n, the splitting (Ki ∩ Ci+1 , Ci ) of Ci+1 is simple. Proof. By definition, (2) and (3) are equivalent. If Si < (K, C) < Si+1 , then 1 < Ki ∩ C < Ki ∩ Ci+1 . Now by the modular property of groups (K∩Ci+1 )C∩Ki = (K∩Ci+1 )(C∩Ki ), and Ci+1 = CK∩Ci+1 = C(K ∩ Ci+1 ). Thus (K ∩ Ci+1 , C ∩ Ki ) is a nontrivial Ci -splitting of Ci+1 ∩ Ki , so (2) implies (1). Conversely suppose (J0 , A0 ) is a nontrivial Ci -splitting of Ki ∩ Ci+1 . Set J = J0 Ki+1 and A = A0 Ci . As (J0 , A0 ) is a Ci -splitting, Ci acts on J0 and A0 , so A is a subgroup of G and Ci+1 = Ci Ki ∩ Ci+1 = Ci (Ki ∩ Ci+1 ) acts on J0 . Thus J is normal in Ki+1 Ci+1 = G. By construction, Ki+1 ≤ J ≤ Ki , and Ci ≤ A ≤ Ci+1 . As Ci is a complement to Ki ∩Ci+1 in Ci+1 and A0 is a complement to J0 in Ki ∩Ci+1 , A is a complement to J0 in Ci+1 . Next J ∩ Ci+1 = J0 Ki+1 ∩ Ci+1 = J0 (Ki+1 ∩ Ci+1 ) = J0 , so as A is a complement to J0 in Ci+1 , J ∩ A = 1 and G = Ki+1 Ci+1 = Ki+1 J0 A = J A. Thus (J, A) is a splitting of G with (Ki , Ci ) < (J, A) < (Ki+1 , Ci+1 ), completing the proof that (1) implies (2). We next define a category we call the splitting category. We first need a lemma and some notation. 2.4. Let α : G → G be a surjective group homomorphism, J = ker(α), and (K, C) a splitting of G. Then (Kα, Cα) is a splitting of G iff (J ∩ K, J ∩ C) is a splitting of J . Proof. As G = KC, G = Gα = KαCα, and as K G, Kα Gα = G . Thus (Kα, Cα) is a splitting of G iff Kα ∩ Cα = 1. Let x ∈ G ; then x ∈ Kα ∩ Cα iff x = kα = cα for some k ∈ K and c ∈ C iff kc−1 ∈ J . Thus (Kα, Cα) is a splitting iff (a) If k ∈ K and c ∈ C with kc−1 ∈ J , then k, c ∈ J . Now observe that (a) is equivalent to (b): (b) J = (K ∩ J )(C ∩ J ). Suppose (a) holds and let j ∈ J . As G = KC, j = ya for some y ∈ K and a ∈ C. Then y, a ∈ J by (a), so (b) holds. Conversely assume (b) holds and kc−1 ∈ J . By (b), kc−1 = zb for some z ∈ J ∩ K and b ∈ J ∩ C, so as K ∩ C = 1, k = z and c−1 = b; that is (a) holds. Finally as K ∩ C = 1, (b) holds iff (K ∩ J, C ∩ J ) is a splitting for J . Thus the lemma is established.

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Write sub(G) for the set of subgroups of G and trip(G) for the set product {G} × sub(G) × sub(G). Given a group homomorphism α : G → G , define αˇ : trip(G) → trip(Gα) by αˇ : (G; K, C) → (Gα; Kα, Cα). The objects in the splitting category are the triples S = (G; K, C) such that (K, C) is a splitting of G. The objects in the splitting category are called splittings. A morphism from S to S = (G ; K , C ) is a map of the form αˇ for some surjective group homomorphism α : G → G such that S αˇ = S . Composition of morphisms is composition of functions. ˇ = S . The The image or cokernel of a morphism αˇ : S → S is the splitting im(α) kernel of αˇ is the triple ker(α) ˇ = (ker(α); K ∩ ker(α), C ∩ ker(α)); by 2.4, ker(α) ˇ is a splitting. Define the set of normal splittings N (S) of S to be the set of splittings (H ; K ∩ H, C ∩ H ) such that H G. Given T = (H, K ∩ H, C ∩ H ) ∈ N(S), define S/T to be the triple (Gπ ; Kπ, Cπ ), where π : G → G/H is the natural map. Call S/T the factor splitting of S modulo T . By 2.4: 2.5 (Homomorphism Theorem for Splittings). Let S = (G; K, C) be a splitting, T = (H ; K ∩ H, C ∩ H ) ∈ N (S), and π : G → G/H the natural map. Then: (1) S/T is a splitting. (2) πˇ : S → S/T is a morphism of splittings. ˇ ∈ N(S). (3) If αˇ : S → S is a morphism of splittings then ker(α) (4) If ker(α) ˇ = T then there exists a unique isomorphism σ : G/H → G such that α = π σ ; moreover σˇ : S/T → S is an isomorphism of splittings. (5) The map R → R πˇ is an injection from the set of splittings R = (G; J, B) with H ≤ J into the set of splittings of S/T . 2.6. Let S = (G; K, C) and T = (G; J, B) be splittings with (K, C) ≤ (J, B), and define ι : G → B by ι : j b → b for j ∈ J and b ∈ B (cf. 3.11.2). Then R = (J ; J, 1) ∈ N (S) and ιˇ : S → (B; K ∩ B, C) is a morphism with kernel R. Let S = (Si : 0 ≤ i ≤ n) be a splitting series for G. For 1 ≤ i ≤ n, define the ith factor of S to be the splitting Fi = (Ci ; Ki−1 ∩ Ci , Ci−1 ). By 2.6 and 2.5.4, Fi ∼ = Si /Ri , where Ri = (Ki ; Ki , 1). Thus for example F1 = (C1 ; C1 , 1) and Fn = Sn−1 . Remark 2.7. Question: Is there a Jordan–Hölder Theorem for splittings? The strongest result definitely does not hold: Different maximal splitting series can have different families of factors. For example we just saw that Sn−1 is a factor in the series S, and in a finite group any maximal proper splitting is the penultimate term in some maximal series. One could ask for something weaker. Let K(S) = (Ki−1 /Ki : 1 ≤ i ≤ n) be the family of factor groups in the series of kernels of S; equivalently K(S) is the family (Ki−1 ∩ Ci : 1 ≤ i ≤ n) of kernels in the factors Si /Ri of S. Is the family K(S)

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independent of the maximal splitting series S? If G is an abelian p-group the answer is yes. Namely G = G1 × · · · × Gn is the direct product of cyclic groups and this decomposition is unique up to isomorphism. Moreover the family of factors in this decomposition is the family K(S) in any maximal splitting series for G. However this property also fails in general. If (K, C) is a simple splitting for G such that C is splitting simple, then, by 2.3 S = 0 < (K, C) < 1 is a maximal splitting series for G with K(S) = {K, C}. If G is the dihedral group of order 8 then up to isomorphism there are two such splittings: G is the split extension of E4 by Z2 and of Z4 by Z2 , and the corresponding series have different kernel families. In Example 2.9 we will see more instances of maximal series with different kernel families. However in all these examples, the length of a maximal splitting series is at least constant. Define G to be splitting solvable if there exists a splitting series S for G with each member of K(S) abelian. Of course splitting solvable groups are solvable, but the converse is not true; cf. Example 2.2. From Example 2.9 below, the Sylow p-groups of general linear groups over fields of characteristic p are splitting solvable. A similar argument shows the Sylow groups of symmetric groups are splitting solvable, as those Sylow groups are direct products of iterated wreath products. 2.8. Let p be a prime, G a p-group, and assume |Z(G)| = p. Then each nontrivial splitting of G with an abelian kernel is simple. Proof. Suppose (K, C) is a nontrivial splitting of G with K abelian; we must show K is C-simple. Assume (J, B) is a nontrivial C-splitting of K. As K is abelian, K = J × B, and as the splitting is C-invariant, CJ (C) = 1 = CB (C). But as K = J × B and G = KC, CJ (C)CB (C) ≤ Z(G), contradicting |Z(G)| = p. Example 2.9. Let L = L1 × · · · × Lm be the direct product of general linear groups Li ∼ = GLni (pei ) for some prime p, and let P ∈ Sylp (L); then P = P1 × · · · × Pm , where Pi = P ∩ Li ∈ Sylp (Li ). Let M be a maximal parabolic of L over P . Then M = KCM , where K is the unipotent radical of M and CM is a Levi factor. Moreover (K, CM ) is a splitting of M; (K, C) is a splitting of P , where C = P ∩ CM ; K is abelian; and CM is the product of general linear groups. Thus proceeding by induction on the Lie rank of L, P is splitting solvable. Indeed the Lie rank of L is m l= li , where li = ni − 1 is the Lie rank of Li . i=1

Further the series S of maximal length we obtain via this process are of length l, and K(S) consists of the radicals of maximal parabolics in Levi factors. Now specialize to the case ei = 1 for all i. In this case we claim the series S of length l are maximal splitting series. As series obtained by pasting together maximal series of factors of a direct product is a maximal series for the product, proceeding by

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induction on the Lie rank of L, we may assume m = 1. Then |Z(P )| = p, so by 2.8, the splitting (K, C) is simple. Then as C is a direct product of general linear groups, the claim follows by induction on the Lie rank. Notice however that in general the families of kernels obtained from different series are different. For example in L4 (p) the series are of length 3 and up to isomorphism, the families are {Ep3 , Ep2 , Ep } and {Ep4 , Ep , Ep }.

3. The homomorphism monoid In this section G is a group and M = End(G) regarded as a monoid under the operation of composition of functions. Let α, β, γ ∈ M. 3.1. .

(1) ker(α) ≤ ker(αβ).

(2) Gαβ ≤ Gβ. (3) For m ≤ n positive integers, ker(α m ) ≤ ker(α n ) and Gα n ≤ Gα m . Proof. As ker(α)αβ = 1β = 1, (1) holds. As Gα ≤ G, (2) holds. Then (3) follows from (1) and (2) by induction on n − m. 3.2. If αβ = β then ker(α) ≤ ker(β) and G = ker(β)Gα. Proof. Suppose αβ = β. Then by 3.1.1, ker(α) ≤ ker(αβ) = ker(β). Also for g ∈ G, g(g −1 )α ∈ ker(β), so G = ker(β)Gα. 3.3. If βα = β then α|Gβ = 1Gβ so Gβ ≤ Gα. Proof. Suppose βα = β. Then for a ∈ Gβ, a = gβ for some g ∈ G and aα = gβα = gβ = a. Recall from Section 1 that β is split in M if β m = β for some m > 1. 3.4. If β is split in M then (ker(β), Gβ) is a splitting of G. Proof. By hypothesis β = ββ m−1 = β m−1 β, so by 3.2, G = ker(β)Gβ m−1 and Gβ m−1 ≤ Gβ by 3.1.3, so G = ker(β)Gβ. By 3.3, ker(β m−1 ) ∩ Gβ = 1, while by 3.1.3, ker(β) ≤ ker(β m−1 ), so ker(β) ∩ Gβ = 1. 3.5. Assume α is an idempotent in M. Then: (1) α|Gα = 1Gα , and (2) (ker(α), Gα) is a spltting of G. Proof. Part (1) follows from 3.3 and (2) from 3.4. 3.6. If Gα is finite then α is split in M iff (ker(α), Gα) is a splitting of G.

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Proof. The first statement implies the second by 3.4. Conversely assume the second statement. By 3.1.2, ϕ = α|Gα ∈ Hom(Gα, Gα), and as (ker(α), Gα) is a splitting, 1 = ker(α) ∩ Gα, so ϕ ∈ Aut(Gα). Thus as Gα is finite, n = |ϕ| is an integer and ϕ n+1 = ϕ. Now for g ∈ G, g = xa with x ∈ ker(α) and a ∈ Gα, so gα = xαaα = aα = aϕ, and similarly gα n+1 = aϕ n+1 = aϕ, so α = α n+1 . 3.7. The following are equivalent: (1) α is an idempotent in M. (2) (ker(α), Gα) is a splitting of G and α|Gα = 1Gα . Proof. By 3.5, (1) implies (2). Assume (2). Adopting the notation of the proof of 3.6, that proof shows that gα = aϕ = a, so gα 2 = aα = a = gα; that is (1) holds. 3.8. .

(1) If ker(α) ≤ ker(β) and α is idempotent then αβ = β.

(2) If α is idempotent then ker(α) ≤ ker(β) iff αβ = β. Proof. Assume the hypotheses of (1). Then by 3.7, G = ker(α)Gα, so as ker(α) ≤ ker(β), G = ker(β)Gα and ker(β) = ker(α)(ker(β) ∩ Gα). By 3.7, α|Gα = 1Gα , so the second equality says ker(β)α ≤ ker(β), and hence ker(β)αβ = 0. Let g ∈ G. By the first equality, g = ka for some k ∈ ker(β) and a ∈ Gα. Then gβ = (ka)β = kβaβ = aβ and gαβ = (kαβ)(aαβ) = aαβ = aβ, so β = αβ. This establishes (1). Then (1) and 3.2 imply (2). 3.9. .

(1) If Gβ ≤ Gα and α is idempotent then βα = β.

(2) If α is idempotent then Gβ ≤ Gα iff βα = β. Proof. Assume the hypotheses of (1); then for g ∈ G, gβα = gβ as α|Gα = 1Gα . Thus (1) holds, while (1) and 3.3 imply (2). Recall from Section 1 that β ≤ α iff αβ = βα = β. 3.10. Assume α is idempotent; then the following are equivalent: (1) β ≤ α. (2) ker(α) ≤ ker(β) and Gβ ≤ Gα. Proof. The lemma follows from 3.8 and 3.9. 3.11. Let be the set of idempotents in M and S the set of splittings of G. (1) The map ξ : α → (ker(α), Gα) is a bijection of with S. (2) ξ is an isomorphism of posets.

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Proof. Let α ∈ . By 3.7, ξ(α) ∈ S. Given S = (K, C) ∈ S, define λ(S) : G → G by (ka)λ = a for k ∈ K and a ∈ C. Then λ(S) ∈ and λ is an inverse for ξ , so (1) is established. By 3.10, the bijection ξ is an isomorphism of posets. 3.12. If G is finite and Gα is splitting simple then either α is nilpotent or α is split in M. Proof. Recall the definition of id(α) from Section 1. Let H = Gα. Since id(α)|H is an idempotent of H , and since H is splitting simple, H id(α) = 1 or H . If H id(α) = 1, then α id(α) = 0, which that id(α) = (id(α))2 = 0, so α is nilpotent. In the second case id(α) ∈ Aut(H ), which implies that α|H ∈ Aut(H ), so ker(α) ∩ Gα = 1, and α is split.

4. Extending and restricting homomorphisms In this section G is a group, A is a subgroup of G, H G, and G = AH . 4.1. Assume α ∈ Hom(A, G) and β ∈ Hom(H, G) such that (1) α|A∩H = β|A∩H , and (2) (ha )β = (hβ)aα for all a ∈ A and h ∈ H . Define ψ : G → G by ψ : ah → aαhβ for a ∈ A and h ∈ H . Then ψ ∈ End(G) extends α and β. Proof. As G = AH , (1) insures that ψ is well defined. Next by (2): (ahbk)ψ = (abhb k)ψ = aαbα(hb )βkβ = aαbα(hβ)bα kβ = aαhβbαkβ = (ah)ψ(bk)ψ, so ψ ∈ End(G). By construction, ψ extends α and β. For g ∈ G, let cg ∈ Aut(H ) be conjugation by g, and AutA (H ) = {ca : a ∈ A}. 4.2. Assume α ∈ Hom(A, G) and β ∈ Aut(H ) such that α|A∩H = β|A∩H and for each a ∈ A, caβ = caα .

(∗)

Define ψ : G → G by ψ : ah → aαhβ for a ∈ A and h ∈ H . Then ψ ∈ End(G) extends α and β. Proof. Condition (∗) says for each h ∈ H : (hβ)aα = (hβ)caα = (hβ)caβ = hca β = (ha )β; that is (*) implies condition (2) of 4.1, so the lemma follows from 4.1.

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4.3. Let β ∈ CAut(H ) (A ∩ H ) ∩ CAut(H ) (AutA (H )). Define ψ : G → G by ψ : ah → ahβ for a ∈ A and h ∈ H . Then ψ ∈ Aut(G). Proof. We apply 4.2 to β and the identity map α on A. As β centralizes AutA (H ), caβ = ca = caα , for each a ∈ A. Further as β centralizes A ∩ H , β|A∩H = α|A∩H . Thus the lemma follows from 4.2. 4.4. Consider the following two conditions: (a) 1A extends to a unique member of End(G). (b) CAut(G) (A) = 1 and there exists no nontrivial splitting (K, C) of G with A ≤ C. Then: (1) Condition (a) implies condition (b). (2) If G is finite then (a) and (b) are equivalent. Proof. Assume (a). Each member of CAut(G) (A) is a member of End(G) extending 1A , so CAut(G) (A) = 1. If (K, C) is a splitting of G with A ≤ C then by 4.1 there is ψ ∈ End(G) with kernel K and ψ|C = 1C , so ψ = 1G and hence K = 1. Conversely assume G is finite and (b) holds. Let θ ∈ End(G) extend 1A , and let ζ = id(θ) in the language of Section 1; thus ζ is an idempotent in End(G). As k = 1 for each positive integer k, ζ θ|A A |A = 1A . Thus by 3.7, (ker(ζ ), Gζ ) is a splitting of G and A ≤ Gζ , so by (b), ζ is an injection and hence an automorphism of G centralizing A. Hence θ = ζ = 1G by (b). 4.5. The following are equivalent: (1) 0A extends to a unique member of End(G). (2) Hom(G/AG , G) = 0. Proof. Let K = AG , π : G → G/K the natural map, and ψ ∈ Hom(G/K, G). Then πψ extends 0A , so if (1) holds then π ψ = 0G , and hence ψ = 0; that is (1) implies (2). Conversely if θ ∈ End(G) extends 0A then A ≤ ker(θ ), so K ≤ ker(θ ). Thus there exists a unique α ∈ Hom(G/K, G) with π α = θ. Hence if (2) holds then α = 0, so θ = 0.

5. (c, d)-embeddings In this section G is a group, p is a prime, and P is a finite subgroup of G. Write 1P for the identity map on P and 0P for the 0-homomorphism mapping each element of P to 1. If P is nilpotent write cl(P ) for the nilpotence class of P .

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Let c, d be nonnegative integers, and define Endc (P ) = {ϕ ∈ End(P ) : P ϕ is nilpotent and cl(P ϕ) ≤ c}, and EndP (G) = {θ ∈ End(G) : P θ ≤ P }. We say P is (c, d)-embedded in G if (1) 1P extends to a unique member of End(G), (2) Each ϕ ∈ Endc (P ) extends to at most one member of End(G), (3) Each ϕ ∈ Endd (P ) extends to at least one member of End(G). A c-embedding is a (c, c)-embedding. Remark 5.1. P is 0-embedded in G iff 1P and 0P extend to unique members of End(G). Remark 5.2. P is nearly closed in G iff P is cl(P )-embedded in G. 5.3. Assume P is (c, 0)-embedded in G and θ ∈ EndP (G) with cl(P θ ) ≤ c. Then: (1) The restriction map ρ : EndP θ (Gθ) → End(P θ ) is injective. (2) P θ is (c, 0)-embedded in Gθ . Proof. Assume ηi ∈ End(Gθ), i = 1, 2, extend φ ∈ End(P θ ). Then ϕ = θ|P φ ∈ End(P ) and θ ηi ∈ End(G) extend ϕ with cl(P θ ηi ) ≤ cl(P θ ) ≤ c. Therefore as P is (c, 0)-embedded in G, θη1 = θη2 . Hence as the image of θ is the domain of ηi , η1 = η2 . Thus (1) holds, and of course (1) implies (2). In the remainder of the section assume P is a finite p-group. 5.4. Assume 0P extends to a unique member of End(G), and G = Z(G)P . Then G = P. Proof. Let Z = Z(G) and e the exponent of P ∩ Z. Then the map β : x → x e is in End(Z) with (P ∩ Z)β = 1. Let α = 0P . Then α and β satisfy the hypotheses of 4.1 with Z, P in the roles of H, A, so by that lemma ψ ∈ End(G), where ψ : az → ze for a ∈ P , z ∈ Z. Thus ψ extends α, so by hypothesis, ψ = 0G . That is e is the exponent of Z. By 4.5, Hom(G/P , G) = 0. But G/P ∼ = Z/(Z ∩ P ) has exponent dividing the power e of p, so if Z/Z ∩P = 1, then Zp is isomorphic both to an image of Z/(Z ∩P ) and to a subgroup of G, so Hom(G/P , G) = 0, a contradiction. 5.5. Assume P is 0-embedded in G. Then: (1) CG (P ) = Z(G). (2) If P is abelian then G = P . (3) If K G with G/K a finite p-group then G = P K.

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Proof. For g ∈ CG (P ) the inner automorphism cg induced on G by g is in CAut(G) (P ), so (1) follows from 4.4. Assume P is abelian. Then P ≤ Z(G) by (1), so G is abelian by a second application of (1). Thus P = G by 5.4, establishing (2). Assume the hypotheses of (3) and suppose G = P K. Then as G/K is a finite p-group, P K ≤ J with |G : J | = p. Then an argument in the final paragraph of the proof of 5.4 supplies a contradiction. Remark 5.6. Notice that 5.5.2 says that Question 2 has a positive answer for finite abelian p-groups. 5.7. Assume P is (1, 0)-embedded in G and θ ∈ EndP (G) with P θ abelian. Then Gθ = P θ . Proof. By 5.3.2, P θ is (1, 0)-embedded in Gθ , so Gθ = P θ by 5.5.2 applied to the embedding of P θ in Gθ.

6. Monoids and (c, d)-embeddings In this section P is a subgroup of a group G. We use the ideals Endc (P ) and Endd (P ) of End(P ) to study (c, d)-embeddings of P . Define EndPc (G) = {ϕ ∈ EndP (G) : P ϕ is nilpotent and cl(P ϕ) ≤ c}. Given a monoid M, an ideal of M is a subset I of M with MI M ⊆ I . Let ρ : ψ → ψ|P be the restriction map from EndP (G) to End(P ). 6.1. .

(1) EndP (G) is a submonoid of End(G).

(2) Endc (P ) and EndPc (G) are ideals of End(P ) and EndP (G), respectively. (3) ρ(EndPc (G)) ≤ Endc (P ). (4) P is nearly closed in G iff ρ : EndP (G) → End(P ) is an isomorphism of monoids; the inverse of ρ is the extension map ex : ϕ → ϕ. ˆ (5) P is (c, d)-embedded in G iff (a) 1G is the unique extension of 1P , (b) ρ : EndPc (G) → Endc (P ) is injective, (c) ρ : EndPd (G) → Endd (P ) is surjective. (6) P is c-embedded in G iff 1G is the unique extension of 1P and ρc = ρ| EndP (G) : EndPc (G) → Endc (P ) is an isomorphism. In that event c exc = ex| Endc (P ) is the inverse of ρc .

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Proof. Straightforward; to illustrate we give a proof of (4). Visibly ρ : EndP (G) → End(P ) is a homomorphism of monoids. Further by definition, P is nearly closed in G iff ρ is a bijection, and of course in that event ex is the inverse of ρ. Using the fact that various group theoretic properties of P , G can be retrieved from the properties of the monoids Endc (P ) and EndPc (G), we can use the fact that ρc is an isomorphism to see that properties of P lift to analogous properties of G. Define a c-embedding (G, P ) (i.e. P is c-embedded in G) to be c-minimal if for each pair (H, Q) with H < G, Q ≤ P , and Q c-embedded in H , we have H = Q. Write ξ for the bijection of 3.11 in the case of both P and G. 6.2. Assume P is c-embedded in G and (PK , PC ) is a splitting of P with cl(PC ) ≤ c. Set α = ξ −1 (PK , PC ). Then: ˆ is the unique splitting of (1) αˆ is an idempotent in EndPc (G) and (K, C) = ξ(α) G with PK ≤ K and PC ≤ C. (2) PC is c-embedded in C. (3) If (G, P ) is c-minimal and PC = P then C = PC so if G is finite and P is a p-group for some prime p, then O p (G) ∩ P ≤ PK . Proof. By definition, PK = ker(α) and PC = P α. As P is c-embedded in G, αˆ ˆ and PC ≤ C = Gα. ˆ As α 2 = α and ex exists by 6.1.6. Then PK ≤ K = ker(α) 2 ˆ so (K, C) is a splitting of G. If (K , C ) were a second is an isomorphism, αˆ = α, −1 such splitting then βˆ = ξ (K , C ) is an idempotent with PK ≤ ker(β) ∩ ker(α) and β|PC = 1PC = α|PC , so α = β and hence (K, C) = (K , C ). Thus (1) holds. Let ϕ ∈ End(PC ) and set ψ = αϕ. Then ψ ∈ Endc (P ), so ψˆ exists. As ψ = αϕ and α is idempotent, αψ = ψ. As P ψ = P αϕ ≤ P α, ψα = ψ. Thus ψ ≤ α, so ψˆ ≤ αˆ by 6.1.6, and hence Gψˆ ≤ Gαˆ by 3.10, so ψˆ |C ∈ End(C). Next for c ∈ PC , c = aα for some a ∈ P and cψˆ = cψ = aαψ = aα 2 ϕ = aαϕ = cϕ, so ψˆ |C extends ϕ on C. Thus ρc : EndPc C (C) → Endc (PC ) is surjective, while ρc is injective by 5.3. This completes the proof of (2). If (G, P ) is c-minimal and PC = P , then C = PC by (2). Thus if G is finite and P is a p-group then O p (G) ≤ K and hence O p (G) ∩ P ≤ K ∩ P ≤ PK . 6.3. Assume P is nearly closed in G. Then: (1) Let (PK , PC ) be a splitting of P and set α = ξ −1 (PK , PC ). Then αˆ is an ˆ is the unique splitting of G with idempotent in EndP (G) and (K, C) = ξ(α) PK ≤ K and PC ≤ C. (2) PC is nearly closed in C. (3) If PC is rigid in some class C containing G then C = PC .

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Proof. Parts (1) and (2) follow from 6.2 and Remark 5.2. Under the hypotheses of (3), as C ≤ G ∈ C, C ∈ C, while by (2), PC is nearly closed in C, so PC = C by the rigidity of PC . 6.4. Assume P = P1 × P2 with P1 = 1 = P2 . Then either of the following imply that G = P: (1) P is c-embedded in G, cl(P ) ≤ c, and (G, P ) is c-minimal. (2) P is nearly closed in G and P1 and P2 are C-rigid for some class C containing G. Proof. Assume hypothesis (1) or (2) holds. Observe Si = (P3−i , Pi ) is a splitting of P for i = 1, 2, so αi = ξ −1 (Si ) is an idempotent in End(P ). Thus by 6.2.1 and 6.3.1, (Ki , Ci ) = ξ(αˆ i ) is a splitting of G with P3−i ≤ Ki , while Pi = Ci by 6.2.3 and 6.3.3. Let K = K1 ∩ K2 . Then G = Ki Ci = Ki Pi and P2 ≤ K1 so K1 = G ∩ K1 = K2 P2 ∩ K1 = KP2 , and hence G = K1 P1 = KP1 P2 = KP . Further K ∩ P = K ∩ P1 P2 ≤ K1 ∩ P1 P2 = P2 (K1 ∩ P1 ) = P2 , and similarly K ∩ P ≤ P1 , so K ∩ P ≤ P1 ∩ P2 = 1. Therefore (K, P ) is a splitting of G, so K = 1 by 4.4.1. Remark 6.5. Notice that 6.4 implies Theorem 1.

7. P -perfect groups In this section G is a finite group, p is a prime, and P is a subgroup of G. Write (G, P ) for the subnormal closure of P in G; that is is the smallest subnormal subgroup of G containing P . We say G is P-nilpotent if G = Op (G)P and G is P-perfect if G = P G . 7.1. Let = (G, P ). Then: (1) is P -perfect. (2) If P ≤ K ≤ G then (K, P ) ≤ . (3) If ≤ K ≤ G then = (K, P ). (4) If α : G → K is a surjective homomorphism then α = (K, P α). (5) If G is nilpotent and P -perfect then G = P .

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(6) If G is nilpotent then = P . Proof. Because P ≤ P , (1) follows from minimality of . Since P ≤ ∩ K K, it follows that (K, P ) ≤ ∩ K, giving (2). Then if ≤ K, (K, P ) is subnormal in , so = (K, P ) by minimality of , establishing (3). Assume the hypotheses of (4). Then α is subnormal in Gα = K, so (K, P α) ≤ α. Then the preimage I in of (K, P α) is a subnormal subgroup of containing P , so I = , and hence α ≤ I α = (K, P α), thus (4) holds. Assume G is nilpotent. Then P is subnormal in G, so (5) and (6) hold. In the remainder of the section assume P is a p-group and set L = O p (G). 7.2. Assume G is P -perfect. Then: (1) G = P L. (2) L = [L, P ]. ¯ = G/L. Then G ¯ is a P¯ -perfect p-group, so P¯ = G ¯ by 7.1.5, and hence Proof. Let G (1) holds. Let K = [L, P ] and G∗ = G/K. Then P ∗ centralizes L∗ so as G∗ = P ∗ L∗ by ∗ (1), P ∗ G∗ . Thus G∗ = P ∗G = P ∗ , so L∗ = O p (G∗ ) = 1. That is (2) holds. 7.3. Assume G is P -perfect and ϕ ∈ Hom(P , G). Let be the set of ϕˆ ∈ End(G) extending ϕ such that Gϕˆ is nilpotent. Then: (1) = ∅ iff P ∩ L ≤ ker(ϕ). (2) If = ∅ then: (a) consists of a unique element ϕ, ˆ (b) Gϕˆ = P ϕ, (c) ker(ϕ) ˆ = L ker(ϕ). Proof. Suppose = ∅ and let ϕˆ ∈ . As G is P -perfect, Gϕˆ is P ϕ-perfect, ˆ so as Gϕˆ is nilpotent, (b) holds by 7.1.5. By (b), L ≤ ker(ϕ), ˆ so P ∩ L ≤ P ∩ ker(ϕ) ˆ = ker(ϕ). This proves half of (1). Also by 7.2.1, G = P L, so ker(ϕ) ˆ = L(P ∩ ker(ϕ)) ˆ = L ker(ϕ), establishing (c). Then as G = LP , (b) and (c) imply ϕˆ is unique, completing the proof of (2). Conversely suppose P ∩L ≤ ker(ϕ). Define ϕˆ : G → G by (al)ϕˆ = aϕ for a ∈ P and l ∈ L. By 4.1 (taking A = P , H = L, α = ϕ and β the trivial homomorphism), ϕˆ ∈ End(G) extends ϕ.

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8. c-embeddings in finite groups In this section G is a finite group, p is a prime, and P is a p-subgroup of G. 8.1. Assume P is 0-embedded in G. Then: (1) G = P G G∞ . (2) If Y G and G/Y is nilpotent then G = Y P . (3) If G is solvable then G is P -perfect. (4) If G is nilpotent then G = P . (5) G = LP . (6) If G is p-nilpotent then G = P . (7) Let P ≤ T ∈ Sylp (G). Then Z(P ) ≤ Z(T ) ≤ Z(G). Proof. Let K = P G and H = G∞ . If KH = G then as G/H is solvable there is a normal subgroup X of G containing KH with |G : X| = q prime. Then as G has a subgroup of order q, Hom(G/K, G) = 0, contrary to 4.5 and the hypothesis that P is 0-embedded in G. This establishes (1). Further (2) follows from (1), since if G/Y is nilpotent and Y P = G then also Y K = G by 7.1.5. Also (3) and (4) follow from (1) and (2). Similarly (5) follows from (2) applied to Y = O p (G). Assume G is p-nilpotent and let Y = Op (G). Then by (2), G = Y P so P is a complement to Y in G. Then (6) follows from 4.4. Part (7) follows from 5.5.1. 8.2. Assume P is 0-embedded in G. Then CAut(L) (P ∩ L) ∩ CAut(L) (AutP (L)) = 1. Proof. Let β ∈ CAut(L) (P ∩ L) ∩ CAut(L) (AutP (L)). Then by 4.3 applied to P and L in the roles of A and H , we get ψ ∈ Aut(G) with ψ|P = 1P and hψ = hβ for h ∈ L. As P is 0-embedded in G, ψ = 1G , so β = 1. 8.3. Assume P is 0-embedded in G and let P ≤ T ∈ Sylp (G). Then: (1) CT (P ∩ L) = CT (L). (2) If P ∩ L is abelian then G = P . Proof. If CT (L) < CT (P ∩ L) then there is g ∈ CT (P ∩ L) − CT (L) with [g, P ] ≤ CT (L). But then 1 = cg ∈ Aut(L) centralizes P ∩ L and Aut P (L), so 8.2 supplies a contradiction. This establishes (1). Assume X = P ∩ L is abelian. Then X ≤ Z(L) by (1). Thus Y = T ∩ L centralizes X, so Y ≤ Z(L) by (1). Hence as L = O p (L) and Y ∈ Sylp (L), L is a p -group. Thus G = P by 8.1.6.

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8.4. Assume P is d-embedded in G, let be a set of subgroups of P such that for each X ∈ , cl(X) ≤ d and X ∩ L is abelian. Let be the set of normal subgroups K of P such that P /K is isomorphic to a member of , and set K. K() = K∈

Then P ∩ L ≤ K(). Proof. Claim it suffices to show for each K ∈ , that there exists ϕˆK ∈ End(G) with K = P ∩ ker(ϕˆK ) and GϕˆK ∼ = P /K. For if so then L ≤ ker(ϕˆK ), so ker(ϕˆK ), L≤ K∈

and hence P ∩L≤

P ∩ ker(ϕˆK ) =

K∈

K = K().

K∈

Thus the claim is established. Next by definition of , for K ∈ there exists X ∈ and ϕ ∈ End(P ) with K = ker(ϕ) and X = P ϕ. As X ∈ , cl(X) ≤ d, so as P is d-embedded in G, ϕ extends to ϕˆ ∈ End(G). Let H = Gϕˆ and LH = O p (H ); by the claim it remains to show H = X. But X ∩ LH ≤ X ∩ L, so X ∩ LH is abelian as X ∈ . Hence H = X by 5.3 and 8.3.2, completing the proof. 8.5. Assume P is 1-embedded in G and let ϕ ∈ End1 (P ). Then: (1) There exists a unique ϕˆ ∈ Hom(G, G) extending ϕ, and Gϕˆ = P ϕ. (2) P ∩ L ≤ [P , P ]. Proof. The first part of (1) is immediate from the definition of 1-embedding, and the second follows from 5.7. P ∗ = R1∗ × · · · × Rn∗ where Ri∗ is cyclic. Let Ki be the Let P ∗ = P /[P , P ]; then preimage in P of j =i Rj∗ and ri ∈ P with Ri∗ = ri∗ . Then |Ri∗ | divides |ri |, so P contains a subgroup Xi ∼ = Ri∗ . Set = {X1 , . . . , Xn }. As Xi is abelian, Xi ∩ L is abelian and cl(Xi ) = 1. Thus P ∩ L ≤ K() by 8.4. Finally P /Ki ∼ = Ri∗ ∼ = Xi , so K() ≤

n

Ki = [P , P ],

i=1

completing the proof. 8.6. Assume P ∈ Sylp (G) and P is 1-embedded in G. Then G = P . Proof. By 8.5.2, P ∩ L ≤ [P , P ] ≤ (P ). Therefore G is p-nilpotent by 9.5, so G = P by 8.1.6.

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8.7. Assume P is 1-embedded in G and [P , P ] is abelian. Then G = P . Proof. By 8.5.2, P ∩ L ≤ [P , P ], so P ∩ L is abelian by hypothesis. Then the lemma follows from 8.3.2. 8.8. Assume P is 1-embedded in G and P is of class at most 3. Then G = P . Proof. As P is of class at most 3, [P , P ] is abelian by 9.4, so the lemma follows from 8.7. Remark 8.9. Notice that 8.6 implies Theorem 2 and 8.7 and 8.8 imply Theorem 3. 8.10. If P is 0-embedded in G of class c and P ∩ L ≤ Lk (P ) for some k ≥ (c − 1)/2, then G = P . Proof. As k ≥ (c − 1)/2, Lk (P ) is abelian by 9.4. Then as P ∩ L ≤ Lk (P ), P ∩ L is abelian, so the lemma follows from 8.3.2. We close this section with a proof of Theorem 4. Assume the hypotheses of Theorem 4 and let S = (Si : 0 ≤ i ≤ n) be a splitting series for P such that each member of K(S) is abelian. Let Sn−1 = (PK , PC ). Then PK is abelian and PC is splitting solvable, so by induction on the order of P , PC is rigid in the class of finite groups. Hence by 6.3, there is a splitting (K, C) of G with PK = P ∩ K and C = PC . In particular P ∩ L ≤ PK , and hence P ∩ L is abelian, so G = P by 8.3.

9. Finite p-groups In this section P is a finite p-group of class c. Let Zi (P ) be the ith term in the ascending central series; that is Z0 (P ) = 1 and for i > 1, Zi (P ) is the preimage in P of Z(P /Zi−1 (P )). Let Li (P ) be the ith term in the descending central series; that is L0 (P ) = P and for i > 0, Li (P ) = [Li−1 (P ), P ]. We record some well known facts: 9.1. Li (P ) ≤ Zc−i (P ). Proof. See for example the proof of 9.6 in [2]. 9.2. [Li (P ), Zj (P )] ≤ Zj −i−1 (P ), where Zr (P ) = 1 for r ≤ 0. Proof. Let P ∗ = P /Zj −i−1 (P ). If i + 1 ≤ j then Zj (P )∗ = Zi+1 (P ∗ ) and Zj −i−1 (P )∗ = 1, while if i +1 ≥ j then P = P ∗ and Zj −i−1 (P ) = 1. Thus replacing P by P ∗ , we may assume j ≤ i + 1 and it remains to show [Li (P ), Zj (P )] = 1. We induct on i; if i = 0 then j ≤ 1 and [Li (P ), Zj (P )] ≤ [P , Zj (P )] ≤ Z0 (P ) = 1, so the induction is anchored and we may assume i is positive. Let L = Li−1 (P ) and U = Zj (P ). We show [L, U, P ] = 1 = [U, P , L] = 1; then by the 3-subgroup lemma, [Li (P ), Zj (P )] = [P , L, U ] = 1.

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Now by induction on i, [L, U ] ≤ Zj −i (P ) ≤ Z(P ) as j ≤ i +1, so [L, U, P ] = 1. Also [U, P ] ≤ Zj −1 (P ) ≤ CP (L) by induction on i as j ≤ i + 1; thus [U, P , L] = 1. 9.3. Li (P ) centralizes Zi+1 (P ). Proof. This is a special case of 9.2. 9.4. If i ≥ (c − 1)/2 then Li (P ) is abelian. Proof. By 9.1, Li (P ) ≤ Zc−i (P ) and if i ≥ (c − 1)/2 then Zc−i (P ) ≤ Zi+1 (P ). Thus the lemma follows from 9.3. 9.5. Assume G is a finite group, P ∈ Sylp (G), and P ∩ O p (G) ≤ (P ). Then G is p-nilpotent. Proof. This is a theorem of Tate in [4]; there is a proof in Theorem 4.7 in [3]. We close this section with a sketch of a proof of the following result: 9.6. Let P be a p-group 2-embedded in a finite group G and assume P is generated by at most three elements of order p. Let L = O p (G). Then: (1) Either G = P ∼ = Q8 ∗ Z4 or P ∩ L ≤ L2 (P ). (2) If cl(P ) ≤ 5 then G = P . We include this discussion to give an indication of how to use 8.4 to extend Theorem 3 to suitable families of p-groups of class greater than 3. The discussion also indicates the limitations of the method and the nature of the obstructions to its application. As 9.6 is so weak, it does not seem worthwhile to give all the details of the proof. We need the following facts which we use without proof; p1+2 denotes the extraspecial p-group of order p 3 generated by elements of order p. Facts. Assume P is of class at least 2 and generated by at most 3 elements of order p. Let be the set of normal subgroups K of P with P /K ∼ = p 1+2 . Then: (1) Either p = 2 and P ∼ K. = Q8 ∗ Z4 or L2 (P ) = (2) There exist

p 1+2

∼ = D ≤ P with D [P , P ].

K∈

Fact 1 can be proved by considering the universal p-group P of class 2 generated by 3 elements of order p. The corresponding group generated by 4 or more elements fails to have this property, so to extend the method, one would have to work with a larger class H of groups then {p1+2 }. Fact 2 is easy, but it would not be easy to establish the analogous statement for the larger class H, if indeed that statement is true. Now here is a proof of 9.6: By 8.7 we may assume [P , P ] is nonabelian and by 8.5, P ∩ L ≤ [P , P ].

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We first prove (1) by applying 8.4 to the set of subgroups X of P such that X∼ = p 1+2 and X L. Define and K() as in 8.4. As P ∩ L ≤ [P , P ], Fact 2 says = ∅. Thus by 8.4 it suffices to show that K() = L2 (P ). This follows from Fact 1 unless P ∼ = Q8 ∗ Z4 . But in that case P = G by 8.7. This completes the proof of (1). By (1) we may assume P ∩ L ≤ L2 (P ). Thus (2) follows from 8.10.

10. Hall–Higman reductions In this section G is a finite group, p is a prime, P is a p-subgroup of G, and L = O p (G). Write E = E (P ) for the set of all (isomorphism types) of embeddings (H, P ) of P in finite groups H . For (Gi , P ) ∈ E , write (G1 , P ) ≤ (G2 , P ) if there exists P ≤ H ≤ G2 such that (G1 , P ) ∼ = (H /K, P K/K) for some normal p -subgroup K of H . Evidently ≤ is a partial order on E . Define D ⊆ E to be a Hall–Higman class if whenever (G, P ) ∈ D and (H, P ) ≤ (G, P ), then also (H, P ) ∈ D. Observe that the intersection of Hall–Higman classes is a Hall–Higman class. An embedding (G, P ) is P-centered if CG (P ) = Z(G). 10.1. The following subclasses of E are Hall–Higman: (1) The class of embeddings (G, P ) such that P is subnormal in G. (2) The class of embeddings (G, P ) such that the subnormal closure in G of P is P -nilpotent. (3) The class of P -centered embeddings. Proof. This is straightforward; for example (1) and (2) follow from 7.1.2 and 7.1.4. However we do (3). Let P ≤ H ≤ G, K a normal p -subgroup of H , and set H ∗ = H /K. Assume G is P -centered. Then CH (P ) ≤ CG (P ) ∩ H ≤ Z(G) ∩ H ≤ Z(H ). As K is a p -group, CH (P )∗ = CH ∗ (P ∗ ) and Z(H )∗ ≤ Z(H ∗ ). Given subclasses D and P of E , define D to be a P -class if D ⊆ P . Example 10.2. Write (G, P ) for the set of all overgroups I of P in G such that I is P -perfect and P -nilpotent, but Op (I ) Op (G). Consider the class PI of extensions (G, P ) such that (G, P ) = ∅. A subclass D of E is P-signalizer closed if D ⊆ PI . Given a subclass P of E , define an embedding (G, P ) to be minimal non-P if (a) (G, P ) ∈ / P , but (b) whenever (H, P ) < (G, P ), (H, P ) ∈ P .

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10.3. Let D and P be subclasses of E , and assume D is a Hall–Higman class. Then either: (a) D is a P -class, or (b) D contains a minimal non-P embedding. Proof. Assume D P . Then there exists (G, P ) ∈ D minimal subject to (G, P ) ∈ / P under the partial order ≤. As D is a Hall–Higman class, (H, P ) ∈ D for each (H, P ) < (G, P ). Thus (H, P ) ∈ P by minimality of (G, P ). That is (G, P ) is a minimal non-P embedding. Define IG (P , p ) to be the set of all P -invariant p -subgroups of G. Define to consist of those X ∈ IG (P , p ) such that X = [X, P ]. Recall the definition of (G, P ) from Section 7.

I− G (P , p )

10.4. Set = (G, P ). Then:

(1) G is P -perfect and P -nilpotent iff G = LP and L ∈ I− G (P , p ).

(2) (G, P ) = ∅ iff [Op (G), P ] is the unique maximal member of I− G (P , p ).

(3) If O p (CG (P )) ≤ Op (G) then (G, P ) = ∅ iff Op (G) is the unique maximal member of IG (P , p ).

(4) contains each P -perfect P -nilpotent subgroup of G and I− G (P , p ) ⊆ .

(5) (, P ) = (G, P ). (6) If is P -nilpotent then (G, P ) = ∅. Proof. Parts (1)–(3) are straightforward. The first statement in (4) follows from 7.1.2; then the second statement in (4) follows from the first and (1). As is subnormal in G, Op () ≤ Op (G), so (4) implies (5). Then (5) implies (6). 10.5. Assume G is minimal non-P -signalizer closed. Then: (1) G is P -perfect. (2) Op (G) = 1.

∗ (3) For each 1 = Y ∈ I− G (P , p ), Y is faithful on F (G) and Y P ∈ (G, P ).

(4) G = I F ∗ (G) for each I ∈ (G, P ), and Op (I ) is faithful on F ∗ (G).

(5) There exists I ∈ (G, P ) such that X = Op (I ) is a q-group of class at most 2 for some prime q, X is of exponent q if q is odd, P centralizes (X), and P is irreducible on X/(X). (6) Either (i) G is solvable and F ∗ (G) = Op (G), or (ii) G is not solvable, F ∗ (G) = Op (G))E(G), Op (I ) centralizes Op (G), and E(G) is the product of components permutated transitively by I .

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Proof. Part (1) follows from 10.4.5 and minimality of G. Let K = Op (G) and G∗ = G/K. If K = 1 then by minimality of G, ∗

(G∗ , P ∗ ) = ∅. Thus [Op (G∗ ), P ∗ ] is the unique maximal member of I− G∗ (P , p ) − − by 10.4.2. But as K is a p -group, IG∗ (P ∗ , p ) = IG (P , p )∗ , so as Op (G∗ ) = 1, (G, P ) = ∅ by another application of 10.4.2, contrary to the choice of G. Thus (2) is established. Let I ∈ (G, P ) and F = F ∗ (G). Then by 10.4.1, I = P X where X = [P , X] is a p -group. Conversely if Y = [Y, P ] is a nontrivial p -group then CY (F ) ≤ Z(F ), so Y is faithful on F by (2). Further [Y ∩ Op (F Y ), F ] ≤ Op (F ) = 1, so Y ∩ Op (F X) = 1. Therefore Y P ∈ (I F, P ), while if G = I F then (I F, P ) = ∅ by minimality of G. Thus (3) and (4) hold. As I is P -nilpotent and X is a p -group, P acts on a Sylow q-group U of X for each q ∈ π(X). Further as X = [X, P ], [U, P ] = 1 for some choice of q. Let V be a supercritical subgroup of U ; as [U, P ] = 1, [V , P ] = 1. Pick W ≤ V minimal subject to 1 = W = [W, P ]. Then P W ∈ (G, P ) by (3), establishing (5). Pick I as in (5) and let R = Op (G). If F ∗ (G) = R then as I and R are solvable and G = I R by (4), G is solvable, so (6i) holds. Thus we may assume L1 is a component of G and let J = LI1 . In particular G is not solvable, so G = RI and hence X ≤ Op (RI ), so X centralizes R. Similarly if G = I J then X centralizes L1 , so as X is faithful on F , it follows that G = I J = P XJ . Thus (6ii) hold, completing the proof of the lemma. 10.6. Assume G is minimal non-P -signalizer closed and G is solvable. Let R = Op (G) and Q = Op (L). Then: (1) G = LP is P -perfect. (2) R = F ∗ (G). (3) L = QX where X and the action of P on X is described in 10.5.5. (4) P Q ∈ Sylp (G). (5) R = QCP (X). (6) [Q, CP (X)] = 1. (7) P X is irreducible on Q/(Q). (8) Either (a) Q is elementary abelian, or (b) Q is special, Q is of exponent p if p is odd, and X centralizes (Q). (9) If CG (P ) = Z(G) then Q is special, Z(Q) = Z(L), and X is elementary abelian. Proof. Part (1) and (2) follow from 10.5. Further by 10.5, G = RXP , where X = [P , X] is described in 10.5.5. Thus L = QX, so (3) holds and (4) follows from (1) and (3).

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By (4), R = R ∩ P Q = Q(R ∩ P ). As P X is p-nilpotent and G = P QX, R ∩ P = CP (X). That is (5) holds. Let M be minimal subject to M ≤ Q, P X ≤ NG (M), and [M, X] = 1. Then as P acts on X, M = [M, X] by minimality of M. If G = P XM then X ≤ Op (P XM) by minimality of G, so X centralizes M, contrary to the choice of M. Thus G = P XM, so M = Q. Then (7) and (8) follow from the minimality of M = Q. Finally assume CG (P ) = Z(G). Let Z = Z(Q). By coprime action, Z = [Z, X] ⊕ CZ (X). As P acts on X it acts on [Z, X], so if [Z, X] = 1 then 1 = C[Z,X] (P ) ≤ Z(G), a contradiction. Thus [Z, X] = 1, so Z = Z(L). As Z(Q) = Z(L) and X is nontrivial on Q, case (b) of (8) holds. By (3) and 10.5.5, (X) ≤ CG (P ) = Z(G), so as X is faithful on Q by (4) and 10.5.4, (X) = 1. This completes the proof of (9). 10.7. If p is odd then the class of solvable P -centered groups is P -signalizer closed. Proof. Assume otherwise. By 10.1.3 and 10.3, there is an embedding (G, P ) which is P -centered and minimal non-P -signalizer closed. Thus G is described in 10.6. Adopt the notation of 10.6 and let Z = Z(Q), G∗ = G/Z, and V = Q∗ . As Q is special, we may regard Z and V as Fp G-modules, and as R = QCP (X) centralizes these modules, they are also modules for G+ = G/R. Further G+ = X+ P + ∼ = XP /CP (X), X centralizes Z, and X+ P + is faithful and irreducible on V . Define f : V × V → Z by f (a ∗ , b∗ ) = [a, b]. Then f is a G-equivariant alternating bilinear map. Let pn be the smallest power of p such that p n ≡ 1 mod q and let F = Fpn . Let AF = A ⊗Fp F for A ∈ {V , Z}. Then V F and Z F are F G-modules and we get a G-equivariant alternating F -bilinear map f F : V F × V F → Z F with f F (1 ⊗ v, 1 ⊗ u) = f (v, u) for u, v ∈ V . Let σ ∈ Aut(F ) be defined by σ : a → a p ; that is = Gal(F /Fp ) = σ ). Then (cf. 25.7 in [2]) AF is an Fp -module via (a × v)σ = aσ ⊗ v for a ∈ F , v ∈ A, and in this representation, commutes with G. Moreover if W is a G-submodule of AF then W = B F for some G-submodule of A. Therefore as G+ is irreducible on V , G is irreducible on V F , so as X+ G+ , V F is a semisimple F X-module. As X is elementary abelian, CV (Y ), V = Y ∈Y

where Y is the set of hyperplanes of Y of X with CV (Y ) = 0. As P X is irreducible on V , P is transitive on Y and for Y ∈ Y, XNP (Y ) is irreducible on CV (Y ). Then CV F (Y ), VF = Y ∈Y

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Michael Aschbacher

and writing Y for the set of weights of X on CV (Y ), CV (Y )λ . CV F (Y ) = λ∈Y

Let vi ∈ CV (Yi ), λi ∈ Yi , and x ∈ X. Then x fixes f F (v1 , v2 ) ∈ Z F , so f F (v1 , v2 ) = f F (v1 , v2 )x = f F (v1 x, v2 x) = f F (λ1 (x)v1 , λ2 (x)v2 ) = λ1 (x)λ2 (x)f F (v1 , v2 ), so f F (v1 , v2 ) = 0 unless λ2 = λ−1 1 . But in the latter case Y1 = ker(λ1 ) = ker(λ2 ) = Y2 , so [CQ (Y1 ), CQ (Y2 )] = 1 if Y1 = Y2 . Thus Q= QY , Y ∈Y

is a central product of the subgroups QY = [CQ (Y ), X]. Therefore QY is special as Q is special and XNP (X) is irreducible on QY /Z(QY ) as XP is irreducible on Q/Z. Suppose NP (Y ) centralizes a ∈ QY − Z. Pick a set C of coset representatives for NP (Y ) in P and let ac. b= c∈C

Then b ∈ CG (P ) − Z(G), a contradiction. Therefore CQY (NP (Y )) ≤ Z. Next X = XY × Y for some NP (Y ) complement XY to Y in X, and XY NP (Y ) is irreducible on QY /Z(QY ), so by the Thompson A × B-Lemma, PY = CP (X/Y ) centralizes QY . Therefore as CQY (NP (Y )) ≤ Z, NP (Y ) = PY . Thus q ≡ 1 mod p and P Y = NP (Y )/PY is cyclic. Then P Y is semiregular on Y , Y = λP Y is an orbit under P Y , and CV F (Y )λ is 1-dimensional. ρ : µ → As λ−1 ∈ Y = λP Y , ρ ∈ P Y , where ρ is the involution such that −1 µ for µ ∈ Y . Let v be a generator of CV (Y )Fλ , = Op (), and u = δ∈ vδ. Then W0 = uOp () is a free module for Op () invariant under , and W1 = W0PY is a sum of free modules for P Y . Then by a remark above W1 = W F for some P Y -submodule of CV (Y ). Further as p is odd, ρ ∈ , so f F (u, uγ ) = 0 for γ ∈ Op ()P Y , and hence f F is 0 on W1 . Thus the preimage QW in QY of W is abelian and indeed as p is odd, Q is of exponent p, so QW is elementary abelian. Thus as P Y is free on QW /Z(QY ), QW splits over Z(QY ) as a P -module, so CQY (NP (Y )) Z, a contradiction. 10.8. Assume p is odd and G is solvable and P -centered. Then Op (G) is the unique maximal p -signalizer for P . Proof. This follows from 10.7 and 10.4.3. Remark 10.9. Notice that 10.8 is just Theorem 5.

On a question of Farjoun

27

Remark 10.10. When p = 2 there is a counter example to 10.8 which is minimal non-P -signalizer closed with X elementary abelian. Take G = P L where L = L1 ×L2 with Li ∼ = SL2 (3), and P is a 2-group acting on L and X ∈ Syl3 (L) with Aut P (L) ∼ = D8 and P ∩ O2 (G) = CP (L). Choose P so that Z2 (P ) ≤ CP (L). For example let P = a, b be metacyclic with |a| = 16, |b| = 4, and a b = a 3 . Then P is of class 4 with Z2 (P ) = a 4 , b2 . Take CP (L) = Z2 (P ) (so that P /CP (L) ∼ = D8 ) and P ∩ L ≤ b2 , a 8 .

References [1] H. Bechtell, Splitting systems for finite solvable groups, Arch. Math. 36 (1981), 295–301. [2] M. Aschbacher, Finite group theory, Cambridge Stud. Adv. Math. 10, Cambridge Univ. Press, Cambridge 1986. [3] B. Huppert, Endliche Gruppen I, Springer-Verlag, Berlin 1967. [4] J. Tate, Nilpotent quotient groups, Topology 3 (1964), 109–111. M. Aschbacher, California Institute of Technology, Pasadena, CA 91125, U.S.A. E-mail: [email protected]

Extensions for finite groups of Lie type: twisted groups Christopher P. Bendel∗, Daniel K. Nakano† and Cornelius Pillen

Abstract. In [BNP1], [BNP2], [BNP3] the authors relate the extensions for modules for finite Chevalley groups to certain extensions for the corresponding reductive groups and Frobenius kernels. In this paper we will show that these results can be extended to the twisted finite groups of Lie type. The applications are most pertinent to the twisted groups of types A, D, and E6 . 2000 Mathematics Subject Classification: Primary 20C, 20G; Secondary 20J06, 20G10

1. Introduction 1.1. Let G be a connected simply connected almost simple algebraic group defined and split over the field Fp with p elements, and let k be the algebraic closure of Fp . Let G(Fq ) be the finite Chevalley group consisting of Fq -rational points of G where q = p r for a positive integer r and let Gr denote the rth Frobenius kernel of G. In a series of papers [BNP1], [BNP2], [BNP3], [BNP4] the authors investigated the connections between the extension theory of modules for the semisimple algebraic group G, the finite Chevalley group G(Fq ) and the Frobenius kernel Gr via the use of spectral sequences involving truncated (bounded) subcategories of G-modules. In particular these methods enabled us to provide explicit formulas for the extensions between simple modules of finite Chevalley groups via algebraic groups and Frobenius kernels. As a direct application, several open problems involving the existence of selfextensions for finite Chevalley groups posed by Humphreys [Hum] were resolved by proving suitable generalizations of Andersen’s work on line bundle cohomology for the flag variety [And1], [And2]. Recent work by Tiep and Zalesskiˇı [TZ, Prop. 1.4] has shown that questions involving extensions between simple modules for finite groups arise naturally when one studies the modular representation theory of finite groups. They prove that the ∗ Research of the first author was supported in part by NSA grant MDA904-02-1-0078. † Research of the second author was supported in part by NSF grant DMS-0102225.

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Christopher P. Bendel, Daniel K. Nakano and Cornelius Pillen

ability to lift certain simple modular representations to characteristic zero can happen only when the simple module admits a self-extension. In the mid 1980s, Smith [Smi] showed that the existence of self-extensions is also relevant in the study of sheaf homology. 1.2. The purpose of this paper is to demonstrate that the ideas from the authors’ earlier work can be used to study the extension theory for the other finite groups of Lie type. Following the description of the classification of the finite Lie groups given in [Car, 1.19], we refer to the untwisted groups as Chevalley groups or split forms of G. If G is of type A, D, or E6 there exist non-trivial automorphisms of the Dynkin diagram of G. Such an automorphism, σ , induces a group automorphism of G that commutes with the Frobenius morphism, which we will also denote by σ . Let Gσ (Fq ) be the finite group consisting of the fixed points of σ composed with the rth Frobenius morphism on G. We will refer to the groups Gσ (Fq ) as twisted groups or quasi-split forms of G [Car, 1.19]. Since our results hold mostly for large primes, we will limit our discussion of the Suzuki and Ree groups to Section 2.4. In the remaining sections the extension theory of modules for the split and quasi-split forms Gσ (Fq ) are compared to those for G and Gr . In our discussion, we will leave out several of the details of earlier proofs which have straight forward generalizations to the twisted groups. Instead we will focus on places where there is care needed in accounting for the twist σ . The new formulas that we obtain involving extensions of simple modules for Gσ (Fq ) do indeed involve using the automorphism σ in the calculations. We also indicate several instances in the text where improvements have been made to our earlier work. 1.3. Notation. The conventions in the paper will follow the ones used in [Jan1]. Throughout this paper G is a connected simply connected almost simple algebraic group defined and split over the finite field Fp with p elements. The field k is the algebraic closure of Fp and G will be considered as an algebraic group scheme over k. Let be a root system associated to G with respect to a maximal split torus T . Moreover, let + (resp. − ) be positive (resp. negative) roots and be a base consisting of simple roots. Let B be a Borel subgroup containing T corresponding to the negative roots. Let X(T ) be the integral weight lattice obtained from contained in the Euclidean space E with the inner product denoted by , . The set X(T ) has a partial ordering given by λ ≥ µ if and only if λ−µ ∈ α∈ Nα for λ, µ ∈ X(T ). Let α ∨ = 2α/α, α be the coroot corresponding to α ∈ . The set of dominant integral weights is defined by X(T )+ = {λ ∈ X(T ) : 0 ≤ λ, α ∨ for all α ∈ }. Furthermore, the set of pr -restricted weights is Xr (T ) = {λ ∈ X(T ) : 0 ≤ λ, α ∨ < pr for all α ∈ }.

Extensions for finite groups of Lie type: twisted groups

31

The Weyl group W is the group generated by the reflections sα : E → E given by sα (λ) = λ − λ, α ∨ α. The group W acts on X(T ) via the “dot action” given by w · λ = w(λ + ρ) − ρ where ρ is the half sum of the positive roots. The longest element in W will be denoted by w0 and the Coxeter number for is h = ρ, α0∨ + 1, where α0 is the maximal short root. 1.4. We adopt the set-up described in [Jan3, 1.3]. Let σ denote an automorphism of the Dynkin diagram of G. The automorphism σ can be extended to a linear bijection on X(T ) that permutes the fundamental weights and preserves the inner product , as well as the partial order on X(T ). Moreover, σ (α0 ) = α0 . The graph automorphism σ induces an automorphism on G which will also be denoted by σ . The group automorphism σ commutes with the standard Frobenius morphism F and is compatible with the action of σ on X(T ). Let Gσ (Fq ) be the group of fixed points of F r σ = σ F r , where q = p r . Throughout the paper Gσ (Fq ) will always either denote a split form, if σ acts trivially on the Dynkin diagram of G, or a quasi-split form of G, if G is of type A, D, or E6 and σ is non-trivial [Car, 1.19]. The Suzuki and Ree groups will be dealt with separately in Section 2.4. For each λ ∈ X(T )+ , let H 0 (λ) = indG B λ. The simple modules for G are labeled by the set X(T )+ and are given by the correspondence λ → L(λ) = socG (H 0 (λ)). Every G-module V becomes a Gσ (Fq )-module by restriction. A complete set of non-isomorphic simple Gσ (Fq )-modules and simple Gr -modules are obtained by restricting {L(λ) : λ ∈ Xr (T )} [Cur], [Ste1]. For any G-module V and simple Gσ (Fq )-module L(λ) we denote by [V : L(λ)]Gσ (Fq ) the multiplicity of L(λ) as a Gσ (Fq )-composition factor of V . One obtains [V : L(λ)]Gσ (Fq ) = [V : L(γ )]G [L(γ ) : L(λ)]Gσ (Fq ) . γ ∈X(T )+

Moreover, twisting a G-module V with the rth Frobenius morphism yields the same result as twisting by σ −1 upon restriction to Gσ (Fq ). According to [Jan3, 1.3] one obtains for all µ ∈ X(T )+ that L(µ)(r) ∼ = L(σ (µ))

(1.4.1)

as a Gσ (Fq )-module.

2. General Ext-formulas 2.1. We briefly review the techniques developed in [BNP1]. For a fixed positive integer r let Cs be the full subcategory of Mod(G) with objects having composition factors whose highest weights lie in πs where πs = {λ ∈ X(T )+ : λ + ρ, α0∨ < 2pr s(h − 1)}.

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Notice that the category C1 differs slightly from Jantzen’s pr -bounded category where it is assumed that λ, α0∨ < 2pr (h − 1) [Jan1, p. 360]. Let FCs be the truncation functor from Mod(G) to Cs which takes M ∈ Mod(G) to the largest submodule of j M in Cs (see [Don, §1.1]). Let Gs = FCs indG Gσ (Fq ) and R Gs be the higher right derived functors of Gs . We remark that if M is a finite-dimensional Gσ (Fq )-module, then Gs (M) is also finite-dimensional. In order to see this fact, for any weight λ in πs , let P (λ) be the projective cover of L(λ) in Cs . Using right adjointness of Gs to the restriction functor we have [Gs (M) : L(λ)] = dim HomG (P (λ), Gs (M)) = dim HomGσ (Fq ) (P (λ), M). The last term is finite because πs is a finite set and Cs is equivalent to the module category for some finite-dimensional quasi-hereditary algebra [BNP1, §4.5]. Since σ preserves the inner product and the order on X(T )+ the arguments in [BNP1, 4.1- 4.4] yield an analog of [BNP1, Thm. 4.4a] as follows. Theorem. For M ∈ Cs and N ∈ Mod(Gσ (Fq )) there exists a first quadrant spectral sequence i,j

i+j

E2 = ExtiG (M, R j Gs (N)) ⇒ ExtGσ (Fq ) (M, N ). 2.2. The rth Steinberg module St r is projective and injective in Mod(Gσ (Fq )) and in Mod(Gr ). For p ≥ 2(h − 1) the projectivity and injectivity extends to the category C1 [Jan2]. The injective hulls of all p r -restricted simple modules in all three categories appear as respective direct summands of tensor products of Str with appropriate p r -restricted simple modules. Moreover, the injective hull Qr (λ) of a simple Gr -module L(λ) lifts to a G-structure and is also injective in C1 [Jan2]. A theorem due to Chastkofsky [Cha] and Jantzen [Jan3] says that the injective hull of L(λ) in Mod(Gσ (Fq )) appears exactly once as a summand of Qr (λ), both for the split forms and for the quasi-split forms. Therefore, the arguments in [BNP1, 4.6–5.2], which were addressing only the split forms extend to the quasi-split forms. One obtains a generalization of [BNP1, 5.2a]. Theorem. Let s ≥ 1, p ≥ 2s(h − 1), M ∈ Mod(Gσ (Fq )), and let L ∈ Cs such that L has only p r -restricted composition factors in its head (as a G-module), then Ext iGσ (Fq ) (L, M) ∼ = ExtiG (L, Gs (M)) for all 0 ≤ i ≤ s. 2.3. For the remainder of the paper we will look at the Ext1 groups. We will simply denote G1 by G, π1 by π , and C1 by C. It turns out that the module G(k) is of particular interest. The theorem below justifies this statement. It is an analog to [BNP2, Thm. 2.2]. The proof in [BNP2] can easily be adapted to the quasi-split case.

Extensions for finite groups of Lie type: twisted groups

33

The Suzuki and Ree groups will be discussed separately in the next section. Notice that the result holds for any prime. Theorem. Let λ, µ ∈ Xr (T ). Then Ext 1Gσ (Fq ) (L(λ), L(µ)) ∼ = Ext1G (L(λ), L(µ) ⊗ G(k)). 2.4. Suzuki and Ree groups. The purpose of this section is to obtain a version of Theorem 2.3 for the Suzuki and Ree groups. Let G be of type C2 or F4 and p = 2 or G of type G2 and p = 3. Then there exists a special isogeny σ on G with σ 2 = F , the Frobenius morphism, which leads to a stronger Steinberg Tensor Product Theorem [Ste1, Section 11], [Sin, 1.4 (2)]. Define Xσ (T ) ⊂ X1 (T ) to be the subset of those weights that are orthogonal to iall long simple roots. Then every λ ∈ X(T )+ has a σ -adic expansion λ = m i=0 σ λi , λi ∈ Xσ (T ). For an odd positive integer m we denote by G(m) the subgroup of G fixed by ) and Ree groups (for types G2 , F4 ). The set σ m . These are the Suzuki (for type C2 i of σ m -restricted weights Xσ m (T ) = { m−1 i=0 σ λi | λi ∈ Xσ (T )} parametrizes the simple modules of G(m). The Steinberg module St(m) with highest weight (σ m −1)ρ is injective and projective. For fixed m we denote by C the subcategory of Mod(G) whose highest weights γ satisfy γ , α0∨ ≤ 2(σ m − 1)ρ, α0∨ . As in Section 2.1 we let FC be the truncation functor from Mod(G) to C which takes M ∈ Mod(G) to the largest submodule of M in C and set G = FC indG G(m) . As before we obtain for any M ∈ C and any N ∈ Mod(G(m)) a spectral sequence i,j

i+j

E2 = ExtiG (M, R j G(N)) ⇒ ExtG(m) (M, N ).

(2.4.1)

In the proof of the following theorem we make use of the fact that the E21,0 -term of the spectral sequence embeds in the E 1 -term. Theorem. Let λ, µ ∈ Xσ m (T ). Then Ext1G(m) (L(λ), L(µ)) ∼ = Ext1G (L(λ), L(µ) ⊗ G(k)). Proof. All simple modules are self-dual and Ext 1G(m) (L(λ), L(µ)) ∼ = Ext1G(m) (L(µ), L(λ)). Therefore, we may assume that µ, α0∨ ≤ λ, α0∨ and furthermore that µ ≯ λ. Set M = St(m)⊗L((σ m −1)ρ −λ). Then M is injective and projective as a G(m)-module and M ⊗ L(µ) is in C. In order to proceed as in the proof of [BNP2, Thm. 2.2] we need the following HomG(m) (M ⊗ L(µ), k) ∼ = HomG(m) (L(λ) ⊗ L(µ), k)

(2.4.2)

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Christopher P. Bendel, Daniel K. Nakano and Cornelius Pillen

and HomG (M, L(λ) ∼ = k.

(2.4.3)

Clearly,

k HomG(m) (L(λ) ⊗ L(µ), k) ∼ = HomG (L(λ) ⊗ L(µ), k) ∼ = 0

for λ = µ, otherwise.

Both (2.4.2) and (2.4.3) follow immediately from the Lemma below. Equation (2.4.2) now replaces [BNP2, (2.2.1)] and (2.4.3) allows us to define the G-module R via 0 → R → M ⊗ L(µ) → L(λ) ⊗ L(µ) → 0. This last equation replaces [BNP2, (2.2.2)]. For the remainder of the proof one can proceed as in the proof of [BNP2, Thm. 2.2] by replacing the spectral sequence [BNP2, (2.1.1)] with (2.4.1). Lemma. Let λ, µ ∈ Xσ m (T ) with µ ≯ λ. Then (a) HomG (St(m) ⊗ L((σ m − 1)ρ − λ), L(λ)) ∼ = k. (b) HomG(m)

(St(m) ⊗ L((σ m

k for λ = µ, ∼ − 1)ρ − λ), L(µ)) = 0 otherwise.

Proof. (a) Set r = (m−1)/2 and let ρ˜ be the sum of all fundamental weights in Xσ (T ). Then St(m) ∼ ˜ (r) . There are unique λ0 ∈ Xr (T ) and λ1 ∈ Xσ (T ) = St r ⊗L((p − 1)ρ) 0 r 1 with λ = λ + p λ . Let Gσ = Ker σ . Then Gσ is an infinitesimal group scheme [Sin, 1.2]. One can argue as follows: HomG (St(m) ⊗ L((σ m − 1)ρ − λ), L(λ)) ∼ HomG (St(m), L((σ m − 1)ρ − λ) ⊗ L(λ)) = ∼ = (HomGr (Str , L((pr − 1)ρ − λ0 ) ⊗ L(λ0 ))G ⊗ HomG (L((p − 1)ρ), ˜ L((p − 1)ρ˜ − λ1 ) ⊗ L(λ1 )) ∼ ˜ L((p − 1)ρ˜ − λ1 ) ⊗ L(λ1 )) (by [Jan1, II.11.9 (2)]) = HomG (L((p − 1)ρ), ∼ ˜ L((p − 1)ρ˜ − λ1 ) ⊗ L(λ1 )))G (by [Sin, 1.8 (1)]). = (HomG (L((p − 1)ρ), σ

The assertion now follows from the fact that L((p −1)ρ) ˜ is projective as a Gσ -module [Sin, 1.7 (1)] and that the multiplicity of L((p − 1)ρ) ˜ in L((p − 1)ρ˜ − λ1 ) ⊗ L(λ1 ) is one. (b) St(m) is projective as a G(m)-module. Therefore dim HomG(m) (St(m) ⊗ L((σ m − 1)ρ − λ) ⊗ L(µ), k)

= dim HomG(m) (St(m), L((σ m − 1)ρ − λ) ⊗ L(µ)) = [L((σ m − 1)ρ − λ) ⊗ L(µ) : St(m)]G(m)

Extensions for finite groups of Lie type: twisted groups

=

35

[L((σ m − 1)ρ − λ) ⊗ L(µ) : L(ν)]G · [L(ν0 ) ⊗ L(σ m ν1 ) : St(m)]G(m)

ν∈X(T )+

=

[L((σ m − 1)ρ − λ) ⊗ L(µ) : L(ν)]G · [L(ν0 ) ⊗ L(ν1 ) : St(m)]G(m) .

ν∈X(T )+

Notice that ν1 = 0 forces ν0 = (σ m − 1)ρ and [L((σ m − 1)ρ − λ) ⊗ L(µ) : St(m)]G(m) = [L((σ m − 1)ρ − λ) ⊗ L(µ) : St(m)]G . The latter is zero unless µ ≥ λ, which implies µ = λ. In this case the multiplicity is one, as desired. Assume that there exists a ν with ν1 = 0 that contributes to the above sum. Observe that σ m ν1 > ν1 . It follows that (σ m − 1)ρ − λ + µ ≥ ν = ν0 + σ m ν1 > ν0 + ν1 . By using [L(ν0 ) ⊗ L(ν1 ) : St(m)]G(m) = [L(ν0 ) ⊗ L(ν1 ) : L(γ )]G · [L(γ0 ) ⊗ L(σ m γ1 ) : St(m)]G(m) γ ∈X(T )+

=

[L(ν0 ) ⊗ L(ν1 ) : L(ν)]G · [L(γ0 ) ⊗ L(γ1 ) : St(m)]G(m) ,

γ ∈X(T )+

we can argue inductively on the size of the weight ν that [L(ν) : St(m)]G(m) = 0 unless ν ≥ (σ m − 1)ρ. But in that case (σ m − 1)ρ − λ + µ ≥ ν = ν0 + σ m ν1 > ν0 + ν1 ≥ (σ m − 1)ρ gives the desired contradiction to µ ≯ λ. 2.5. From now on all finite groups considered are either split or quasi-split forms of G. The remainder of Section 2 is devoted to collecting information on G(k). Observe that if ν ∈ π is expressed as ν = ν0 + pr ν1 for weights ν0 ∈ Xr (T ) and ν1 ∈ X(T )+ , then the weight ν1 satisfies ν1 , α0∨ < 2(h − 1). We will show that the highest weight of any composition factor of G(k) is contained in a prescribed finite set which is independent of p and r. For a positive integer m, we define the set

m = {ν ∈ X(T )+ : ν, α0∨ < m}. Proposition. If L(ν0 + p r ν1 ) is a composition factor of G(k) then ν0 , ν1 ∈ 2(h−1) . Proof. We know already that ν1 ∈ 2(h−1) . Assume that [G(k) : L(ν0 + p r ν1 )]G = 0.

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Christopher P. Bendel, Daniel K. Nakano and Cornelius Pillen

Then [G(k) : L(ν0 )]Gr = 0. This implies that HomGr (Qr (ν0 ), G(k)) = 0. Set ν) [Jan1, II.11.9]. ν = (pr − 1)ρ + w0 ν0 . Then Qr (ν0 ) is a Gr -summand of Str ⊗L( One obtains ν), G(k)) = HomGr (Str , G(k) ⊗ L( ν)∗ ) = 0. HomGr (Str ⊗L( ν)∗ is not zero. Therefore The St r -isotypical component of the Gr -socle of G(k) ⊗ L( there exists a weight γ such that ν)∗ ) = HomG (Str ⊗L(γ )(r) ⊗ L( ν), G(k)) = 0. HomG (Str ⊗L(γ )(r) , G(k) ⊗ L( The G-module Str ⊗L(γ )(r) is simple with highest weight (p r − 1)ρ + p r γ . This weight has to be less than or equal to any highest weight µ of G(k) ⊗ L( ν)∗ . But any ∨ ∨ r such µ satisfies µ + ρ, α0 < (3p − 1)(h − 1). Hence, γ , α0 < 2(h − 1). Since G(k) is the truncation of indG Gσ (Fq ) (k) it follows that HomG (Str ⊗L(γ )(r) ⊗ L( ν), IndG Gσ (Fq ) (k)) = 0. and by adjointness ν), k) = 0. HomGσ (Fq ) (Str ⊗L(γ )(r) ⊗ L( As a Gσ (Fq )-module L(γ )(r) ∼ = L(σ (γ )) and one obtains ν), Str ) = 0. HomG(Fq ) (L(σ (γ )) ⊗ L( It follows from [Jan2, Satz 1.5] that ν), Str ) dim HomGσ (Fq ) (L(σ (γ )) ⊗ L( = [L(σ (γ )) ⊗ L( ν) : Str ]Gσ (Fq ) = [L(σ (γ )) ⊗ L( ν) ⊗ L(σ (µ)) : Str ⊗L(µ)(r) ]G . µ∈X(T )+

ν = −w0 ν0 + pr µ − σ (µ) for some Therefore σ (γ ) ≥ (p r − 1)ρ + pr µ − σ (µ) − µ. Since σ preserves the inner product one obtains 2(h − 1) > γ , α0∨ ≥ ν0 , α0∨ .

2.6. An immediate consequence of Theorem 2.3 and Proposition 2.5 is that any extension between simple modules for the finite group Gσ (Fq ) has to come from some extension between a finite set of modules for the algebraic group. For large primes this will be made more explicit later. Corollary. Let λ, µ ∈ Xr (T ). If Ext1Gσ (Fq ) (L(λ), L(µ)) = 0 then there exist weights

ν0 , ν1 ∈ 2(h−1) such that Ext1G (L(λ) ⊗ L(ν0 ), L(µ) ⊗ L(ν1 )(r) ) = 0.

Proof. By Theorem 2.3 we have Ext1Gσ (Fq ) (L(λ), L(µ)) ∼ = Ext 1G (L(λ), L(µ)⊗G(k)). Our discussion in 2.1 and Proposition 2.5 show that G(k) is finite-dimensional with

37

Extensions for finite groups of Lie type: twisted groups

simple composition factors of the form L(γ + p r ν1 ) where γ , ν1 ∈ 2(h−1) . If Ext 1Gσ (Fq ) (L(λ), L(µ)) = 0 then there exist γ , ν1 ∈ 2(h−1) such that Ext 1G (L(λ), L(µ)⊗L(γ )⊗L(ν1 )(r) ) ∼ = Ext 1G (L(λ)⊗L(−w0 γ ), L(µ)⊗L(ν1 )(r) ) = 0. Set ν0 = −w0 γ and the assertion follows. 2.7. Proposition 2.5 gives useful information only for q > 2(h − 1). In that case we are also able to describe the G-socle of G(k). Lemma. Let p r ≥ 2(h − 1) then socG G(k) =

L(σ (ν) + p r (−w0 ν)).

ν∈ 2(h−1)

Proof. The condition p r ≥ 2(h − 1) forces any weight in 2(h−1) to be p r -restricted. By Proposition 2.5 and right adjointness of G to the restriction functor one obtains HomG (L(ν0 + p r ν1 ), G(k)) ⊗ L(ν0 + p r ν1 ) socG G(k) = ν0 ,ν1 ∈ 2(h−1)

=

HomGσ (Fq ) (L(ν0 ) ⊗ L(ν1 )(r) , k) ⊗ L(ν0 + p r ν1 )

ν0 ,ν1 ∈ 2(h−1)

=

HomGσ (Fq ) (L(ν0 ) ⊗ L(σ (ν1 )), k) ⊗ L(ν0 + p r ν1 )

ν0 ,ν1 ∈ 2(h−1)

=

(by (1.4.1))

HomGσ (Fq ) (L(ν0 ), L(−w0 σ (ν1 ))) ⊗ L(ν0 + p r ν1 )

ν0 ,ν1 ∈ 2(h−1)

=

L(−w0 σ (ν) + p r ν)

ν∈ 2(h−1)

=

L(σ (ν) + p r (−w0 ν)).

ν∈ 2(h−1)

2.8. The nicest and most explicit extension results can be obtained when G(k) is semisimple. The semisimplicity of G(k) was shown for the split forms in [BNP1, 7.4]. We present a slightly different proof that involves only the algebraic group and works for split and quasi-split forms of G. Proposition. Let p ≥ 3(h − 1), then G(k) is semisimple. Moreover, G(k) = L(σ (ν) + p r (−w0 ν)). ν∈ 2(h−1)

Proof. For type A1 we refer to [BNP1, Thm. 7.4], so we may therefore assume that G is not of type A1 . Assume further that L(λ0 + pr λ1 ) and L(ν0 + pr ν1 ) are composition

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Christopher P. Bendel, Daniel K. Nakano and Cornelius Pillen

factors of G(k). We look at the first three terms of the five-term exact sequence corresponding to the Lyndon–Hochschild–Serre spectral sequence [Jan1, I.6.6]: 0 → Ext1G/Gr (L(λ1 )(r) , HomGr (L(λ0 ), L(ν0 )) ⊗ L(ν1 )(r) ) → Ext 1G (L(λ0 + p r λ1 ), L(ν0 + p r ν1 )) → HomG/Gr (L(λ1 )(r) , Ext1Gr (L(λ0 ), L(ν0 )) ⊗ L(ν1 )(r) ) → ··· . By Proposition 2.5 we have λ0 , λ1 , ν0 , ν1 ∈ 2(h−1) . The size of p forces all these weights to be inside the lowest alcove. Clearly we have Hom G1 (L(λ0 ), L(ν0 )) ∼ = HomG (L(λ0 ), L(ν0 )) and by the Linkage Principle Ext 1G (L(λ1 ), L(ν1 )) = 0. Therefore, the first term of the sequence vanishes. Since λ0 and ν0 are inside the lowest alcove it follows from [BNP3, 5.4 Cor B] that Ext 1G1 (L(λ0 ), L(ν0 )) = 0. Thus the third term in the sequence also vanishes. We conclude that there are no extensions between simple composition factors of G(k). The module is semisimple and the assertion follows from Lemma 2.7. 2.9. The semisimplicity of G(k) for p ≥ 3(h − 1) gives us an extremely useful Ext1 formula for simple Gσ (Fq )-modules. For the split forms this was shown in [BNP2, 2.5]. Theorem. For p ≥ 3(h − 1) and λ, µ ∈ Xr (T ), Ext 1Gσ (Fq ) (L(λ), L(µ)) ∼ Ext1G (L(λ) ⊗ L(ν)(r) , L(µ) ⊗ L(σ (ν))). = ν∈ h

Proof. For λ, µ ∈ Xr (T ), we have Ext1Gσ (Fq ) (L(λ), L(µ)) ∼ = Ext1G (L(λ), L(µ) ⊗ G(k)) (by Thm. 2.3) ∼ Ext1G (L(λ), L(µ) ⊗ L(σ (ν) + p r (−w0 ν))) = ν∈ 2(h−1)

∼ =

ν∈ 2(h−1)

∼ =

ν∈ h

(by Prop. 2.8) Ext1G (L(λ) ⊗ L(ν)(r) , L(µ) ⊗ L(σ (ν)))

Ext1G (L(λ) ⊗ L(ν)(r) , L(µ) ⊗ L(σ (ν))) (by [BNP2, Prop. 2.4 c]).

Extensions for finite groups of Lie type: twisted groups

39

3. Extensions for r ≥ 2 The formula given in the previous theorem involves extensions between a simple pr bounded G-module and the tensor product of two simple p r -restricted G-modules, one of them having a small highest weight. It turns out that for r ≥ 2 the extension and the tensor product can be calculated separately. This allows for even more explicit formulas. Also, for r ≥ 3, if 3.1. For γ , δ ∈ X(T )+ , let S(γ , δ) = HomG (L(γ ), L(δ)). r−2 i−1 i λ where λ ∈ X (T ), set λ ¨ p = λi and for λ ∈ Xr (T ) and λ = r−1 i i 1 i=0 i=1 p r = 2 set λ¨ = 0. Theorem 3.1 is a generalization of [BNP3, Thm. 2.4b]. It follows from Theorem 2.9 by reidentifying the G-extensions as done in [BNP3] using [BNP2, Thm. 3.2] (equivalently [BNP3, Thm. 2.1]) and [BNP3, Prop. 2.3]. Theorem. Let p ≥ 3(h − 1), r ≥ 2, and λ, µ ∈ Xr (T ). Then Ext 1Gσ (Fq ) (L(λ), L(µ)) ∼ = Ext1G (L(λ), L(µ)) ⊕ R where R=

ν∈ h −{0}

Ext1G (L(λr−1 ) ⊗ L(ν)(1) , L(µr−1 )) ¨ ⊗ HomG (L(λ0 ), L(µ0 ) ⊗ L(σ (ν))) ⊗ S(λ¨ , µ).

or equivalently HomG (L(ν)(1) , Ext 1Gr (L(λr−1 ), L(µr−1 )) R= ν∈ h −{0}

¨ ⊗ HomG (L(λ0 ), L(µ0 ) ⊗ L(σ (ν))) ⊗ S(λ¨ , µ). 3.2. The next theorem, which is an analog of [BNP3, Thm. 3.1], identifies the extensions between a pair of simple Gσ (Fq )-modules with the G-extensions of a single λ pair of p r -restricted simple G-modules. Note that if n = r − 1, then the weights and µ defined below simply equal λ and µ respectively. Theorem. Let p ≥ 3(h − 1) and λ, µ ∈ Xr (T ). Set n = min({i | λi = µi } ∪ {r − 1}) and define p r -restricted weights λ=

r−n−2 i=0

p i σ (λn+1+i ) +

r−1 i=r−n−1

pi λi−r+n+1 ,

40

Christopher P. Bendel, Daniel K. Nakano and Cornelius Pillen

and

µ=

r−n−2 i=0

r−1

p i σ (µn+1+i ) +

p i µi−r+n+1 .

i=r−n−1

(a) If r ≥ 3, then λ), L( µ)). Ext1Gσ (Fq ) (L(λ), L(µ)) ∼ = Ext1G (L( (b) If r = 2 and λ0 = µ0 or λ1 = µ1 , then λ), L( µ)). Ext 1Gσ (Fq )) (L(λ), L(µ)) ∼ = Ext1G (L( (c) If r = 2 and λi = µi for i = 0, 1, set λˆ = σ (λ1 )+pλ0 and µˆ = σ (µ1 )+pµ0 . Then ˆ Ext 1Gσ (Fq )) (L(λ), L(µ)) ∼ = Ext1G (L(λ), L(µ)) ⊕ Ext1G (L(λˆ ), L(µ)). Proof. The Frobenius morphism is an automorphism on Gσ (Fq ). Hence, Ext 1Gσ (Fq ) (L(λ), L(µ)) ∼ = Ext1Gσ (Fq ) (L(λ)(r−1−n) , L(µ)(r−1−n) ), where n = min({i | λi = µi } ∪ {r − 1}). Steinberg’s tensor product theorem together with (1.4.1) implies that L( λ) ∼ µ) ∼ = L(λ)(r−1−n) and L( = L(µ)(r−1−n) as Gσ (Fq )modules. Therefore, we may identify λ), L( µ)). Ext 1Gσ (Fq ) (L(λ), L(µ)) ∼ = Ext1Gσ (Fq ) (L( (a) For r ≥ 3, Theorem 3.1 and [BNP3, Thm. 2.4a] imply that Ext1Gσ (Fq ) (L(λ), L(µ)) = Ext1G (L( λ), L( µ)) = 0 unless there exists an i with 0 ≤ i ≤ r − 1 and λi = µi . We assume such an i exists and apply Theorem 3.1 to the pair of weights µr−1 which forces Ext1G1 (L( λr−1 ), L( µr−1 )) = 0 by λ and µ. Observe that λr−1 = [And1]. The remainder term R in Theorem 3.1 vanishes and λ), L( µ)) ∼ λ), L( µ)). Ext 1Gσ (Fq ) (L(λ), L(µ)) ∼ = Ext1Gσ (Fq ) (L( = Ext1G (L( This proves part (a). λr−1 = µr−1 and one can (b) Assume r = 2. If either λ0 = µ0 or λ1 = µ1 , then argue as above. (c) Assume r = 2 and λi = µi for i = 0, 1. By Theorem 3.1, we have Ext1Gσ (Fq ) (L(λ), L(µ)) ∼ = Ext1G (L(λ), L(µ)) ⊕ R

Extensions for finite groups of Lie type: twisted groups

where R=

ν∈ h −{0}

41

Ext 1G (L(λ1 ) ⊗ L(ν)(1) , L(µ1 )) ⊗ HomG (L(λ0 ), L(µ0 ) ⊗ L(σ (ν))).

Since λ0 = µ0 , we have R= Ext1G (L(λ1 ) ⊗ L(ν)(1) , L(µ1 )) ⊗ HomG (L(λ0 ), L(µ0 ) ⊗ L(σ (ν))). ν∈ h

Since σ is an automorphism on G, we have Ext 1G (L(λ1 ) ⊗ L(ν)(1) , L(µ1 )) ∼ = Ext1G (L(σ (λ1 )) ⊗ L(σ (ν))(1) , L(σ (µ1 ))). We can re-write R as R= Ext1G (L(σ (λ1 )) ⊗ L(ν)(1) , L(σ (µ1 ))) ⊗ HomG (L(λ0 ), L(µ0 ) ⊗ L(ν)). ν∈ h

By [BNP3, Thm. 2.4a], the remainder term R is exactly Ext 1G (L(λˆ ), L(µ)). ˆ 3.3. Various conjectures have been made about the dimensions of Ext1 -groups. Here it is shown that in most cases the dimensions of the Ext 1 groups between simple modules for the finite groups are bounded by the dimensions of Ext1 groups for the reductive algebraic groups. The corollary below is a generalization of [BNP3, Thm. 3.3]. The proof follows along the same lines and details are left to the reader. Corollary. Let p ≥ 3(h − 1), r ≥ 2, and λ, µ ∈ Xr (T ). Then max{dimk Ext1Gσ (Fq ) (L(λ), L(µ)) | λ, µ ∈ Xr (T )} = max{dimk Ext1G (L(λ), L(µ)) | λ, µ ∈ Xr (T )}, unless, r = 2, σ = 1, and the underlying root system is of type Cn , n ≥ 1. In that case max{dimk Ext1Gσ (Fq ) (L(λ), L(µ)) | λ, µ ∈ Xr (T )} ≤ 2 max{dimk Ext1G (L(λ), L(µ)) | λ, µ ∈ Xr (T )}.

4. Self-extensions Let p ≥ 3(h − 1). In [BNP2] it was shown that self-extensions, i.e. extensions of a simple G(Fq )-module by itself, do not exist unless q = p and the root system of the underlying algebraic group is of type Cn . Since there are no quasi-split forms of type Cn it is not surprising that the result generalizes to the twisted case.

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Christopher P. Bendel, Daniel K. Nakano and Cornelius Pillen

4.1. Self-extensions, r ≥ 2. Theorem 3.2 gives a nice one-to-one correspondence between extensions of simple modules for the finite group Gσ (Fq ) and extensions of p r -restricted simple modules for the algebraic group. Since it is well-known that self-extensions do not exist for algebraic groups it is an immediate consequence of Theorem 3.2 that self-extensions also vanish for simple Gσ (Fq )-modules as long as q > p and p ≥ 3(h − 1). Theorem. For p ≥ 3(h − 1), r ≥ 2, and λ ∈ Xr (T ), Ext1Gσ (Fq ) (L(λ), L(λ)) = 0. 4.2. Self-extensions, r = 1. In the case p = q the analysis in [BNP2] was considerably harder since self-extensions were known to exist for groups of type Cn [Hum], [TZ, Rem. 3.18], [Pil]. It turns out, however, that the quasi-split forms do not add to this list. One obtains the following result. Theorem. Let p ≥ 3(h − 1) and λ ∈ X1 (T ). If either (a) G does not have underlying root system of type Cn (n ≥ 1) or

(b) λ, αn∨ = p−2−c 2 , where αn is the unique long simple root and c is odd with 0 < |c| ≤ h − 1, then Ext 1Gσ (Fp ) (L(λ), L(λ)) = 0. Proof. By Theorem 2.9 one has Ext1Gσ (Fp ) (L(λ), L(µ)) ∼ =

ν∈ h

Ext 1G (L(λ) ⊗ L(ν)(r) , L(µ) ⊗ L(σ (ν))),

where h = {ν ∈ X(T )+ |ν, α0∨ < h}. The only complication in the quasi-split case versus the split case is that the weights ν and σ (ν) may be distinct. However, Corollary 4.9 in [BNP2] says that Ext1G (L(λ) ⊗ L(ν1 )(r) , L(µ) ⊗ L(σ (ν2 ))) vanishes for any ν1 , ν2 ∈ h as long (a) or (b) holds.

5. First cohomology for simple modules In this section we consolidate our results on the first cohomology for simple groups that can be found in [BNP1] [BNP3] and extend them to the quasi split forms of G. Notice that Theorem 5.1 offers some improvements on the condition of the prime (compare to [BNP1, Thm. 7.4]) and the size of the weights ( h−1 versus h ). For technical reasons we exclude the well-known results for groups of type A1 [AJL]. The most striking result is a one-to-one correspondence between p r -bounded simple Gmodules with non-vanishing cohomology and simple Gσ (Fq )-modules with non-zero cohomology. The correspondence is simply the identity for pr -restricted G-modules, while non-restricted G-modules correspond to simple Gσ (Fq )-modules whose pr restricted highest weights are not linked to 0. Naturally, this gives also a one-to-one

Extensions for finite groups of Lie type: twisted groups

43

correspondence between the cohomology of simple modules for the quasi-split and the split forms over a given G. p we denote the extended affine Weyl group generated by the ordinary Weyl 5.1. By W group and the translation by elements in pX(T ). Two weights λ and ν are said to be G1 -linked if λ = w · ν + pγ for some w ∈ W and γ ∈ X(T ) or, equivalently, if there p such that λ = u · ν. In this section we make use of the exists an element u ∈ W strong linkage principle [Jan1, II.6.13] and replace the order relation “≤” on X(T ) by the stronger relation “↑” as defined in [Jan1, II.6.4]. Theorem. Let p ≥ 3(h−1), λ ∈ Xr (T ), and h−1 = {ν ∈ X(T )+ | ν, α0∨ < h−1}. Assume that G is not of type A1 . (a) If λ is not G1 -linked to any weight in h−1 , then H1 (Gσ (Fq ), L(λ)) = 0. (b) If λ is G1 -linked to the zero weight, then H1 (Gσ (Fq ), L(λ)) ∼ = H1 (G, L(λ)). (c) If r = 1 and λ is G1 -linked to some element in h−1 , then there exist unique p such that λ = u · ν and ν ∈ h−1 and u ∈ W H1 (Gσ (Fq ), L(λ)) ∼ = H1 (G, L(u · 0 + pσ −1 (ν))). In addition, if pσ −1 (ν) ↑ u · 0 then k if λ = pωα + sα · (−w0 σ (ωα ))where α ∈ 1 H (Gσ (Fq ), L(λ)) = and ωα is the corresponding fundamental weight, 0 else. (d) If r > 1, then H1 (Gσ (Fq ), L(λ)) = 0 unless there exist unique ν ∈ h−1 and p such that λ = pr−1 u · 0 + ν. In that case u∈W H1 (Gσ (Fq ), L(λ)) ∼ = H1 (G, L(u · 0 + pσ −1 (ν))). In addition, if pσ −1 (ν) ↑ u · 0 then r r−1 k if λ = p ωα − p α − w0 σ (ωα ) where α ∈ H1 (Gσ (Fq ), L(λ)) = and ωα is the corresponding fundamental weight, 0 else. Proof. Since p ≥ 3(h − 1) all weights ν ∈ h are in the interior of the lowest alcove. Consequently, from Jantzen’s translation principle [Jan1, II. 7.9], L(ν) ∼ = T0ν (k). One

44

Christopher P. Bendel, Daniel K. Nakano and Cornelius Pillen

can now argue as follows: H1 (G(Fq ), L(λ)) ∼ = Ext1G(Fq ) (k, L(λ)) ∼ Ext1G (L(ν)(r) , L(λ) ⊗ L(σ (ν))) (by Theorem 2.9) = ν∈ h

∼ =

ν∈ h

∼ =

ν∈ h

∼ =

ν∈ h

∼ =

ν∈ h

Ext1G (L(−w0 σ (ν)), L(λ) ⊗ L(−w0 ν)(r) ) Ext1G (L(ν), L(λ) ⊗ L(σ −1 (ν))(r) ) Ext1G (T0ν (k), L(λ + pr σ −1 (ν))) Ext1G (k, Tν0 L(λ + p r σ −1 (ν))) ([Jan1, II.7.6])

Assume that Tν0 L(λ+p r σ −1 (ν)) is not zero and denote by λν the unique pr -restricted weight such that L(λν + p r σ −1 (ν)) ∼ = Tν0 L(λ + p r σ −1 (ν)). We have Ext 1G (k, Tν0 L(λ + p r σ −1 (ν))) ∼ = Ext1G (k, L(λν + p r σ −1 (ν))) ∼ = Ext 1 (L(−w0 σ −1 (ν))(r) , L(λν )). G

Define Q via the exact sequence of G-modules 0 → L(λν ) → H 0 (λν ) → Q → 0 and obtain the exact sequence HomG (L(−w0 σ −1 (ν))(r) , Q) → Ext1G (L(−w0 σ −1 (ν))(r) , L(λν )) → Ext 1G (L(−w0 σ −1 (ν))(r) , H 0 (λν )). We conclude that Ext1G (k, Tν0 L(λ+p r σ −1 (ν))) = 0 unless either −w0 p r σ −1 (ν) ↑ λν by [Jan1, II.6.6] or λν = pr ωα − p i α and −w0 σ −1 (ν) = ωα , where α is a simple root and ωα the corresponding fundamental weight. The last assertion follows from [BNP3, Prop. 4.3 b]. In the first case we conclude that pr −w0 σ −1 (ν), α0∨ = pr ν, α0∨ ≤ (p r − 1)(h − 1). In either case ν, α0∨ < h − 1, unless G is of type A1 . Obviously the cohomology vanishes unless λ is G1 -linked to some ν ∈ h−1 . This implies part (a) of the Theorem. Next assume that λ is G1 -linked to two elements ν1 and ν2 in h−1 . Then there exist w ∈ W and γ ∈ X(T ) such that w · ν1 + pγ = ν2 . This implies for any simple root α that ν2 + ρ, α ∨ − w(ν1 + ρ), α ∨ = pγ , α ∨ . Since νi , α0∨ ≤ h − 2, the absolute value of the left hand side is at most h − 2 + 1 + 2h − 3 = 3(h − 1) − 1 < p. Thus γ = 0, which forces w = 1 and ν1 = ν2 . Consequently, if λ is G1 -linked to some p and ν ∈ h−1 such that λ = u · ν. element in h−1 , then there exist unique u ∈ W

Extensions for finite groups of Lie type: twisted groups

45

This implies that λν = u·0 and H1 (Gσ (Fq ), L(λ)) ∼ = Ext1G (k, L(u·0+p r σ −1 (ν))) ∼ = 1 r −1 H (G, L(u · 0 + p σ (ν))). Now part (b) and the first part of (c) follow immediately. Assume further that r = 1 and pσ −1 (ν) ↑ u · 0. From our earlier observations we know that the cohomology vanishes unless u · 0 = pωα − α = pωα + sα · 0 for some simple root α and ν = −w0 σ (ωα ). This forces λ = pωα + sα · (−w0 σ (ωα )) and H1 (Gσ (Fq ), L(λ)) ∼ = Ext1G (k, L(pωα −α+p(−w0 ωα )) ∼ = Ext1G (L(ωα )(1) , L(pωα − α)). The remainder of part (c) now is a consequence of [BNP3, Prop. 4.5]. Finally, we assume that r > 1 and that λ is G1 -linked to some non-zero ν ∈ p , and H1 (Gσ (Fq ), L(λ)) ∼

h−1 . Then ν is unique, λ = w · ν with unique w ∈ W = 1 r −1 H (G, L(w · 0 + p σ (ν))). From [BNP3, Thm. 2.4a] one obtains that H1 (G, L(w · p . In this case [BNP3, 0+p r σ −1 (ν))) = 0, unless w·0 = pr−1 u·0 for a unique u ∈ W 1 1 ∼ Thm. 2.4a] implies further that H (Gσ (Fq ), L(λ)) = H (G, L(w · 0 + pr σ −1 (ν))) ∼ = H1 (G, L(u · 0 + pσ −1 (ν))). Moreover, λ = ν + pr−1 u · 0. Now assume that pσ −1 (ν) ↑ u · 0. Again it follows from [BNP3, Prop. 4.5] that 1 H (G, L(u · 0 + pσ −1 (ν))) = k, if u · 0 = pωα − α for some simple root α and ν = −w0 σ (ωα ). This implies that λ = pr ωα − p r−1 α − w0 σ (ωα ). In all other cases the cohomology vanishes. The modules L(pωα + sα · (−w0 σ (ωα ))) and L(pr ωα − p r−1 α − w0 σ (ωα )) are of interest because for appropriate choices of α they are the smallest simple modules with H1 (Gσ (Fp ), L(λ)) = 0 and H1 (Gσ (Fq ), L(λ)) ∼ = H1 (G, L(λ)), respectively (see [Jan4, Prop. 2.2], [BNP3, 4.8]).

References [And1]

H. H. Andersen, Extensions of modules for algebraic groups, Amer. J. Math. 106 (1984), 498–504.

[And2]

H. H. Andersen, Extensions of simple modules for finite Chevalley groups, J. Algebra 111 (1987), 388–403.

[AJL]

H. H. Andersen, J. Jørgensen, P. Landrock, The projective indecomposable modules of SL(2, pn ), Proc. London Math. Soc. (3) 46 (1983), 38–52.

[BNP1] C. P. Bendel, D. K. Nakano, C. Pillen, On comparing the cohomology of algebraic groups, finite Chevalley groups, and Frobenius kernels, J. Pure and Appl. Algebra 163 (2001), 119–146. [BNP2] C. P. Bendel, D. K. Nakano, C. Pillen, Extensions for finite Chevalley groups I, Adv. Math. 183 (2004), 380–408. [BNP3] C. P. Bendel, D. K. Nakano, C. Pillen, Extensions for finite Chevalley groups II, Trans. Amer. Math. Soc. 354 (2002), 4421–4454. [BNP4] C. P. Bendel, D. K. Nakano, C. Pillen, Extensions for Frobenius kernels, J. Algebra 272 (2004), 476–511.

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[Car]

R. W. Carter, Finite Groups of Lie Type. Conjugacy Classes and Complex Characters, Wiley & Sons, Chichester 1993.

[Cha]

L. Chastkofsky, Characters of projective indecomposable modules for finite Chevalley groups, Proc. Symp. Pure Math. 37 (1980), 359–362.

[Cur]

C. W. Curtis, Representations of Lie algebras of classical type with applications to linear groups, J. Math. Mech. 9 (1960), 307–326.

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S. Donkin, On Schur algebras and related algebras I, J. Algebra 104 (1986), 310–328.

[Hum]

J. E. Humphreys, Non-zero Ext1 for Chevalley groups (via algebraic groups), J. London Math. Soc. 31 (1985) 463–467.

[Jan1]

J. C. Jantzen, Representations of Algebraic Groups, Academic Press, Orlando 1987.

[Jan2]

J. C. Jantzen, Darstellungen halbeinfacher Gruppen und ihrer Frobenius-Kerne, J. reine angew. Math 317 (1980), 157–199.

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J. C. Jantzen, Zur Reduktion modulo p der Charaktere von Deligne und Lusztig, J. Algebra 70 (1981), 452–474.

[Jan4]

J. C. Jantzen, First cohomology groups for classical Lie algebras, Progr. Math. 95, Birkhäuser Verlag 1991, 289–315.

[Pil]

C. Pillen, Self-extensions for the finite symplectic groups, preprint.

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P. Sin, Extensions of simple modules for special algebraic groups, J. Algebra 170 (1994), 1011–1034.

[Smi]

S. D. Smith, Sheaf homology and complete reducibility, J. Algebra 95 (1985), 72–80.

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R. Steinberg, Endomorphisms of linear algebraic groups, Mem. Amer. Math. Soc. 80, Amer. Math. Soc., Providence 1968.

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R. Steinberg, Representations of algebraic groups, Nagoya Math. J. 22 (1963), 33–56.

[TZ]

P. H. Tiep, A. E. Zalesskiˇı, Mod p reducibility of unramified representations of finite groups of Lie type, Proc. London Math. Soc. 84 (2002), 439–472.

Christopher P. Bendel, Department of Mathematics, Statistics and Computer Science, University of Wisconsin-Stout, Menomonie, WI 54751, U.S.A. E-mail: [email protected] Daniel K. Nakano, Department of Mathematics, University of Georgia, Athens, GA 30602, U.S.A. E-mail: [email protected] Cornelius Pillen, Department of Mathematics and Statistics, University of South Alabama, Mobile, AL 36688, U.S.A. E-mail: [email protected]

On the classifying space and cohomology of Thompson’s sporadic simple group David J. Benson∗

Abstract. This paper examines the classifying space of Thompson’s sporadic simple group. Two theorems are proved. The first gives a homotopy colimit decomposition for the 2-completion of the classifying space. The second shows that the mod two cohomology contains a copy of the Dickson invariants of rank five, over which it is finitely generated as a module.

1. Introduction In this paper, we consider the classifying space and the cohomology of John Thompson’s sporadic simple group Th at the prime two. We prove the following theorem. Theorem 1.1. Let C and N be maximal 2-local subgroups of Th of shape 21+8 A9 and 25 L5 (2) respectively, intersecting in a group K of shape 21+8 A8 = 25 24 L4 (2). Then the obvious map of groups from the amalgamated free product C ∗K N to Th is a mod two cohomology isomorphism. The corresponding map on classifying spaces from the homotopy colimit (double mapping cylinder) of the diagram BK BN

/ BC

to BTh is a homotopy equivalence after 2-completion. The proof of this theorem is straightforward, and really just involves combining the appropriate results from the literature. One consequence of this theorem is that the Mayer–Vietoris sequence of the amalgamated free product reduces to a short exact sequence 0 → H ∗ (Th, F2 ) → H ∗ (C, F2 ) ⊕ H ∗ (N, F2 ) → H ∗ (K, F2 ) → 0. In other words, H ∗ (C, F2 ) and H ∗ (N, F2 ) are subrings of H ∗ (K, F2 ) whose linear span is the whole ring, and whose intersection is H ∗ (Th, F2 ). Note that K contains a Sylow 2-subgroup of Th. ∗ The author is partly supported by a grant from the NSF.

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We need to work harder to prove the following theorem, which is really the main theorem of the paper. Theorem 1.2. There is a polynomial subring of H ∗ (Th, F2 ) of the form R = F2 [x16 , x24 , x28 , x30 , x31 ] with the following properties. (i) If E is any rank five elementary abelian 2-subgroup of Th (note that all maximal elementary abelian 2-subgroups in Th have rank five), then the restriction of R to H ∗ (E, F2 ) is injective, and the image is equal to the rank five Dickson invariants. (ii) H ∗ (Th, F2 ) is finitely generated as an R-module. This theorem continues a thread relating the Dickson invariants to sporadic groups, described in the papers [2], [5], [6], [8]. This began when Adem, Maginnis and Milgram computed the cohomology of the Mathieu group M12 and noticed that there was a copy of the rank three Dickson invariants, over which H ∗ (M12 , F2 ) was a finitely generated free module. There is a maximal 2-local subgroup N of M12 which is a nonsplit extension 23 L3 (2), and these Dickson invariants restrict to the invariants of N/O2 (N ) in the cohomology of O2 (N). Benson and Wilkerson [8] proved that this inclusion was induced by a map of classifying spaces BM12 → BG2 , and speculated on higher rank cases. In the paper [5], a homotopy colimit decomposition of the classifying space B Co3 was found. Using this, a map of classifying spaces B Co3 → BDI(4) was constructed, realizing a copy of the rank four Dickson invariants as a subring of H ∗ (Co3 , F2 ) over which it is finitely generated as a module. Here, DI(4) is the finite loop space at the prime two constructed by Dwyer and Wilkerson [15]. This map corresponds to the existence of a maximal 2-local subgroup which is a nonsplit extension 24 L4 (2). In the article [6], some observations were made concerning the relationship between BDI(4) and some nonexistent sporadic simple groups considered by Solomon [33]. These observations and some unpublished work of Puig were among the motivations leading to the notion of a p-local finite group; just enough local information is given to be able to build a p-completed classifying space, and the p-local finite group can be recovered from its p-completed classifying space. For more information, see Broto, Levi and Oliver [11]. In terms of this language, Levi and Oliver [19] made more precise the connections between BDI(4) and Solomon’s groups. We complete the picture by displaying a copy of the rank five Dickson invariants as a subring of H ∗ (Th, F2 ), corresponding to the maximal 2-local subgroup 25 L5 (2). It is a theorem of Lin and Williams [20] that there can be no space whose cohomology is the rank five Dickson invariants, so there is no obvious way to realize the inclusion of the Dickson invariants topologically. Furthermore, there is no nonsplit extension of the form 26 L6 (2), so there is no next case to consider, and the game stops here. It is also worth observing that if we allow the generators to be raised to the power of a high enough p-power, there is always a copy of the Dickson invariants over which the cohomology of a finite group with coefficients in Fp is finitely generated

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as a module. See for example Theorem 2.1 of [8], Theorem 8.1.8 of Neusel [25], or Theorem 8.2 of [7]. The Landweber–Stong conjecture, proved by Bourguiba and Zarati [9] (see also Henn [17]), says in this situation that the depth of the cohomology ring can be computed by seeing how many of the Dickson invariants form a regular sequence, taken in ascending order of degree. A simpler proof of the Landweber– Stong conjecture, but restricted to the context of cohomology of finite groups, can be found in [7]. Since the triality group 3D4 (2) : 3 is a maximal subgroup of Th, and has the same 2-rank, the following is an easy corollary of Theorem 1.2. Corollary 1.3. Both H ∗ (3D4 (2), F2 ) and H ∗ (3D4 (2) : 3, F2 ) contain a copy of the rank five Dickson invariants R = F2 [x16 , x24 , x28 , x30 , x31 ] as a subring over which they are finitely generated as a module. We remark that this can also be proved in a different way. Namely, 3D4 (2) : 3 is a finite subgroup of the compact Lie group F4 . Since H ∗ (BF4 ; F2 ) is known from the work of Borel to be a polynomial ring F2 [y4 , y6 , y7 , y16 , y24 ] (see for example Vavpetiˇc and Viruel [35]), it is not hard to find a suitable copy of the Dickson invariants as a subring over which this ring is a finitely generated module, and then restrict. One can show that the restriction maps from H ∗ (BF4 ; F2 ) to H ∗ (3D4 (2), F2 ) and H ∗ (3D4 (2) : 3, F2 ) are injective, by restricting further to a rank five elementary abelian 2-subgroup. Information on Thompson’s simple group of order 215 .310 .53 .72 .13.19.31 can be found in Thompson [34], Markot [24], Reifart [29], Smith [31], Parrott [26], Lyons [22], Holt [18], Yoshiara [38] and Havas, Soicher and Wilson [16]. The character table can be found in the Atlas [12], and the maximal subgroups were found by Wilson [37] and Linton [21]. Some authors write E or F3 or F3|3 for this group. Throughout this paper, the word “space” means “simplicial set.” When we refer to the p-completion of a space X, we mean Bousfield–Kan Fp -completion, denoted R∞ X (where R = Fp ) in Bousfield and Kan [10]. We note that the classifying spaces we are considering are Fp -good, so that their Fp -completions are Fp -complete. Specifically, it is shown in Ch. VII §§3–5 of [10] that spaces with finite fundamental group and spaces with perfect fundamental group are Fp -good; most of our groups are finite, and C ∗K N is perfect. Background material on the Dickson invariants can be found in Wilkerson’s primer [36], or in §8.1 of [4].

2. Local structure and the classifying space In this section, we give two closely related proofs of Theorem 1.1, and prepare the ground for the cohomological computation in the next section. We begin by recalling from Wilson [37] some details of the 2-local structure of Thompson’s sporadic simple group Th. All the involutions (elements of order two) in Th are conjugate. If t is

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an involution then the centralizer C = CTh (t) is a nonsplit extension 21+8 A9 where the normal subgroup P = O2 (C) is an extraspecial group 21+8 of “plus” type (i.e., of 2-rank five) and the quotient is an alternating group A9 . The 8-dimensional 2modular representation of C/P ∼ = A9 on P /Z(P ) is not the deleted permutation representation, but is one of the two representations which may be obtained from this one by applying a triality automorphism of + 8 (2). So in the permutation action of the quotient C/P ∼ = A9 on the maximal elementary abelian subgroups of P , there is one orbit of size nine, which may be put in one-one correspondence in a natural way with the nine points permuted by A9 . A maximal elementary abelian subgroup of P is said to be nice if it is in this orbit. Of the 135 maximal elementary abelian subgroups of P , nine are nice and 126 are not. The 126 which are not nice also form a single orbit under the action of A9 , with stabilizer (A5 × A4 ) : 2. The centralizer of a maximal elementary abelian 2-subgroup which is not nice has shape (24 × Q8 ). The normalizer modulo the centralizer is GL2 (4) extended by a field automorphism, namely L2 (4) = 24 (A5 × 3) : 2. Each of the 270 noncentral involutions in P in is in a unique nice subgroup of P . These involutions are all conjugate in C, and if we choose one then the four group V4 it generates with t has centralizer in Th of shape 22 [29 ]L3 (2). An involution in C/P ∼ = A9 of cycle type (22 15 ) does not lift to an involution in C, so the only other conjugacy class of four group in Th has a representative V4 generated by t and an involution in C whose image in A9 has cycle type (24 1). The centralizer of V4 has order 210 .3. If we choose a four group V4 in P which does not contain t and is not contained in a nice elementary abelian subgroup of P (all such four groups are conjugate in C), then it is conjugate in Th to V4 . The involutions in the centralizer in C of V4 all lie in P . From this, Wilson deduces the following. Lemma 2.1. Every elementary abelian 2-subgroup of Th is conjugate to a subgroup of P . Every elementary abelian subgroup of P is either contained in a nice elementary abelian subgroup of P , or it is contained in a unique conjugate in Th of P , and its normalizer is contained in C. Remark 2.2. Note that an elementary abelian 2-subgroup which is not nice can be contained in more than one conjugate of C. Let Q be a nice elementary abelian subgroup of P . Then N = NTh (Q) is a nonsplit extension 25 L5 (2) where the normal subgroup is the elementary abelian subgroup Q = O2 (N ) ∼ = (Z/2)5 and the quotient is the group L5 (2) of five by five invertible matrices over the field of two elements. The group N is called the Dempwolff group [13]. There are exactly two conjugacy classes of maximal 2-local subgroups of Th, and they are represented by C and N . Proof of Theorem 1.1. We give two proofs. The first proof goes as follows. Let A2 (Th) be the poset of nontrivial elementary abelian 2-subgroups of Th. Denote by P the subposet of A2 (Th) consisting of the conjugates of Z(C) = Z(P ) ∼ = Z/2 and

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of O2 (N ) ∼ = (Z/2)5 . Ryba, Smith and Yoshiara showed that the reduced Lefschetz module of P is projective. Denote by F the kernel of the map C ∗K N → Th. By the Kurosh subgroup theorem, this is a free group. The exact sequence coming from the simplicial cochains on |P | is 0 → F2 → F2 [Th /C] ⊕ F2 [Th /N ] → F2 [Th /K] → H 1 (F, F2 ) → 0. This sequence is split exact, because C, N and K all contain a Sylow 2-subgroup of Th. This shows that the reduced Lefschetz module of P is isomorphic to H 1 (F, F2 ). It follows that H 1 (F, F2 ) is a projective F2 Th-module. Lemma 3.1 of [8] then proves the theorem. For the second proof, it is proved in Smith andYoshiara [32] that the inclusion of P into A2 (Th) is a homotopy equivalence. This also follows directly from Lemma 2.1. Since the inclusion is also a Th-equivariant map, the theorem then follows from Theorem 1.10 of Dwyer [14]. The above description of the 2-local structure also allows us to describe the cohomology variety VTh , namely the maximal ideal spectrum of H ∗ (Th, k), where k is an algebraically closed field of characteristic two. Namely, there are just two conjugacy classes of maximal elementary abelian 2-subgroups in Th. They both have rank five. One is the nice subgroups, represented by O2 (N ), and the other is represented by a rank five elementary abelian subgroup of P which is not nice. Using the work of Quillen [27], [28] we see that VTh has two irreducible components, both of dimension five. Their intersection has dimension four.

3. The cohomology Next, we describe in a little more detail the structure of the subgroup C of Th of shape 21+8 A9 , and produce an element w of degree 16 in H ∗ (BC, F2 ) with nontrivial restriction to Z(C) = t . We shall show that for a suitable choice of such an element w, x16 = Tr C,Th (w) ∈ H 16 (Th, F2 ) plays the role of the lowest Dickson invariant. The remaining Dickson invariants are obtained from this one by applying Steenrod operations. We start by describing a different but closely related group manufactured from the extraspecial group P . Since P is an extraspecial 2-group of order 21+8 and 2-rank five, it may be generated by involutions a1 , . . . , a4 , b1 , . . . , b4 which commute with the exception of the following nontrivial commutators: [a1 , b1 ] = [a2 , b2 ] = [a3 , b3 ] = [a4 , b4 ] = t, where t is the central involution. The ordinary irreducible representations of P can all be written over the reals. They consist of 28 one dimensional representations and

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one irreducible representation on a real vector space V of dimension 16 (note that |P | = 28 + 162 ). The representation V may be described as an induced representation RP ⊗RQ V where V is the one dimensional representation of the subgroup Q = t, a1 , a2 , a3 , a4 in which t → (−1) and ai → (1) for 1 ≤ i ≤ 4. In terms of matrices, V is a representation in which t is a diagonal matrix of −1’s, a1 , a2 , a3 and a4 act as diagonal matrices with eight 1’s and eight −1’s, and b1 , b2 , b3 and b4 act as permutation matrices. + The automorphism group of P has shape 28 SO+ 8 (2), where SO8 (2) is the corresponding special orthogonal group (which has a simple subgroup of index 2 denoted in the Atlas [12] by O8+ (2) and elsewhere by + 8 (2)). Given any automorphism α of P , the conjugate of V by α is isomorphic to V . So there is a 16 by 16 matrix Aα , conjugation by which has the same effect as the action of α. In other words, if ρ : P → Aut(V ) is the representation of P on V , then for all g ∈ P we have ρ(α(g)) = Aα ρ(g)A−1 α . The matrix Aα is uniquely determined up to multiplication by scalars (by Schur’s lemma), and if we demand that Aα have determinant one then it is unique up to sign. It follows that the group formed by taking both possibilities of Aα for each automorphism α is a double cover of Aut(P ) of shape 21+8 SO+ 8 (2). There are three conjugacy classes of subgroups of + (2) isomorphic to A9 , per8 muted transitively by the triality automorphism. For one of them the restriction of the natural eight dimensional representation is the deleted permutation representation. The other two are conjugate in SO+ 8 (2). Choosing a representative of one of the latter two conjugacy classes, we obtain a group C0 of shape 21+8 A9 . This is still not isomorphic to the subgroup C of Th, but they are isomorphic modulo centers. To obtain C from C0 , we form the Baer sum of this central extension with the central extension pulled back from the nontrivial double cover of A9 . In other words, take C0 × C1 , where C1 is the double cover of A9 , pass to the subgroup consisting of pairs with the same image in A9 , and then quotient out the diagonal central element of order two. The smallest faithful complex representation of C is 128 dimensional, and is obtained by tensoring V with an irreducible faithful eight dimensional real representation of C1 . Let w16 ∈ H 16 (C0 , F2 ) be the degree 16 Stiefel–Whitney class of the real representation V . Since the restriction of V to Z(C0 ) is equal to a direct sum of 16 copies of the sign representation, the formula for the Stiefel–Whitney classes of a direct sum shows that w16 restricts to z016 ∈ H ∗ (BZ(C0 ), F2 ) = F2 [z0 ]. It follows that in the spectral sequence H ∗ (C0 /Z(C0 ), H ∗ (Z(C0 ), F2 )) ⇒ H ∗ (C0 , F2 ) the element z016 on the fiber is a universal cycle. The element x = d2 (z0 ) ∈ H 2 (C0 /Z(C0 ), F2 ) classifies the central extension by Z(C0 ), and Serre’s transgression theorem (§II.9c of [30]) shows that Sq8 Sq4 Sq2 Sq1 x = 0.

On the classifying space and cohomology of Thompson’s sporadic simple group

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Next, we examine C1 ∼ = 2A9 . The additive structure and a large part of the multiplicative structure of the cohomology of this group was calculated by Adem, Karagueuzian, Milgram and Umland [1]. We do not need to make use of this description, because all we need is a single Stiefel–Whitney class. The deleted permutation representation of A9 of dimension eight displays it as a subgroup of SO(8). Let 1 → γ → Spin(8) → SO(8) → 1 be the double cover map, where γ is a central element of order two. The inverse image of A9 in Spin(8) is isomorphic to C1 , so we regard C1 as a subgroup of Spin(8) in this way. Now if we let be one of the two spin representations of Spin(8) of dimension 8, then the Stiefel–Whitney class w8 () is an element of H 8 (B Spin(8); F2 ) whose restriction to H ∗ ( γ , F2 ) is the eighth power of the degree one generator. It follows that the restriction u = resSpin(8),C1 (w8 ()) ∈ H 8 (C1 , F2 ) is an element whose restriction to Z(C1 ) is nontrivial. So in the spectral sequence H ∗ (C1 /Z(C1 ), H ∗ (Z(C1 ), F2 )) ⇒ H ∗ (C1 , F2 ), the element z18 ∈ H ∗ (Z(C1 ), F2 ) = F2 [z1 ] on the fiber is a universal cycle. The element y = d2 (z1 ) ∈ H 2 (C1 /Z(C1 ), F2 ) classifies the central extension by Z(C1 ). Again using Serre’s transgression theorem [30], we have Sq4 Sq2 Sq1 y = 0, Now C/Z(C) ∼ = C1 /Z(C1 ) ∼ = A9 . = C0 /Z(C0 ), of shape 28 A9 , and C/O2 (C) ∼ So there is an inflation map inf : H ∗ (C1 /Z(C1 ), F2 ) → H ∗ (C/Z(C), F2 ), and the element x + inf(y) ∈ H 2 (C/Z(C), F2 ) classifies the central extension of C/Z(C) by Z(C). This is because the Baer sum of central extensions of a group corresponds to the sum of the degree two cohomology classes. See the exercises to Section IV.4 of Mac Lane [23], for example. Since the Steenrod operations are F2 -linear, and commute with the inflation map, we have Sq8 Sq4 Sq2 Sq1 (x + inf(y)) = Sq8 Sq4 Sq2 Sq1 (x) + inf(Sq8 Sq4 Sq2 Sq1 (y)) = 0 in H ∗ (C/Z(C), F2 ). Again using Serre’s transgression theorem [30], it follows that z16 is a universal cycle in the spectral sequence H ∗ (C/Z(C), H ∗ (Z(C), F2 )) ⇒ H ∗ (C, F2 ). From this, we deduce that there is an element w ∈ H 16 (C, F2 ) with the property that the restriction to Z(C) is z16 .

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Lemma 3.1. With this definition of w, the transfer from C to Th of w is an element x16 ∈ H ∗ (Th, F2 ) with the property that the restriction to Z(C) of x16 is equal to z16 . Proof. To compute the restriction of the transfer, we use the Mackey double coset formula. Double cosets where the intersection of C with the conjugate of Z(C) is trivial do not contribute to the sum. The fact that Z(C) is equal to the center of a Sylow 2-subgroup of C implies that the remaining double cosets apart from the one corresponding to the identity element correspond to conjugacy classes of involutions in C of even size, which therefore contribute zero in characteristic two. Lemma 3.2. The restriction resTh,O2 (N) (x16 ) is equal to the degree 16 Dickson invariant in H ∗ (O2 (N), F2 )N/O2 (N) . Proof. The restriction must be nonzero by Lemma 3.1, since Z(C) ≤ O2 (N ). It must also be invariant under the action of N/O2 (N). There is only one nonzero invariant in this degree, namely the degree 16 Dickson invariant. We can now create all the Dickson invariants, using the action of the Steenrod algebra. Namely, we set x24 = Sq8 (x16 ), x28 = Sq4 (x24 ), x30 = Sq2 (x28 ) and x31 = Sq1 (x30 ). Since the Steenrod operations commute with the restriction map, the formulae given in Wilkerson [36] show that the restrictions of x16 , x24 , x28 , x30 and x31 are the generators for the rank five Dickson invariants in H ∗ (O2 (N ), F2 ). In particular, these elements are algebraically independent. Lemma 3.3. Let E be an elementary abelian subgroup of P of rank five which is not nice. Then H 16 (E, F2 )NTh (E) is two dimensional, and is spanned by the degree 16 Dickson invariant and the element u formed by taking the product of the 16 elements of H ∗ (E, F2 ) whose restrictions to Z(C) are nonzero. Proof. Set H = NTh (E)/CTh (E) = 24 (A5 × 3)2. We work our way up a normal series for H . The invariants of O2 (H ) is a polynomial ring on four generators of degree one and a generator of degree 16. The invariants of A5 on the polynomial ring generated by the four generators of degree one with the given action of A5 (namely, a two dimensional representation over F4 with the scalars restricted) were computed by Adem and Milgram [3]. From Theorem 3.2 of that paper, it is not hard to read off the degree 16 invariants, together with the action of the extra elements of order three and two. There are five dimensions of A5 -invariants, of which only one dimension is (A5 × 3)2-invariant, namely the square of the degree eight generator. The remaining four dimensions are given by multiplying elements of degrees 5, 8 and 3. The fixed space of this four dimensional space is zero, even under the extra element of order three. This shows that the degree 16 invariants of 24 (A5 × 3)2 on H ∗ (O2 (N ), F2 ) are two dimensional. Since the degree 16 Dickson invariant and u have different restrictions to subgroups of order two in N other than Z(C), it follows that they span this two dimensional space.

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Lemma 3.4. Let E be any elementary abelian 2-subgroup of Th of rank five. Then the restrictions of x16 , x24 , x28 , x30 and x31 to E are equal to the Dickson invariants. Proof. If E is nice, this follows from Lemma 3.2. So suppose that E is not nice. By Lemma 2.1, we can assume that E is contained in P . The restriction of x16 is certainly invariant under NTh (E), and so by Lemma 3.3, it is in the two dimensional space spanned by the degree 16 Dickson invariant and u. The element x16 has nonzero restriction to every involution in Th. The element u restricts to zero on a cyclic subgroup of E other than Z(C), and u plus the Dickson invariant restricts to zero on Z(C). So the only possibility is that the restriction of x16 is equal to the degree 16 Dickson invariant. Applying the Steenrod operations shows that x24 , x28 , x30 and x31 restrict to the remaining Dickson invariants. Proposition 3.5. The cohomology ring H ∗ (Th, F2 ) is finitely generated as a module over the subring F2 [x16 , x24 , x28 , x30 , x31 ]. Proof. It follows from Quillen’s F-isomorphism theorem [27, 28] that it suffices to check that the cohomology of every elementary abelian 2-subgroup of Th is finitely generated over the restriction of this polynomial subring. There are two conjugacy classes of maximal elementary abelian 2-subgroups, both of rank five, and these are dealt with in Lemma 3.4. Remark 3.6. It would be very interesting to know whether H ∗ (Th, F2 ) is a free module over F2 [x16 , x24 , x28 , x30 , x31 ]; in other words, whether H ∗ (Th, F2 ) is a Cohen– Macaulay ring. We have no information on this question.

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, The spectrum of an equivariant cohomology ring, II, Ann. of Math. 94 (1971), 573–602.

[29] A. Reifart, A characterization of Thompson’s sporadic simple group, J. Algebra 38 (1976), 192–200. [30] J.-P. Serre, Homologie singulière des espaces fibrés, Ann. of Math. 54 (1951), 425–505. [31] P. E. Smith, A simple subgroup of M? and E8 (3), Bull. London Math. Soc. 8 (1976), 161–165. [32] S. D. Smith and S. Yoshiara, Some homotopy equivalences for sporadic geometries, J. Algebra 192 (1997), 326–379. [33] R. Solomon, Finite groups with Sylow 2-subgroups of type .3, J. Algebra 28 (1974), 182–198. [34] J. G. Thompson, A simple subgroup of E8 (3), Finite Groups, ed. by N. Iwahori, Japan Soc. for the Promotion of Science, 1976, 113–116. [35] A. Vavpetiˇc andA. Viruel, On the homotopy type of the classifying space of the exceptional Lie group F4 , Manuscripta Math. 107 (2002), 521–540. [36] C. W. Wilkerson, A primer on Dickson invariants, in Proceedings of the Northwestern Homotopy Theory Conference, Contemp. Math. 19, American Math. Society, Providence,RI 1983, 421–434. [37] R. A. Wilson, Some subgroups of the Thompson group, J. Aust. Math. Soc. Ser. A 44 (1988), 17–32. [38] S. Yoshiara, The radical 2-subgroups of the sporadic simple groups J4 , Co2 and Th, J. Algebra 233 (2000), 309–341. Dave Benson, Department of Mathematics, University of Georgia, Athens GA 30602, U.S.A. E-mail: [email protected]

New versions of Schur–Weyl duality Stephen Doty

Abstract. After reviewing classical Schur–Weyl duality, we present some other contexts which enjoy similar features, relating to Brauer algebras and classical groups. 2000 Mathematics Subject Classification: 20

1. Classical Schur–Weyl duality 1.1. Schur’s double-centralizer result. Consider the vector space V = Cn . The symmetric group Sr acts naturally on its r-fold tensor power V ⊗r , by permuting the tensor positions. This action obviously commutes with the natural action of GLn = GLn (C), acting by matrix multiplication in each tensor position. So we have a C GLn CSr bimodule structure on V ⊗r . (Here CG denotes the group algebra of a group G.) In 1927, Schur [34] proved that the image of each group algebra under its representation equals the full centralizer algebra for the other action. More precisely, if we name the representations as follows ρ

σ

C GLn −−−−→ End(V ⊗r ) ←−−−− CSr

(1)

then we have equalities ρ(C GLn ) = EndSr (V ⊗r )

(2)

σ (CSr ) = EndGLn (V

(3)

⊗r

).

(Here, for a given set S operating on a vector space T through linear endomorphisms, EndS (T ) denotes the set of linear endomorphisms of T commuting with each endomorphism coming from S.) Results of Carter–Lusztig [7], de Concini and Procesi [8], and J. A. Green [19] (and others) show that all the above statements remain true if one replaces C by an arbitrary infinite field K. 1.2. Schur algebras. The finite-dimensional algebra in (2) above, for any K, is known as the Schur algebra, and often denoted by SK (n, r) or simply S(n, r). The Schur algebra “sees” the part of the rational representation theory of the algebraic

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group GLn (K) occurring (in some appropriate sense) in V ⊗r . More precisely, there is an equivalence between r-homogeneous polynomial representations of GLn (K) and SK (n, r)-modules. In characteristic 0, those representations (as r varies) determine all finite-dimensional rational representations, while in positive characteristic they still provide a tremendous amount of information. The representation σ in (1) is faithful if n r, so σ induces an isomorphism KSr EndGLn (V ⊗r ) = EndSK (n,r) (V ⊗r )

(n r).

(4)

This leads to intimate connections between polynomial representations of GLn (K) and representations of KSr , a theme that has been exploited by many authors in recent years. Perhaps the most dramatic example of this is the result of Erdmann [18] (building on previous work of Donkin [13] and Ringel [33]) which shows that knowing decomposition numbers for all symmetric groups in positive characteristic will determine the decomposition numbers for general linear groups in the same characteristic. Conversely, James [24] had already shown that the decomposition matrix for a symmetric group is a submatrix of the decomposition matrix for an appropriate Schur algebra. Thus the (still open) general problem of determining the modular characters of symmetric groups is equivalent to the similar problem for general linear groups (over infinite fields). 1.3. The enveloping algebra approach. Return to the basic setup, over C. One may differentiate the action of the Lie group GLn (C) to obtain an action of its Lie algebra gln . Replacing the representation ρ in (1) by its derivative representation dρ : U(gln ) → End(V ⊗r ) leads to the following alternative statement of Schur’s result: dρ(U(gln )) = EndSr (V ⊗r )

σ (CSr ) = Endgln (V ⊗r ).

(5) (6)

In particular, the Schur algebra (over C) is a homomorphic image of U(gln ). All of this works over an arbitrary integral domain K if we replace U(gln ) by its “hyperalgebra” UK := K ⊗Z UZ obtained by change of ring from a suitable Z-form of U(gln ); see [11]. (One can adapt the Kostant Z-form, originally defined for the enveloping algebra of a semisimple Lie algebra, to the reductive gln .) 1.4. The quantum case. Jimbo [25] extended the results of 1.3 to the quantum case (where the quantum parameter is not a root of unity). One needs to replace Sr by the Iwahori–Hecke algebra H(Sr ) and replace U(gln ) by the quantized enveloping algebra U(gln ). The analogue of the Schur algebra in this context is known as the q-Schur algebra, often denoted by S(n, r) or Sq (n, r). Dipper and James [9] have shown that q-Schur algebras are fundamental for the modular representation theory of finite general linear groups.

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As many authors have observed, the picture in 1.1 can also be quantized. For that one needs a suitable quantization of the coordinate algebra of the algebraic group GLn . There is a completely different (geometric) construction of q-Schur algebras given in [1]. 1.5. Integral forms. The Schur algebras SC (n, r) admit an integral form SZ (n, r) such that SK (n, r) K ⊗Z SZ (n, r) for any field K. In fact SZ (n, r) is simply the image of UZ (see 1.3) under the surjective homomorphism U(gln ) → SC (n, r). Similarly, the quantum Schur algebra SQ(v) (n, r) admits an integral form defining all specializations via base change. One needs to replace Z by A = Z[v, v −1 ]; then the integral form SA (n, r) is the image of the Lusztig A-form UA under the surjection U(gln ) → SQ(v) (n, r). (To match this up with various specializations in the literature, one often has to take q = v 2 .) 1.6. Generators and relations. Recently, in joint work with Giaquinto (see [16]), a very simple set of elements generating the kernel of the surjection U(gln ) → SC (n, r) was found. A very similar set of elements generates the kernel of the surjection U(gln ) → SQ(v) (n, r). These elements are expressible entirely in terms of the Chevalley generators for the zero part of U(gln ) or U(gln ). Thus we obtain a presentation of SC (n, r) and SQ(v) (n, r) by generators and relations, compatible with the usual Serre (Drinfeld–Jimbo) presentation of U(gln ) (resp., U(gln )). As a result, we find a certain subset of the integral PBW-basis for U(gln ) or U(gln ) the image of which gives an integral basis for SZ (n, r) or SA (n, r). This basis yields a similar basis in any specialization. Moreover, a subset of it provides a new integral basis of H(Sn ).

2. The Brauer algebra From now on I will assume, unless stated otherwise, that the underlying field is C (it could just as well be any field of characteristic zero). One expects that many statements will be valid over an arbitrary infinite field, via some appropriate integral form, similar to what happens in type A. 2.1. The algebra Br(x) . Let R be a commutative ring, and consider the free R[x](x) module Br with basis consisting of all r-diagrams. An r-diagram is an (undirected) graph on 2r vertices and r edges such that each vertex is incident to precisely one edge. One usually thinks of the vertices as arranged in two rows of r each, the top and bottom rows. (See Figure 1.) Edges connecting two vertices in the same row (different rows) are called horizontal (resp., vertical). We can compose two such diagrams D1 , D2 by identifying the bottom row of vertices in the first diagram with the top row of vertices in the second diagram. The result is a graph with a certain number, δ(D1 , D2 ), of

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interior loops. After removing the interior loops and the identified vertices, retaining the edges and remaining vertices, we obtain a new r-diagram D1 ◦ D2 , the composite diagram. Multiplication of r-diagrams is defined by the rule D1 · D2 = x δ(D1 ,D2 ) (D1 ◦ D2 ). (x)

One can check that this multiplication makes Br into an associative algebra; this is the Brauer algebra. (See Figures 1–3 for an illustration of the multiplication in the Brauer algebra.) Note that if we take x = 1 then the set of r-diagrams is a monoid under diagram (1) composition, and Br is simply the semigroup algebra of that monoid.

Figure 1. Two Brauer diagrams D1 , D2 for r = 5.

Figure 2. Computing the composite of D1 and D2 .

Figure 3. The composite diagram D1 ◦ D2 .

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For any x the group algebra R[x]Sr may be identified with the subalgebra of (x) Br spanned by the diagrams containing only vertical edges. Such Brauer diagrams provide a graphical depiction of permutations. The group algebra R[x]Sr of Sr (x) also appears as a quotient of Br , the quotient by the two-sided ideal spanned by all diagrams containing at least one horizontal edge. Label the vertices in each row of an r-diagram by the indices 1, . . . , r. For any 1 i = j r let ci,j be the r-diagram with horizontal edges connecting vertices i, j on the top and bottom rows. All other vertices in the diagram ci,j are vertical, connecting vertex k on the top and bottom rows, for all k = i, j . Brauer observed that (x) Br is generated by the permutation diagrams together with just one of the ci,j . (x)

2.2. Schur–Weyl duality. Brauer [5] introduced the algebra Br in 1936 to describe the invariants of symplectic and orthogonal groups acting on V ⊗r . (Brauer’s conventions were slightly different; we are here following the approach of Hanlon and Wales (−n) is isomorphic with the algebra defined by Brauer [23], who pointed out that Br to deal with the symplectic case.) Let G be Spn or On , where n is even in the first instance. By restricting the action ρ considered in 1.1 we have an action of G on V ⊗r . (n) One can extend the action of Sr to an action of Br (over C) on V ⊗r , where = −1 if G = Spn and = 1 if G = On . To do this, it is enough to specify the action of the diagram ci,j . This acts on V ⊗r as one of Weyl’s contraction maps contracting in tensor positions i and j . So we have (commuting) representations ρ

σ

(n)

CG −−−−→ End(V ⊗r ) ←−−−− Br

(7)

which satisfy Schur–Weyl duality; i.e., the image of each representation equals the full centralizer algebra of the other action: ρ(CG) = EndB (n) (V ⊗r ) r

σ (Br(n) ) = EndG (V ⊗r ).

(8) (9)

The algebras in equality (8) are the symplectic and orthogonal Schur algebras (see [12], [14], [15]). If n r−1 the representation σ in (7) is faithful [6]; thus it induces an isomorphism (n) Br EndG (V ⊗r ). 2.3. Schur–Weyl duality in type D. In type Dn/2 (n even) the orthogonal group On is not connected, and contains the connected semisimple group SOn (special orthogonal group) as subgroup of index 2. In order to handle this situation, Brauer (see also [20]) (n) defined a larger algebra Dr , spanned by the usual r-diagrams previously defined, together with certain partial r-diagrams on 2r vertices and r − n edges, in which n vertices in each of the top and bottom rows are not incident to any edge, and showed (n) (n) that the action of Br can be extended to an action of this larger algebra Dr on V ⊗r .

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Thus we have representations ρ

σ

(n)

C SOn −−−−→ End(V ⊗r ) ←−−−− Dr (n)

Brauer showed that the actions of SOn and Dr

.

(10)

on V ⊗r satisfy Schur–Weyl duality:

ρ(C SOn ) = EndD (n) (V ⊗r )

(11)

σ (Dr(n) ) = EndSOn (V ⊗r ).

(12)

r

The algebra in (11) is a second Schur algebra in type D, a proper subalgebra of the algebra EndB (n) (V ⊗r ) appearing in (8) above. r

2.4. Generators and relations. One can formulate the above statements of Schur– Weyl duality using enveloping algebras, analogous to 1.3. This leads to a presentation (see [17]) of the symplectic and orthogonal Schur algebras which is compatible with (a slight modification of) the usual Serre presentation of the enveloping algebra U(g), where g = spn (n even) or son . 2.5. The quantum case. There is a q-version of the Schur–Weyl duality considered in this section, although not as developed as in type A. One needs to replace the Brauer algebra by its q-analogue, the Birman–Murakami–Wenzl (BMW) algebra (see [4], [31]), and replace the enveloping algebra by a suitable quantized enveloping algebra. One can think of the BMW algebra in terms of Kauffman’s tangle monoid; see [26], [22], [30]. (Roughly speaking, tangles are replacements for Brauer diagrams, in which one keeps track of under and over crossings, subject to certain natural relations.) There are applications of the BMW algebra to knot theory, as one might imagine. This leads to a q-analogue of the symplectic Schur algebras, in particular, which have been studied by Oehms [32]. (n) To the best of my knowledge, a q-analogue of the larger algebra Dr (n even) considered in 2.3 remains to be formulated.

3. The walled Brauer algebra (x) 3.1. The algebra Br,s . This algebra was defined in 1994 in [2]. It is the subalgebra (x) of Br+s spanned by the set of (r, s)-diagrams. By definition, an (r, s)-diagram is an (r + s)-diagram in which we imagine a wall separating the first r from the last s columns of vertices, such that: (a) all horizontal edges cross the wall; (b) no vertical edges cross the wall. An edge crosses the wall if its two vertices lie on opposite sides of the wall. The (x) (x) multiplication in Br,s is that of Br+s .

New versions of Schur–Weyl duality

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Label the vertices on the top and bottom rows of an (r, s)-diagram by the numbers 1, . . . , r to the left of the wall and −1, . . . , −s to the right of the wall. Let ci,−j (1 i r; 1 j s) be the diagram with a horizontal edge connecting vertices i and −j on the top row and the same on the bottom row, and with all other edges connecting vertex k (k = i, −j ) in the top and bottom rows. It is easy to see that the walled Brauer algebra is generated by the permutations it contains along with just one of the ci,−j . (Note that ci,−j is the (r + s)-diagram denoted by ci,r+j in 2.1.)

(x)

3.2. Dimension. What is the dimension of Br,s ? One way to answer that question is to consider the map, flip, from (r + s)-diagrams to (r + s)-diagrams, defined by interchanging the top and bottom vertices to the right of the imaginary wall. For example, Figure 4 shows a (4, 2)-diagram (to the left) and its corresponding 6-diagram, obtained from the left diagram by applying flip. Note that flip is involutary: applying it twice gives the original diagram back again.

Figure 4. A (4, 2)-diagram and its corresponding permutation, after applying flip.

One easily checks that the map flip carries (r, s)-diagrams bijectively onto the set of (r +s)-diagrams with all edges vertical. Such diagrams correspond with permutations (x) of r + s objects, so the dimension of Br,s is (r + s)!.

(x) 3.3. Another view of Br,s . The above correspondence between (r, s)-diagrams and (x) permutations gives another way to think of the multiplication in Br,s . Given two (r, s)-diagrams D1 , D2 let D1 , D2 be their corresponding permutations obtained by applying flip. Define a new (rather bizarre) composition on permutations as follows. Given any two permutation diagrams D1 , D2 (with r + s columns of vertices) identify the first r vertices of the bottom row of D1 with the first r vertices of the top row of D2 , and identify the last s vertices of the top row of D1 with the last s vertices of the bottom row of D2 . After removing loops and identified vertices this gives a new permutation diagram D3 in which the vertices in the top (resp., bottom) row are the remaining top (bottom) row vertices from the original diagrams.

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Let δ(D1 , D2 ) be the number of loops removed in computing the composite permutation diagram D3 . Define multiplication of permutation diagrams by the rule

D1 · D2 = x δ(D1 ,D2 ) D3 In other words, we are multiplying permutations by composing maps “on the right” on one side of the wall, and “on the left” on the other side (roughly speaking). For example, Figure 5 below shows the computation of the composite diagram in the walled Brauer algebra (left column) and the computation in terms of the corresponding permutations (right column). Figure 6 shows the resulting diagrams after the single loop and identified vertices have been removed.

Figure 5. Composition of diagrams and permutations. The diagrams on the left correspond under flip with the permutations on the right.

One can check that D3 coincides with (D1 ◦ D2 ) and δ(D1 , D2 ) = δ(D1 , D2 ). (x) In other words, flip defines an algebra isomorphism between the algebra Br,s and the (x) r,s spanned by permutation diagrams with the multiplication defined above. algebra B (x) (R[x]Ss )opp . (x) R[x]Sr and B Note that in particular B r,0 0,s

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New versions of Schur–Weyl duality

Figure 6. The corresponding diagrams resulting from Figure 5. The two diagrams correspond under flip.

3.4. Schur–Weyl duality. Consider mixed tensor space V r,s := V ⊗r ⊗V ∗⊗s , where V ∗ is the usual linear dual space of V . Mixed tensor space is naturally a module for (n) GLn , and one obtains an action of Br,s on V r,s simply by restricting the action of (n) Br+s , which acts the same on V r,s as it does on V ⊗(r+s) , since on restriction to On we have V V ∗ . Thus we have the following commutative diagram ρ

C GLn −−−−→ ι

End(V r,s )

ρ

σ

(n)

←−−−− Br,s ι σ

(13)

(n)

C On −−−−→ End(V ⊗(r+s) ) ←−−−− Br+s (n)

in which the vertical maps ι, ι are inclusion. By [2], the actions of GLn and Br,s on V r,s in the first row of the diagram satisfy Schur–Weyl duality: ρ(C GLn ) = EndB (n) (V r,s )

(14)

(n) ) = EndGLn (V r,s ). σ (Br,s

(15)

r,s

The algebra in (14) is another Schur algebra S(n; r, s) in type A, studied in [10]. These Schur algebras provide us with a new family of quasihereditary algebras, generalizing the classical Schur algebras, since S(n; r, 0) S(n, r). In fact, the S(n; r, s) provide a new class of generalized Schur algebras in the sense of Donkin [11]. For fixed n, the family of S(n; r, s)-modules as r, s vary constitutes the family of all rational representations of GLn . Whence the name rational Schur algebras for the S(n; r, s). When n r + s, the representation σ in the top row of (13) above is faithful, so (n) induces an isomorphism Br,s EndGLn (V r,s ).

3.5. The quantum case. Quantizations of the walled Brauer algebra have been defined and studied in work of Halverson [21], Leduc [29], Kosuda–Murakami [28], and Kosuda [27].

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4. The deranged algebra 4.1. The problem. One might wonder if there are versions of Schur–Weyl duality in which the natural module V is replaced by some other representation. Perhaps the first choice would be to replace V with the adjoint module, i.e., an algebraic group acting on its Lie algebra via the adjoint representation. The simplest instance of this would be type A, where we consider the module sl⊗r n as an SLn -module, and ask for its ⊗r centralizer algebra EndSLn (sl⊗r ) = End (sl GL n n n ). (It makes no difference whether ⊗r we regard sln as module for SLn or for GLn . We look at sl⊗r n rather than gln since gln is not simple as a GLn -module or SLn -module.) 4.2. Relation with Brauer algebras. Even though this is a question about type A, its solution is intimately connected with the walled Brauer algebra. Here is a brief outline of the solution to this problem, recently obtained in [3]. The main idea is to utilize the decomposition gln = sln ⊕C (as SLn or GLn module) to write ⊗r ⊗t (16) gl⊗r n = sln ⊕ 0t

(n) e C eBr,r

(n 2r)

(17)

where e = i (1 − n−1 ci,−i ) (notation of 3.1). The algebra in (17) has a basis consisting of all elements of the form eDe as D ranges over the set of (r, r)-diagrams with no horizontal edges connecting i to −i. Such diagrams correspond under flip, after inverting the signs labeling the bottom row, with derangements of 2r objects. (A derangement is a permutation having no fixed points.) Let N(k) = the number of derangements of k objects, then we have by the Inclusion–Exclusion Principle (see [35, 2.2.1 Example]) k k−j k (−1) j! (18) N (k) = j j =0

which is the nearest integer to k!/e (e = 2.7182818 . . . ). Thus the dimension of (n) the centralizer algebra eBr,r e is given by N (2r). Because of this connection with (n) derangements, the algebra eBr,r e is known as the deranged algebra. In [3] explicit formulas are obtained for the number of times a given simple module L appears as a summand of sl⊗r n . In particular, it is shown that N(r) = the multiplicity of the trivial module in sl⊗r n

New versions of Schur–Weyl duality

69

and N (r − 1) = the multiplicity of sln in sl⊗r n demonstrating that the combinatorics of derangement numbers is inherent in this theory. 4.3. Schur–Weyl duality. At least when n 2r, the actions of GLn and C on sl⊗r n satisfy Schur–Weyl duality: ρ(C GLn ) = EndeB (n) e (sl⊗r n )

(19)

(n) e) = EndGLn (sl⊗r σ (eBr,r n ).

(20)

r,r

This will almost certainly hold for all n, r. Acknowledgements. These notes are based on a lecture, various versions of which I have given in the past year, in a number of locations, including Stuttgart, Birmingham, Queen Mary (London), Lancaster, Manchester, Oxford, and Cambridge. I’m grateful to the organizers of those events for the opportunity to present these ideas.

References [1]

Be˘ılinson, A. A., Lusztig, G., MacPherson, R. A geometric setting for the quantum deformation of GLn , Duke Math. J. 61 (1990), 655–677.

[2]

Benkart, G., Chakrabarti, M., Halverson, T., Leduc, R., Lee, C., Stroomer, J., Tensor product representations of general linear groups and their connections with Brauer algebras, J. Algebra 166 (1994), 529–567.

[3]

Benkart, G., Doty, S., Derangements and tensor powers of adjoint modules for sln , J. Algebraic Combin. 16 (2002), 31–42.

[4]

Birman, J. S., Wenzl, H., Braids, link polynomials and a new algebra, Trans. Amer. Math. Soc. 313 (1989), 249–273.

[5]

Brauer, R., On algebras which are connected with the semisimple continuous groups, Ann. of Math. 38 (1937), 857–872.

[6]

Brown, William P., An algebra related to the orthogonal group, Michigan Math. J. 3 (1955), 1–22; The semisimplicity of ωfn , Ann. of Math. 63 (1956), 324–335.

[7]

Carter, R. W., Lusztig, G., On the modular representations of the general linear and symmetric groups, Math. Z. 136 (1974), 193–242.

[8]

de Concini, C., Procesi, C., A characteristic free approach to invariant theory, Adv. in Math. 21 (1976), 330–354.

[9]

Dipper, R., James, G., The q-Schur algebra, Proc. London Math. Soc. (3) 59 (1989), 23–50; q-tensor space and q-Weyl modules, Trans. Amer. Math. Soc. 327 (1991), 251–282.

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[10] Dipper, R., Doty, S., Rational Schur algebras, in preparation. [11] Donkin, S., On Schur algebras and related algebras I, J. Algebra 104 (1986), 310–328. [12] Donkin, S., Good filtrations of rational modules for reductive groups, in: The Arcata Conference on Representations of Finite Groups (Arcata, Calif., 1986), Proc. Sympos. Pure Math. 47, Part 1, Amer. Math. Soc., Providence, RI 1987, 69–80. [13] Donkin, S., On tilting modules for algebraic groups, Math. Z. 212 (1993), 39–60. [14] Doty, S., Polynomial representations, algebraic monoids, and Schur algebras of classical type, J. Pure Appl. Algebra 123 (1998), 165–199. [15] Doty, S., Representation theory of reductive normal algebraic monoids, Trans. Amer. Math. Soc. 351 (1999), 2539–2551. [16] Doty, S., Giaquinto, A., Presenting Schur algebras, Internat. Math. Res. Notices 2002, 1907–1944. [17] Doty, S., Giaquinto, A., Sullivan, J., Presenting Schur algebras in types B, C, D, in preparation. [18] Erdmann, K., Decomposition numbers for symmetric groups and composition factors of Weyl modules, J. Algebra 180 (1996), 316–320. [19] Green, J. A., Polynomial representations of GLn , Lecture Notes in Math. 830, SpringerVerlag, Berlin, New York 1980. [20] Grood, C., Brauer algebras and centralizer algebras for SO(2n, C), J. Algebra 222 (1999), 678–707. [21] Halverson, T., Characters of the centralizer algebras of mixed tensor representations of GL(r, C) and the quantum group Uq (gl(r, C)), Pacific J. Math. 174 (1996), 359–410. [22] Halverson, T., Ram, A., Characters of algebras containing a Jones basic construction: the Temperley–Lieb, Okasa, Brauer, and Birman–Wenzl algebras, Adv. Math. 116 (1995), 263–321. [23] Hanlon, P., Wales, D., On the decomposition of Brauer’s centralizer algebras, J. Algebra 121 (1989), 409–445; Eigenvalues connected with Brauer’s centralizer algebras, J. Algebra 121 (1989), 446–476; Computing the discriminants of Brauer’s centralizer algebras, Math. Comp. 54 (1990), 771–796. [24] James, G. D., The decomposition of tensors over fields of prime characteristic, Math. Z. 172 (1980), 161–178. [25] Jimbo, M., A q-analogue of U (gl(N + 1)), Hecke algebra, and the Yang–Baxter equation, Lett. Math. Phys. 11 (1986), 247–252. [26] Kauffman, L. H., An invariant of regular isotopy, Trans. Amer. Math. Soc. 318 (1990), 417–471. [27] Kosuda, M., Representation of q-analogue of rational Brauer algebras, Tsukuba J. Math. 21 (1997), 707–728. [28] Kosuda, M., Murakami, J., The centralizer algebras of mixed tensor representations of Uq (gln ) and the HOMFLY polynomial of links, Proc. Japan Acad. Ser. A Math. Sci. 68 (1992), 148–151; Centralizer algebras of the mixed tensor representations of quantum

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group Uq (gl(n, C)), Osaka J. Math. 30 (1993), 475–507. [29] Leduc, R., A two-parameter version of the centralizer algebra of the mixed tensor representations of the general linear group and quantum general linear group, Ph.D. Dissertation, University of Wisconsin-Madison 1994. [30] Morton, H. R.; Wassermann, A. J., A basis for the Birman–Wenzl algebra, Preprint 1989, with foreword and references updated Jan. 2000. [http://www.liv.ac.uk/~su14/ knotprints.html] [31] Murakami, J., The Kauffman polynomial of links and representation theory, Osaka J. Math. 24 (1987), 745–758; The representations of the q-analogue of Brauer’s centralizer algebras and the Kauffman polynomial of links, Publ. Res. Inst. Math. Sci. 26 (1990), 935–945. [32] Oehms, S., Symplektische q-Schur-Algebren, Dissertation, Universität Stuttgart, Stuttgart, 1997. Berichte aus der Mathematik, Verlag Shaker, Aachen 1997. [33] Ringel, C., The category of modules with good filtrations over a quasi-hereditary algebra has almost split sequences, Math. Z. 208 (1991), 209–223. [34] Schur, I., Über die rationalen Darstellungen der allgemeinen linearen Gruppe (1927). Reprinted in Schur, I., Gesammelte Abhandlungen, Vol. III, pp. 68–85, Springer-Verlag, Berlin, 1973. [35] Stanley, R. P., Enumerative combinatorics, Vol. 1, Cambridge Stud. Adv. Math. 49, Cambridge Univ. Press, Cambridge 1997. Stephen Doty, Department of Mathematics and Statistics, Loyola University Chicago, Chicago, IL 60626, U.S.A. E-mail: [email protected]

Just infinite periodic Lie algebras Norberto Gavioli, Valerio Monti and Carlo M. Scoppola

Abstract. In this paper we give a characterization of just infinite Lie subalgebras of a loop algebra in terms of the existence of certain ideals and in terms of the existence of certain nontrivial L-endomorphisms of positive degree. Such an algebra has a faithful representation of finite degree over the ring of polynomials with coefficients in the ground field, and is shown to be ideally r-constrained for a suitable r. An upper bound for r is given in the solvable case. 2000 Mathematics Subject Classification: 17B70

1. Introduction The study of graded Lie algebras has received considerable attention in recent years (see [3], [10], [5], and [9] for complete references), beyond its obvious importance as a tool in the study of pro-p-groups. Indeed, it is possible to attach a Lie ring to any given pro-p-group G (see e.g. [8]), defining a suitable Lie bracket on the sum of the factors of a strongly central series of G. One can choose among different central series for this construction, and the properties of the corresponding Lie ring reflect this choice. In many interesting cases, the Lie ring turns out to be a Lie algebra over a finite field. An effective study of these classes of (groups and) algebras is based, of course, on a variety of technical features. Among these, an increasingly prominent role seems to be taken by the concept of periodicity: various kinds of repeating patterns in the structure of both the groups and the corresponding Lie rings or algebras have indeed been observed (see [6, III.17.9], [7], [1]). In this paper we focus on the study of just infinite(-dimensional) graded Lie algebras [9], and we propose a notion of periodicity for them. In Section 2 we begin the study of the ideal structure in just infinite Lie algebras, with a particular attention to abelian ideals, and with some remarks on the solvable case. Section 3 is then devoted to just infinite subalgebras of some Kac–Moody algebras. Our main result is their characterization in terms of the existence of proper ideals containing no non-trivial homogeneous elements, and also in terms of existence of certain homomorphisms that induce a periodicity on the algebra structure. We define here the notion of depth, that proves to be very handy in the computations. Then we

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show that these algebras are r-constrained (see [4]). Furthermore this section contains the construction of a faithful representation of a periodic just infinite Lie algebra over the ring of polynomials with coefficients in the ground field. The paper ends with some remarks on the solvable case.

2. Just infinite Lie algebras Lie algebras denoted by Latin capital letters appearing in this paper will be finitely generated N-graded Lie algebras generated by their homogeneous component of degree 1. We shall write g-algebra to mean graded algebra. We shall use curly capital letters to denote Lie algebras without any extra structure. In case of graded algebras we shall write g-ideal and g-module respectively to mean a homogeneous ideal or a finitely generated graded module.

If M is a g-module and x ∈ M we say that x = ni=0 xji is the homogeneous decomposition of x if {ji }ni=0 is a strictly increasing sequence and, for i = 0, . . . , n, the element xji is non-trivial and homogeneous of degree ji . The elements xj0 and xjn are called respectively the head and the

tail of x and are denoted by hd x and tl x. We shall use the notation M i to denote j ≥i Mj . Note that if L is a Lie g-algebra then Li is the usual i-th Lie power of L. We shall make repeated use of the following easy lemma without any further mention. Lemma 2.1. Let M be an L-g-module and N a g-submodule of finite codimension. Then there exists an index j such that M j ≤ N . Definition 2.2. Let L be a g-algebra and let M be an L-g-module. The g-module M is just infinite if it is infinite-dimensional and, for every non-trivial g-submodule N, dim(M/N ) is finite. A (necessarily infinite-dimensional) g-algebra L is just infinite if L is just infinite as an L-g-module. Lemma 2.3. Let M be a just infinite L-g-module. Then every non-trivial ( possibly non-graded ) submodule of M has finite codimension in M. Proof. Let N be a non-trivial submodule

of M. If x is a non-trivial element of N whose tail has degree i, then x − tl x ∈ j

of (N + j

above that some Ki For i ≥ 0, set Ki := (N + j

i Mi for i ≥ k, that

Mj ≥ Mi . By an easy induction we get N

+ j

so that N + j

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As a consequence we have the following easy remarks which we shall often use without explicit reference: Remark 2.4. The intersection of two non-trivial submodules of a just infinite g-module is non-trivial. Remark 2.5. A just infinite g-module has no non-trivial finite-dimensional submodule. Lemma 2.6 (Schur). Let M be a just infinite L-g-module. If N is a submodule of M and f : N → M is a non-trivial L-homomorphism then f is injective. Proof. By Remark 2.5, f (N) has infinite dimension. As f (N) ∼ = N/ ker f , ker f has infinite codimension in N and, a fortiori, in M, so that ker f = 0 by Lemma 2.3. Remark 2.7. The centre of a just infinite Lie g-algebra L is trivial. Proof. If Z (L) = 0, then the vector space generated by a non-trivial element of Z (L) would be a non-trivial ideal with infinite codimension. The following application of Schur’s Lemma will be useful in the following: Lemma 2.8. Let I be a non-trivial ideal of a just infinite Lie g-algebra L. If CL (I ) = 0 then every ideal J which does not centralize I contains an element y (which we may choose to be homogeneous if J is a g-ideal ) such that ad y induces an injective L-endomorphism of I (in particular y does not centralize any non-trivial element of I ). Proof. We first assume that CL (I ) contains a non-trivial g-ideal: by Lemma 2.1, / CL (I ) but L/CL (I ) is nilpotent. We may then choose y ∈ J such that y ∈ [y, L] ≤ CL (I ). Moreover, we may choose y homogeneous if J is a g-ideal. It can be easily checked that ad y induces a non-trivial L-endomorphism of I , which is injective by Schur Lemma. We now come to the general case and claim that CL (I ) contains a non-trivial gideal, so that the previous proof applies. We consider the subspace gr I generated by the heads of the non-trivial elements of I : an easy check shows that gr I is a g-ideal centralized by the heads of the non-trivial elements of CL (I ). This means that gr I is a non-trivial ideal whose centralizer contains a non-trivial g-ideal, so that by the first part of the proof every ideal which does not centralize gr I contains an element y which does not centralize any non-trivial element of gr I . In particular assume that I does not centralize gr I , so that we may choose y in I : this is impossible since y centralizes at least the elements of CL (I ) ∩ gr I , which is non-trivial, since it is the intersection of two non-trivial ideals. We have then proved that I centralizes gr I , i.e., CL (I ) contains a non-trivial g-ideal, as claimed.

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We recall that the nil radical of a (not necessarily graded) Lie algebra is the sum of all nilpotent ideals. The nil radical is not in general nilpotent: this is true, however, for a just infinite Lie g-algebra L, since the non-trivial ideals of L have finite codimension so that the nil radical can be expressed as a finite sum of nilpotent ideals and is, therefore, nilpotent. We may actually say much more about the nil radical of a just infinite Lie g-algebra: Proposition 2.9. Let L be a just infinite Lie g-algebra and let M be the nil radical of L. Let I be a non-trivial ideal of L. The following conditions are equivalent: (i) I is abelian; (ii) I is nilpotent; (iii) I ≤ M; (iv) CL (I ) = 0. If one (and hence all) of the previous conditions is verified then CL (I ) = M. Proof. Clearly (i) implies (ii) and (ii) implies (iii). Since we already noted that M is nilpotent, its centralizer is non-trivial, so that the same is true for the ideals contained in M: (iii) implies (iv). To prove that (iv) implies (i) note that if I were not abelian then we could apply Lemma 2.8 and find an element y in I which does not centralize any non-trivial element of I : this is impossible, since y centralizes at least one non-trivial element of I , that is y itself. Now suppose that I satisfies the conditions above: CL (I ) is then a non-trivial ideal with non-trivial centralizer, i.e. it satisfies condition (iv), so that, by (iii), CL (I ) ≤ M. On the other hand, M satisfies trivially condition (iii): therefore it is abelian and then centralizes every ideal contained in it, so that CL (I ) ≥ M. Corollary 2.10. The nil radical M of a just infinite Lie g-algebra is an abelian g-ideal. Proof. If M = 0 there is nothing to prove. Otherwise M satisfies condition (iii) of Proposition 2.9 so that it is abelian and it is the centralizer of the non-trivial ideals contained in it. In the proof of Lemma 2.8 we showed that a non-trivial centralizer contains a non-trivial g-ideal: in particular M contains a non-trivial g-ideal, say I . Clearly CL (I ) is a g-ideal, and it is exactly M by Proposition 2.9. Corollary 2.11. For a just infinite Lie g-algebra L the following conditions are equivalent: (i) L is solvable; (ii) L contains a non-trivial abelian ideal; (iii) the nil radical of L is non-trivial. Proof. Clearly (i) implies (ii) and (ii) implies (iii). If (iii) holds, let M be the nil radical of L: then M is abelian (hence solvable) by Corollary 2.10, and L/M is nilpotent (hence solvable) by Lemma 2.1. Therefore L is solvable.

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3. Periodic algebras If L is a (non graded) finite-dimensional Lie algebra over the field F , the loop algebra1 Loop(L) is the tensor product L ⊗F (sF [s]) where F [s] is the ring of polynomials in the indeterminate s. The algebra Loop(L) is graded over N and it is not necessarily generated by its homogeneous component of degree 1. Definition 3.1. A non-trivial ideal I of a Lie g-algebra L is completely inhomogeneous if no non-trivial homogeneous element of L belongs to I . We have the following easy characterization of completely inhomogeneous ideals in just infinite Lie g-algebras: Remark 3.2. A non-trivial ideal I of a just infinite Lie g-algebra L is completely inhomogeneous if and only if L/I is not nilpotent. Proof. If L/I is nilpotent then I contains a term of the lower central series of L, that is a non-trivial homogeneous ideal. Conversely, if I contains a non-trivial homogeneous element, say x, then I contains the g-ideal generated by x, so that Lemma 2.1 implies that I contains a term of the lower central series of L and L/I is nilpotent. In the following theorem, the hypothesis on the codimension of I is needed only to construct the loop algebra Loop(L/I ). Proposition 3.3. Let L be a Lie g-algebra and let I be a completely inhomogeneous ideal of finite codimension in L. Then L is isomorphic to the g-subalgebra of Loop(L/I ) generated by L1I+I · s. Proof. We define a linear map ϕ : L → Loop(L/I ) by setting ϕ(x) := (x + I )s i for x ∈ Li . If x ∈ Li and y ∈ Lj then [ϕ(x), ϕ(y)] = [(x + I )s i , (y + I )s j ] = [(x + I ), (y + I )]s i+j = ([x, y] + I )s i+j = ϕ([x, y]). Thus ϕ is a graded Lie algebra homomorphism. By definition of ϕ, a homogeneous element k of ker ϕ must belong to I . Since I is completely inhomogeneous this means that k = 0 and ϕ is injective. As L is generated by L1 we have that ϕ(L) is generated by ϕ(L1 ) = L1I+I · s. As a partial converse we have: Proposition 3.4. An infinite-dimensional g-subalgebra L of a loop algebra contains a completely inhomogeneous ideal. 1 Strictly speaking, we are defining the positive part of a loop algebra, since the loop algebra is generally defined as L ⊗F (F [s, s −1 ]) where F [s, s −1 ] is the ring of the Laurent polynomials in the indeterminate s.

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Proof. If L ⊆ Loop(L), the intersection of L with the kernel of the homomorphism s=1

L ⊗ sF [s] −−→ L is a non-trivial ideal of L whose only homogeneous element is 0. Proposition 3.4 does not hold in general if we drop the hypothesis that L is infinite-dimensional. As a trivial example consider a finite-dimensional abelian algebra L. Then L · s is a subalgebra of Loop(L), all of whose non-trivial elements have degree 1, and then it does not contain any completely inhomogeneous ideal. We define the depth of a non-trivial element x of a Lie g-algebra as the difference between the degree of the tail of x and the degree of the head of x. Clearly x is homogeneous if and only if it has depth 0. We shall need the following lemma. Lemma 3.5. Let I be a non-trivial ideal of a Lie g-algebra L. Let d be the minimal depth of non-trivial elements of I and let J be the subspace of L generated by the heads of the elements of I of depth d. Then: (i) every homogeneous element of J is the head of a unique element of I of depth d; (ii) J is a g-ideal; (iii) there exists a non-trivial L-homomorphism t : J → L which sends homogeneous elements of degree i into homogeneous elements of degree i + d. Proof. Let H be the subset of L formed by the heads of the elements of I of depth d. If h ∈ H and x and x¯ are elements of I of depth d whose head is h, then x − x¯ is either 0 or it has depth less than d: by minimality of d the latter case cannot hold, so that x = x. ¯ This proves the uniqueness part of (i). Set b(h) := x. Let Hˆ := H ∪ 0. We define a map t : Hˆ → L by setting ht := tl b(h) if h = 0 and 0t := 0. If h ∈ Hˆ and z is a homogeneous element of L then either [h, z] = 0 or [h, z] is the head of [b(h), z], which is an element of I of depth at most d and hence exactly d. In both cases [h, z] belongs to J , thus proving (ii). Moreover, an easy inspection of the possible cases shows that [ht, z] = [h, z]t: then to prove (iii) we only need to show that t can be extended to a linear map to the whole of J . For every positive integer i set Hi := H ∩ Li . If h and k belong to Hi and α and β are scalars, then αh + βk is either 0 or it is the head of αb(h) + βb(k) which is an element of I of depth at most d, and then, once again, exactly d. We have then shown that Hˆ i := Hi ∪ 0 is a subspace of L and therefore it is equal to J ∩ Li , thus proving the existence part of (i). Moreover, it can be easily checked that t is linear on Hi , so that it can be extended to a linear map on J thus completing the proof of (iii). Definition 3.6. A just infinite Lie g-algebra L is periodic if there exists a g-ideal M and a non-trivial L-homomorphism t : M → L which sends homogeneous elements of degree i into homogeneous elements of degree i + d, with d positive integer. The homomorphism t is called an M-period of degree d.

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By Schur Lemma an M-period t of degree d is necessarily injective. As M is a non-trivial g-ideal it contains a term of the lower central series of L so that t is defined on Lj for all sufficiently large j , thus inducing an L-homomorphism between Lj and Lj +d . Since also Mt is a non-trivial g-ideal, it contains a term of the lower central series of L so that the homomorphisms above are surjective for all sufficiently large j . Remark 3.7. An M-period t of degree d induces an L-isomorphism between Lj and Lj +d for all sufficiently large j . By definition an M-period t does not necessarily send M into itself. We may, however, restrict t to a suitable g-ideal N (e.g. a term of the lower central series of L contained in M) and get: Remark 3.8. Given an M-period t there exists a g-ideal N contained in M such that the restriction of t to N is an N -period sending N into itself. Remark 3.9. A solvable just infinite Lie g-algebra L is periodic. Proof. By Corollaries 2.10 and 2.11 the nil radical M of L is an abelian non-trivial g-ideal. By Lemma 2.8 there exists a homogeneous element of L whose adjoint induces an M-period. We can give some characterizations of periodic algebras: Theorem 3.10. For a just infinite Lie g-algebra L the following are equivalent: (i) L is isomorphic to a g-subalgebra of a loop algebra; (ii) L has a completely inhomogeneous ideal; (iii) L is periodic. Proof. We have already seen the equivalence of (i) and (ii) in Propositions 3.3 and 3.4. Assume now (ii), and choose a completely inhomogeneous ideal I of L. The minimal depth of non-trivial elements of I is then a positive number d, so that Lemma 3.5 provides us with a g-ideal J and a J -period of degree d. Conversely, if t is an M-period of a periodic just infinite Lie g-algebra L then M(t − 1) is a non-trivial completely inhomogeneous ideal of L. The previous theorem shows that a periodic just infinite Lie g-algebra L is isomorphic to a g-subalgebra of a loop algebra Loop(L): note that, while L is not necessarily unique, it is necessarily not nilpotent for, otherwise, L would be finite-dimensional. However, it is possible to choose L to have: Proposition 3.11. A periodic just infinite Lie g-algebra L is isomorphic to a g-subalgebra of a loop algebra Loop(L) where L has the following properties: (i) every proper quotient of L is nilpotent;

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(ii) L has a unique minimal ideal ; (iii) is a differentiably simple algebra. Proof. Let I be an ideal of L which is maximal among the completely inhomogeneous ideals of L: by Proposition 3.3, L is isomorphic to a g-subalgebra of Loop(L), where L := L/I . By Remark 3.2, our choice of I implies (i). Since L is not nilpotent and finite-dimensional, there exists a non-trivial ideal such that = Li for all sufficiently large i. By (i), a non-trivial ideal of L contains a term of the lower central series of L, so that it contains : this proves (ii). If J is an ideal of invariant under Der( ) then J is clearly an ideal of L: minimality of implies (iii). In the previous proposition we chose an ideal maximal among completely inhomogeneous ideals of a periodic just infinite Lie g-algebra L: such an ideal is not unique. In fact, let t be an M-period: by Remark 3.8, we may assume that t sends M to itself, so that powers of t are defined. It can be easily checked that M(1 + t + t 2 ) and M(t + t 2 ) are completely inhomogeneous ideals of L. Since their sum contains the g-ideal M, there is no completely inhomogeneous ideal containing both of them, and therefore they are contained in different maximal completely inhomogeneous ideals. Definition 3.12. Let L be a Lie g-algebra and r a positive integer. As in [4] we shall say that L is ideally r-constrained if for every g-ideal I of L and every positive integer j one of the following holds: (1) I ≤ Lj , (2) I ≥ Lj +r−1 . Remark 3.13. In [2] it is proved that a similar condition holds in a pro-p-group G for some r if and only if G has finite obliquity (see [7] for the definition of obliquity). The definition of obliquity has been adapted to Lie g-algebras in [4], where it is proved that a Lie g-algebra is ideally r-constrained for some r if and only if it has finite obliquity. If L is an ideally r-constrained Lie g-algebra then every non-trivial g-ideal contains a term of the lower central series of L. As the terms of the lower central series have finite codimension in L (we are assuming that L is finitely generated) this implies that L is either finite-dimensional or just infinite. In the periodic case we have a converse: Proposition 3.14. A periodic just infinite Lie g-algebra L is ideally r-constrained for some integer r. Proof. For every positive integers i and j define the subset of Lj : Vi,j := {x ∈ Lj | [x, L1 , . . . , L1 ] = Li+j }. i times

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By choosing a system of affine coordinates in Lj and Li+j we see that Vi,j is described by the vanishing of a finite number of determinants in the coordinates of Lj , so that Vi,j is an affine subvariety of Lj . a g-ideal conSince L is just infinite, every non-trivial element x in Lj generates taining Li+j for a suitable i: this means that x ∈ / Vi,j , so that i Vi,j = 0. Clearly Vi+1,j ≤ Vi,j : as a decreasing sequence of affine varieties stabilizes after a finite number of steps there exists a minimum integer r(j ) such that Vr(j ),j = 0. A g-ideal I which is not contained in Lj +1 contains a non-trivial homogeneous element in Lj and then it contains Lj +r(j ) . We only need to show that the sequence of integers {r(j )}j is bounded above. By Remark 3.7 the period t induces an L-isomorphism between Lj and Lj +deg(t) for j ≥ k. This implies that r(j + deg(t)) = r(j ) for j ≥ k, so that sup{r(j )} = max{r(j ) | j ≤ k + deg(t) − 1}. Definition 3.15. Let M be an L-g-module. An L-endomorphism of M is graded of degree d if, for every i, it sends homogeneous elements of degree i into homogeneous element of degree i + d. Proposition 3.16. An L-endomorphism t of a just infinite L-g-module M can be expressed as the sum of a finite number of graded L-endomorphisms of nonnegative degree (in particular there is no graded L-endomorphism of negative degree). If t is not graded of degree 0 then t acts transcendentally on M, M is a finitely generated free module over the polynomial ring F [t] and L acts over M as an algebra of F [t]-endomorphisms. Proof. If m is a homogeneous element of M of degree i, say, we define mtd as the homogeneous component of degree i + d of mt. It can be easily checked that td is linear on Mi for every i, so that it can be extended to a linear map on the whole of M, and a further simple inspection shows that td is an L-endomorphism of M. For a given m in M we have clearly mtd = 0 for all but a finite number of d’s. If we moreover choose m to be a non-trivial homogeneous element of minimal degree we have mtd = 0 for d < 0. Schur Lemma then implies that td = 0 for all but a finite number of d’s and that td = 0 for d < 0. This proves that t is the sum of a finite number of graded L-endomorphisms of nonnegative degree. We now assume that t is not graded of degree 0: the maximum integer d such that td = 0 is then positive. Since Mtd is a non-trivial g-submodule of M, there exists a subspace V of M, generated by a finite number of homogeneous elements, such that M = V + Mtd . We claim that V generates M as an F [t]-module: to this aim we show that every homogeneous element m of M belongs to V F [t]. By our choice of V , there exist v ∈ V and l ∈ Mtd both homogeneous of the same degree of m, say i, and such that m = v + l. As l ∈ Mtd , there exists u in M such that l = utd : since td is injective, by Schur Lemma, and it is graded, u is a homogeneous element of degree less than i. By definition of td , utd − ut is a sum of homogeneous elements of degree less than i: arguing by induction on i, we may then assume that both u and utd − ut belong to V F [t]. Therefore m = v + (utd − ut) + ut belongs to V F [t].

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We have then shown that M is finitely generated as an F [t]-module. This implies that t is transcendental over F for, otherwise, F [t] would be finite-dimensional over F , so that M would be finite-dimensional too. A non-trivial polynomial p(t) in t is injective by Schur Lemma: therefore M is a torsion-free F [t]-module. As F [t] is a principal ideal domain, M is a finitely generated free F [t]-module. Clearly L acts over M as a Lie algebra of F [t]-endomorphisms since t centralizes the action of L. We cannot directly apply the previous proposition to an M-period t of a just infinite Lie g-algebra L, since t needs not to be an endomorphism of M. However, thanks to Remark 3.8, this problem can be overridden by taking the restriction of t to a suitable g-ideal: Corollary 3.17. If t is an M-period of a just infinite Lie g-algebra L such that Mt ≤ M then M is a non-trivial finitely generated free F [t]-module. Corollary 3.18. Let L be a just infinite solvable g-algebra with nil radical M. If x is a (not necessarily homogeneous) element of L \ M such that [x, L] ≤ M then M is a finitely generated free F [t]-module, where t is induced by ad x, and L acts on M as a Lie algebra of F [t]-endomorphisms. Proposition 3.19. A periodic just infinite Lie g-algebra L has a faithful representation τ of finite degree n over F [t] such that the kernel K of the ordinary representation τ

t=1

→ gl(n, F [t]) −−→ gl(n, F ) L− is a completely inhomogeneous ideal. Proof. We fix an M-period t such that Mt ≤ M and a homogeneous element x of L such that [x, L] ≤ M: moreover, if L is solvable, we ask that M is the nil radical of L (as in the proof of Remark 3.9) and that x ∈ / M (this is not necessarily the element whose adjoint induces the period t). By Corollary 3.17, N := F [t] ⊕ M is a finitely generated free F [t]-module. We now define τ : L → gl(N, F [t]) by setting: τ (l)

(f (t), m) −−→ (0, [x, l]f (t) + [m, l]). It can be easily verified that this actually defines a representation. We show that τ is faithful. If l ∈ / CL (M) then (0, m)τ (l) = 0 for some m ∈ M. If l is a non-trivial element in CL (M) then, by Corollary 2.11 and Proposition 2.9, L is solvable and M = CL (M): therefore l belongs to M and ad x induces a non-trivial L-endomorphism of M. Schur Lemma then implies that x does not centralize any element of M: in particular [x, l] = 0, so that (1, 0)τ (l) = (0, [x, l]) = 0. Now we show that K is a completely inhomogeneous ideal. Let k be a homogeneous element in K. Therefore N τ (k) ≤ N (1 − t). In particular, for every homogeneous m ∈ M, we have that (1, 0)τ (k) and (0, m)τ (k) belong to N(1 − t). By

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definition of τ this means that [x, k] and [m, k] belong to M(1 − t). As M(1 − t) is a completely inhomogeneous ideal, while [x, k] and [m, k] are homogeneous elements, this yields [x, k] = [m, k] = 0. For the arbitrariness of m we then get that k centralizes M + F x. Since M + F x is an ideal which is not abelian, Proposition 2.9 implies that k = 0, that is K does not contain any non-trivial homogeneous element. We now apply the above results to the solvable case. Corollary 3.20. Let L be a just infinite solvable g-algebra L with nil radical M. If L is not metabelian then char(F ) is non-zero and it divides the rank of the free F [t]-module M. Proof. Just note that if L is not metabelian, then, in the construction given in Proposition 3.19, we may choose x in the derived subalgebra L2 of L and we may assume that x is the element whose adjoint induces the period t so that the matrix 0 0 ... 0 0 t 0 τ (x) = . . . . . . t 0 0 must have zero trace. On the other hand, if n is the rank of M, Tr(τ (t)) = nt so that char(F ) divides n. Definition 3.21. A periodic just infinite Lie g-algebra is linear if it is isomorphic to a g-subalgebra of Loop(gl(n, F )) for some n. Corollary 3.22. A periodic just infinite Lie g-algebra L has a graded faithful representation and therefore it is a linear algebra. Proof. This is an easy consequence of the Ado–Iwasawa theorem, however we provide here a different self-contained proof. Proposition 3.19 ensures us that L has a completely inhomogeneous ideal K such that L/K is a subalgebra of gl(n, f ). Therefore Loop(L/K) is a g-subalgebra of Loop(gl(n, f )). By Proposition 3.3 L is isomorphic to the g-subalgebra of Loop(gl(n, f )) generated by L1K+K . In Proposition 3.14 we proved that a periodic just infinite Lie g-algebra is ideally r-constrained for a suitable r. In the solvable case we may give an estimate for such an r. To do this we need the following: Definition 3.23. Let L be a g-algebra and M an L-g-module. The width of M is defined as wd(M) := sup∞ i=0 dim(Mi ) Note that wd(M) is not necessarily finite. Clearly a periodic algebra (in particular a just infinite solvable g-algebra) has finite width. We are now ready to state and prove the:

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Theorem 3.24. Let L be a just infinite solvable Lie g-algebra with nil radical M. Let c be the nilpotency class of L/M and w be the width of M. Set r := wc + 1. Then L is ideally r-constrained. Proof. If I is a g-ideal of L not contained in Lj , we need to show that I contains Lj +wc . Clearly Ii = 0 for every i ≥ j − 1. In particular Ij +c = 0. Take a homogeneous element x of degree c which is not in M: like in Remark 3.9 ad x induces a M-period t of degree c. The period t sends Ii into Ii+c for i ≥ c + 1, so that ∞ dim(Ij +kc ) k=1 is a non-decreasing sequence of positive integers bounded above by ). This means w. Therefore there exists k¯ ≤ w such that dim(Ij +kc ¯ ) = dim(Ij +(k+1)c ¯ j +(k+1)c−1 ¯ . If we set I˜ := j +kc Ii we have that [I˜, L1 ] ≤ I˜ + I˜t that Ij +kc ¯ t = Ij +(k+1)c ¯ ¯ ∞ i so that Iˆ := 0 I˜t is a non-trivial g-ideal. As L is just infinite, Iˆ contains a term of the lower central series of L so that there exists h ≥ 0 such that Iˆ ¯ =L ¯ . j +(k+h)c

j +(k+h)c

h h Since Iˆj +(k+h)c = Ij +kc ¯ ¯ t and t induces an injective linear map from Lj +kc ¯ into ¯

j +kc ≥ Lj +wc , as required. , we have that Ij +kc Lj +(k+h)c ¯ ¯ = Lj +kc ¯ . Therefore I ≥ L

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Norberto Gavioli, Dipartimento di Matematica Pura ed Applicata, Università degli Studi di L’Aquila, Via Vetoio, 67010 Coppito (L’Aquila) AQ, Italy E-mail: [email protected] Valerio Monti, Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate, Università degli Studi di Roma ‘La Sapienza’, via Scarpa, 16, 00161 Roma, Italy E-mail: [email protected] Carlo M. Scoppola, Dipartimento di Matematica Pura ed Applicata, Università degli Studi di L’Aquila, Via Vetoio, 67010 Coppito (L’Aquila) AQ, Italy E-mail: [email protected]

On the Schur indices of cuspidal unipotent characters Meinolf Geck

Abstract. In previous work of Gow, Ohmori, Lusztig and the author, the Schur indices of all unipotent characters of finite groups of Lie type have been explicitly determined except for six cases in groups of type F4 , E7 and E8 . In this paper, we show that the Schur indices of all cuspidal unipotent characters for type F4 and E8 are 1, assuming that the group is defined over a field of “good” characteristic. This settles four out of the six open cases. For type E7 , we show that the Schur indices are at most 2. 2000 Mathematics Subject Classification: Primary 20C15; Secondary 20G40

1. Introduction By the work of Feit, Gow, Ohmori, Turull, . . . , one expects that the Schur index of an irreducible character of a finite simple group is 1 or 2. This paper is a contribution to a solution of this problem, as far as the unipotent characters of a finite group of Lie type are concerned. By Lusztig [16], the Schur indices of all rational-valued unipotent characters are generally 1. However, there are exceptions in the finite unitary groups (Ohmori [20]) and the Ree groups 2F4 (q) ([6]) where the Schur indices are 2. In [6], we were also able to compute the Schur indices of some non-rational valued unipotent characters, but not all. This paper deals with some of the remaining cases, which arise in groups of type F4 , E7 and E8 . Our principal tool are Kawanaka’s generalized Gelfand–Graev characters [11]. These are characters induced from certain unipotent subgroups. It is known that every unipotent character occurs with “small” multiplicity in such an induced character. We shall show here that certain generalized Gelfand–Graev representations can actually be realized over Q (in type F4 or E8 ) or over a quadratic extension of Q (in type E7 ). The method for doing this is inspired by Ohmori [19], who considered the case of ordinary Gelfand–Graev representations. Furthermore, using the explicit multiplicity formulas obtained by Kawanaka [11] and Lusztig [14], we check that the unipotent characters that we wish to consider all occur with multiplicity 1 in those generalized Gelfand–Graev representations.

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Combining the previous work by Ohmori, Lusztig and the author with the results obtained in this paper, the known results on Schur indices of unipotent characters can be summarized as follows. Let G be a simple algebraic group and F : G → G be an endomomorphism such that GF is a finite (twisted or untwisted) group of Lie type. Let ρ be a unipotent irreducible character of GF . Then the Schur index of ρ is given as follows. (1) Assume that ρ is cuspidal and rational-valued. (a) If GF is of type 2An−1 where n = s(s + 1)/2 for some s 1, then the Schur index of the unique cuspidal unipotent character of GF is 1 if n is congruent to 0 or 1 modulo 4, and 2 otherwise; see Ohmori [20]. (b) If GF is a Ree group of type 2F4 , then the Schur index of the unique cuspidal unipotent character which occurs with even multiplicity in all Deligne–Lusztig generalized characters RT ,1 is 2; see [6], Theorem 1.6. (c) In all other cases, the Schur index of ρ is 1; see Lusztig [16], Theorem 0.2. (2) Assume that ρ is cuspidal and not rational-valued. Then the Schur index of ρ is 1, except possibly for the two cuspidal unipotent characters in type E7 (where the Schur index√ is at most 2) or the two cuspidal unipotent characters with character field Q( −1) in type E8 over a field of characteristic 5; see Table 1 and Corollary 1.5 in [6], and Sections 2, 3 in the present paper. See Gow [9], p. 119, for groups of type 2B 2 and 2G2 . (3) Assume that ρ is unipotent but not necessarily cuspidal. Then ρ occurs with non-zero multiplicity in the Harish-Chandra induction RLG (ψ) where L is an F -stable Levi complement in some F -stable proper parabolic subgroup of G and ψ is a cuspidal unipotent character of LF . In this situation, the Schur index of ρ equals that of ψ; see [6], Proposition 5.6. Thus, there are two remaining cases to be dealt with: • the characters E7 [±ξ ] in type E7 ; √ • the characters E8 [± −1] for a group of type E8 in characteristic 5. For each of these cases, we show that the determination of the Schur index can be reduced to the problem of computing explicitly the values of certain induced characters. One of these problems is discussed in [7].

2. On the rationality of generalized Gelfand–Graev representations The aim of this section is to show that certain generalized Gelfand–Graev representations of a finite group of Lie type can be realized over Q. For this purpose, we have to recall in some detail the construction of these representations. We shall freely

On the Schur indices of cuspidal unipotent characters

89

use standard results and notations concerning (connected reductive) linear algebraic groups and their Lie algebras (see Carter [2] and the references there). Let G be a connected reductive group defined over the finite field Fq , with corresponding Frobenius map F : G → G. We fix an F -stable Borel subgroup B ⊆ G and write B = U.T where U is the unipotent radical of B and T is an F -stable maximal torus. Let X = Hom(T , k × ) be the character group of T and ⊆ X be the root system of G with respect to T . Then B determines a positive system + ⊆ and a corresponding set of simple roots ⊆ + . We have G = T , Xα | α ∈ and U = Xα α∈+

where Xα is the root subgroup corresponding to α ∈ . (Here, it is understood that the product is taken over some fixed order of the roots.) For each α ∈ , let α ∨ ∈ Y = Hom(k × , T ) be the corresponding coroot. Given λ ∈ X and µ ∈ Y , let λ, µ be the unique integer such that γ λ,µ = (λ ◦ µ)(γ ) for all γ ∈ k × . This defines a non-degenerate pairing , : X × Y → Z; the matrix C := α, β ∨ α,β∈ is the Cartan matrix of G. Now, for any α ∈ , there is an isomorphism of algebraic groups xα : k + → Xα such that txα (ξ )t −1 = xα (α(t)ξ )

for all t ∈ T and ξ ∈ k.

Since we will only need to consider this case, let us assume that F is of split type, such that F (xα (ξ )) = xα (ξ q ) for all α ∈ , ξ ∈ k, F (t) = t q for all t ∈ T . We consider the conjugacy classes of unipotent elements in G. For this purpose, we assume throughout that the characteristic p of Fq is a good prime for G. Recall that this means that p is good for each simple factor involved in G, and that the conditions for the various simple types are as follows. An : Bn , Cn , Dn : G2 , F4 , E6 , E7 : E8 :

no condition, p = 2, p = 2, 3, p = 2, 3, 5.

Then it is known (see Kawanaka [11] and the references there; see also Premet [21]) that one can naturally attach to each unipotent class of G a so-called weighted Dynkin diagram, that is, an additive map d : → Z such that

d(α) ∈ {0, 1, 2} for all α ∈ .

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The assignment from unipotent classes to additive maps as above is injective, but not surjective in general. Complete lists of weighted Dynkin diagrams for the various types of simple algebraic groups can be found in Carter [2], §13.1. Given such a weighted Dynkin diagram d, the corresponding unipotent class is determined as follows. We set Ld := T , Xα | α ∈ , d(α) = 0 and Ud,i := Xα α∈+ d(α)i

for i = 1, 2, 3, . . . . Then Pd := Ud,1 .Ld is a parabolic subgroup of G, with unipotent radical Ud,1 and Levi complement Ld . By Kawanaka [11], Theorem 2.1.1, there is a unique unipotent class C in G such that C ∩ Ud,2 is dense in Ud,2 ; furthermore, C ∩ Ud,2 is a single Pd -conjugacy class and we have CG (u) ⊆ Pd

for all u ∈ C ∩ Ud,2 .

Then C is the unipotent class attached to the given weighted Dynkin diagram d. In order to define the generalized Gelfand–Graev representation associated with F , we need to introduce some further notation. Let g be the Lie an element u ∈ C ∩ Ud,2 algebra of G over k = Fq . Then g is also defined over Fq and we have a corresponding Frobenius map F : g → g. Let t ⊆ g be the Lie algebra of T . We have a Cartan decomposition g=t⊕ keα where F (t) = t and F (eα ) = eα for all α ∈ . α∈

We shall set cα := κ(eα∗ , eα ) for any α ∈ + , where κ : g×g → k is a non-degenerate G-invariant, associative bilinear form and x → x ∗ is an opposition Fq -automorphism of g, that is, an automorphism such that t∗ = t and eα∗ ∈ Fq e−α for all α ∈ (see Kawanaka [11], §3.1). × Finally, we fix a non-trivial homomorphism χ : F+ q →C . F ; write Definition 2.1 (Kawanaka). Consider a unipotent element u ∈ C ∩ Ud,2 F u∈ xα (ηα ) · Ud,3 where ηα ∈ Fq .

(∗)

α∈+ d(α)=2

F → C× by With this notation, we define a map ϕu : Ud,2 ϕu xα (ξα ) = χ cα ηα ξα where ξα ∈ Fq . α∈+ d(α)2

α∈+ d(α)=2

Since κ(eα , eβ ) = 0 unless α = −β, this definition coincides with Kawanaka’s original definition in [11], (3.1.5); see also [5], Chapter 1. The map ϕu actually is a F . Inducing that character from group homomorphism, that is, a linear character of Ud,2

On the Schur indices of cuspidal unipotent characters

91

F to GF , we obtain Ud,2 F

F F 1/2 (ϕu ) = [Ud,1 : Ud,2 ] · u IndG UF d,2

where u is the character of the generalized Gelfand–Graev representation associated F : U F ] is an even with u; see Kawanaka [11], (3.1.12). Note that the index [Ud,1 d,2 power of q, so the square root exists. F , Remark 2.2. In the above setting, let us choose another element u ∈ C ∩ Ud,2 F xα (ηα ) · Ud,3 where ηα ∈ Fq . u ∈ α∈+ d(α)=2

Then we can also apply the above constructions to u . Thus, we obtain a corresponding generalized Gelfand–Graev character u such that F

F F 1/2 (ϕu ) = [Ud,1 : Ud,2 ] · u . IndG UF d,2

Using the defining formula for u , it is straightforward to check that u = u if u and u lie in the same PdF -orbit. (Indeed, if u = gug −1 where g ∈ PdF , then F and so the induced characters are equal.) Now ϕu (x) = ϕu (g −1 xg) for all x ∈ Ud,2 let C be the G-conjugacy class of u. Then C F splits into orbits under the finite group GF , and it is well-known that these GF -orbits are parametrized by the F -conjugacy classes of CG (u)/CG (u)◦ . Since CG (u) ⊆ Pd , a complete set of representatives for F . the GF -orbits in C F can be found inside C ∩ Ud,2 Assume now that G is simple of adjoint type and write = {α1 , . . . , αl } where l is the rank of G. Then we have X = Z. Let {ω1 . . . , ωl } be the dual basis of Y , that is, we have αi , ωj = δij for 1 i, j l. Now we can state the following result which generalizes the arguments in Ohmori [19], Lemma 2 and Proposition 1(i). (See also Example 2.4 below.) Proposition 2.3. Let G be simple of adjoint type and assume that there exist integers nj ∈ Z (1 j l) such that l

nj α, ωj = 1 for all α ∈ + such that d(α) = 2 and ηα = 0,

j =1

where the coefficients ηα are defined as in Definition 2.1 (∗). Then the character F : U F ]1/2 · can be realized over Q. If, moreover, we also have [Ud,1 u d,2 l

nj α, ωj = 0 for all α ∈ + such that d(α) = 0,

j =1 F : U F ]1/2 · can be realized over Q, for every u ∈ C ∩ U F . then [Ud,1 u d,2 d,2

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Proof. Let ν be a generator for the multiplicative group of Fq . We claim that there exists an element t ∈ T F such that for all α ∈ + with d(α) = 2 and ηα = 0.

α(t) = ν

(†)

This is seen as follows. By [2], Proposition 3.1.2, the map h : k × × · · · × k × → T ,

(x1 , . . . , xl ) →

l

ωi (xi ),

i=1

l factors

is an Fq -isomorphism. In particular, we have T F = {h(x1 , . . . , xl ) | xi ∈ F× q }. Now set t = h(ν n1 , . . . , ν nl ) ∈ T F . Let α ∈ + be such that d(α) = 2 and ηα = 0. Then we have l l l α,ω α(t) = xj j = ν nj α,ωj = ν j =1 nj α,ωj = ν, j =1

j =1

as required. Thus, (†) is proved. F . Now let H := t. Then H is a group of order q − 1 which normalizes Ud,2 F to the semidirect product U F .H and denote the induced Let us induce ϕu from Ud,2 d,2 character by ψu . We will show that ψu is a rational-valued irreducible character. First F . Furthermore, by Mackey’s note that ψu has non-zero values only on elements in Ud,2 F is given by h h −1 formula, the restriction of ψu to Ud,2 h∈H ϕu where ϕu (x) = ϕu (hxh ) F for all x ∈ Ud,2 . Now let F x := xα (ξα ) ∈ Ud,2 for any ξα ∈ Fq . α∈+ d(α)2

Then we have txt −1 =

txα (ξα )t −1 =

α∈+ d(α)2

xα (α(t)ξα )

α∈+ d(α)2

and so, using (†), ϕut (x) = χ

cα ηα α(t)ξα = χ ν cα ηα ξα .

α∈+ d(α)=2

α∈+ d(α)=2

Similarly, for any 1 i q − 1, we obtain i

ϕut (x) = χ(ν i γx )

where γx :=

α∈+ d(α)=2

cα ηα ξα ∈ Fq .

On the Schur indices of cuspidal unipotent characters

93

i

This shows, first of all, that the characters ϕut (1 i q − 1) are pairwise different and, hence, the character ψu is seen to be irreducible. Furthermore, we obtain that q−1 q−1 i q − 1 if γx = 0, ϕuh (x) = ϕut (x) = χ (ν i γx ) = ψu (x) = −1 if γx = 0. h∈H i=1 i=1 F . Thus, ψ is a rational-valued In particular, this shows that ψu (x) ∈ Z for all x ∈ Ud,2 u F .H . In order to show that ψ can be realized over Q, we irreducible character of Ud,2 u note that (again by Mackey’s formula), the restriction of ψu to H is the character of the regular representation of H . Hence the trivial character 1H occurs with multiplicity 1 in that restriction. By Frobenius reciprocity, ψu itself occurs with multiplicity 1 in F .H obtained by inducing 1 . Since the latter character clearly is the character of Ud,2 H realized over Q, a standard argument on Schur indices shows that ψu can be realized over Q (see Isaacs [10], Corollary 10.2). Finally, by the transitivity of induction, the character F

F F 1/2 (ϕu ) = [Ud,1 : Ud,2 ] · u IndG UF d,2

can be realized over Q, as claimed. Now assume that the integers nj satisfy the additional equations described above F . Thus, we have u = gug −1 where and consider an arbitrary element u ∈ C ∩ Ud,2 g ∈ Pd . Let F xα (ηα ) · Ud,3 where ηα ∈ Fq . u ∈ α∈+ d(α)=2

We claim that if ηα = 0, then α = α1 + β where d(β) = 0, d(α1 ) = 2 and ηα1 = 0. To see this, it is enough to consider the special case where g ∈ T or where g ∈ Xβ with d(β) 0. The assertion is clear if g ∈ T . On the other hand, if g ∈ Xβ where d(β) 0, then Chevalley’s commutator relations show that gug −1 will lie in a product of root subgroups Xiβ+α1 where d(α1 ) 2 and i = 0, 1, 2, . . . , and the assertion also follows. We can now conclude that the integers nj automatically have the property that l j =1

nj α, ωj = 1 for all α ∈ + such that d(α) = 2 and ηα = 0.

Thus, we can apply the above arguments to u as well and conclude that the character F : U F ]1/2 · can be realized over Q. [Ud,1 u d,2 Example 2.4 (cf. Ohmori [19], p. 151). Assume that G is simple of adjoint type. The weighted Dynkin diagram corresponding to the conjugacy class of regular unipotent elements in G is the map d0 : → Z such that d0 (αi ) = 2 for all 1 i l. In this

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case, we have Pd0 = B, Ld0 = T and Ud0 ,1 = Ud0 ,2 = U ;

furthermore, u = li=1 xαi (ηi ), where ηi ∈ F× q , is a representative in that class. Now the system of equations in Proposition 2.3 is simply given by l

nj αi , ωj = 1 for all 1 i l.

j =1

Thus, we have nj = 1 for 1 j l. Consequently, u (which is just an ordinary Gelfand–Graev character) can be realized over Q. Example 2.5. Assume that G is simple of adjoint type and that d is a weighted Dynkin diagram of the following special type: There exists some i0 ∈ {1, . . . , l} such that d(αi0 ) = 2

and d(αi ) = 0

for i ∈ {1, . . . , l} \ {i0 }.

We claim that, in this case, the generalized Gelfand–Graev character u can be realized F . This is seen as follows. First note that U over Q, for any u ∈ C ∩ Ud,2 d,1 = Ud,2 . Now let ηα be defined as in Definition 2.1(∗) and consider the two sets of equations in Proposition 2.3. The second set of equations reads l

nj αi , ωj = 0

for all i ∈ {1, . . . , l} \ {i0 }.

j =1

This yields ni = 0 for all i = i0 . Then the first set of equations reads ni0 α, ωi0 = 1

for all α ∈ + such that d(α) = 2 and ηα = 0.

However, writing any such α as a linear combination of simple roots, we see that αi0 always occurs with coefficient 1 and so the above equations hold with ni0 = 1. Thus, the systems of equations in Proposition 2.3 have a unique solution. We close this section with a general remark concerning the Schur indices of unipotent characters. This remark shows that it will be enough to consider the unipotent characters of simple groups of adjoint type. Remark 2.6. Assume that G is simple of adjoint type. Let G1 be an algebraic group over Fq such that G1 /Z(G1 ) is simple of the same type as G. We denote the Frobenius map on G1 again by F . Then we have a surjective homomorphism of algebraic groups ϕ : G1 → G which is defined over Fq and such that ker(ϕ) = Z(G1 ). Let χ be a unipotent character of GF . Denote by χ1 the character of GF1 which is obtained by first restricting χ to ϕ(GF1 ) and then pulling back to GF1 via ϕ. By [4], Proposition 7.10, the map χ → χ1 defines a bijection between the unipotent characters of GF and GF1 , respectively. One can express this by saying that the unipotent characters are “insensitive” to the center of G.

On the Schur indices of cuspidal unipotent characters

95

For a unipotent character χ ∈ Irr(GF ), denote by Q(χ ) = Q(χ (g) | g ∈ GF ) the character field of χ and by mQ (χ) the Schur index of χ ; we use similar notations for a unipotent character of GF1 . We claim that Q(χ) = Q(χ1 )

and

mQ (χ ) = mQ (χ1 ).

Proof. First note that we clearly have Q(χ1 ) ⊆ Q(χ ). To show that we have equality, let E ⊇ Q be a finite Galois extension such that any unipotent character of GF and of GF1 can be realized over E. Given any τ ∈ Gal(E/Q) and any unipotent character χ ∈ Irr(GF ), we denote by χ τ the irreducible character obtained by algebraic conjugation; it is still a unipotent character of GF (see the remarks in [6], (5.1)). Now the map χ → χ1 certainly is compatible with field automorphisms. Thus, we have χ1τ = (χ τ )1 . Since the map χ → χ1 is a bijection, we conclude that χ1τ = χ1 if and only if χ τ = χ . This implies that Q(χ) = Q(χ1 ), as required. Now consider the Schur indices. First note that we certainly have mQ (χ1 ) mQ (χ ). (Indeed, if χ can be realized over some extension field of Q, then χ1 is automatically realized over the same field.) To show the reverse inequality, we argue as follows. The character χ1 can be realized over an extension field E1 ⊇ Q such that [E1 : Q] = mQ (χ1 ). We now regard χ1 as a character of ϕ(GF ). By Frobenius reciprocity, χ occurs with multiplicity 1 in the character obtained by inducing χ1 from ϕ(GF ) to GF . Since that induced character can also be realized over E1 , a standard argument on Schur indices ([10], Corollary 10.2) shows that mQ (χ ) mQ (χ1 ), as required.

3. Simple groups of type G2 , F4 and E8 Throughout this section, let G be simple of (adjoint) type G2 , F4 or E8 , defined over the finite field Fq with corresponding Frobenius map F : G → G. Then F is of split type, in the sense of the previous section. The cuspidal unipotent characters of GF and the previously known results on their Schur indices are listed in Table 1. Now let us assume that G is defined over a field of good characteristic. All the cuspidal unipotent characters of GF have the same “unipotent support” in the sense of Lusztig [14]; see also [8], Proposition 4.2. Hence there is a unique unipotent class C0 in G such that all the cuspidal unipotent characters of GF have unipotent support C0 . Let e0 = dim Bu (the dimension of the variety of Borel subgroups containing an element u ∈ C0 ). Then, by [8], Theorem 3.7, q e0 is the exact power of q dividing the degrees of the unipotent characters of GF having unipotent support C0 . By inspection of the tables in [2], Chapter 9, we find that e0 = 1, 4 or 16 for G of type G2 , F4 or E8 , respectively. Furthermore, for G of type G2 or F4 , the class C0 is uniquely determined by this condition. In type E8 , we use the additional information that CG (u)/CG (u)◦ must be non-trivial (see once more the formula in [8], Theorem 3.7). Then C0 is uniquely determined. The weighted Dynkin diagram d0 associated to C0 is specified

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Table 1. Schur indices of cuspidal unipotent characters in type G2 , F4 and E8 (notation of Carter [2], §13.9; see also [6], Table 1)

Type G2 F4

E8

Character G2 [1], G2 [−1] G2 [θ], G2 [θ 2 ] F4I [1], F4I I [1], F4 [−1] F4 [i], F4 [−i] F4 [θ], F4 [θ 2 ] E8I [1], E8I I [1], E8 [−1] E8 [−θ], E8 [−θ 2 ], E8 [θ], E8 [θ 2 ] E8 [i], E8 [−i] E8 [ζ ], E8 [ζ 2 ], E8 [ζ 3 ], E8 [ζ 4 ] i :=

√ −1,

Schur index 1 (see Lusztig [16]) 1 (see [6]) 1 (see Lusztig [16]) ? (1 in char. 2, 3; see [6]) 1 (see [6]) 1 (see Lusztig [16]) ? (1 in char. 2, 3, 5; see [6]) ? (1 in char. 2, 3, 5; see [6]) ? (1 in char. 2, 3; see [6]) 1 (see [6])

θ := exp(2π i/3),

ζ := exp(2π i/5).

in Table 2. Furthermore, we have CG (u)/CG (u)◦ ∼ = S3 , S4 or S5 for G of type G2 , F4 or E8 , respectively.

Table 2. Weighted Dynkin diagrams for the unipotent supports of cuspidal unipotent characters in type G2 , F4 and E8

0 w < α1

G2

E8

0 w α1

2 w α2

0 w α1

F4

0 w α3

α4

0 w

2 w α5

2 w > α2 0 w α6

0 w α7

0 w α3

0 w α3 0 w α8

0 w α2 Now let us fix a cuspidal unipotent character ρ of GF . By Lusztig, [14], Theorem 11.2, there exists some u ∈ C0 ∩UdF0 ,2 such that ρ occurs with “small” multiplicity in u . In the present situation, Kawanaka actually obtained an explicit multiplicity formula. To state that formula, we need to recall some facts about the parametrization of unipotent characters; see Lusztig [12], Main Theorem 4.23.

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97

Let F0 be the unique family of unipotent characters which contains the cuspidal unipotent characters. Let G0 be the finite group attached to that family; we have G0 ∼ = CG (u)/CG (u)◦ in the present situation. Let M0 be the set of all pairs (x, σ ) where x ∈ G0 (up to conjugacy) and σ ∈ Irr(CG0 (x)). Then we have a bijection M0 → F0 ,

(x, σ ) → ρ(x,σ ) .

(which satisfies some further conditions which we do not need to recall here). On the other hand, there is a well-defined “split” element u0 ∈ C0 ∩ UdF0 ,2 ; see Shoji [22], Remark 5.1. Having fixed that element, there is a canonical parametrization of the GF -conjugacy classes contained in C0F by the conjugacy classes of G0 . We shall denote by uy the generalized Gelfand–Graev character with respect to a unipotent element uy ∈ C0 ∩ UdF0 ,2 in the GF -class parametrized by y ∈ G0 . Having fixed this notation, we can now state the following result. Theorem 3.1 (Kawanaka). Recall that G is a group of type G2 , F4 or E8 in good characteristic. Let ρ(x,σ ) , uy be the multiplicity of ρ(x,σ ) in the generalized Gelfand– Graev character uy , where x, y ∈ G0 and σ ∈ Irr(CG0 (x)). Then we have σ (1) if x = y (mod G0 ), ρ(x,σ ) , uy = 0 otherwise. Proof. Kawanaka obtained an explicit formula for the values of all the generalized Gelfand–Graev characters of GF ; see [11], Corollary 3.2.7 and Lemma 3.3.10. Furthermore, in [11], §4.2, he obtained explicit formulas for the values of the unipotent characters on unipotent elements. (In the latter reference, it is actually assumed that the characteristic is large. But this hypothesis is only used in reference to results on the Green functions of GF , which are now known to hold in general; see Shoji [23], Theorem 5.5.) The above multiplicity formula is obtained by a simple formal rewriting of Kawanaka’s formulas. In large characteristic, that formula could also be deduced from Lusztig’s more general results in [14]. Corollary 3.2. Recall that G is a group of type G2 , F4 or E8 in good characteristic. Then every cuspidal unipotent character of GF has Schur index 1. Proof. By inspection of the tables in [2], §13.9, all cuspidal unipotent characters of GF are labelled by pairs (x, σ ) ∈ M0 where σ (1) = 1. Thus, by Theorem 3.1, every such character occurs with multiplicity 1 in some generalized Gelfand–Graev character attached to an element in C0 ∩ UdF0 ,2 . Now we notice that the weighted Dynkin diagrams in Table 2 are of the type considered in Example 2.5. Thus, all the generalized Gelfand–Graev characters associated with unipotent elements in C0 ∩UdF0 ,2 can be realized over Q. Hence, the assertion follows by a standard argument on Schur indices (see Isaacs [10], Corollary 10.2).

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The above result also covers some of the cases (at least in good characteristic) already dealt with by Lusztig [16] or the author [6]. In combination with the previously known results listed in Table 1, we see that there is only one case left that remains to be considered: the characters E8 [±i] for a group of type E8 in characteristic 5. We shall now outline a strategy for dealing with this case. That strategy is analogous to that in [6], (6.5). For the remainder of this section, let G be a simple group of type E8 over Fq where q is odd and p ≡ 1 mod 4. Let us label the simple roots of G as in Table 2 and let {ω1 , . . . , ω8 } be the dual basis of Y . Then we have T = {h(x1 , . . . , x8 ) | xi ∈ k × } as in the proof of Proposition 2.3. Now let ν be a generator of the multiplicative group of Fq and consider the semisimple element s := h(1, 1, 1, 1, 1, ν (q−1)/4 , 1, 1) ∈ T F . Thus, s is an element of order 4. Its centralizer is given by G1 = T , Xα | α ∈ 1 where 1 = {α ∈ | α(s) = 1}. It is readily checked that a system of simple roots of 1 is given by 1 = ( \ {α6 }) ∪ {−α0 }, where α0 = 2α1 + 3α2 + 4α3 + 6α4 + 5α5 + 4α6 + 3α7 + 2α8 is the unique highest root in . The root system 1 is of type D5 × A3 . Now, by Lusztig [13], Proposition 21.3, there is a well-defined unipotent class C1 ⊆ G1 which supports a “cuspidal local system”. That unipotent class is determined explicitly by [17], Corollary 4.9, as far as the D5 -factor is concerned. (We necessarily have the class of regular unipotent elements in the A3 -factor; see [13], §18.) Using [2], §13.9, we see that the weighted Dynkin diagram d1 of C1 is given as follows: D5 ×A3

2 w α1

2 w α3

α4

0 w

2 w α5

2 w α7

2 w α8

2 w −α0

2 w α2 By [1], Theorem 11.3.2, we can explicitly identify an adjoint group of type D5 with a 10-dimensional orthogonal group. Determining the Jordan normal form of the matrix corresponding to the element u1 := xα1 (1)xα5 (1)xα2 (1)xα3 +α4 (1)xα4 +α5 (1) · xα7 (1)xα8 (1)x−α0 (1), ◦ (u ) ∼ Z/4Z; see [6], we see that u1 ∈ C1 ∩ UdF1 ,2 . Now, we have CG1 (u1 )/CG 1 = 1 Table 2, and the references there. Let ψ be one of the two faithful irreducible characters of this group and define (C1 ,ψ) as in [14], (7.5). Thus, (C1 ,ψ) is a certain linear combination (where the coefficients involve the values of ψ) of the generalized Gelfand–Graev characters of GF1 associated with the various unipotent elements in C1 ∩ UdF1 ,2 . Now we can state the following result.

On the Schur indices of cuspidal unipotent characters

99

Proposition 3.3. In the above setting, assume that the following conditions hold: (a) The values of (C1 ,ψ) are given by Lusztig’s formula [14], Corollary 7.6. (This is known to be true if the characteristic is “large”.) (b) The assertion in [15], Theorem 0.8, holds for cuspidal character sheaves in G. (This is known to be true if the characteristic is good; see [23], Theorem 4.1.) Then the Schur index of the cuspidal unipotent character E8 [±i] is 1. Proof. We begin by showing that the generalized Gelfand–Graev characters of GF1 which are associated with the unipotent elements in C1 ∩ UdF1 ,2 can be realized over Q. For this purpose, we use an argument analogous to that in Proposition 2.3. We shall consider a system of equations as in Proposition 2.3 (but with respect to the unipotent element u1 ∈ G1 specified above): 8 1 for α ∈ {α1 , α5 , α2 , α3 + α4 , α4 + α5 , α7 , α8 , −α0 }, nj α, ωj = 0 for all α ∈ + such that d1 (α) = 0. j =1 One readily checks that this system has a unique solution: (n1 , n2 , . . . , n8 ) = (1, 1, 1, 0, 1, −5, 1, 1). Thus, t := h(ν, ν, ν, 1, ν, ν −5 , ν, ν) ∈ T F is a semisimple element of order q − 1. Now let u ∈ C1 ∩ UdF1 ,2 . Then, as in the proof of Proposition 2.3, we see that α(t) = ν (if α is involved in the expression of u as a product of root subgroup elements) or α(t) = 1 (if d1 (α) = 0). Let H = t and denote by ψu the character obtained by inducing ϕu from UdF1 ,2 to UdF1 ,2 .H . Following exactly the same kind of arguments as in the proof of Proposition 2.3, we see that ψu is an irreducible character which can actually be realized over Q. Consequently, the generalized Gelfand–Graev character u of GF1 can also be realized over Q, as claimed. We now follow the line of argument in [6], (6.5). First note that the character values of E8 [±i] generate the field Q(i); see [6], Table 1. Thus, we must show that E8 [±i] can be realized over Q(i). Let be the G-conjugacy class of su1 . Then supports a “cuspidal local system” E ; see [6], Table 2 and the references there. Using [12], Main Theorem 4.23, the explicit description of the Fourier matrix for type E8 , and our assumption (b), we see that E8 [±i], χ(,E ) =

1 ε 4

where ε ∈ C, |ε| = 1;

here, χ(,E ) is a characteristic function of (, E ) and , denotes the usual hermitian product on the space of complex-valued class functions on GF . On the other hand, ◦ (u ) ∼ Z/4Z, the set C ∩ U F splits into four classes in GF , since CG1 (u1 )/CG 1 = 1 1 d1 ,2 1 with representatives u1 , u2 , u3 , u4 ∈ C1 ∩UdF1 ,2 say. Then, our assumption (a) implies

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that we have 4 χ(,E ) =

4

F

r=1 λ∈Irr(Z1 )

ξr,λ IndG (λ ϕur ) Z ×U F d1 ,2

1

where Z1 := Z(G1 ) = s and ξr,λ ∈ Z[i]; see the analogous equation in [6], (6.5), and note that Z(G1 )◦ = {1}. Now, since Z1 ∩ H = {1}, it is easily shown that (Z1 ×UdF ,2 ).H

IndZ

1

F 1 ×Ud ,2 1

UdF ,2 .H

(λ ϕur ) = λ IndU F1

d1 ,2

(ϕur ) = λ ψur

can be realized over Q(i). Thus, we have 4 χ(,E ) = α rα ρα where rα ∈ Z[i] and ρα are characters of GF which can be realized over Q(i). Since E8 [±i], 4χ(,E ) = ε has absolute value 1, we conclude that the greatest common divisor of the multiplicities E8 [±i], ρα must be 1. Hence, a standard argument on Schur indices ([10], Corollary 10.2) implies that E8 [±i] can be realized over Q(i), as desired. In principal, the hypothesis on (C1 ,ψ) might be verified by an explicit computation for a group of type D5 ×A3 (in the same way as Kawanaka [11] verified that hypothesis for groups of exceptional type).

4. Simple groups of type E7 Let G be a simple group of adjoint type E7 in good characteristic. There are precisely √ two cuspidal unipotent characters of GF denoted by E7 [±ξ ] where ξ = −q; see the table in [2], §13.9. By [6], Table 1, the character values of E7 [±ξ ] lie in the field Q(ξ ) and the Schur index of E7 [±ξ ] is 1, 2 or 4. Furthermore, by [6], Example 6.4, we already know that the Schur index is 1 if p ≡ 1 mod 4 or if q is not a square, where p is the characteristic of Fq . Thus, the remaining task is to determine the Schur index when q is a square and p ≡ 1 mod 4. Let C0 be the common unipotent support of the two cuspidal unipotent characters. Arguing as in the previous section (that is, checking the exact power of q dividing the degree of E7 [±ξ ] and comparing with the tables in [2], Chapter 9), we find that the corresponding weighted Dynkin diagram d0 is given as follows. E7

1 w α1

0 w α3

α4 0

1 w

w α2

0 w α5

1 w α6

0 w α7

On the Schur indices of cuspidal unipotent characters

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Lemma 4.1. Assume that q is an even power of p. Let u ∈ C0 ∩ UdF0 ,2 and recall the definition of the linear character ϕu from Section 2. Then there exists a field K ⊇ Q F (ϕu ) can be realized over K. such that [K : Q] 2 and IndG UF d0 ,2

Proof. The following argument is inspired by Ohmori [19], Proposition 1. By Mizuno [18], Lemma 28, a representative u0 ∈ C0 ∩ UdF0 ,2 is given by u0 = x20 (1)x21 (1)x24 (1)x28 (1)x30 (1), where the subscripts correspond to the following roots in + : 21 : α1 + α3 + α4 + α5 , 20 : α1 + α2 + α3 + α4 , 28 : α2 + α3 + 2α4 + α5 , 24 : α2 + α4 + α5 + α6 , 30 : α3 + α4 + α5 + α6 + α7 . We consider a set of equations similar to that in Proposition 2.3: 7 2 for α ∈ + labelled by 20, 21, 24, 28 and 30, nj α, ωj = 0 for all α ∈ + such that d0 (α) = 0. j =1 One readily checks that this system has a unique solution: (n1 , n2 , . . . , n7 ) = (1, 0, 0, 1, 0, 1, 0). Now let ν be a generator of the multiplicative group of Fp ⊆ Fq . Since q is an even power of p, we can find a square root of ν in Fq , which we denote by ν 1/2 ∈ Fq . Then the element t := h(ν n1 /2 , . . . , ν n7 /2 ) = h(ν 1/2 , 1, 1, ν 1/2 , 1, ν 1/2 , 1) ∈ T F has order 2(p − 1) and, as in the proof of Proposition 2.3, we see that α(t) = ν for all α ∈ + such that d0 (α) = 2 and ηα = 0 where the coefficients ηα are defined as in Definition 2.1(∗) (with respect to the unipotent element u ∈ C0 ∩ UdF0 ,2 ). Now let H = t. Then H is a group of order 2(p − 1) which normalizes UdF0 ,2 . Let us induce ϕu from UdF0 ,2 to the semidirect product UdF0 ,2 .H and denote the induced character by ψu . Arguing as in the proof of Ohmori [19], Proposition 1(ii), we see that ψu is the sum of two rational-valued irreducible characters ψ1 and ψ2 , where one of them, ψ1 say, can be realized over Q. By the Brauer–Speiser theorem (see [3], 74.27), the character ψ2 has Schur index at most 2. Thus, there exists a field K ⊇ Q such that [K : Q] 2 and ψ = ψ1 + ψ2 can be realized over K. Consequently, by the transitivity of induction, the character F

F

d0 ,2

d0 ,2 .H

(ϕu ) = IndG IndG UF UF can also be realized over K.

(ψu )

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Remark 4.2. We have remarked at the beginning of this section that it is enough to compute the Schur index of E7 [±ξ ] in the case where q is a square √ and p ≡ 1 mod 4. In this case, the character values of E7 [±ξ ] lie in the field Q( −1). Now Ohmori [19], p. 154, points out that the character ψ2 arising in the above proof has non-trivial local√Schur indices at ∞ and at the prime p. This implies √ that p ≡ 1 mod 4. Indeed, if we had K ⊆ Q( K is not contained in Q( −1) if √ √ −1), then ψ2 could be realized over Q( −1). Since p ≡ 1 mod 4, the field Q( −1) is contained in the field of p-adic numbers. Hence, ψ2 could be realized over the latter field, contradicting the fact that the local Schur index at p is not one. Corollary 4.3. Recall that G is a group of type E7 in good characteristic. Then the two cuspidal unipotent characters of GF have Schur index at most 2. Proof. We have already remarked at the beginning of this section that the Schur index of E7 [±ξ ] is 1 if q is not a square. So let us now assume that q is an even power of p. Let u0 ∈ C0 ∩ UdF0 ,2 . Then it is known that CG (u0 )/CG (u0 )◦ ∼ = Z/2Z (see F the tables in [2], §13.1). Thus, C0 splits into two classes in the finite group GF . Let u1 ∈ C0 ∩ UdF0 ,2 be a representative of the second GF -conjugacy class contained in C0F . Combining [8], Theorem 3.7 and Remark 3.8, we have that E7 [±ξ ], u0 + E7 [±ξ ], u1 = 1. (This could also be deduced from Kawanaka’s explicit multiplicity formulas [11].) Thus, E7 [±ξ ] occurs with multiplicity 1 in u for some u ∈ C0 ∩UdF0 ,2 . Consequently, we have F (ϕu ) = [UdF0 ,1 : UdF0 ,2 ]1/2 =: m0 , E7 [±ξ ], IndG UF d0 ,2

where the number m0 on the right hand side is odd since we are in good characteristic. Let K ⊇ Q be as in Lemma 4.1. Then a standard result on Schur indices (see Isaacs [10], Lemma 10.4) shows that the Schur index of E7 [±ξ ] divides [K : Q] m0 . However, as we have already remarked at the beginning of this section, the Schur index of E7 [±ξ ] must be 1, 2 or 4. Thus, since m0 is odd, we can now conclude that the Schur index of E7 [±ξ ] divides [K : Q] 2. Remark 4.4. The Schur index of E7 [±ξ ] equals 1 or 2, according to whether F E7 [±ξ ], IndG (ψ1 ) is odd or even, U F .H d0 ,2

where ψ1 is defined in the proof of Lemma 4.1. Thus, the problem is reduced to the computation of the above scalar product, and this will be discussed in [7]. Acknowledgements. I wish to thank the organizers of the “Finite Groups 2003” conference, C. Y. Ho, P. Sin, P. H. Tiep and A. Turull, for the invitation and the University of Gainesville for its hospitality.

On the Schur indices of cuspidal unipotent characters

103

References

[1]

R. W. Carter, Simple groups of Lie type, Wiley, New York 1972; reprinted 1989 as Wiley Classics Library Edition.

[2]

R. W. Carter, Finite groups of Lie type: Conjugacy classes and complex characters, Wiley, New York 1985; reprinted 1993 as Wiley Classics Library Edition.

[3]

C. W. Curtis and I. Reiner, Methods of representation theory, vol. II, Wiley, New York 1987; reprinted 1994 as Wiley Classics Library Edition.

[4]

P. Deligne and G. Lusztig, Representations of reductive groups over finite fields, Ann. of Math. 103 (1976), 103–161.

[5]

M. Geck, Verallgemeinerte Gelfand-Graev Charaktere und Zerlegungszahlen endlicher Gruppen vom Lie-Typ, Dissertation, RWTH Aachen 1990.

[6]

M. Geck, Character values, Schur indices and character sheaves, Represent. Theory 7 (2003), 19–55.

[7]

M. Geck, The Schur indices of the cuspidal unipotent characters of the finite Chevalley groups E7 (q), Osaka J. Math., to appear.

[8]

M. Geck and G. Malle, On the existence of a unipotent support for the irreducible characters of finite groups of Lie type, Trans. Amer. Math. Soc. 352 (2000), 429–456.

[9]

R. Gow, Schur indices of some groups of Lie type, J. Algebra 42 (1976), 102–120.

[10] I. M. Isaacs, Character Theory of Finite Groups, Academic Press, New York 1976; corrected reprint by Dover Publ., New York 1994. [11] N. Kawanaka, Generalized Gelfand–Graev representations of exceptional algebraic groups I, Invent. Math. 84 (1986), 575–616. [12] G. Lusztig, Characters of reductive groups over a finite field, Ann. of Math. Stud. 107, Princeton University Press, Princeton 1984. [13] G. Lusztig, Character sheaves, IV, Adv. Math. 59 (1986), 1–63. [14] G. Lusztig, A unipotent support for irreducible representations, Adv. Math. 94 (1992), 139–179. [15] G. Lusztig, Remarks on computing irreducible characters, J. Amer. Math. Soc. 5 (1992), 971–986. [16] G. Lusztig, Rationality properties of unipotent representations, J. Algebra 258 (2002), 1–22. [17] G. Lusztig and N. Spaltenstein, On the generalized Springer correspondence for classical groups, in Algebraic groups and related topics, Adv. Stud. Pure Math. 6, Kinokuniya and North-Holland, Tokyo and Amsterdam, 289–316. [18] K. Mizuno, The conjugate classes of unipotent elements of the Chevalley groups E7 and E8 , Tokyo J. Math. 3 (1980), 391–459. [19] Z. Ohmori, On the Schur indices of certain irreducible characters of reductive groups over finite fields, Osaka J. Math 25 (1988), 149–159.

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[20] Z. Ohmori, The Schur indices of the cuspidal unipotent characters of the finite unitary groups, Proc. Japan Acad. Ser. A Math. Sci. 72 (1996), 111–113. [21] A. Premet, Nilpotent orbits in good characteristic and the Kempf–Rousseau theory, J. Algebra 260 (2003), 338–366. [22] T. Shoji, Green functions of reductive groups over a finite field, Proc.Symp. Pure Math. 47, Amer. Math. Soc., Providence, RI 1987, 289–302. [23] T. Shoji, Character sheaves and almost characters of reductive groups II, Adv. Math. 111 (1995), 314–354. Meinolf Geck, Institut Girard Desargues, Université Lyon 1, 21 av Claude Bernard, 69622 Villeurbanne cedex, France E-mail: [email protected]

An extension of Thompson’s Replacement Theorem by algebraic group methods George Glauberman

1. Introduction Thompson’s Replacement Theorem was proved in 1969. Starting with two abelian subgroups A, B of a finite p-group, one constructs another abelian subgroup A∗ , “better” than A, that can replace A for some applications. While the proof is relatively short, the existence of A∗ seems somewhat mysterious. The purpose of this expository article is to interpret the theorem by methods from algebraic group theory and to give some extensions and open problems suggested by these methods. It is a pleasure to have this opportunity to thank John Thompson for forty years of help, encouragement, and inspiration. Throughout this paper, p denotes an arbitrary prime and S an arbitrary finite pgroup. We let (S) be the set of abelian subgroups A of S of maximal order, i.e., such that |A| ≥ |B| for every abelian subgroup B of S. We let e (S) be the set of elementary abelian subgroups of maximal order (and thus of maximal rank) in S. Finally, for each subgroup T of S, let |T ∩ N |, f (T ) = NS

i.e., the product of the orders |T ∩ N | as N ranges over all the normal subgroups of S. (This function is convenient for stating results, although not for the actual proofs.) The following is a combination of part of Thompson’s Replacement Theorem, its proof, and its corollaries ([HB] III, pp. 19–22): Theorem 1 (Thompson). Suppose A is in (S) or e (S), B is an abelian subgroup of S, and A normalizes B, but B does not normalize A. Then there exists an abelian subgroup A∗ of AB such that (a) |A∗ | ≥ |A|; (b) f (A∗ ) > f (A); (c) A∗ normalizes A; and (d) if p is odd and A is elementary abelian, then A∗ is elementary abelian.

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Moreover, there exists an element x in B such that [x, A] normalizes A, but x does not normalize A. For such an element x, let A1 = [x, A]. Then we may take A∗ = A1 CA (A1 ). For p = 2, there is a counterexample to part (d) of the theorem in which S is a dihedral group of order 16, A is a noncyclic subgroup of order 4, and B is the cyclic subgroup of order 8 in S. Here A∗ must be the cyclic subgroup of order 4 in S. Corollary to Theorem 1. Assume A is chosen in (S) with f (A) maximal. Then whenever A normalizes an abelian subgroup B of S, then B normalizes A. One might suspect in the corollary that A is normal in S. However, J. Alperin showed in 1965 ([H], p.349; [G1], pp. 324–325) that for each choice of p there is an example in which no element of (S) or of e (S) is normal in S.

2. Extensions of the Replacement Theorem About ten years ago, I attempted to extend Theorem 1 by allowing B to have nilpotence class two. A counterexample for p = 2 shows that one must assume p is odd. I found a natural candidate for A∗ , but also an example where the candidate is not abelian. However, detailed study of this putative counterexample revealed an awkward gap, namely, an unnatural candidate for A∗ that worked. This led to the following result: Theorem 2 (1997, [G1], Theorems 1 and 2). Assume p is odd. (1) Suppose A is in (S), B is a subgroup of S of nilpotence class at most two; and A normalizes B, but B does not normalize A. Take x as in Theorem 1. Let i Aˆ = Ax | all integers i ,

A1 = [[A, x], x],

ˆ A2 = a[a , x]2 | a, a ∈ A and x centralizes a[a , x] (mod Z(A)) and A∗ = A1 A2 . Then A and A∗ satisfy conditions (a)–(c) of Theorem 1. (2) Suppose p ≥ 5. Take A in (S) with f (A) maximal. Then A AS . Remark. Here AS denotes the normal closure of A in S, i.e., the subgroup of S generated by all the conjugates Ax as x ranges over S. The group A∗ appeared “unnatural” because of the appearance of the squared terms [a , x]2 . After Theorem 2 was proved, Alperin mentioned to me an analogous result for complex Lie algebras (discussed in Section 4 below) that had been proved by using algebraic group theory. This led to an extension of Theorem 2 for elementary abelian subgroups, and later for all abelian subgroups:

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107

Theorem 3 (1998, 2004). (1) ([G2], Theorem D) Suppose A is an abelian subgroup of S; B is a subgroup of S of nilpotence class at most p − 1; and A normalizes B, but B does not normalize A. Then there exists an abelian subgroup A∗ of AB that satisfies conditions (a)–(d) of Theorem 1. (2) ([AG], Theorem 5.5) Suppose p ≥ 5. Take A in e (S) with f (A) maximal. Then A AS . Remark. Part (1) was first proved in [AG] for the case in which A is elementary abelian. The case in Part (1) in which p is 2 or 3 follows from Theorems 1 and 2. Theorems 2 and 3 suggest the following open problems: Problem 1. Suppose p is 2 or 3. Does there exist A in (S) such that A is normal in AS ? Problem 2. Suppose p = 3. Does there exist A in e (S) such that A is normal in AS ? We do not allow p = 2 in Problem 2 because there is no A when S is a dihedral group of order 2n for n ≥ 5.

3. Algebraic group methods in proofs To illustrate the use of algebraic group methods in the proofs of Theorems 2 and 3, we consider the special case in which B is abelian. This gives us a new proof and a new interpretation of Thompson’s Replacement Theorem. Unfortunately, we have to assume that p is odd. Assume the hypothesis of the theorem, and suppose p is odd. It is easy to find an i x as in the theorem. Take Aˆ = Ax | all i , as in Theorem 2. Then [[A, x], x] ≤ [[A, B], B] ≤ [B, B] = 1

and

[[A, x], [A, x]] = 1.

(3.1)

and Aˆ has nilpotence class at most 2 (since A, It is now easy to show that Aˆ = x ˆ A A ). Since p is odd, we may apply an idea of R. Baer and M. Lazard ([HB], ˆ let y+z = y 1/2 zy 1/2 and [y, z] = y −1 z−1 yz pp. 347–349; [Kh], p. 126). For y, z in A, (as usual). Then Aˆ is a Lie ring with respect to the operations + and [ , ]. AAx

Moreover, let p e be the exponent of A and let R be the ring Z/pe Z. Then Aˆ is a Lie algebra over R. Now let t be an indeterminate over R and form the polynomial ring R[t]. Then the tensor product L = Aˆ ⊗R R[t] becomes a Lie algebra over R[t]. ˆ it induces an automorphism α on Aˆ by conjugation. Since x normalizes the group A, Then α respects the operations + and [ , ], and hence is an automorphism of Aˆ as a Lie algebra. By letting α act trivially on the polynomial ring R[t], we extend α to an automorphism of L.

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Let δ = α − 1. By (3.1), δ2 = 0

and

ˆ Aδ] ˆ = [Lδ, Lδ]. 0 = [Aδ,

(3.2)

For every integer r, α r = (1 + δ)r = 1 + rδ. Now we “raise α to the power t” in some sense by defining τ = 1 + tδ. By (3.2) and some calculation, we may show that τ is an automorphism of the Lie algebra L over R[t]. As A is an abelian subgroup of S, we have [A, A] = 0. Since τ is an automorphism of L, [τ (A), τ (A)] = 0.

(3.3)

Let L0 be τ (A)R[t], the R[t]-submodule of L spanned by τ (A). By (3.3), [L0 , L0 ] = 0. For any nonzero elements f and g of L0 , consider the leading terms, say, f = an t n + · · ·

and g = bm t m + · · · .

Since 0 = [f, g], we obtain [an , bm ] = 0. Finally, we let A∗ consist of the zero element, together with the leading coefficients of all of the nonzero elements of L0 . Then a few calculations show that this group is the same as the group A∗ constructed in the Replacement Theorem. This proof is generalized to prove Theorems 2 and 3. We take x and Aˆ as in Theorem 2, so that i i Aˆ = Ax | all i and Ax Aˆ for all i. Since B has class at most p − 1, we can eventually show that Aˆ has class at most p − 1. Therefore, Aˆ becomes a Lie algebra by a beautiful result of Lazard ([Kh], p. 124). We take R, t, L, and α as before. Here (α − 1)p = 0. Let δ = log α =

p−1 i=1

(α − 1)i (−1)i+1 . i

Then p

δ =0

and α = exp δ =

p−1 δi i=0

Let τ = α t = exp tδ. Now define L0 and A∗ from τ as above.

i!

.

An extension of Thompson’s Replacement Theorem by algebraic group methods

109

4. Other open problems Theorem 3 was originally inspired by a result on nilpotent complex Lie algebras by B. Kostant that uses methods of algebraic group theory ([AG], p. 534). Kostant’s method also easily gives the following result: Suppose G is a unipotent algebraic group over an algebraically closed field K of characteristic 0. Let (G) be the set of all closed abelian subgroups of G of maximal dimension. Then some element of (G) is normal in G. I do not know whether the analogous proposition for K of prime characteristic is valid. The examples of Alperin mentioned above show that the analogous proposition for finite p-groups is false. Thus, we cannot generally regard finite p-groups as analogues of algebraic groups of characteristic 0. Rather, it appears that the finite analogues of unipotent algebraic groups of characteristic 0 form a proper subclass of the finite p-groups. What is this subclass? We may regard the question as an imprecise open problem, since it is a matter of taste as well as proofs. Lazard’s work strongly suggests that this subclass includes all finite p-groups of class at most p − 1, since there is a Lie algebra associated to each of these groups and to each algebraic group. Thompson’s Replacement Theorem and many other results concerning J (S) or Je (S) are proved by showing that certain large subgroups of S lie in a given normal subgroup of S. They suggest the following imprecise conjecture: If p is odd, then S has a “large” characteristic subgroup N that is analogous to a unipotent algebraic group of characteristic 0. As a step in this direction, we ask: Problem 3. (a) Which p-groups S can be written as, or uniquely associated with, Lie algebras? (b) Can S be so written or associated if there is an algebraic group G defined over Fp such that S is the set of all Fp -rational points of G? (c) Which p-groups arise as in (b)? Thompson’s Replacement Theorem is actually broader than Theorem 1; it remains valid if [B, A], rather than B, is abelian. Problem 4. Extend the proof in Section 3 to cover the cases in which [B, A] is abelian or p = 2. (Here one should keep in mind the counterexample to part (d) of Theorem 1 for p = 2.)

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References [AG] J. Alperin and G. Glauberman, Limits of abelian subgroups in finite p-groups, J. Algebra 203 (1998), 533–566. [G1]

G. Glauberman, Large abelian subgroups of finite p-groups, J. Algebra 196 (1997), 301–338.

[G2]

G. Glauberman, Large subgroups of small class in finite p-groups, J. Algebra 272 (2004), 128–153.

[H]

B. Huppert, Endliche Gruppen I, Springer-Verlag, Berlin 1967.

[HB] B. Huppert and N. Blackburn, Finite Groups II, III, Springer-Verlag, Berlin 1982. [Kh]

E. I. Khukhro, p-Automorphisms of Finite p-Groups, London Math. Soc. Lecture Note Ser. 246, Cambridge University Press, Cambridge 1998.

[T]

J. G. Thompson, A replacement theorem for p-groups and a conjecture, J. Algebra 13 (1969), 149–151.

George Glauberman, Department of Mathematics, University of Chicago, 5734 S. University Ave., Chicago, IL 60637, U.S.A. E-mail: [email protected]

Simple connectedness of the geometry of nondegenerate subspaces of a symplectic space over arbitrary fields Ralf Gramlich∗

Abstract. K. M. Das [4], [5] has shown that the geometry of nondegenerate subspaces of a symplectic space over a finite field is simply connected. The purpose of the present article is to provide a short, direct and general proof of that result for arbitrary fields. Some standard amalgam-theoretic consequences for the groups Sp2n (F) are given as well.

1. Introduction Let F be a field and let V be a 2n-dimensional vector space over F endowed with a nondegenerate alternating bilinear form. Let (n, F) be the geometry of rank n − 1 whose elements are the nondegenerate proper subspaces of V with natural incidence. Then the following holds. Theorem 1. Let n ≥ 4 and let F be a field. Then the geometry (n, F) is 2-simply connected. The rank of (3, F) is two, whence it cannot be simply connected (the incidence graph of (3, F) is not a tree), so the above result is sharp. The group Sp2n (F) acts flag-transitively on (n, F). Choose a canonical basis e1 , f1 , …, en , fn of V . Then e1 , f1 , e1 , f1 , e2 , f2 , …, e1 , f1 , . . . , en−1 , fn−1 is a chamber (i.e., a maximal flag) of (n, F). The maximal parabolics Pi , 1 ≤ i ≤ n − 1 with respect to this chamber are of the form Sp2i (F) × Sp2n−2i (F). A consequence of Theorem 1 is the following. Theorem 2. Let n ≥ 4 and let F be a field. Then Sp2n (F) is the universal completion of the amalgam of at least three maximal parabolics Pi . Denote by A2 (n, F) the amalgam of 1- and 2-parabolics of Sp2n (F) with respect to the chamber e1 , f1 , e1 , f1 , e2 , f2 , …, e1 , f1 , . . . , en−1 , fn−1 . That is, A2 (n, F) is the union of all intersections i∈J Pi where J is a subset of {1, . . . , n − 1} of co-cardinality at most two. Then we have the following. ∗ The author would like to express his gratitude to the organizers for the funding of his trip to the USA and his stay at the conference.

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Theorem 3. Let n ≥ 4 and let F be a field. Then Sp2n (F) is the universal completion of the amalgam A2 (n, F). For finite fields Theorem 1 has been proved independently first by K. M. Das ([4] and [5], for all finite fields) and later by C. Hoffman, S. Shpectorov and the author ([7], for all finite fields of size at least four). While Das achieved this result as a consequence of his study of Quillen complexes of finite symplectic groups based on work of Aschbacher [1] and Quillen [14], the proof by Hoffman, Shpectorov and the author makes use of the simple connectedness of another finite geometry that is closely related to (n, F). The purpose of this paper is to present a short, direct and general proof of the simple connectedness of (n, F) for an arbitrary field F. The investigation of geometries like (n, F) is motivated by the desire for a systematic approach to Phan-type theorems similar to the results of [12] and [13]. For an outline of the idea, using chamber systems and buildings, as well as precise definitions of flips and flipflop geometries refer to [3]. (Note that those concepts are not relevant for the understanding of the present article.) Example 1b and Example 2b of [3] give a description of similar geometries. However one big difference remains: while the examples of [3] can be described using flips and opposite chamber systems, one cannot obtain the geometry (n, F) in this way; for, a flip induced by a symplectic form does not admit a chamber that is mapped onto its opposite, so the resulting flipflop geometry is empty. The most remarkable property of (n, F) is that the simple connectedness holds and can be shown completely independent of the field. Neither the geometry from Example 1b nor the geometry from Example 2b of [3] have this property; they are not simply connected for low rank over the field of two elements. Theorem 2 and Theorem 3 dealing with the presentation of Sp2n (F) by certain amalgams nowadays are standard corollaries of Theorem 1. A classification of those amalgams is not included in the present article. For a closely related characterization of Sp2n (F) refer to Theorem 4.4.25 of [6] or to the results of [8].

2. Connectedness properties of Let F be a field and let V be a 2n-dimensional vector space over F endowed with a nondegenerate alternating bilinear form. Perpendicularity with respect to that form is denoted by ⊥. Let = (n, F) be the pregeometry of rank n − 1 whose elements are the nondegenerate proper subspaces of V with symmetrized containment as incidence. The nondegenerate subspaces of V of dimension two are called points, those of dimension four lines and so on. Evidently, is a geometry (i.e., maximal flags are chambers, which means they have size n − 1). Moreover, by induction on n using Witt’s theorem, the geometry is flag-transitive with the group Sp2n (F) as a flag-transitive group of automorphisms of ; see Lemma 7.3 of [7] for a proof. Lemma 2.1. Let n ≥ 2 and let l, m be nondegenerate two-dimensional subspaces of V with l ∩ m = {0} and dim(l ⊥ ∩ m) ≥ 1. Then l, m is nondegenerate.

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Proof. The statement is obvious, if m ⊆ l ⊥ . If m ⊆ l ⊥ , then m ∩ l ⊥ is a onedimensional subspace of V , say p. Choose a canonical basis a, b of l and choose q ∈ m, such that p, q form a canonical basis of m. Then the Gram matrix of the symplectic form restricted to l, m with respect to the basis a, b, p, q has the shape 0 1 0 ∗ −1 0 0 ∗ 0 0 0 1 , ∗ ∗ −1 0 which evidently has full rank, whence l, m is nondegenerate. Proposition 2.2. Let n ≥ 3. Then the collinearity graph of has diameter two. In particular, is connected. Proof. Let l, m be nondegenerate two-dimensional subspaces of V . Let p be an arbitrary one-dimensional subspace of l, m⊥ and let q be an arbitrary one-dimensional subspace of V that intersects p ⊥ trivially. Then p, q is a point of (as q ⊆ p ⊥ ) which is collinear to l and m (by Lemma 2.1). Corollary 2.3. Let n ≥ 3. Then the geometry is residually connected. Recall the definition of the fundamental group of a connected geometry .1 A path of length k in the geometry is a sequence of elements x0 , . . . , xk of such that xi and xi+1 are incident, 0 ≤ i ≤ k − 1. We do not allow repetitions; hence xi = xi+1 . A cycle based at an element x is a path in which x0 = xk = x. Two paths are homotopically equivalent if one can be obtained from the other via the following operations (called elementary homotopies): inserting or deleting a return (i.e., a cycle of length 2) or a triangle (i.e., a cycle of length 3). The equivalence classes of cycles based at an element x form a group under the operation induced by concatenation of cycles. This group is called the fundamental group of and denoted by π1 (, x). A geometry is called simply connected if its fundamental group is trivial.2 Notice that in order to prove that is simply connected it is enough to prove that any cycle based at x is homotopically equivalent to the cycle of length 0. A cycle with this property is called null-homotopic, or homotopically trivial. Let us go back to the geometry (n, F). We pick the base element x to be a point of . Lemma 2.4. Let n ≥ 4. Every cycle of based at x is homotopically equivalent to a cycle passing only through points and lines. 1 Note that there are two concurrent notions of a fundamental group. One notion of a fundamental group

considers the (incidence graph of the) geometry as a one-dimensional simplicial complex (triangles are not null-homotopic), the other notion considers the (incidence graph of the) geometry as a two-dimensional simplicial complex (triangles are null-homotopic). In the present paper we use the latter notion. 2 Standard topology shows that a geometry is simply connected if and only if it does not admit any proper simplicial cover. For combinatorial proofs see [9], [10] or [15]; alternatively, see [16].

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Proof. This follows immediately from the residual connectedness of . See Lemma 5.1 of [7] for a proof; alternatively, see [2]. We can therefore restrict our attention to the point-line incidence graph of . However, as is not a partially linear geometry (there exist nondegenerate twodimensional subspaces of V that are contained in more than one nondegenerate fourdimensional subspace of V , so a pair of collinear points does not necessarily admit a unique joining line), it is not immediately clear that we can restrict ourselves to the collinearity graph of . The following lemma takes care of that problem. Lemma 2.5. Let n ≥ 4. Any digon p, l, q, m of consisting of points p = q and lines l = m is homotopically trivial. Proof. The dimension of l, m equals five, because l and m intersect in a threedimensional space. The space x = l ⊥ ∩ l, m is the radical of l, m. Choosing any vector y ∈ l ⊥ \x ⊥ , we obtain a nondegenerate six-dimensional space l, m, y that contains the digon p, l, q, m. Therefore the digon is homotopically trivial. The above lemma allows us to restrict our attention to the collinearity graph of , since the homotopy type of a path γ passing through points and lines is independent of the particular choice of the joining lines of the pairs of collinear points occurring in γ . In order to determine the fundamental group of we have to distinguish between good and bad triangles, i.e., triangles a, b, c whose points span a nondegenerate subspace

a, b, c of V and triangles a, b, c whose points do not span a nondegenerate subspace

a, b, c of V . Obviously, a good triangle is null-homotopic as it spans a line or a plane of the geometry . For bad triangles we have the following lemma. Lemma 2.6. Let n ≥ 4. Any triangle in the collinearity graph of can be decomposed into good triangles. Proof. Let a, b, c be the nondegenerate two-dimensional subspaces of V that constitute the three points of some triangle. Choose some line l of containing a and b. The space l ⊥ is nondegenerate and has dimension at least four. Therefore, to any onedimensional subspace p ⊆ l, c⊥ , we can find a one-dimensional subspace q of l ⊥ such that p, q =: d ⊥ l ⊇ a, b is nondegenerate and intersects c trivially. By Lemma 2.1, the space c, d is nondegenerate, and we have decomposed the triangle a, b, c into triangles which admit two points that are perpendicular. So now assume we have a triangle a, b, c with a ⊥ c. If a ∩ b = ∅ and b ∩ c = ∅, then b ⊂ a, c, and there is nothing to prove. Hence, by symmetry, we can assume that b ∩ c = ∅. As above choose some line l of containing a and b. The space l ⊥ is nondegenerate and has dimension at least four. To any one-dimensional subspace p ⊆ l, c⊥ , we can find a one-dimensional subspace q of l ⊥ such that p, q =: d ⊥ l ⊇ a, b is nondegenerate and intersects b, c trivially. This is possible as the dimension of b, c ∩ l ⊥ is at most two (since ∅ = b ∩ b⊥ ⊃ b ∩ l ⊥ ). Notice that the triangles a, b, d and a, c, d are good (indeed, d ⊥ l ⊃ a, b and a ⊥ c, d). Moreover,

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for hyperbolic bases xb , yb of b and xc , yc of c and p, yd of d, we obtain the following Gram matrix for the alternating form on b, c, d with respect to xb , yb , xc , yc , p, yd : 0 1 ∗ ∗ 0 0 −1 0 ∗ ∗ 0 0 ∗ ∗ 0 1 0 ∗ ∗ ∗ −1 0 0 ∗ . 0 0 0 0 0 1 0 0 ∗ ∗ −1 0 Evidently, this matrix has full rank if and only if its submatrix consisting of the first four rows and columns has full rank. But that submatrix is the Gram matrix of the form restricted to the nondegenerate space b, c. Therefore we have decomposed the original triangle into good triangles. Lemma 2.7. Let n ≥ 4. Any quadrangle in the collinearity graph of can be decomposed into triangles. Proof. Let a, b, c, d be the nondegenerate two-dimensional subspaces of V that constitute the four points of some quadrangle in the collinearity graph of . Consider W := a, b⊥ and U := c, d⊥ . If W ⊥ U , then c, d⊥ = U ⊆ W ⊥ = a, b. Since the dimension of c, d⊥ is at least four and the dimension of a, b at most four, equality holds. But this implies that a ⊥ c and b ⊥ d, yielding an immediate decomposition of the quadrangle into triangles. If W ⊥ U , we can find one-dimensional subspaces p ⊆ W , q ⊆ U with p ⊥ q. The resulting nondegenerate two-dimensional space e := p, q is collinear to a, b, c, d by Lemma 2.1. (Notice that e cannot intersect a, b, c, or d nontrivially since a, b ⊥ W ⊆ p and c, d ⊥ U ⊆ q, but p ⊥ q.) Lemma 2.8. Let n ≥ 4. Any pentagon in the collinearity graph of can be decomposed into triangles and quadrangles. Proof. Let a, b, c, d, e be the nondegenerate two-dimensional subspaces of V that constitute the five points of some pentagon in the collinearity graph of . Choose a line l of passing through c and d. Then l ⊥ is nondegenerate and has dimension at least four. Let p ⊆ a, l⊥ be an arbitrary one-dimensional space. There exists a onedimensional subspace of l ⊥ that together with p spans a nondegenerate space f that intersects a trivially. We have f ⊥ l ⊇ c, d. Moreover, f also spans a nondegenerate space with a, by Lemma 2.1, decomposing the pentagon. Any cycle of greater length decomposes by the bound on the diameter of the collinearity graph of from Lemma 2.2. Therefore is simply connected. An induction argument as in [7] implies that is 2-simply connected, and Theorem 1 follows. The proof of Theorem 2 is identical to the proof of Theorem 4 of [7]. Theorem 3 follows by Tits’ lemma (Corollaire 1 of [17]) and a standard induction argument as in the proof of Theorem 1 of [7].

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References [1]

M. Aschbacher, Simple connectivity of p-group complexes, Israel J. Math. 82 (1993), 1–43.

[2]

C. D. Bennett and S. Shpectorov, A new proof of a theorem of Phan, J. Group Theory 7 (2004), 287–310.

[3]

C. D. Bennett, R. Gramlich, C. Hoffman and S. Shpectorov, Curtis–Phan–Tits theory, in Groups, Combinatorics and Geometries (Durham, 2001), ed. by A. A. Ivanov, M. W. and J. Saxl, World Sci. Publishing, River Edge, NJ, 2003, 13–29.

[4]

K. M. Das, Some results about the Quillen complex of Sp2n (q), J. Algebra 209 (1998), 427–445.

[5]

K. M. Das, The Quillen complex of groups of symplectic type: the characteristic 2 case, J. Algebra 223 (2000), 556–561.

[6]

R. Gramlich, On graphs, geometries, and groups of Lie type, PhD thesis, TU Eindhoven 2002.

[7]

R. Gramlich, C. Hoffman and S. Shpectorov, A Phan-type theorem for Sp(2n, q), J. Algebra 264 (2003), 358–384.

[8]

R. Gramlich, On the hyperbolic symplectic geometry, J. Combin. Theory Ser. A 105 (2004), 97–110.

[9]

R. Gramlich, G. W. Hofmann, K.-H. Neeb, Semi-edges, reflections, and Coxeter groups, preprint.

[10] R. Gramlich, Phan theory, Habilitationsschrift, TU Darmstadt, 2004. [11] B. Mühlherr, On the simple connectedness of a chamber system associated to a twin building, preprint. [12] K. W. Phan, On groups generated by three-dimensional special unitary groups, I. J.Austral. Math. Soc. Ser. A 23 (1977), 67–77. [13] K. W. Phan, On groups generated by three-dimensional special unitary groups. II, J. Austral. Math. Soc. Ser. A 23 (1977), 129–146. [14] D. Quillen, Homotopy properties of the poset of nontrivial p-subgroups of a group, Adv. Math. 28 (1978), 101–128. [15] H. Seifert, W. Threlfall, Lehrbuch der Topologie, Chelsea Publishing Company, NewYork 1934. [16] D. B. Surowski, Covers of simplicial complexes and applications to geometry, Geom. Dedicata 16 (1984), 35–62. [17] J. Tits, Ensembles Ordonnés, immeubles et sommes amalgamées, Bull. Soc. Math. Belg. Sér. A 38 (1986), 367–387. Ralf Gramlich, TU Darmstadt, FB Mathematik / AG 5, Schloßgartenstraße 7, 64289 Darmstadt, Germany E-mail: [email protected]

Classification of 2F -modules, II Robert M. Guralnick and Gunter Malle∗

Abstract. We classify the irreducible 2F -modules for the quasi-simple groups of Lie type with the modules in the natural characteristic (leaving three open cases). In concert with the results of our first paper, this gives the complete classification of such modules that arise in the context of the Aschbacher-Smith classification of quasithin groups. As a direct application we also obtain the list of irreducible F -modules of finite quasi-simple groups. 2000 Mathematics Subject Classification: 20G05, 20D06

1. Introduction and statement of results In this paper we study the following situation: Let G be a finite group such that F ∗ (G) is quasi-simple. For an absolutely irreducible faithful KG-modules V over a field K of characteristic and a subgroup A ≤ G of G we let f (A) := f V (A) := |A|2 · |CV (A)|. We want to classify those modules V for which G has a nontrivial elementary abelian -subgroup A satisfying |V | ≤ f (A).

(1.1)

Modules satisfying this and similar conditions have been studied for quite a while – going back to Thompson, Glauberman, Mason and others (see the remarks and references in [6]). Usually, there are conditions imposed in addition to (1.1) involving an assumption of quadratic or cubic action of the subgroup A on V . These modules play a crucial role in a new approach to the revision of the classification of finite simple groups. In particular, for = 2, the results of this paper and [6] are used by Aschbacher and Smith in the proof of the classification of quasithin simple groups [2]. The Aschbacher–Smith result is the final paper in the original proof of the classification of finite simple groups and is also part of the second generation proof. These modules and variations also come up in Meierfrankenfeld’s program to study finite simple groups. ∗ The first author was partially supported by NSF grant DMS 0140578.

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In [6], we essentially found all examples of such modules with F ∗ (G) quasi-simple aside from the case that F ∗ (G) is a group of Lie type and V is a module in the natural characteristic for G. In this paper, we consider this case and classify all but three cases. In particular, we classify all such modules that arise in [2]. We expect to complete the classification of the remaining situations in a future article. We feel that it is important to have the results needed for the quasithin classification in the literature in a timely manner. Theorem A. The absolutely irreducible 2F -modules for quasi-simple groups of Lie type in their defining characteristic are among those given in Table 1. We also shows that all modules in Table 1 except possibly those in the last three rows actually satisfy the 2F -condition. Here, for the natural module Yn of Spinn (a ), has to be odd when n is odd. Table 1. 2F -modules in defining characteristic

G SLn (a ) SLn (a ) SLn (a ) SLn (2a ) SL6 (a ) SUn (a ) Sp2n (a ) Sp2n (a )

d n n

2 n+1 2

n2 20 n 2n 2n 2 − 1 − δ,n

V Yn 2 ∧ (Yn )

2 (Yn ) (a) Yn ⊗ Y n ∧3 (Y6 ) Yn Y2n 2 ˜ ∧ (Y2n )

f a a a a a 2a a a

(a)

Sp4 (2a ) (±) Spinn (a ) Spin2n+1 (a ) a Spin+ 2n ( ) a Spin− 2n ( ) 2B (22a+1 ) 2 G2 (a ) F4 (2a ) E6 (a )sc

16 n 2n 2n−1 2n−1 4 7 − δ,2 26 27

Y4 ⊗ Y4 Yn Zn Zn−1 Zn−1 M(λ1 ), M(λ2 ) M(λ2 ) M(λ1 ), M(λ4 ) M(λ1 ), M(λ6 )

a a a a 2a 2a + 1 a a a

F4 (a ) 2E (a ) 6 sc E7 (a )sc

26 − δ,3 27 56

M(λ4 ) M(λ1 ), M(λ6 ) M(λ7 )

a 2a a

conditions n≥4 ≥3

n = 2, 3 or = 2, n = 4 n≥6 n = 3, 4, 5 n = 4, 5, 6 n = 4, 5 a>1

>2

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In this table, the modules are labeled by their highest weights, and the fundamental weights λi are labeled as in Table 4, §3.1. Furthermore, Yn denotes the natural module (of dimension n), Zn is the 2n -dimensional spin module of the spin group Spin2n+1 (q) with highest weight λ1 (this includes the case of symplectic groups in even characteristic), and Zn−1 denotes either one of the two 2n−1 -dimensional half-spin modules of (±) spin groups Spin2n (q) with highest weight λ1 or λ2 . Next, ∧2 , ∧3 , and 2 denote the alternating square, the alternating cube, and the symmetric square, respectively, and M (a) is the a -Frobenius twist of the module M. Also, recall that for a given type X of finite groups of Lie type there are several isogenous variants which share the same non-abelian composition factor and the same order but have slightly different structures. In particular, the adjoint group Xad is almost simple, whereas the simplyconnected type group Xsc is quasi-simple (for example, 2E6 (q)sc is quasi-simple with center of order (3, q + 1)). The three open cases are E7 (q)sc , 2E6 (q)sc and F4 (q) on their smallest modules (the last only for odd characteristic). Note that in the first case f (A) = q 55 = |V |/q for A an elementary abelian subgroup of order q 27 . There are two other open cases remaining from the first part — the two largest Conway groups on their smallest modules in characteristic 2. Again, in those two cases there are elementary abelian subgroups A with f (A) = |V |/2. Our approach is to classify the modules satisfying (1.1) and not impose other conditions (of course, one obtains corollaries to our main results by then imposing extra conditions). Given the results in [6], it thus remains to consider groups of Lie type G in the same characteristic . By a basic lemma, we know log (|G|) grows quadratically with the rank. The irreducible KG-modules of this dimension are known. We quote a recent paper of Lübeck [11] for this (which depends upon a result of Premet [15]). This follows from the beautiful theory of representations of Chevalley groups in the natural characteristic (see for example [5, Sec. 2.8] for background on this). Liebeck [10] has obtained some weaker bounds than those given in [11], but these results are not sufficient for our purposes. We should point out that the background material that we use is a bit different than that of Aschbacher–Smith [2] or Gorenstein–Lyons–Solomon [5]. In the first part [6], we assumed the results from the modular Atlas [8] about lower bounds for the sporadic groups, of James about representations of alternating groups and of Landazuri–Seitz–Zalesskii for cross characteristic representations. As we remarked above, in the equicharacteristic case, we are using the results of Premet and Lübeck. One could produce independent proofs of these results for the cases of interest in [2] to make that result more self-contained. For the exceptional groups we also use results of Lawther [9] on Jordan blocks of exceptional groups on certain small modules and some information on the order relation on unipotent classes, see Spaltenstein [16]. This does not concern any of the modules considered in [2]. In the present paper we treat the absolutely irreducible representations of finite quasi-simple groups of Lie type G in their defining characteristic . In fact, we will

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not consider here the non-simple groups Sp4 (2) and G2 (2) in characteristic 2, and 2G (3) in characteristic 3, since these were considered as the automorphism groups 2 S6 , U3 (3).2 and L2 (8).3 of simple groups in non-defining characteristic in the first part [6, Props. 3.1, 4.3, 4.1]. On the other hand, we do treat here the following pairs (G, ) left open in the first part: (A5 ∼ = SL2 (4), 2),

(2.A5 ∼ = SL2 (5), 5),

(A8 ∼ = SL4 (2), 2),

(L2 (7) ∼ = SL3 (2), 2),

(2.U4 (2) ∼ = Sp4 (3), 3),

(S4 (3) ∼ = SU4 (2), 2),

(2.A6 ∼ = SL2 (9), 3), (2.L3 (2) ∼ = SL2 (7), 7) (G2 (2) ∼ = SU3 (3), 3),

as well as the ‘sporadic group’ (2F4 (2) , 2). Note that exceptional Schur multipliers of groups of Lie type are always of order a power of the defining characteristic, so we need not worry about them and just have to consider quotients of simply connected groups of Lie type by central subgroups. By a slight abuse of notation, in the sequel we shall weaken the faithfulness-requirement a little and say that V is a 2F -module for G if it is so for some quotient by a central subgroup contained in G . The explicit description of representations in the defining characteristic via their highest weight makes it easy to work out the kernel of such a representation in any individual case. Since F -modules are, by definition, also 2F -modules we also obtain a classification of F -modules of finite quasi-simple groups as a rather easy application of our results on 2F -modules: Theorem B. The absolutely irreducible F -modules of finite quasi-simple groups are given in Table 2. Table 2. F -modules for quasi-simple groups

G SLn (a ) SLn (a ) SUn (a ) Sp2n (a ) (±) Spinn (a ) Spin7 (a ) a Spin+ 2n ( ) G2 (2a ) A2n 3.A6 A7

d n n 2

n 2n n 8 2n−1 6 2n − 2 3 4

V Yn 2 ∧ (Yn ) Yn Y2n Yn Z3 Zn−1 M(λ2 )

2 2 2

f a a 2a a a a a a 1 2 1

conditions n≥4 n≥4 n≥6 n = 4, 5

Here, for the natural module Yn of Spinn (a ), has to be odd when n is odd.

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This result is already in the literature. The earlier papers used very heavily the fact that by the Thompson replacement lemma, one can assume that the subgroup A acts quadratically. In particular, see Aschbacher [1], Meierfrankenfeld-Stroth [12], [13] for alternating, sporadic and odd characteristic Lie-type groups in characteristic 2, and Meixner [14] for odd characteristic Lie-type groups in odd characteristic. Our proof is independent of these references. We now give an outline of the contents of the paper. In order to study modules in defining characteristic and determine precisely which of them satisfy (1.1), very frequently we need to deal with tensor products, symmetric squares and exterior powers of smaller representations. Upper bounds for centralizers on such modules are derived in Section 2.2. In Section 3.1 we recall some basic results about representations in defining characteristic. In 3.3 we determine the maximal size of a subgroup of a group of Lie type consisting only of long root elements, and in 3.4 we collect information on the -ranks of classical groups. In Sections 4–8 we treat the families of linear and unitary groups, of symplectic groups, of orthogonal groups, and of exceptional groups of Lie type, with Section 5 containing some results on the particularly difficult case of spin representations. Note that, in contrast to the situation in the first part [6], we don’t reach a complete classification of 2F -modules for non-perfect almost quasi-simple groups. Still, we prove the following partial result: if F ∗ (G) is quasi-simple but not an orthogonal group in even dimension, then all composition factors of the restriction to F ∗ (G) of a 2F -module for G are again 2F -modules. We hope to come back to the case of the orthogonal groups and more generally to the complete determination of 2F -modules for almost quasi-simple groups in a future paper.

2. Auxiliary results 2.1. 2F -modules. Let K be a finite field of characteristic , G a finite group and V a finite-dimensional KG-module. For x ∈ G we call the codimension of CV (x) the rank of x (on V ) (i.e., the rank of x − 1). For a subgroup A of G write f (A) := f V (A) := |A|2 · |CV (A)|. V is called a 2F -module for G with offender A = 1 if dim(V ) > 1 and |V | ≤ f (A). Similarly we define f1 (A) := f1V (A) := |A| · |CV (A)| and say that V is an F -module for G with offender A = 1 if dim(V ) > 1 and |V | ≤ f1 (A). We also say that V is an F+ -module for G if f1 (A) > |V | for some A ≤ G. The following well-known observation will be useful (see [3], [6, Lemma 2.1]): Lemma 2.1. Let A, B ≤ G with f (B) maximal among all subgroups of G and let C := A, B. Then f (C) ≥ f (A).

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In particular, if V is a 2F -module for G with offender A = 1, and G is the normal closure of A then f (A) ≤ f (G), Moreover, if V is irreducible then f (A) ≤ |G|2 . Also it follows that: Lemma 2.2. Suppose C ≤ G is an elementary abelian -subgroup containing A and B, with f (A) = f (B) maximal among all subgroups of C. Then f (AB) = f (A). This shows that when searching for examples satisfying (1.1) we may sometimes assume that A is maximal elementary abelian. The following can be used to obtain upper bounds on the centralizer of an subgroup: Lemma 2.3. Let B ≤ A ≤ G with dim CV (A) = dim CV (B). Then CV (Ag | g ∈ CG (B)) = CV (B) Proof. Let g ∈ CG (B). Then CV (A) = CV (B) = CV (B g ) = CV (Ag ). We will often use this in the following form. Let B be an elementary abelian -subgroup, g an -element in CG (B) \ B and A := B, g. If Ah | h ∈ CG (B) has smaller centralizer in V than B, then A has smaller centralizer than B. For example, if CG (B) is an extension of B by a simple group H , then all classes of -elements in CG (B) outside B generate H . But H contains -elements and their centralizer can be read off from the Brauer character of V . The following result is very useful for inductive arguments along parabolic subgroups (see [6, Prop. 2.4]): Proposition 2.4. Let V be a 2F -module for G. Let H ≤ G such that H contains an offender. The either there exists an offender A ≤ U := O (H ), or CV (U ) is a 2F -module for H /U . The next result is used for treating automorphism groups (see [6, Prop. 2.7]): Proposition 2.5. Let G be a finite group with a normal subgroup H ≤ G of index . Let W be an absolutely irreducible Ff H -module of dimension dim W ≥ 2 whose induction V = IndG H W to G is again irreducible. If V is a 2F -module for G with offender A, then one of the following holds: (i) W is a 2F -module for H with offender B := A ∩ H ; (ii) = f = 2, dim CW (B) = 1, dim W ≤ r where |A| = 2r ; (iii) |A| = = 2, dim W = 2, f ≤ 2. Note that mutatis mutandis all the results above remain true for F -modules and f1 (A). The following result will be applied to parabolic subgroups. Lemma 2.6. Let P = U L be a semidirect product with U a normal elementary abelian -subgroup of maximal size in P , with complement L. Then U is not an F+ -module for L. Moreover, the following are equivalent:

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Classification of 2F -modules, II

(i) U is not an F -module for L; (ii) U is the unique elementary abelian subgroup of P with order |U |. Proof. If A ≤ L is an elementary abelian -subgroup, then |A||CU (A)| ≤ |U | (because U is the maximal order of an elementary abelian -subgroup), whence U is not an F+ -module. The same argument shows that if U is an F -module, then ACU (A) has order |U | for any offender A (offender in the sense of F -modules). Thus, U an F -module implies that U is not unique in P . Conversely, if W is another elementary abelian subgroup of P of order |U |, then take A = W U/U = 1 and B = A ∩ U ≤ CU (W ). So |A||CU (A)| = |U | and A is an offender. 2.2. Centralizers on tensor products. It will become obvious from the description of the irreducible modules of groups of Lie type in defining characteristic in Section 3.1 that in order to identify possible 2F -modules we need some information on centralizer spaces of unipotent subgroups on tensor products and alternating and symmetric powers. We start by considering indecomposable representations of cyclic groups C of prime order over a field K of characteristic . Here the following result of Hughes and Kemper [7] will be useful to estimate dimensions of centralizer spaces on symmetric and alternating squares: Lemma 2.7. Let G = C be the cyclic group of order and V an indecomposable KG-module of dimension n (so n ≤ ). Then dim(∧k (V ))G =

1 2

ζ kn−2i1 −···−2ik ,

(1 + ζ ) ζ ∈M2

0≤i1 <···

where the first sum runs over the set M2 of all 2th complex roots of unity. This follows for example from [7, Th. 2.5 and Prop. 3.2]. Lemma 2.8. Let G = C be the cyclic group of order and V , W be indecomposable KG-modules of dimension n, m respectively. Then (a) dim(V ⊗ W )G = min{n, m}, when > 2, (b) dim 2 (V )G = n+1 2 (c) dim ∧2 (V )G = n2 . Proof. By assumption both V , W are uniserial. Thus the centralizer of G on V ⊗ W has dimension dim(V ⊗ W )G = dim HomG (1, V ⊗ W ) = dim HomG (V ∗ , W ) = min{n, m}. The third part follows easily by evaluation of the formula in Lemma 2.7. If > 2 then V ⊗ V is the direct sum of ∧2 (V ) and 2 (V ), so the second part is immediate from (a) and (c).

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Let us next consider arbitrary modules for the cyclic group. Lemma 2.9. Let G = C and V , W be KG-modules of dimension n, m respectively, with z := dim V G , y := dim W G . Then we have (a) dim(V ⊗ W )G ≤ min{ny, zm, (n − z)(m − y) + zy}, n−z z+1 for > 2, (b) dim 2 (V )G ≤ min z(n+1) 2 , 2 + 2 n−z+1 + 2z . (c) dim ∧2 (V )G ≤ min zn 2 , 2 Proof. Part (a) has been proved in Lemma 2.8 when V , W are indecomposable. Now assume for example that V = V1 ⊕ V2 as KG-module, so that V ⊗ W = V1 ⊗ W ⊕ V2 ⊗ W. Since both taking fixed points as well as our stipulated bounds are clearly additive in both components, we get the desired result by induction. In part (b), our formula holds again for indecomposable V by Lemma 2.8. If V = V1 ⊕ V2 then

2 (V ) ∼ = 2 (V1 ) ⊕ V1 ⊗ V2 ⊕ 2 (V2 ). Writing ni := dim Vi , zi := dim ViG , this shows that the centralizer space has dimension at most equal to z1 (n1 + 1)/2 + (z1 n2 + z2 n1 )/2 + z2 (n2 + 1)/2 = (z1 + z2 )(n1 + n2 + 1)/2 by induction respectively by part (a), and similarly for the second bound. In case (c), assume that V = V1 ⊕ V2 . Then we have

∧2 (V ) ∼ = ∧2 (V1 ) ⊕ V1 ⊗ V2 ⊕ ∧2 (V2 ). In the notation introduced in the previous part this gives a centralizer space of dimension at most z1 n1 /2 + (z1 n2 + z2 n1 )/2 + z2 n2 /2 = (z1 + z2 )(n1 + n2 )/2. The second bound is proved analogously by using the second type of inequality from part (a). We are now ready to deal with arbitrary elementary abelian -groups G. Assume that G is generated by g1 , . . . , gr say. Let V be a KG-module. Since G is abelian, gi stabilizes CV (gj ) for all j , and this allows to apply induction on the number r of generators. Lemma 2.10. Let G = g1 , . . . , gr be an elementary abelian -group and V , W finite g dimensional KG-modules. Define V0 := V , Vi := Vi−1i (1 ≤ i ≤ r), ni := dim Vi (0 ≤ i ≤ r), and similarly Wi of dimension mi . Then

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(a) dim(V ⊗ W )G r

≤

min{(ni−1 − ni )(mi−1 − mi ), (ni−1 − ni )mi , ni (mi−1 − mi )} + nr mr , i=1

(b)

dim 2 (V )G

r nr + 1 (ni−1 − ni ) min{ni , ni−1 − ni − 1} + ≤ for > 2, 2 2 i=1

(c)

dim ∧2 (V )G

r

≤ i=1

nr (ni−1 − ni + 1) . min{ni , ni−1 − ni } + 2 2

Proof. The proof will be by induction on r. The case r = 1 is Lemma 2.9. First consider part (a). By induction, for G := g1 , . . . , gr−1 we have dim(V ⊗ W )

G

r−1

≤

(ni−1 − ni )(mi−1 − mi ) + nr−1 mr−1 . i=1

On the other hand, clearly V G ⊗ W G ≤ (V ⊗ W )G , and this subspace is stabilized by gr . Thus we obtain an upper bound for dim(V ⊗ W )G as r−1

G

dim(V ⊗ W ) ≤

(ni−1 − ni )(mi−1 − mi ) + dim(V G ⊗ W G )gr

i=1 r−1

≤

(ni−1 − ni )(mi−1 − mi ) + (nr−1 − nr )(mr−1 − mr ) + nr mr i=1

by the case r = 1 in Lemma 2.9, as claimed. In (b) and (c) we may argue in precisely the same way, using that 2 (V G ) ≤ 2

(V )G and ∧2 (V G ) ≤ ∧2 (V )G . The previous result allows to conclude that Lemma 2.9 in fact holds for arbitrary elementary abelian -groups. Corollary 2.11. Let G be an elementary abelian -group, V , W two KG-modules of dimension n, m, with centralizer spaces of dimension z := dim V G , y := dim W G respectively. Then (a) dim(V ⊗ W )G ≤ min{ny, zm, (n − z)(m − y) + zy}, n−z z+1 for > 2, (b) dim 2 (V )G ≤ min z(n+1) 2 , 2 + 2 n−z+1 + 2z . (c) dim ∧2 (V )G ≤ min zn 2 , 2

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Proof. We have dim(V ⊗ W )G = dim HomG (1, V ⊗ W ) = dim HomG (V ∗ , W ). Now any G-homomorphism from V ∗ to W is determined by the image of the head of V ∗ , isomorphic to the socle V G of V . So dim HomG (V ∗ , W ) ≤ dim V G dim W = zm. The second bound in (a) follows by symmetry, the third with Lemma 2.10. In the notation there we have n = n0 , m = m0 , z = nr , y = mr , and r

G

dim(V ⊗ W ) ≤

(ni−1 − ni )(mi−1 − mi ) + nr mr i=1 r

≤

(n0 − nr )(mi−1 − mi ) + nr mr = (n − z)(m − y) + zy. i=1

Parts (b) and (c) follow in precisely the same way from Lemma 2.10 (b) and (c). For convenient reference, let us reformulate this result in terms of the codimension c := n − z of V G (resp. d := m − y of W G ): codim(V ⊗ W )G ≥ max{nd, mc, (n − c)d + (m − d)c}, # "c for > 2, (2.1) codim 2 (V )G ≥ max (n + 1), c(n − c + 1) " 2n # (c − 1), c(n − c − 1) . codim ∧2 (V )G ≥ max 2 It would be possible with a little bit more work to obtain similar results as above for centralizers on 3 (V ) and ∧3 (V ). But in the present work we need such results only for ∧3 of at most 10-dimensional modules, where ad-hoc arguments can be used.

Table 3. Ranks on exterior squares and cubes

dim V J (x)

3

21n−2 22 1n−4 23 1n−6 31n−3

1 − − 2

4

5

6

7

8

4

5

∧2 (V ) 2 2 − 4

3 4 − 6

4 5 6 6 8 10 6 9 12 8 10 12

6

7

8

9 10

∧3 (V ) 1 2 − 2

3 4 − 6

6 10 15 21 8 14 22 32 10 16 25 37 12 20 30 42

28 44 52 56

For the convenience of the reader we have collected the ranks, (i.e., the codimension of the centralizer, as defined in Section 2.1) of some unipotent elements x with simple Jordan types, on exterior square and cube of low dimensional vector spaces V in Table 3. The first column describes the Jordan type J (x) of x on V , the next six

Classification of 2F -modules, II

127

columns give the rank on ∧2 (V ) for 3 ≤ dim V ≤ 8, and the last seven columns give lower bounds for the rank on ∧3 (V ), with 4 ≤ dim V ≤ 10. The following result gives the centralizer dimension for elements interchanging the factors of a tensor product: Lemma 2.12. Let G be a group, V a finite dimensional KG-module and σ ∈ Aut(G). Let V σ := {vσ | v ∈ V } denote a copy of V with KG-module structure defined by vσ.g σ := (v.g)σ for g ∈ G, so V ⊗ V σ becomes a Gσ -module in the obvious way. Then

dim V + 1 σ σ dim(V ⊗ V ) = . 2 Proof. Choose a basis B of V . Then σ acts asa permutation on the basis {v ⊗ wσ | v, w ∈ B} of V ⊗ V σ , with precisely dim 2V +1 orbits.

3. Groups of Lie type in defining characteristic In this section, we first recall the description of the representations of groups of Lie type in their defining characteristic. In the second part we use orbit closure arguments to show that most of the time long root elements have largest centralizer on a given module. We next determine the maximal size of subgroups consisting of long root elements only. In the fourth part we prove results on centralizers of subspaces in the natural representation of classical groups. Finally, we outline the approach to the classification of 2F -modules in the individual cases. 3.1. Representations in defining characteristic. We start by recalling the wellknown description of absolutely irreducible representations of groups of Lie type in their defining characteristic, due to Steinberg, as expounded for example in [5, 2.8]. ¯ be a simple simply-connected algebraic group over the algebraic closure Let first G K of a finite field Fa and its weight lattice. To any dominant weight λ ∈ there is ¯ The modules M(λ), for λ associated an irreducible highest weight module M(λ) for G. running over the dominant weights, form a complete set of non-isomorphic irreducible ¯ ¯ →G ¯ be a Frobenius morphism G-modules over K (see [5, 1.14.6]). Now let F : G defining a rational structure over Fa with corresponding finite group of fixed points ¯ F . Then M(λ) remains irreducible upon restriction to G if λ is a restricted G := G weight for G, that is, all coefficients of λ when expressed as a linear combination of fundamental weights are less than a . Moreover, the (restrictions of the) M(λ), where λ is restricted, constitute a complete set of non-isomorphic irreducible G-modules over K [5, 2.8.2]. These can be described in more detail by Steinberg’s tensor product theorem. A highest weight representation M(λ) is called basic if λ is a basic weight for G, that is, all coefficients of λ when expressed as a linear combination of fundamental weights

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¯ ¯ →G ¯ is any endomorphism, are less than . If V is a G-module over K and F : G ¯ we adopt the following notation for the G-module obtained by composing F with the representation on V . If F is the untwisted Frobenius morphism defining a rational ¯ is of type B2 , G2 , F4 in characteristic structure over Fi , then we write V (i) . If G = 2, 3, 2 respectively, and F is an exceptional Frobenius morphism of order two with ¯ F defined over F2a+1 then we write V (a+1/2) . corresponding group of fixed points G d a Now first assume that G = Gn ( ) is not a Suzuki group. Let M(λ) be an or Ree i λ , where the λ are basic. irreducible KG-module with highest weight λ = a−1 i i i=0 Then M(λ) is isomorphic to the tensor product (1)

V0 ⊗ V1

(a−1)

⊗ · · · ⊗ Va−1

(3.1)

of different Frobenius twists of basic modules Vi ∼ = M(λi ), 0 ≤ i ≤ a − 1, for G. 2 2a+1 ) is a Suzuki or Ree group then any irreducible KG-module is If G = Gn ( isomorphic to a tensor product (1/2)

V0 ⊗ V1

(a)

⊗ · · · ⊗ V2a

(3.2)

of different Frobenius twists of basic modules Vi , 0 ≤ i ≤ 2a, for G. In both cases this tensor decomposition is unique [5, 2.8.6]. We next describe the fields of definition for these representations (see [5, 2.8.8]). First assume that G is untwisted. Then the field of definition of a tensor product as in (3.1) has degree s over F , where s denotes the index of the centralizer of (V0 , V1 , . . . , Va−1 ) in the cyclic permutation group generated by (0, 1, . . . , a − 1). In particular, it has index at most r in Fa if the tensor product involves precisely r ≥ 1 nontrivial factors. Thus all non-trivial basic modules have field of definition Fa . The corresponding statement, with a replaced by 2a + 1, holds for the Suzuki and Ree ¯ of groups. If G is a Steinberg twist, with corresponding graph automorphism ρ of G order t, then the preceding description remains true if all factors Vi , that is, their basic weights, are invariant under ρ. Otherwise the field of definition is the extension of degree t of the field described above. An effective tool in the investigation of groups of Lie type in their defining characteristic will be Proposition 2.4 in conjunction with the following result of Smith and Timmesfeld (see [5, Thm. 2.8.11] and the subsequent remark): ¯ be a simple algebraic group over the algebraic closure K of a Theorem 3.1. Let G ¯ F a corresponding finite group of Lie type, P ≤ G a parabolic finite field, G = G subgroup with Levi decomposition P = U L. If V is an irreducible KG-module then ¯ then the L acts irreducibly on CV (U ). Moreover, if V has highest weight λ for G highest weight for L on CV (U ) is the restriction of λ. We label the fundamental weights for classical groups according to the convention given in Table 4.

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Classification of 2F -modules, II Table 4. Labeling of Dynkin diagrams of classical type

An

n≥1

Cn

n≥2

1 t

2 t

2 1 t> t

3 t 3 t

p p p

p p p

n t n t

Bn

n≥3

Dn

n≥4

1 2 t< t t1 @

3 @ @t

t

3 t

p p p

n t

4 t

p p p

n t

2

3.2. Orbit closures and centralizers. In this section we show that root elements are the non-trivial unipotent elements with largest centralizer dimension on any given module. ¯ be a connected reductive algebraic group over an algebraically Lemma 3.2. Let G ¯ and T¯ ≤ B¯ a maximal torus. Then the closed field K. Let B¯ be a Borel subgroup of G, closure of the T¯ -orbit of any unipotent element 1 = u ∈ B¯ contains a root element. ¯ with respect to (T¯ , B) ¯ and + ⊂ the set Proof. Let be the root system of G + of positive roots. Then there exist tα ∈ K (α ∈ ) with u = α uα (tα ), where ¯ Assume that tα , tβ = 0 for at least two {uα (t) | t ∈ K} are the root subgroups in B. + roots α = β. Since no two roots in are proper multiples of each other, there exists a subtorus T¯ of T¯ of co-dimension 1 centralizing uβ (t) but acting non-trivially on the root subgroup for α. In particular the closure of the T¯ -orbit of u contains an element u whose product decomposition involves at least one factor less than the one for u, but which involves uβ (tβ ). Induction now shows the claim. ¯ be a connected reductive algebraic group over an algebraically Corollary 3.3. Let G closed field K. Then the closure of any non-trivial unipotent class contains a long root element, respectively a long or short root element if G has a factor Cm or F4 in characteristic 2, or a factor G2 in characteristic 2 or 3. Proof. If all roots of G have the same length, this follows immediately from the previous lemma. Now assume that G has two different root lengths, that is, G has a factor of type Bm , Cm , m ≥ 2, G2 or F4 . Direct computation with the commutator relations shows that the closure of the class of short root elements contains long root elements in B2 = C2 , G2 and F4 , except in the characteristics given in the statement. Thus we may conclude with Lemma 3.2. Corollary 3.4. Let G be a finite group of Lie type in characteristic and V an F Gmodule over a field F of characteristic . Then the centralizer dimension on V of any non-trivial unipotent element is bounded above by the centralizer dimension of a long root element (respectively of a long or short root element if G is of type Cm or F4 in characteristic 2 or of type G2 in characteristic 2 or 3).

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¯ denote the algebraic group corresponding to G over the algebraic closure Proof. Let G K of the ground field. Then the representation of G on V extends to a representation ¯ on V¯ := V ⊗ K, see Section 3.1. Thus it suffices to prove the assertion for G. ¯ of G Now note that the property of u having centralizer dimension z on V¯ is equivalent to the fact that all (n − z + 1) × (n − z + 1)-minors of a matrix representing u − 1 vanish, where n = dim V . This closed condition is clearly inherited by elements in the closure of the conjugacy class of u. Thus the assertion follows from Corollary 3.3. We extend the previous result to outer elements in extensions of algebraic groups ¯ be a connected reductive algebraic group over by graph automorphisms. For this let G ¯ of an algebraically closed field of characteristic and σ a graph automorphism of G ¯ and order . More precisely, fix a Borel subgroup B¯ and a maximal torus T¯ ≤ B¯ of G ¯ assume (as we may) that σ stabilizes the set of simple roots with respect to (T¯ , B). ¯ σ be as above. Then σ lies in the closure of any unipotent class Lemma 3.5. Let G, ¯ in the coset σ G. ¯ any unipotent class in σ G ¯ has a representative σ u in σ B. ¯ Proof. Since σ stabilizes B, + ¯ ¯ ¯ ¯ Let T ≤ T be the centralizer in T of σ , a subtorus. If α ∈ vanishes on all of T , then o(σ )−1

α(t) = −

i

α(t σ )

for all t ∈ T ,

i=1

o(σ )−1 i whence i=0 α σ = 0. But σ stabilizes the set of positive roots + , hence this is not possible. We may now copy the argument from the proof of Lemma 3.2 to conclude that the ¯ T -orbit of σ u contains σ . Since representations of groups of Lie type extended by a graph automorphism extend to the algebraic group, we obtain as above: Corollary 3.6. Let G be a finite group of Lie type in characteristic , σ a graphautomorphism of G of order and V an F G, σ -module over a field F of characteristic . Then the centralizer dimension on V of any non-trivial unipotent element in σ G is bounded above by the centralizer dimension of σ on V . The following result is well known. Proposition 3.7. Let G be a finite group of Lie-type of simply connected type and P < G a proper parabolic subgroup, P − the opposite parabolic subgroup. Let U and U − denote the unipotent radicals. Then G = U, U − . Proof. Let L be the common Levi subgroup of P and P − . Set H = U, U − . Then L normalizes U and U − and so H . So P and P − each normalize H . Clearly G = P , P − (since they are contained in no common proper parabolic subgroup and the only overgroups of P are parabolic subgroups). If G is simple (modulo its center),

Classification of 2F -modules, II

131

the result follows (since the center is contained in the Frattini subgroup). Note that in the cases that G is not simple, it is still true that G is the normal closure of its Sylow -subgroup. In particular, this shows that H = G if G is simple or G is rank one (for then U is a Sylow -subgroup). This leaves only the cases 2F4 (2), Sp4 (2) and G2 (2). One checks directly that the unipotent radical of a maximal parabolic subgroup is not contained in the derived subgroup (for example, see [4]). 3.3. Long root elements. We also need an upper bound for abelian subgroups consisting entirely of long root elements. Lemma 3.8. Let G be a simple algebraic group over an algebraically closed field. Let α, γ be two different long roots (with respect to the choice of a maximal torus T of G) with α + γ not a root. If the product Xα Xγ of the corresponding root subgroups contains a long root element outside of Xα ∪ Xγ , then γ − α is a root. Proof. After conjugation with a suitable element of the Weyl group we may assume that γ is the highest root. Then Xγ is normalized by the corresponding Borel subgroup B = U T . Let 1 = x ∈ Xα , 1 = y ∈ Xγ with v := xy a long root element. Since G is simple, all long root subgroups are conjugate and there exists g ∈ G with v g ∈ Xγ . Write g = uwb with u ∈ U , w in the Weyl group and b ∈ B according to the Bruhat decomposition, then v uw ∈ Xγ , so v u ∈ Xγ˜ with γ˜ w = γ . On the other hand v ∈ Xα Xγ , so v u ∈ Xαu Xγ ⊂ U . By the Chevalley commutator relations all non-trivial elements in Xαu have lowest term xα (t) for some t = 0. On the other hand v u = xγ˜ (t ) for some t = 0, so γ˜ = α. Note that Xαu = Xα . But then again by the commutator relations there must exist a root β with α + β = γ . ¯ be a simple algebraic group defined over Fq , G the set of fixed Proposition 3.9. Let G points under the corresponding Frobenius map, not of Suzuki- or Ree-type. Let A ≤ G be an abelian subgroup all of whose non-identity elements are long root elements. Let X be a long root subgroup. (a) Then logq |A| is bounded above by the values given in Table 5; and (b) A is conjugate to a subgroup of the unipotent radical of the parabolic subgroup NG (X). Proof. Let x, y be two commuting long root elements and assume that v = xy is again a long root element. Let X, Y be long root subgroups containing x, y respectively. Then [X, Y ] = 1 by [5, Prop. 3.2.9]. Moreover, since both lie in a Borel subgroup, and any two Borel subgroups have a maximal torus in common, we may assume that both are standard root subgroups: X = Xγ , Y = Xα . If X = Y then by Lemma 3.8 the difference γ − α is again a long root. Thus an upper bound for A is obtained by determining the maximal size of a set of long roots such that no sum of two of them is a root, but any difference is. This is easily done. It turns out that the maximal rank is realized by the maximal type A-subsystem consisting of long roots.

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Moreover, we can conjugate and assume that X = Xγ with γ the highest root. Write NG (X) = LQ with Q the unipotent radical and L the Levi complement. Suppose that Y = Xα . If Y is not contained in Q, then Y ≤ L and also X−α ∈ L. Thus, X−α commutes with X and so γ − α is not a root, a contradiction to Lemma 3.8. Table 5. Maximal ranks of subgroups of commuting long root elements

G logq |A|

An n

Bn (n ≥ 2) Cn n−1 1

Dn (n ≥ 4) n−1

G2 2

F4 3

E6 5

E7 7

E8 8

3.4. -ranks and centralizers. For the investigation of possible 2F -modules we sometimes need quite precise information on the -ranks of subgroups of classical groups with given centralizer on the natural module. This is a straightforward generalization of results in [5, Thm. 3.3.3]: Proposition 3.10. Let A ≤ SLn (a ) be an elementary abelian -subgroup with centralizer dimension dim CYn (A) ≥ n − c on the natural module Yn . Then |A| ≤ min{ac(n−c) , an

2 /4

}.

Moreover, for each c there is a subgroup A for which this bound is attained. In particular the -rank of SLn (a ) equals an2 /4. Proof. Since SLn (a ) acts transitively on the set of subspaces of Yn of fixed dimension we may assume that CYn (A) is spanned by the first n − c standard basis vectors. Thus I 0 A ≤ R := . ∗ ∗ The subgroup R is generated by full root subgroups of G. The result now follows precisely as in the proof of [5, Thm. 3.3.3], where the case c = n was treated. An example of an elementary abelian -subgroup of maximal rank is given by the group of lower unitriangular matrices with non-zero entries only in the lower left n/2×n/2-block if n is even (respectively (n+1)/2×(n−1)/2-block if n is odd). For the other classical subgroups of GLn (q) we make use of the invariant form on the natural module Yn : Proposition 3.11. Let A ≤ SUn (a ), n ≥ 2, be an elementary abelian -subgroup with centralizer dimension dim CYn (A) ≥ n − c on the natural module Yn . Then n 2 , 2 n even, ac arn |A| ≤ min{ , }, with rn := 2 2 n−1 + 1, n odd. 2

In particular the -rank of SUn

(a )

is at most arn .

Classification of 2F -modules, II

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Proof. The precise -rank of SUn (a ) is given in [5, Table 3.3.1], and a short examination shows that it is bounded above by arn , proving the second bound. If c > n/2 then the first bound is not smaller than the second, so we may now assume c ≤ n/2. We will argue by induction on n, the induction base being given by the trivial case n = 2. Let W = R ⊕ X be the decomposition of the centralizer of A into its radical and the nonsingular part. Then c := dim R ≤ c and X has dimension n − c − c . Now A acts faithfully on X ⊥ of dimension c + c . Since X ⊥ ∩ W = R has dimension c the centralizer of A on X⊥ has dimension at most n − c. If c + c < n 2 then |A| ≤ ac by induction. Thus it remains to consider the case c + c = n, so c = c = n/2. In particular, n is even. Then the second bound gives the desired conclusion. We next consider the maximal -rank of elementary abelian -subgroups of Sp2n (q) with given centralizer on the natural module Y2n : Proposition 3.12. Let A ≤ Sp2n (a ) be an elementary abelian -subgroup with centralizer dimension dim CY2n (A) ≥ 2n − c on the natural module Y2n . Then c+1 n+1 |A| ≤ min a ( 2 ) , a ( 2 ) , and for each c there is a subgroup A for which this bound is attained. In particular the -rank of Sp2n (a ) equals a n+1 2 . n Moreover, if c = n and |A| > a (2) then CY (A) is maximal totally isotropic. 2n

Proof. Let A be an elementary abelian unipotent subgroup of G := Sp2n (q), q := a , trivial on a space of dimension 2n − c. If c ≥ n, then already a maximal abelian unipotent subgroup of G fixes a subspace of dimension n ≥ 2n − c, so assume that c ≤ n. Let W be the centralizer of A and R the radical of W . Then W = R ⊕ X as A-module with X nonsingular. Since A is trivial on X, it leaves X⊥ invariant. Suppose that R has dimension c – note c ≤ c (since R is contained in W ⊥ ). Now X has dimension 2n − c − c and so X⊥ has dimension c + c . But A is faithful on X⊥ and so by the result [5, Thm. 3.3.3] on maximal abelian subgroups, |A| ≤ qd(d+1)/2 where d = (c + c )/2 ≤ c, whence |A| ≤ q c(c+1)/2 – of course we may have c = c and obtain equality. Finally, if c = n and W is not totally isotropic, then c < c, d ≤ n−1, and thus |A| ≤ q n(n−1)/2 . Lemma 3.13. Assume that n ≥ 2. Let A be an elementary abelian unipotent subgroup of G := Sp2n (q) of order q n(n+1)/2 . Then A is the unipotent radical of the stabilizer of some totally singular subspace of dimension n in the natural representation for G unless q is even and n = 2. In that case then the result is true up to a graph automorphism. Proof. Let P be the stabilizer of a maximal totally singular subspace with Q the unipotent radical of P . We may assume that A ≤ P . By Lemma 2.6 it suffices to

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show that Q is not an F -module for L. For q odd, Q is the symmetric square of the natural module of GLn (q), and the claim follows from (2.1) in conjunction with Proposition 3.10 (see also Theorem 9.5). Suppose now that q is even. Then Q contains an L-submodule R isomorphic to the wedge square of the natural module for L. It follows from Proposition 3.14 that R is not an F + module for n > 3, whence Q is not an F -module. If n = 3, we still see that Q is not an F -module since its composition factors are the dual of the natural module and the Frobenius twist and the same argument applies. If n = 2, the result follows by an easy inspection (or by considering abelian sets of roots). Finally, the same investigation for orthogonal groups: (±)

Proposition 3.14. Let A ≤ SOn (a ), n ≥ 4, be an elementary abelian -subgroup with centralizer dimension dim CYn (A) ≥ n − c on the natural module Yn . Then n−1 2 + 1, n ≥ 9 odd, > 2, 2 c n − 2, n = 5, 7, > 2, |A| ≤ min{a (2) , arn }, with rn := n 2 , n ≥ 8 even, 2 n2 + 1, n = 4, 6. 2 (±)

In particular the -rank of SOn (a ) is at most arn . (±)

Proof. The precise -rank of SOn (a ) is given in [5, Table 3.3.1], and a short examination shows that it is bounded above by arn , proving the second bound. If c > n/2 then the first bound is not smaller than the second, so we may now assume c ≤ n/2. We will argue by induction on n, the induction base being given by the trivial case n = 2. As above let W = R ⊕ X be the decomposition of the centralizer of A into its radical and the nonsingular part. Then c := dim R ≤ c and X has dimension n − c − c . Our group A acts faithfully on X⊥ of dimension c + c . centralizer of A on X⊥ has dimension at Since X ⊥ ∩ W = R has dimension c the a (2c ) most n − c. If c + c < n then |A| ≤ by induction. Thus it remains to consider the case c + c = n, so c = c = n/2, R totally isotropic. If n = 4, 6 then again the second bound gives the desired conclusion. For n = 4, 6 it is easy to write out explicit matrices for the centralizer of a maximal isotropic subspace in orthogonal group the and verify (with the argument from loc.cit.) that its -rank is n/2 2 . The following result will allow to deal with the 27-dimensional modules for E6 (q). Lemma 3.15. Use the notation of 3.3.4 in [5]. Suppose that V has a unique tame subalgebra I of maximal order and moreover V = V (A) where A is the unipotent radical of a parabolic subgroup P . Then A is the unique elementary abelian subgroup of U of order |A|.

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Proof. Suppose that B is an elementary abelian unipotent subgroup of U of order |A|. By [5, 3.3.4] it follows that V (B) has the same order and so by hypothesis, V (B) = V (A). Let be the set of fundamental roots corresponding to U . Now A is just the product of the root subgroups, Uλ for λ ∈ \ where is the set of roots corresponding to the Levi subgroup L of P . Let = ∩ . Choose the linear functional f (as in loc. cit.) so that f (α) > f (γ ) where α ∈ \ and γ ∈ . With this ordering, it is clear that V (B) ≤ V (A) implies that B ≤ A. In particular, V (B) = V (A) implies that B = A, whence the result. Lemma 3.16. Let A be an elementary abelian unipotent subgroup of Spin+ 14 (q) of order |A| > q 20 . Then A is contained in the unipotent radical of the stabilizer of a totally singular subspace of maximal dimension. Proof. By Proposition 3.14 the tame abelian homogeneous subalgebra I (A) must belong to an abelian set of roots of maximal size. Direct computation shows that such sets correspond to unipotent radicals as in the claim, so I (A) ≤ I (Q) for such a radical Q. Since these roots correspond to a radical, we see that for some appropriate ordering of the roots, A ≤ U (I (Q)) = Q. 3.5. The general strategy. We can now describe the general approach to the determination of 2F -modules for groups of Lie type in their defining characteristic. Let ˜ = t G(q), q = a , a quasi-simple group of Lie ˜ be a finite group with G := F ∗ (G) G ˜ type, and (G : G) of -power order. Note that G is a central quotient of the group of fixed points under a Frobenius map of a simple simply connected algebraic group over the algebraic closure of F , or G = 2F4 (2) . Let V be a faithful absolutely irreducible ˜ Thus, the restriction of V to G is a direct sum of absolutely irreducible F¯ G-module. F¯ G-modules as described in Section 3.1, or G = 2F4 (2) . The 2-modular irreducible representations of the latter are contained in [8]. We distinguish two cases. If V remains irreducible upon restriction to G, then by Steinberg’s tensor product theorem it is a product V = V1 ⊗ · · · ⊗ Vr of r ≤ a Frobenius twists of nontrivial basic representations. Furthermore, by the description of fields of definition, it is defined over a subfield of Fq t of index at most rt. So by Lemma 2.1 we arrive at the necessary condition r ! i=1

dim Vi ≤ 2r logq |Aut(G)|

(3.3)

for V to be a 2F -module. If V is basic this reduces to dim V ≤ 2 logq |G|, and all such modules have been classified by Lübeck [11]. For any given type we can thus reduce to a short list of possibilities which then have to be investigated further. Alternatively, if V is a product of at least two non-trivial factors, by Lemma 2.10 the centralizer on V of a possible subgroup A has dimension at most dim V /d1 where d1 is the minimum of the {di := dim Vi | 1 ≤ i ≤ r}. If b denotes the -rank of G,

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we arrive at the inequality d2 . . . dr ≤ 2rb/a.

(3.4)

For small values of d1 this may be better than (3.3). If A involves outer automorphisms then we may replace b by the -rank of the centralizer of outer -elements, which gives even stronger conditions. Similarly, if G is twisted, then the field of definition of V may be larger and we get better inequalities. On the other hand, if V restricted to G is reducible, that is, if V is induced, then Proposition 2.5 applies. Moreover, any outer -element has centralizer space of dimension at most dim(V )( − 1)/ on V . Since dim V ≥ 3, A has to contain nontrivial inner elements as well, which have at least d2 . . . dr -dimensional commutator on the centralizer of outer elements. Together this gives the condition r . (3.5) d2 . . . dr (d1 − d1 + 1) ≤ 2 rb + a

4. Linear and unitary groups 4.1. The case SLn (q). In this subsection we consider the case of Ln (q), q = a . Let Yn denote the natural module for SLn (q) of dimension n, with right SLn (q)-action. Let λ1 , . . . , λn−1 denote the fundamental weights for SLn , ordered along the Dynkin diagram of type An−1 . Clearly, an SLn (q)-module V is a 2F -module if and only if its dual V ∗ is. Similarly, if V is a 2F -module, then so are all of its Frobenius-twists V (a) . It thus suffices to consider representations up to taking duals and up to Frobenius-twists. Now if V a λ has highest weight i i , then the contragredient representation has highest weight i a λ . We therefor only need to consider highest weights up to the symmetry n−i i i induced by the graph automorphism. It turns out to be convenient to treat the 2-dimensional linear groups separately. This will also serve as an induction base for the investigation of the other groups of Lie type. Proposition 4.1. Let V be an absolutely irreducible 2F -module in defining characteristic for a group G with F ∗ (G) = SL2 (a ). Then all composition factors of the restriction V |F ∗ (G) occur in Table 7. In particular, their highest weights are one of i λ1 ,

2i λ1 ( > 2),

(i + i+a/2 )λ1 (if 2 | a)

where 0 ≤ i < a. Proof. The basic SL2 (a )-modules, with highest weights tλ1 , 0 ≤ t ≤ − 1, are the symmetric powers t (Y2 ) of the natural module Y2 . On these, -elements of SL2 (a ) act indecomposably. 1. We first consider G = SL2 (a )-modules V . Let A be an offender on V . Then A is contained in the unipotent radical U of a Borel subgroup B of G. Since the maximal

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torus of B has only gcd( − 1, 2) non-trivial orbits on U , Lemma 2.1 shows that U itself is an offender. Hence we may assume A = U . By the Theorem 3.1 of Smith the torus then acts irreducibly on CV (A), so dim CV (A) = 1. If V is defined over the field with f elements this gives the condition |V | = df ≤ |A|2 |CV (A)| = 2a f , where d = dim V , hence f (d − 1) ≤ 2a. By the description in Section 3.1 V is a tensor product V1 ⊗ · · · ⊗ Vr of Frobenius twists of basic modules Vi of dimension di = dim Vi ≥ 2, with field of definition of index at most r in Fa . Thus we have f ≥ a/r and hence 2r − 1 ≤ 2r. This immediately forces r ≤ 2. In the case r = 1, that is, V is a Frobenius twist of a basic module, our inequality shows that d = d1 ≤ 3. This leads to Examples 4.1 and 4.3 with highest weights λ1 and 2λ1 . If r = 2 the assumption f = a violates the first inequality, since d ≥ 4. Hence f = a/2, the two factors are conjugate and we arrive at Example 4.4 with highest weight (1 + a/2 )λ1 . 2. The only outer automorphisms of S = L2 (a ) of order are field automorphisms, and these exist if and only if | a. So for the rest of the proof let G = SL2 (a )., where | a. The -rank of the centralizer in S of a field automorphism of order is a/. We first consider extendable representations V = V1 ⊗ · · · ⊗ Vr of G. Then necessarily | r and our standard inequality (3.4) reads d1 . . . dr−1 (dr − 1) ≤ 2r/ + 2r/a. This is satisfied only when di = 2 and either r = = a = 3 or r = = 2, a ≤ 2. But for SL2 (27) the 4-dimensional centralizer of the outer automorphism of order 3 is not centralized by an inner 3-element, so this case is out. The second possibility gives an example for the highest weight (1 + a/2 )λ1 . 3. Finally assume that V is induced from a representation W of SL2 (q) which is not invariant under the field automorphism of order . The field automorphism has centralizer of dimension dim(W ) on the induced module V = W ↑G S , and on this centralizer, non-trivial inner -elements have at least d1 . . . dr−1 (dr − 1)-dimensional commutator when W = V1 ⊗ · · · ⊗ Vr with di = dim Vi . Since V is defined over a subfield of index at most r of Fa , we arrive at the condition d1 . . . dr−1 (dr − 1) ≤ 2r + 2r/a. This forces r ≤ 2, = 2, di = 2. The case r = 1 occurs only when a = 2, 4 and leads to an example from the natural module. When r = 2 then moreover a = 2. But the group L2 (4) does not have non-extendable representations of degree 4. Note that the conclusion of Proposition 4.1 even holds for the solvable groups SL2 (2) and SL2 (3). For the remaining groups of Lie type we make use of the results of Liebeck [10] classifying the basic modules of Chevalley groups G in defining characteristic of small dimension (see also the more extensive lists of Lübeck [11]).

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Lemma 4.2. Let n ≥ 3, M(λ) a basic highest weight module for SLn (a ) of dimension dim M(λ) ≤ n2 +1. Then M(λ) is ( possibly the dual of ) one of the modules in Table 6. Table 6. The basic SLn (a )-modules of dimension at most n2 + 1

λ λ1 λ2 2λ1 λ1 + λn−1

M(λ) Yn ∧2 (Yn )

2 (Yn ) ˜ n) A(Y

λ3 3λ1 λ1 + λ 2

∧3 (Yn )

3 (Y3 )

dim M(λ) n n 2 n+1

n≥4 >2 n≥3

2

n2 − 1 − δ,n n 3

10 16

6≤n≤8 n = 3, > 3 n = 4, = 3

This is Theorem 2.2 in [10]. Thus generically (i.e., for n ≥ 9) only four types of modules occur. Moreover, the two smallest non-trivial dimensions are n and n2 . We need an extension of Lemma 4.2 giving lower bounds for the dimension of modules with symmetric highest weight for small n, as well as for modules with weights supported on the end nodes only; both follow for example from [11, Thm. 4.4]: Lemma 4.3. Let λ = t1 (λ1 + λn−1 ) + t2 (λ2 + λn−2 ) + · · · with 0 ≤ ti < be a symmetric basic weight for SLn (a ), n ≥ 3. Then dim M(λ) ≥ 2n2 unless λ = λ1 + λn−1 , Lemma 4.4.

or λ = λn/2 for n = 4, 6, 8,

or λ = 2λ2 for n = 4.

(a) We have dim M(λ1 + 2λn−1 ) ≥ 2n2 for n ≥ 4.

(b) We have dim M(2λ1 + 2λn−1 ) ≥ 2n2 for n ≥ 3. Lemma 4.5. Let n ≥ 3, λ = t1 λ1 + t2 λn−1 with t1 , t2 ∈ {i , 2i , i + a/2+i | 0 ≤ i < a/2}. Then the highest weight module M(λ) is not a 2F -module for SLn (a ). Proof. 1. For λ one of λ1 +i tλn−1 , 2λ1 +i tλn−1 or t (λ1 +i λn−1 ) with t = 1+a/2 the corresponding highest weight module M(λ) is a tensor product of dimension at least n(n2 − 2), n3 , (n2 − 2)2 , respectively. Only in the latter case M(λ) is defined over a proper subfield of Fq , of index 2. In any case the dimension exceeds the bound (2n2 − 1)a/f . Thus we may assume that t1 , t2 ∈ {i , 2i }, and then, by taking a suitable Frobenius twist, that t1 ∈ {1, 2}, t2 = 2i if t1 = 2. 2. Now let V = M(λ) for λ = λ1 + i λn−1 . Thus V is the tensor product ˜ n ) for Yn ⊗ (Yn∗ )(i) , except when i = 0 in which case V is the adjoint module A(Y a SLn ( ), a submodule of possibly a quotient by a 1-dimensional subspace of Yn ⊗ Yn∗ .

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Now first note that the assertion of Lemma 2.10 (a) for the centralizer remains true ˜ n ). Indeed, if A is cyclic, then all indecomposable modules are uniserial, so for A(Y the centralizer on a quotient cannot increase. The general statement was derived by considering successive actions of cyclic groups on previous centralizers, so the proof remains valid. Returning to V , if the centralizer of A on the natural module or its dual has dimension less than (n − 1)/2 then the first bound from Corollary 2.11 (a) gives a contradiction. If A contains non-trivial elements with distinct centralizers, then Lemma 2.10 (a) shows that the centralizer on V is too small. If A contains no transvections, then its rank is again too small. Thus A consists of transvections all fixing the same hyperplane in Yn , and all fixing the same hyperplane in (Yn∗ )(i) . But this forces |A| = q, and now Corollary 2.11 (a) gives the final contradiction. 3. Thus, if t2 = 2 then M(λ) is a tensor product defined over Fq of dimension at 2 least n n+1 2 , which is larger than 2n −1. So we are left with (t1 , t2 ) ∈ {(1, 2), (2, 2)}. For λ = 2λ1 +2λn−1 , the corresponding highest weight module has dimension at least 2n2 by Lemma 4.4 (b), hence this is also impossible. Also, the weight λ1 + 2λn−1 for n ≥ 4 is excluded by Lemma 4.4 (a). 4. So finally we consider λ = λ1 + 2λ2 in the case of G = SL3 (q). For this, we investigate possible offenders. Let + = {α, β, α + β} denote the set of positive roots for G with respect to the choice of a maximal torus and a Borel subgroup B containing it. For a root ν ∈ + let Xν = {xν (t) | t ∈ Fq } be the corresponding root subgroup. If there exists an offender A in the unipotent radical U of a maximal parabolic subgroup, then we may assume A = U by Lemma 2.1. But U together with the unipotent radical of the opposite parabolic subgroup generate G, by Lemma 3.7, so the centralizer dimension of U on M(λ) is at most dim(M(λ))/2. This implies that dim(M(λ)) ≤ 8, which is not the case by Lemma 4.2. Hence A contains elements involving xα (t) and xβ (u) for some non-zero t, u ∈ Fq . Then A also contains an element involving the product xα (t)xβ (u). But the normal closure of such an element in the Borel subgroup B is the whole unipotent radical U of B. Thus again by Lemma 2.1 we may and will assume that A = U . We have |A| = 3a , dim CV (A) = 1 by Theorem 3.1, and so dim(M(λ)) ≤ 6 if A is an offender on V . This is again a contradiction to Lemma 4.2. Lemma 4.6. The exterior cube ∧3 (Y7 ) of the natural module, with highest weight λ3 , is not a 2F -module for SL7 (q). Proof. 1. Let U be the unipotent radical of a Borel subgroup of SL7 (q) consisting of all lower unitriangular matrices. We may assume that a possible offender A on V = ∧3 (Y7 ) is contained in U . By Table 3 any transvection has at most 25-dimensional centralizer on V , so by Lemma 3.4 the same is true for any non-trivial unipotent element. Moreover, if A contains more than q elements, then dim CV (A) ≤ 21, by arguments similar to those in the proof of Lemma 2.10. Thus, |A| ≥ q (36−21)/2 > q 7 . Let P1 = U1 L1 be the end-node parabolic above U containing GL6 (q) on the first six basis vectors. Its unipotent radical U1 is elementary abelian of order q 6 . Thus, by our previous argument, we can’t have A ≤ U1 .

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2. Now assume that A ∩ U1 is trivial. Then A is isomorphic to its image B in the Levi subgroup of P1 of type GL6 (q). The composition factors of V for GL6 (q) are ∧3 (Y6 ) and ∧2 (Y6 ). Any unipotent element of GL6 (q) has rank at least 6 on ∧3 (Y6 ) and at least 4 on ∧2 (Y6 ). Moreover, any unipotent subgroup of order larger than q has centralizer codimension at least 9 on ∧3 (Y6 ) and at least 7 on ∧2 (Y6 ) by Lemma 2.10. Thus, |B| ≥ q 8 . Then either B is contained in the unipotent radical of a GL4 (q)-parabolic of GL6 (q), which can be ruled out by the standard argument, or it contains elements with at least three different centralizers, which decreases both centralizer dimensions at least by 1, and |B| = q 9 . Then by Proposition 3.10 B is the unipotent radical of a GL3 (q) × GL3 (q)-parabolic of GL6 (q), whose centralizer is too small by Theorem 3.1. Hence A ∩ U1 is non-trivial, so the centralizer of A on ∧2 (Y6 ) has codimension at least 10, and since A ≤ U1 , the centralizer on ∧3 (Y6 ) has codimension at least 6. Thus |A| ≥ q 8 . 3. Next assume that |A ∩ U1 | ≤ q. Since B := AU1 /U1 can have order at most q 9 by Proposition 3.10, we have q 8 < |A| ≤ q 10 . Let P2 = U2 .GL5 (q) be the end-node parabolic of L1 containing GL5 (q) on the first five basis vectors. By Theorem 3.1, CV (U1 ) is ∧3 (Y6 ) for GL6 (q), and on this the Levi factor GL5 (q) has composition factors ∧3 (Y5 ) and ∧2 (Y5 ). If |B ∩ U2 | ≤ q then the image of B in the Levi factor GL5 (q) has maximal possible order q 6 . But then its centralizer is only 3-dimensional on ∧3 (Y5 ) by Theorem 3.1, and we get a contradiction. On the other hand, if |B ∩ U2 | > q, then the centralizer of A ∩ U1 U2 has codimension at least 10 + 9 = 19 on V , whence |A| > q 9 . Then the image of A in GL5 (q) has order larger than q 3 , and thus a centralizer of codimension at least 5 on ∧3 (Y5 ). Then |A| ≥ q 12 , so A is the unipotent radical of a GL3 (q) ◦ GL4 (q)-parabolic, which has at most 2-dimensional centralizer by Theorem 3.1, impossible. 4. We may thus assume that |A ∩ U1 | > q. Then A ∩ U1 has centralizer of codimension at least 14 on V by direct computation, and the image B has centralizer of codimension at least 6 on CV (U1 ) = ∧3 (Y6 ). Thus |A| ≥ q 10 . But then necessarily B ∩ U2 = 1. The case that A is contained in U1 U2 was ruled out before, so the image C of A in GL5 (q) is non-trivial. Then we obtain a centralizer of codimension at least 20+3 = 23 on V , whence |A| > q 11 . First assume that |B ∩U2 | ≤ q. Then |C| > q 4 , which has centralizer of codimension at least 5 on ∧3 (Y5 ), contradicting the bound. On the other hand, if |B ∩ U2 | > q, then the centralizer of A has codimension at least 14 + 9 + 3 = 26 on V , again impossible. As a first step we treat the case where G does not involve outer automorphisms of S = Ln (q). Proposition 4.7. The absolutely irreducible 2F -modules in defining characteristic for SLn (q), q = a , n ≥ 3, are as given in Table 7. In particular, their highest weights are (up to Frobenius twists and taking the dual) one of λ1 ,

2λ1 ,

(1 + a/2 )λ1 ,

λ2 ,

λ3 (for n = 6).

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Proof. We proceed by induction, the induction base being given by the case n = 2 in Proposition 4.1. Note that for basic 2F -modules, (3.3) shows that dim V ≤ 2n2 − 1. 1. Let V be a 2F -module for G = SLn (q) of dimension d, defined over Ff . Let P = U.L = q n−1 .GLn−1 (q) be a maximal parabolic subgroup of G corresponding to an end node on the Dynkin diagram. First assume that there exists an offender A ≤ U for V . Since U is the natural module for L, hence irreducible, Lemma 2.1 allows to conclude that in this case U itself is an offender. On the other hand, by Proposition 3.7 the centralizer dimension of U on V is at most d/2. Thus the 2F condition gives df/2 ≤ 2(n − 1)a, hence d ≤ 4(n − 1)a/f . By Steinberg’s theorem, V is a tensor product V = V1 ⊗ · · · ⊗ Vr of Frobenius twists of non-trivial basic modules Vi of dimension di := dim Vi ≥ n, and defined over a subfield of Fq of index at most r. We obtain nr ≤ d ≤ 4(n − 1)a/f ≤ 4(n − 1)r. Since n ≥ 3 this forces r ≤ 2. Moreover, if r = 2 then di = n, and V1 , V2 are Galois (a/2) conjugate, so V = Yn ⊗ Yn up to Frobenius twists and taking the dual. This is Example 4.4. When r = 1 then d ≤ 4(n − 1), so by Lemma 4.2 we are in one of ˜ n ) with highest weight λ1 + λn−1 , the cases of Example 4.1–4.3 or 4.5, or V = A(Y which was excluded in Lemma 4.5. 2. Thus, by Proposition 2.4 we may assume that CV (U ) is a 2F -module for L (hence for L = SLn−1 (q)) for both end node parabolics. Hence by induction the highest weight λ of V or its dual is one of t1 λ1 + t2 λn−1

(t1 , t2 ∈ {0, i , 2i , i + a/2+i }),

i λ2 , tλ2 i λ3

(t ∈ {2i , i + a/2+i }, n = 4), (n = 6, 7).

Indeed, if λ has more than one non-zero coefficient, then there are precisely two, and these lie at the two ends, since by induction all 2F -weights have support only on one node. Moreover the entries are at most those for SL2 (q). Similarly for the cases with just one non-zero coefficient. The weights of the form t1 λ1 + t2 λn−1 not occurring in the conclusion were ruled out in Lemma 4.5; the possibility λ = λ3 for n = 7 was excluded in Lemma 4.6. 3. For n = 4, λ = (1 + a/2 )λ2 , the module M(λ) is the tensor product of V1 with (a/2) V1 , where V1 is the natural module for SO+ 6 (q) (namely, the exterior square of the natural module for G). The precise bounds for the rank of a subgroup of SO+ 6 (q) with given centralizer in Proposition 3.14 together with Corollary 2.11 show that this case does not occur. 4. Finally, the highest weight module M(λ) of G = SL4 (q) with highest weight λ = 2λ2 is the heart of 2 (∧2 (Y4 )), the symmetric square of the alternating square of the natural module. By Lemma 2.10 and Corollary 3.4 any non-trivial unipotent

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element of G has centralizer of dimension at most 4 on ∧2 (Y4 ). But then the centralizer ˜ 4 ) in the proof of Lemma 4.5 we on 2 (∧2 (Y4 )) has dimension at most 11. As for A(Y conclude that the same is true on M(λ). Hence an offender has order at least q 4 , so is equal to the unipotent radical of a GL2 (q)◦GL2 (q)-parabolic. But by Theorem 3.1 the latter only has centralizer dimension 3 on M(λ). Thus M(λ) is not an example. Proposition 4.8. Let V be an absolutely irreducible 2F -module in characteristic for a group G with F ∗ (G) = SLn (q), q = a , n ≥ 3. Then all composition factors of the restriction V |F ∗ (G) are 2F -modules for F ∗ (G). Proof. Let V be a 2F -module for G. 1. First assume that the restriction of V to F ∗ (G) is not irreducible. Let W be one of the irreducible constituents. By Proposition 2.5 either W itself is a 2F -module for a proper subgroup of G containing F ∗ (G), or = 2, W is defined over F22 and dim W ≤ k, where k denotes the -rank of an offender A. If a ≤ 2 this forces dim W ≤ n2 /4 + 2, so W is a twist of a basic module, and Lemma 4.2 shows that W = Yn , or n = 4, W = ∧2 (Y4 ), up to twists and duals. These are Examples 4.1 and 4.2. If a ≥ 3 then W has to be a product of at least a/2 factors, of dimension at least n, which gives na/2 ≤ an2 /8 + 2. This does not lead to new cases. Hence it remains to consider modules V with irreducible restriction to F ∗ (G), that is, highest weight modules whose highest weight is invariant under G. 2. Assume that G = SLn (a ). involves a field automorphism of order (so in particular | a). We may assume that A is not contained in F ∗ (G), so neither of the unipotent radicals of end node parabolics contains an offender. Theorem 3.1 yields that W := CV (U ) is an example for SLn−1 (a ).. By induction (with induction base SL2 (q)) this forces V or its dual to have highest weight λ ∈ {tλ1 , t (λ1 + λn−1 ), tλ2 (n = 4) | t = i + i+a/2 }. In particular = 2 and 2 | a. The first case is Example 4.4. The second one is a ˜ n ) ⊗ A(Y ˜ n(a/2) ), hence has dimension at least (n2 − 2)2 . This Frobenius twist of A(Y violates the upper bound 2(2n2 −1). The third module is a Frobenius twist of the tensor (a/2) product ∧2 (Y4 ) ⊗ ∧2 (Y4 ) for n = 4. In particular, the centralizer of an offender A has codimension at least 2 24 , so the 2F -condition reads 4(4 − 1) ≤ 2(42 /4 + 1), contradiction. 3. Next consider the case that = 2 and G = SLn (2a ).2 is obtained by adjoining the graph automorphism γ . The centralizer in SLn (2a ) of a graph automorphism of order 2 is contained in a symplectic group Sp n (2a ). Thus by Proposition 3.12 its 2 2-rank equals an(n + 2)/8 for n even, a(n2 − 1)/8 for n odd. By step 1 we may restrict to modules V with highest weight invariant under the graph automorphism. 3A. First consider modules with highest weight of the form t (λ1 + λn−1 ). Since ˜ n ). If = 2 these are tensor products of Frobenius twists of the adjoint module A(Y there are at least two factors, the dimension becomes too large. Now assume that V is ˜ n ). Let A be an offender on V for G. If B := A ∩ SLn (q) has centralizer a twist of A(Y

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of codimension c on the natural module, then Corollary 2.11(a) Lemma 2.12 show and that the centralizer on V has codimension at least 2c(n−c)+ n−c 2 . This is minimal for c = 1, and comparing with the rank of A we deduce n ≤ 4. If n = 4 then necessarily B has rank at least 3a/2, thus contains elements with distinct centralizers. This gives a contradiction with Lemma 2.10 (a). In the case n = 3 we have that Y3 ⊗Y3∗ = V ⊕1, that is, the centralizer on V is strictly smaller than on the tensor product Y3 ⊗ Y3∗ . This leads to a stronger inequality and thus to a contradiction. Since for n = 3 the only invariant weights are of the form t (λ1 + λ2 ), we may conclude that n ≥ 4. 3B. By Proposition 2.4 we may assume that W := CV (U ) is an example for the Levi complement, extended by γ , of any γ -stable parabolic subgroup of G. For n odd, induction starting from the case n = 3 in step 3A shows that only weights of the ˜ n ). form t (λ1 + λn−1 ) are possible, that is, tensor products of Frobenius twists of A(Y But these were ruled out in step 3A. For n even we first study basic modules. For n = 4 by step 3A the only basic highest weight to consider is λ1 + λ2 + λ3 . But this belongs to the Steinberg module for SL4 (2) (by [5, Thm. 2.8.7]), so has dimension 64, bigger than 2n2 − 1. For n = 6, our induction argument together with step 3A only leaves the weight λ1 + λ3 + λ5 , which is ruled out by Lemma 4.3. Now induction shows that there are no cases for even n ≥ 8. Still for n even, if V is an invariant tensor product of r ≥ 2 factors, then necessarily all of its factors have to be invariant. We obtain the condition n(n + 2) 2r d2 . . . dr ≤ r + . 4 a 4 This leads to r = 2, n = 4, d = 2 . Moreover, the two factors have to be Galois conjugate, so 2 | a. The subgroup A ∩ SL4 (q) has centralizer of dimension at most 4 on ∧2 (Y4 ), so its centralizer on V has dimension at most 20 by Corollary 2.11 (a). This forces a = 1, but we just saw that a has to be even in this case. This completes the consideration of the extension by the graph automorphism. 4. Next consider the case that = 2 and G = SLn (2a ).2 is obtained by adjoining a graph-field automorphism γ (so a is even). Here, the centralizer of γ of order 2 2 in SLn (2a ) is SUn (2a/2 ). So by Proposition 3.11 it has 2-rank n2 a/2 if n is even, 2 + 1 a/2 if n is odd. Basic representations do not extend, so respectively n−1 2 V is a non-trivial tensor product. The only possibility for invariant tensor products is V = Yn ⊗ Yn∗ (a/2) . But this is defined over Fa , so Corollary 2.11(a) yields a contradiction. 5. Finally assume that = 2, 2 | a, G = SLn (2a ).22 contains both the graph and the field automorphism of order 2. Here again basic representations are not invariant. The standard inequality (3.5) shows that the case of tensor products does not lead to examples in this situation. 4.2. The case SUn (q). The reasoning for the unitary groups is very similar to the one for linear groups.

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Proposition 4.9. Let G = SU6 (q) and V the exterior cube of the natural module. Let A be a nontrivial elementary abelian unipotent subgroup of G, then f (A) < |V |. Proof. Let A be an elementary abelian unipotent subgroup with f (A) maximal among such subgroups and assume that f (A) ≥ |V |. We actually work in GU6 (q). Let Q be the unipotent radical of the parabolic subgroup P fixing a totally singular three dimensional space in the natural module for G. So |Q| = q 9 . Write P = QL where L ∼ = GL3 (q 2 ) is the Levi complement. We note that f (Q) = q 19 by Theorem 3.1. In particular, since Q is an irreducible P -module, it follows that f (A) < |V | if A ≤ Q. Set B = A ∩ Q. We note that if 1 = x ∈ A, then CV (A) is at most of dimension 14 and moreover if x is not a long root element, then CV (A) has dimension at most 12, by Table 3. Since the only long root elements in A must be contained in Q, it follows that CV (A) has dimension at most 12. In particular, |A| ≥ q 4 . Consider A/B (as a subgroup of L). Note that as a P -module, we have a filtration of V : 0 < C = CV (P ) < W < M < V where W/C and M/W are 9-dimensional Fq L-modules, with W/C isomorphic to the natural module for L tensor its Frobenius twist and M/W the dual of W/C. First suppose that A/B contains a regular unipotent element of L (note that this implies that > 2). Choose x ∈ A with xB this regular unipotent element. It follows that CV (A) is at most 8-dimensional, whence |A| ≥ q 6 . In particular, |B| ≥ q 2 and does not consist of long root elements. Thus, CV (B)/CW (B) has dimension at most 3. This implies that dim CV (A) ≤ 7. In particular, |B| > q 2 and is contained in CQ (x). It follows by inspection that B contains a rank 3 element in Q. Thus, CV (B)/CW (B) is at most 1-dimensional. Thus, CV (A) has dimension at most 1 + 3 + 1 = 5. So f (A) ≤ q 19 , a contradiction. So A/B consists of long root elements in L. Suppose that A/B embeds in a long root subgroup X of L (this is not a long root subgroup of G). If |A/B| > q, then |CS (A/B)| ≤ q 4 where S = Q or S is one of the two nontrivial L-composition factors on V . If B = 1, this implies that f (A) ≤ q 14 . If B = 1, then CV (A) ≤ W and so CV (A) has dimension at most 9. Then |A| > q 5 and so |B| > q 3 . This allows to improve the estimate on CV (A) to be of dimension at most 8, whence |A| = q 6 (since that is an upper bound). In particular, this implies that B has order q 4 and so must be CQ (X). A straightforward computation shows that CV (B) is 4-dimensional. This implies that f (A) ≤ q 16 . The remaining possibility is that A/B is not contained in a long root subgroup but is contained in the unipotent radical of either the stabilizer of 1-space or 2-space in the natural representation of L. In the latter case, we observe that CW/C (A/B) is 1-dimensional. If |B| ≤ q, then CV (A) has dimension at most 2 + 5 = 7, a contradiction. If |B| > q, then CV (A) has dimension at most 2 + 3 = 5 and |A| ≤ q 2 |CQ (A/B)| ≤ q 6 , whence f (A) < |V |. So A/B centralizes a 2-space in the natural representation for L but is not contained in a root subgroup. It follows that |CQ (A/B)| = q 4 , |CW/C (A/B)| = q 4 and

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CM/W (A/B) is 1-dimensional. Thus, CV (A) is at most 7-dimensional and so |A| ≥ q 6 . This implies that B = 1 and so CV (A) is at most 6-dimensional. Thus, |A| = q 7 (since it has order at most q 7 ). In particular, we see that B = CQ (X) where X is the full unipotent radical containing A/B. We then compute that CV (B) is 4-dimensional, whence f (A) ≤ q 18 . Theorem 4.10. The absolutely irreducible 2F -modules in defining characteristic for SUn (q), q = a , n ≥ 3, are as given in Table 7 (see p. 147). Proof. 1. The unitary group SUn (q) is a subgroup of the linear group SLn (q 2 ) defined over the field of q 2 elements. By the general results presented in Section 3.1, the irreducible representations of SUn (q) are the restrictions of those of SLn (q 2 ). We distinguish two cases. 2. First assume that the restriction V to SUn (q) of the SLn (q 2 )-representation is defined over a proper subfield (necessarily of index 2). Then its highest weight has to be invariant under the graph automorphism γ . In particular, each tensor factor is itself invariant γ , so has dimension at least d0 := n2 − 2 for n = 4, 6, respectively nunder d0 := n/2 for n = 4, 6, by Lemma 4.2. We first show that in this case V is a twist ˜ n ), or of ∧k (Yn ) for n = 2k ∈ {4, 6, 8}. of A(Y 2A. If V is a tensor product V1 ⊗ · · · ⊗ Vr of at least two factors, of dimensions di = dim(Vi ) ≥ d0 , then our fundamental inequality (3.3) shows that d0r ≤ r(2n2 −1). With the lower bound for d0 mentioned above this only leaves the case n = 4, r = 2, Vi Galois conjugate twists of ∧2 (Yn ). This is the natural module for O− 6 (q), and Proposition 3.14 in conjunction with Corollary 2.11(a) allows to exclude this. 2B. Hence we may now assume that V is (a twist of) a basic module, of dimension at most 2n2 − 1. By Lemma 4.3 the only possibility to consider is n = 4, λ = 2λ2 , with dim M(λ) ≥ 19. Let P1 = U1 .L1 be a maximal parabolic subgroup with Levi complement L1 of type GU2 (q), so U1 is special of order q 1+4 . First suppose that there exists an offender A ≤ U1 . Since both Z(U1 ) and U1 /Z(U1 ) are irreducible L1 -modules, we may assume by Lemma 2.1 that either A = Z(U1 ) or A = U1 . In the first case, A is a root subgroup, 7 conjugates of which generate G, so the 2F -condition gives d/7 ≤ 2, which is not the case. In the second case A is a unipotent radical, which by Theorem 3.1 has centralizer of dimension 3 on M(λ), too small. So we may assume that A is not contained in U1 . Let P2 = U2 .L2 be the maximal parabolic subgroup with Levi complement L2 = GL2 (q 2 ). Then L2 acts irreducibly on the elementary abelian unipotent radical U2 of order |U2 | = q 4 . Thus, if A is contained in U2 , we may assume that A = U2 . Since two conjugates of U2 generate SU4 (q) by Proposition 3.7, we get dim M(λ) ≤ 16, which is not the case. We may thus assume that A is not contained in a conjugate of U1 or U2 . But then the normal closure of A in the unipotent radical U of a Borel subgroup is U itself, of order q 6 , and with centralizer of dimension 1. This yields the final contradiction for M(2λ2 ).

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˜ n ) and ∧k (Yn ), n = 2k ≤ 8. The case k = 2 is 2C. It remains to deal with A(Y ˜ Example 4.7. For V = A(Yn ) let A be an offender with centralizer of codimension c on the natural module Yn . Then by Proposition 3.11 the 2F -condition becomes 2c(n − c) − 2 ≤ 2c2 . This shows that c ≥ (n − 1)/2, so the stronger inequalities from Lemma 2.10 (a) apply and we obtain a contradiction. The wedge cube ∧3 (Y6 ) is not a 2F -module by Proposition 4.9. Let P = U L be a maximal parabolic subgroup of SU8 (q) with Levi complement of type GU6 (q) and U special of order q 1+12 . If ∧4 (Y8 ) is a 2F -module for SU8 (q), then the previous result implies that an offender must be contained in U . But this is ruled out immediately by Theorem 3.1 and Proposition 3.9. 3. Now assume that the representation of SLn (q 2 ) is defined over the same field as the restriction to SUn (q). Then any example for SUn (q) is a fortiori an example for SLn (q 2 ), hence one of the cases in Table 7. Of these, all except ∧2 (Y4 ), ∧3 (Y6 ), and the fourth example are defined over Fq 2 . The first example, which is the natural module, is Example 4.6. For V = ∧2 (Yn ), n ≥ 5, let A be an offender with centralizer of codimension c on the natural module. With Proposition 3.11 and (2.1) we obtain the inequality 2c(n − c − 1) ≤ 2c2 , which forces c ≥ (n − 1)/2. But this can be ruled out with the stronger assertion from Lemma 2.10 (c). The symmetric square is excluded in the same way. Proposition 4.11. Let V be an absolutely irreducible 2F -module in characteristic for a group G with F ∗ (G) = SUn (q), q = a , n ≥ 3. Then all composition factors of the restriction V |F ∗ (G) are 2F -modules for F ∗ (G). Proof. The case of 2F -modules V with reducible restriction to F ∗ (G) may be dealt with precisely as in the proof of Proposition 4.8. So now assume that V is irreducible for SUn (q). 1. First assume that is odd and G = SUn (a ). involves a field automorphism of order . Since basic representations are not invariant under field automorphisms of odd order, V is a tensor product of at least 2 factors. If all factors Vi are already defined over Fq , then dim Vi ≥ n2 − 2 for n = 4, 6, respectively dim Vi ≥ 6, 20 when n = 4, 6. Our standard inequality (3.3) shows that in this case V must be the product of two 6-dimensional representations for SU4 (q). But then the field of definition does not drop, and the inequality is not satisfied. If at least one factor is defined over Fq 2 then the improved inequality again rules out this case. 2. It remains to consider the case that = 2 and G is obtained from SUn (2a ) by adjoining the graph automorphism of order 2. Since V restricts irreducibly to SUn (2a ), all factors must have symmetric highest weight. If V is basic, then all coefficients of ˜ n ). For the highest weight are at most 1, since = 2. For n = 3 this only leaves A(Y ˜ n ), ∧2 (Yn ) and the Steinberg module with weight λ1 +λ2 +λ3 . The n = 4 we find A(Y dimension 64 of the latter is too large. By Theorem 3.1 we may assume that CV (U ) is an example for the Levi complement of any stable parabolic subgroup P = U.L. ˜ n ). For n ≥ 6 Induction now proves that for n = 5 we only have to consider A(Y n/2 ˜ Lemma 4.3 only leaves A(Yn ), or ∧ (Yn ) with n = 6, 8.

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˜ n ). Indeed, if A is an offender, First note that V cannot be the adjoint module A(Y a with B = A ∩ SUn (2 ) with centralizer of dimension n − c on the natural module, then

c+1 2c(n − c) ≤ 2 + 2/a 2 by Proposition 3.12. This implies c ≥ 2/3n − 1, but then the upper bound for the right hand side from Proposition 3.12 eliminates this case. Let V := ∧3 (Y6 ) for the extension G of SU6 (q) with the graph automorphism. The centralizers in SU6 (q) of both classes of outer involutions are contained in Sp6 (q), for which V has two natural composition factors Y6 (exchanged by outer elements of G) and one spin module Z3 . In particular the centralizer of B := A ∩ Sp6 (q) is contained in a module with composition factors Y6 and Z3 , and outer involutions have rank at least 6. Thus |A| ≥ q 3 , and |B| ≥ q 2 . By Propositions 3.12 and 3.14 such subgroups centralize at most a 4 + 6-dimensional subspace of Y6 ⊕ Z3 , whence |A| ≥ q 5 , |B| ≥ q 4 . Again, such a subgroup has at most 3 + 4-dimensional centralizer on Y6 ⊕ Z3 , forcing |B| = q 6 maximal possible. But then B is conjugate to the stabilizer of a totally isotropic subspace on the natural module, by Lemma 3.13, which has just 1-dimensional centralizer on Y6 . This would imply |B| > q 6 , impossible. The case of ∧4 (Y8 ) is now ruled out by induction. If V is a product of at least two factors, then the usual argument only leaves the possibility n = 4, d1 = d2 = 6, that is, the Vi are natural modules for SO− 6 (q). This can be ruled out with Proposition 3.14 and (2.1). 4.3. The examples for SLn (q) and SUn (q). We proceed to verify that the cases left open in the previous proofs do in fact all lead to examples of 2F -modules. Table 7 contains the examples of 2F -modules V for SLn (q) and SUn (q) of dimension d Table 7. 2F -modules for linear and unitary groups in defining characteristic

G SLn (a ) SLn (a ) SLn (a ) SLn (2a ) SL6 (a ) SUn (a ) SU4 (a )

d n n

V f Yn a 2 ∧ (Yn ) a 2 n+1 a

2 (Yn ) 2 (a) 2 n Yn ⊗ Y n a 3 20 ∧ (Y6 ) a n Yn 2a 2 6 ∧ (Yn ) a

conditions n≥4 ≥3

log |A| a (n − 1)a (n − 1)a 2n2 /4a 5a a a

defined over Ff , up to Frobenius twists and taking duals, together with the size of one possible offender A.

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Case 4.1: On its natural module Yn , SLn (q) acts as a transvection group, so we get an example with |A| = q. Case 4.2: When n ≥ 4 the alternating square ∧2 (Yn ) of the natural module is irreducible for SLn (q). The subgroup 1 1 (4.1) A := ai ∈ Fq 1 a1 . . . an−1 1 on ∧2 (Yn ), so this of order q n−1 visibly has centralizer of dimension at least n−1 2 yields an example. Case 4.3: For > 2 the symmetric square 2 (Yn ) is irreducible. Again the subgroup A in (4.1) gives an example: its centralizer has dimension at least n2 . (a) Case 4.4: Let q = 2a and Yn the a-th Frobenius twist of the natural module. (a) Then Yn ⊗ Yn is irreducible for SLn (q) and defined over the subfield Fa of Fq of index 2. Choosing for A a maximal elementary abelian -subgroup, condition (1.1) reads 2 n −1 2 n a − za ≤ 2 · 2a 4 where z ≥ 1 denotes the dimension of the centralizer of A. Thus this gives an example. Case 4.5: For n = 6 the alternating cube ∧3 (Y6 ) gives an example for SLn (q) with the offending subgroup A from (4.1): its rank equals 5a, and it centralizes at least the alternating cube of a 5-dimensional subspace, of dimension 10. Case 4.6: The special unitary group SUn (q) acts as a transvection group on its natural module Yn over Fq 2 , thus we get an example with |A| = q. Case 4.7: The 4-dimensional unitary group SU4 (q) is isomorphic to the 6-dimensional orthogonal group of minus-type. The latter is a bi-transvection group, so we get an example with |A| = q.

5. Spin modules In order to treat the symplectic and orthogonal groups we now collect some facts on the 2n -dimensional spin module Zn of the orthogonal group Spin2n+1 (q), with highest weight λ1 (this includes the case of symplectic groups in even characteristic), and of (±) the two 2n−1 -dimensional half-spin modules Zn−1 of orthogonal groups Spin2n (q) with highest weight λ1 or λ2 . 5.1. Ranks on spin modules. For the following lemma we make use of the known classification and closure relation on the set of unipotent classes, which is described

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for example in the book of Spaltenstein [16]. Let K be the algebraic closure of a finite field Fq . Lemma 5.1. Let n ≥ 5, 1 = x ∈ G := Spinn (K) be unipotent, V a (half ) spin module for G. Then (a) dim CV (x) ≤ (3/4) dim V , and (b) dim CV (x) ≤ (5/8) dim V unless n = 8 or x is a long root element. Proof. Since the spin module for Spin2k−1 (K) is the restriction of the half-spin module for Spin2k (K) and the class of root elements of Spin2k (K) intersects Spin2k−1 (K) in the class of long root elements, it is sufficient to prove the assertion for n even. Clearly root elements x of Spin4 (K) ∼ = SL4 (K) have dim CV (x) = (3/4) dim V on the natural module V for SL4 (K). Now the half-spin module of Spin2k (K) splits into the two different half-spin modules of Spin2k−2 (K) upon restriction, so the first claim follows. For the remaining part let us assume that q is even (the case of odd characteristic is similar). Then the unipotent classes of GO2k (K) are parameterized by pairs (λ, ) of a partition λ of 2k in which each even part occurs an even number of times (giving the Jordan blocks in the natural representation) and a map from the parts of λ to {0, 1}. Furthermore, if the class parameterized by (λ1 , 1 ) lies in the closure of the class parameterized by (λ2 , 2 ) then λ1 ≤ λ2 in the dominance order on partitions, plus a condition on the i (see [16, I.2.5]). Now long root elements lie in the class parameterized by 202 12k−2 , and there are just two unipotent classes directly above this one, parameterized by 22 12k−4 and 204 12k−8 . Let us first consider k = 4. Then the class 204 splits into two SO2k (K)-classes, and all three classes are fused under triality. By our remarks above, the class 22 14 has rank 2 on the natural module while the other two classes have rank 4. Via triality this implies that 22 14 has rank 4 on the half-spin modules, while the other two classes have rank 2 on one of the half-spin modules, rank 4 on the other. Since all other unipotent classes of SO2k (K) lie above all three of these classes, the assertion is proved for k = 4. For k ≥ 5 the two classes above 202 12k−2 don’t split in SO2k (K). The rank can be computed inductively by restricting to SO2k−2 (K), and we obtain that 22 12k−4 has rank (1/2) dim V while 204 12k−8 has rank (3/8) dim V . Lemma 5.2. Let V be a half-spin module for Spin+ 2n (q) with n ≥ 3. If |A| > q, then dim CV (A) ≤ (5/8) dim V . Proof. By Lemma 5.1 it suffices to consider A consisting of root elements. By Lemma 3.9 we see that A ≤ U1 , the unipotent radical of NG (X) where X is a root subgroup (this is the stabilizer of a two dimensional totally singular subspace in the natural representation). We may assume that there is a nontrivial element x ∈ A ∩ X. Since |A| > q, we may choose y ∈ A \ X. Then x, y is trivial on a space of + codimension 4 in the natural representation and so embed in Spin+ 8 (q) × Spin2n−8 (q).

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Thus, it suffices to consider n = 4 (since the spin module for G is a direct sum of copies of the spin module when restricted to Spin+ 8 (q)). We see by Proposition 3.14 that the fixed space of x, y on the spin module is at most 5 dimensional. The key to studying the spin modules is the existence of maximal parabolic subgroups whose unipotent radical has large centralizer space. Indeed, let G = (±) Spinn (q), and P = U L a maximal parabolic subgroup with U elementary abelian (±) of order |U | = q n−2 and L of type Spinn−2 (q). Then upon restriction the (half-) spin module V for G splits into two (distinct half-)spin modules for L. Moreover, W := CV (U ) is a (half-)spin module for L. Let A ≤ P be a unipotent subgroup of G. Then we have f1 (A) = |A| |CV (A)| ≤ |AU/U | |CW (AU/U )| |A ∩ U | |CV (A ∩ U )/W | = f1W (AU/U )|A ∩ U | |CV (A ∩ U )/W |,

(5.1)

and the corresponding equation for f (A). 5.2. Spin modules for even-dimensional orthogonal groups 9 Proposition 5.3. On the half spin modules V of Spin+ 10 (q) we have |A| |CV (A)|< |V | for all unipotent elementary abelian subgroups A = 1. 7

9 Proof. Let 1 = A ≤ Spin+ 10 (q) with f0 (A) := |A| |CV (A)| maximal and assume 7

36

that f0 (A) ≥ |V |. Root elements have rank 4 on V , so |A| ≥ q 7 > q 5 . In particular A contains non-root elements, of rank at least 6, whence |A| > q 7 . Let P = U L be a maximal parabolic subgroup containing A with U abelian of order q 8 and L of type Spin+ 8 (q). Since CV (U ) has dimension 8 we have A ≤ U . Assume first that AU/U consists only of root elements, so |AU/U | ≤ q 3 and |A ∩ U | > q 4 . Then CV (A) 72 has codimension at least 8 by (5.1), forcing |A| ≥ q 7 > q 10 , larger than possible. Thus |AU/U | > q 3 . Via triality we may interpret the spin module W := CV (U ) for L as the natural module, and then by Proposition 3.14 the centralizer of AU/U has codimension at least 4 on W . Since A ∩ U = 1, its centralizer has codimension at least 4 as well, whence |A| > q 10 , the final contradiction. Proposition 5.4. The half spin modules M(λ1 ), M(λ2 ) of Spin+ 2n (q) are not F modules for n ≥ 6. Proof. Let V denote one of the two half-spin modules for Spin+ 12 (q), A a possible offender on V , and P = U L a maximal parabolic subgroup containing A with U abelian of order q 10 , L of type Spin+ 10 (q). The ranks in Lemma 5.1 and Proposition 3.9 show that |A| ≥ q 12 . Thus neither A ≤ U nor A ∩ U = 1. Assume first that |A ∩ U | > q 5 . Then A ∩ U contains not only root elements, whence CV (A) has codimension at least 4 + 12 = 16 by (5.1) and Lemma 5.1. Thus |A| ≥ q 16 , too large. Hence we have that |A ∩ U | ≤ q 5 , and |AU/U | ≥ q 7 . Then (5.1) gives centralizer codimension at least 6 + 8 = 14. We conclude that |A| ≥ q 14 , so |AU/U | ≥ q 9 . But

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now Proposition 5.3 applied to AU/U ≤ L on W shows that |CW (AU/U )| < q 9 , so |A| > q 15 , again too large. The case of n ≥ 7 follows by induction. Proposition 5.5. The half spin modules M(λ1 ), M(λ2 ) of Spin+ 2n (q) are not 2F modules for n ≥ 7. Proof. Let V denote one of the two half-spin modules for Spin+ 14 (q), A a possible offender on V , and P = U L the stabilizer of a singular vector containing A with U of order q 12 , L of type Spin+ 12 (q). We first claim that for any subgroup B ≤ U with |B| > q we have f1 (B) ≤ q 44 . Indeed, choose such a subgroup with f1 (B) maximal and subject to that with |B| maximal. Since q < |B| we have |CV (B)| ≤ q 40 by Lemma 5.2, so f1 (B) ≤ q 43 if |B| ≤ q 3 . By the maximality assumption, B is a vector space over Fq (this will not change CV (B)), so |B| ≥ q 4 . Note that either B contains a totally singular 2-space (considering B as an L-module) or B contains a nondegenerate 2-space. Let g ∈ L stabilize this 2-space. We see that f1 (BB g ) ≥ f1 (B) (since |B ∩ B g | ≥ q 2 ) and so by the choice of B, B g = B for any such g. Witt’s theorem implies that either B = U or the 2-space is the radical of B — now choose a different 2-space to see that B = U . Thus, f1 (B) = f1 (U ) = q 12+32 = q 44 . Now let W := CV (U ) and set B := A ∩ U . As usual we see from Lemma 5.1 that |A| ≥ q 12 . Proposition 3.14 implies that B = 1. If A = B, then f (A) = f (U ) = q 56 < |V |. So B = 1 and AU/U = 1. Now f (A) ≤ |A| |AU/U | |CW (AU/U )| |A ∩ U | |CV (A ∩ U )/W | = |A|f1W (AU/U ) f1 (B)/|W |, so Proposition 5.4 implies that f (A) < |A|f1 (B) ≤ |A|q 44 . Thus, |A| > q 20 . This implies by Lemma 3.16 that A ≤ Q where Q is the unipotent radical of the stabilizer of a maximal totally isotropic space. Thus, f (A) ≤ f (Q). However, CV (Q) has dimension either 1 or 7 (depending upon which class of maximal totally isotropic subspaces we take). Thus, f (Q) ≤ q 49 , a contradiction. The case of n ≥ 8 follows by induction. Proposition 5.6. The half spin modules M(λ1 ), M(λ2 ) of Spin− 2n (q) are not F modules for n ≥ 4. Proof. The half-spin module V for Spin− 6 (q) is the natural module for SU4 (q), defined over Fq 2 . Proposition 3.11 shows that an offender A with f1 (A) ≥ |V | necessarily has |A| = q 4 maximal possible and thus f1 (A) = |V |. Now let G = Spin− 8 (q). Long root elements have rank 2 on the half-spin module V , 2 so an offender A has |A| ≥ q 4 . Application of Lemma 5.2 to the overgroup Spin+ 8 (q ) 6 shows that dim CV (A) ≤ 5, whence |A| = q is maximal possible. Consider a maximal parabolic subgroup P = U L containing A with U abelian of order q 6 and Levi complement L of type Spin− 6 (q). If A ≤ U , then A = U , and Theorem 3.1 gives

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a contradiction to the F-condition. Otherwise, W := CV (U ) is a half-spin module for L, so from (5.1) and the first part of the proof we have f1 (A) < q 6 q 2 f1W (AU/U ) ≤ q 16 . The case of n ≥ 5 now follows by induction using Theorem 3.1. Proposition 5.7. The half spin modules M(λ1 ), M(λ2 ) of Spin− 2n (q) are not 2F modules for n ≥ 6. Proof. The half-spin modules V of dimension 32 for Spin− 12 (q) are defined over Fq 2 . As usually we see from Lemma 5.1 that |A| ≥ q 12 . Let U be the unipotent radical of a maximal parabolic subgroup containing A and with Levi subgroup of type Spin− 10 (q). Note that AU/U = 1, so dim CW (AU/U ) ≤ 12. If A ∩ U contains not only long root elements then dim CV (A ∩ U )/W ≤ 4 and |A| ≥ q 16 , too large. Thus |A ∩ U | ≤ q 4 , so |AU/U | ≥ q 8 , dim CW (AU/U ) ≤ 10,

and

dim CV (A ∩ U )/W ≤ 8,

whence |A| ≥ q 14 . But then |AU/U | = q 10 is maximal possible and only centralizes half of W . This forces |A| ≥ q 16 , again too large. The case of n > 6 follows by induction using Theorem 3.1. 5.3. Spin modules for odd-dimensional orthogonal groups Proposition 5.8. The spin module M(λ1 ) is not an F -module for Spin2n+1 (q), n ≥ 4, q odd. Proof. On the spin module for Spin9 (q), q = a odd, long root elements have rank 4, and all other non-trivial elements have rank at least 6 by Lemma 5.1. Thus an offender A has |A| ≥ q 6 . Consider a maximal parabolic subgroup P = U L with U abelian of order q 7 and L of type Spin7 (q). The case that A ≤ U is excluded by Theorem 3.1. Let W := CV (U ), the spin module for L. If A∩U consists of long root elements only, then |AU/U | ≥ q 3 by Table 5. But such a group has |CW (AU )| ≤ q 5 by Lemma 5.1, while |CV (A ∩ U )/W | ≤ q 4 . So (5.1) gives |A| ≥ q 7 , whence |AU/U | ≥ q 4 . This has centralizer codimension at most 4 on W , forcing |A| ≥ q 8 , too large. Otherwise, |CW (AU/U )| ≤ q 6 and |CV (A ∩ U )/W | ≤ q 2 , which is again impossible. By induction this also excludes the spin module for Spin2n+1 (a ), n ≥ 5. Proposition 5.9. The spin module M(λ1 ) is not a 2F -module for Spin13 (q). Proof. The half-spin modules of Spin+ 14 (q) restrict irreducibly to the spin module of the subgroup Spin13 (q). Thus the result follows from Proposition 5.5.

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5.4. Spin modules for symplectic groups. Our Lemma 5.1 also applies to the spin module of Sp2n (2a ), but note that long roots of Spin2n+1 (2a ) become short roots of Sp2n (2a ). Proposition 5.10. The spin module M(λ1 ) is not an F -module for Sp2n (q), q even, when n ≥ 4. Proof. Let V = M(λ1 ) be the spin module for Sp8 (q) of dimension 16. From Lemma 5.1 we find that |A| ≥ q 4 . By Proposition 3.9 (viewing our group as a group of type Bn where the long roots are short roots in type Cn ), A can’t consist of short root elements only, so there are elements of rank at least 6 and |A| ≥ q 6 . Let P = U L be a maximal parabolic subgroup containing A with U abelian of order q 7 and L of type Sp6 (q), the normalizer of a long root subgroup X ≤ U . If A ≤ U then f1 (A) ≤ f1 (U ) = q 7 q 8 < |V |. On the other hand, if B := A ∩ U = 1 then |AU/U | = |A| = q 6 is maximal possible. By Lemma 3.13 A is the centralizer in Sp6 (q) of a maximal isotropic subspace on Y6 , which has 1-dimensional centralizer on the spin module Z3 , a contradiction. So B is a nontrivial proper subgroup of A. Let W := CV (U ), the spin module for L. This embeds Sp6 (q) into Spin+ 8 (q) on W , which may be viewed as the natural (q) by triality. In particular Proposition 3.14 gives upper bounds for module for Spin+ 8 unipotent subgroups of L with given centralizer on W . Suppose that |B| ≤ q. Then |AU/U | ≥ q 5 and so CW (AU/U ) has dimension at most 4 and CV (A)/CW (A) has dimension at most 4. Thus, |A| ≥ q 8 and |A/B| ≥ q 7 , too large. So |B| > q. Then Lemma 5.2 implies that |CW (AU/U )| ≤ q 6

and

|CV (A ∩ U )/W | ≤ q 2 ,

so |A| ≥ q 8 and |AU/U | ≥ q. But U/X is the natural module for L, so a subgroup of order q centralizes at most a subspace of size q 6 of U . Thus |A ∩ U | ≤ q 6 , implying |AU/U | ≥ q 2 . But this only centralizes a 5-dimensional subspace, whence |AU/U | ≥ q 3 . This in turn has at most 5-dimensional centralizer on W , so |A| ≥ q 9 . This forces |AU/U | ≥ q 4 , with at most 4-dimensional centralizer on W . Hence |A| ≥ q 10 and |AU/U | ≥ q 5 , but then |A ∩ U | ≤ q 4 , a contradiction. The case of n ≥ 5 follows by induction applying Theorem 3.1 to the maximal parabolic subgroup stabilizing a 1-space.

6. Symplectic groups Let Y2n denote the natural 2n-dimensional symplectic module for Sp2n (q) and Zn the 2n -dimensional spin module. We need some information on small basic modules from Lübeck [11, Thm. 4.4]:

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Lemma 6.1. Let M(λ) be a basic highest weight module for Sp2n (a ), n ≥ 2, of dimension dim M(λ) ≤ 4n2 + 2n + 1. Then M(λ) is one of the modules in Table 8. Here the fundamental weights are numbered according to Table 4. Table 8. The basic Sp2n (a )-modules of dimension at most 4n2 + 2n + 1

λ λn

M(λ) Y2n

λn−1

∧˜ (Y2n )

2λn

2 (Y2n )

λ1 λ1 + λ 2 2λ1 3λ2 λ1 λ1 λ2 λ3

Zn

2

3 (Y

2)

dim M(λ) 2n 2n 2 − 1 − δ,n 2n+1 2 2n

16 − 4δ,5 14 − δ,5 20 14 42 − δ,3 48 − 8δ,3 110 − 10δ,2

>2 n ≤ 8, n=2 n = 2, n = 2, n = 3, n = 4, n=4 n=5

=2 >2 >3 >2 >2

6.1. The case Sp4 (q). It is convenient to handle the case of Sp4 (q) ∼ = Spin5 (q) separately. (Note that the isomorphism interchanges the two fundamental weights λ1 and λ2 .) Proposition 6.2. Let V be an absolutely irreducible 2F -modules in characteristic for a group G with F ∗ (G) = Sp4 (q), q = a = 2. Then the composition factors of V |F ∗ (G) are as given in Table 9 (see p. 163). Proof. 1. We start by considering modules for G = Sp4 (q). The two end-node parabolic subgroups P1 = U1 .Sp2 (q)(q − 1) and P2 = U2 .GL2 (q) have special respectively abelian unipotent radical of order |Ui | = q 3 . Let U0 denote the invariant 1-dimensional subspace of U1 . 1A. If there exists an offender A ≤ U0 , then by Lemma 2.1 we may assume that A = U0 . This is a full long root subgroup, five of which generate G. Hence the 2F -condition for a module V of dimension d defined over Ff reads df/5 ≤ 2a, that is, d ≤ 10a/f . On the other hand, if A is contained in U1 but not in U0 , then we may choose A = U1 . Now two conjugates of A generate G, so we arrive at the condition df/2 ≤ 6a. The same argument applies when A ≤ U2 . Hence in any case d ≤ 12a/f . If V = V1 ⊗ · · · ⊗ Vr then using that di = dim(Vi ) ≥ 4 and f ≥ a/r we

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get (4)r ≤ 12r. This is only satisfied when r = 2, d1 d2 ≤ 24 and V1 , V2 are Galois conjugate, or when r = 1, d ≤ 12. In the first case necessarily d1 = d2 = 4, which is Example 6.4. If r = 1 and d = 12 then A ∈ {U1 , U2 } and dim CV (A) = 6. But Lemma 6.1 shows that V = M(λ1 + λ2 ), whence dim CV (A) ≤ 4, too small. On the other hand, if d ≤ 11, then V is one of the modules in Lemma 6.1 which we will consider in 1B. 1B. If there exists an offender contained in none of U1 , U2 , then it must involve root elements corresponding to both simple roots. But the normal closure of such a subgroup inside the Borel subgroup is the whole unipotent radical. Then |A| = q 4 , dim CV (A) = 1, so (d − 1)f ≤ 8a. This forces d ≤ 9, and again we are in one of the cases of Lemma 6.1. So V has highest weight λ1 , λ2 or 2λ2 . The first two are Examples 6.1 and 6.2. The highest weight module for 2λ2 , = 2, is the symmetric square of Y4 . Let A be an offender with centralizer of codimension c on Y4 . Then the centralizer on 2 (Y4 )hascodimension at least c(5 − c) by (2.1). Since the rank of A 3a is bounded above by c+1 2 a by Proposition 3.12, we conclude that c = 2, |A| = . But then the stronger inequality in Lemma 2.10 (b) gives a contradiction. 2. Now assume that G = Sp4 (a ). involves a field automorphism, so | a. Let first V be a module with irreducible restriction to Sp4 (a ). Since basic representations are not invariant this implies that V is a product of r ≥ 2 factors. Then (3.4) yields that r = = 2 and d1 = d2 = 4. Let A be a candidate offender with centralizer of dimension 4−c on the natural module. For these, by Lemma 2.10 and Proposition 3.12 we obtain c = a = 2. But in this case the centralizer of an outer involution is isomorphic to S6 × 2, so has 2-rank 4, and our inequality is violated. If V˜ is induced from V , then (3.5) yields that r = 1. If d = 4 then by Proposition 3.12 we have d( − 1) + c ≤ c2 + c + 2/a for a subgroup A with centralizer dimension 4 − c. This is satisfied if d = 4, the natural module. Otherwise A has at most d − 2-dimensional centralizer on V˜ , and we have d( − 1) ≤ 6, which by Lemma 6.1 gives no further example. 3. Finally we assume that = 2, 2 a and G = Sp4 (2a ).2 involves a graphautomorphism. Since we treated Sp4 (2) = S6 in the first part [6, Prop. 3.1], we may assume that a ≥ 3. Our standard inequalities allow to rule out this case immediately.

6.2. Some fundamental representations. We first recall properties of the fundamental representations of the symplectic group. In characteristic 0, the fundamental representations occur as direct summands of exterior powers of the natural representation as follows:

∧n+1−k (Y2n ) = M(λk ) ⊕ M(λk+2 ) for 1 ≤ k ≤ n as Sp2n (C)-modules (where M(λk+2 ) is the zero-module for k + 2 > n). In positive characteristic, the fundamental representations still occur inside the exterior powers of the natural representation. If the dimensions of the M(λk ) are known, then the composition factors of ∧k (Y2n ) can be worked out inductively, using the restriction

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formula

∧k (Yn )|GLn−1 = ∧k (Yn−1 ) ⊕ ∧k−1 (Yn−1 ). Proposition 6.3. The highest weight module M(λ1 ) of Sp2n (q), q odd, is not a 2F module for n ≥ 3. Proof. Let V = M(λ1 ) be the irreducible Sp6 (q)-module of dimension 14. Note that ∧3 (Y6 ) = V ⊕ Y6 . Thus, by Table 3, long root elements have rank 5, elements in Sp4 (q) which are not long root elements have rank 6, and elements with 3 Jordan blocks of size 2 have rank 7. Let P be the stabilizer of a totally isotropic 3-dimensional space in the natural module. So P = QL where L = GL3 (q) and |Q| = q 6 . Consider an L-composition series of V : 1 < F < W < M < V where F and V /M are trivial 1-dimensional modules, W/F is the symmetric square of a natural module for L and M/W is its dual. Moreover, Q ∼ = (W/F )∗ as L-modules. In particular, if 1 = x ∈ L, then dim CW/F (x) ≤ 3 by (2.1). Let A be a nontrivial elementary abelian unipotent subgroup with f (A) maximal and assume that f (A) ≥ |V |. If A ≤ Q then f (A) ≤ f (Q) = q 13 < |V |, impossible. Thus both B := A ∩ Q = 1 and AU/U = 1. Since Q ∼ = (W/F )∗ this forces |B| ≤ q 3 . Moreover, since CV (B) ≤ M we find that dim CV (A) ≤ 7, and |A| > q 3 . In particular, |B| > q, and B contains short root elements. But then dim CV (A)/CW (A) = dim CM (A)/CW (A) ≤ 2, so dim CV (A) ≤ 6 and |A| ≥ q 4 . We note also that we can view A as a vector space over Fq (since we may replace the nilpotent part of any element of A by a scalar multiple – these still generate an elementary abelian unipotent subgroup and centralize the same space). In particular, |A| is a power of q. First assume that AU/U doesn’t consist of root elements of L only. Since CV (B) ≤ M this implies that CV (A) has dimension at most 5. Thus |A| > q 4 , |B| > q 2 , contradicting the fact that AU/U only centralizes a 2-dimensional subspace of Q. So A/B consists of root elements of L. If |B| = q 3 , we see that CV (B) is at most 6-dimensional (because CV (B)/CW (B) has dimension at most 2 and CW (B) has dimension at most 4). Suppose that CV (A) = CV (B). Then B := Ag | g ∈ Q also centralizes CV (A). Since [A, Q] = CA (Q) (since A does not act quadratically on Q), B = B. The fact that the centralizers are the same on W implies that they generate the same Fq -subspace of Q and so CW (B) has dimension less than 4. Thus, CV (A) has dimension at most 5 and |A/B| = q 2 . It follows that A/B is the full unipotent radical of a maximal parabolic subgroup of L (and the one that centralizes a 3-dimensional subspace of Q). Thus, A/B centralizes a 1-dimensional subspace on W/F and so CV (A) has dimension at most 4. Thus, CV (A) has dimension 4 and |A| = q 5 . We leave this case for the moment and return to it later. So |B| < q 3 and |A| < q 5 , whence CV (A) has dimension at least 6 and so CW (A) has dimension at least 4. So |B| = q 2 and |A/B| = q 2 . It follows that CV (A) = CV (B) is six dimensional – a contradiction as above.

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We go back to the remaining case and consider the other parabolic U J = R = NG (X) where X is a long root subgroup. We see that B does contain a long root subgroup and so we may take X ≤ A < R. Let B1 = A ∩ U . If |B1 | > q 2 , then A ≤ C(B1 ) is conjugate to Q, a case we have already dealt with. Otherwise, |B1 | ≤ q 2 and so since |A| = q 5 , we have |B1 | = q 2 and A/B1 of order q 3 in J . Taking a composition series for J consisting of the orthogonal module, the natural module and the orthogonal module, we see that since X ≤ A, the centralizer of A is contained in the first two. Since |A/B| = q 3 this is the unipotent radical of the stabilizer of a totally singular 2-space in the natural module of J ; this fixes a 2-space in the natural module and a 1-space in the orthogonal module. It follows that CV (A) is 3-dimensional. Thus, f (A) = q 13 . Thus, V is not an 2F -module. The case of n ≥ 4 follows by induction. Proposition 6.4. The highest weight module M(λ2 ) of Sp8 (q) is not a 2F -module. Proof. In characteristic different from 3 we have ∧3 (Y8 ) = M(λ2 ) ⊕ Y8 . Thus from Table 3 we see that transvections have rank 14 on V , and bitransvections (and hence by [16, p. 237] any other non-transvections) have rank at least 20. Since A can’t consist of transvections only by Proposition 3.9, this forces |A| = q 10 , maximal possible. But this means that A is the unipotent radical of a GL4 (q)-parabolic, which leads to a contradiction with Theorem 3.1. In characteristic 3 the wedge cube ∧3 (Y8 ) has composition factors M(λ2 ) and two copies of Y8 . Let P = U L be a maximal parabolic subgroup with Levi factor of type Sp6 (q), and special unipotent radical U of order q 1+6 . Then the restriction of V to the Levi factor has composition factors M(λ1 ) and two copies of M(λ2 ) (the heart of the exterior square ∧2 (Y6 )) for Sp6 (q). Clearly the centralizer of any element of Sp6 (q) on V is at most as large as on the semisimplification of V . For the latter, Table3 shows that transvections have rank at least 13. Hence |A| > q 6 and A has to contain non-transvections. Any other non-trivial unipotent elements have rank at least 18, whence |A| ≥ q 9 . Then |A ∩ U | ≥ q 3 , so this group contains non-transvections. These contribute rank at least 18 on V . On the other hand the image of A in L, of order at least q 5 , has centralizer codimension at least 4 on the ˜ 2 (Y6 ). This forces |A| ≥ q 11 , impossible. Sp6 (q)-module CV (U ) = ∧ Proposition 6.5. The highest weight module M(λ3 ) of Sp10 (q) is not a 2F -module. Proof. Let first q be odd. Then ∧3 (Y10 ) has composition factors V := M(λ3 ) and M(λ1 ) = Y10 . As Y10 is self-dual for Sp10 (q) we have ∧3 (Y10 ) = M(λ3 ) ⊕ Y10 . Thus from Table 3 and Corollary 3.4 we see that transvections have rank 27 on V , and any other non-trivial unipotent element has rank at least 42. With Proposition 3.12 this implies that an offender on V has to consist of long root elements, but this is a contradiction with Proposition 3.9.

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In characteristic 2 the wedge cube ∧3 (Y10 ) has composition factors V := M(λ3 ) and two copies of Y10 . In order to determine the centralizers of transvections and bitransvections on V , we restrict to Sp8 (q). As an Fq Sp8 (q)-module, V has composition factors M(λ2 ) and two copies of M(λ3 ). Clearly the centralizer of any element of Sp8 (q) on V is at most as large as on the semisimplification. Using the results in the proof of Proposition 6.4 we find that transvections have rank at least 26 on V , and bitransvections (and hence by [16, p. 237] any other non-transvections) have rank at least 40. We may now conclude as before. Let G := Sp2n (q), n ≥ 2. Let En denote the codimension 1 submodule of the exterior square of the natural module for G. This is either an irreducible module or contains a 1-dimensional trivial module (the latter occurring precisely when | n). Let V be the nontrivial irreducible composition factor of En . In particular, dim V = 2n2 − n − δ where δ = 1 unless | n in which case δ = 2. Let P be the parabolic subgroup of G fixing a 1-space in the natural representation. Then P = NG (X) where X is a long root subgroup. Let U be the unipotent radical of P . Write P = LU where L is the Levi complement. Note that U/X is the natural L-module and that U is special if is odd and is abelian if q is even. Let W = CV (U ), W = CV (U − ) and M = CV (X). We see that W and W are natural L-modules and that M/W is isomorphic to ∧2 (W ) as an [L, L]-module if does not divide n and is isomorphic to En−1 otherwise. By (2.1) any nontrivial unipotent element of G has rank at least 2n − 2 on V . Lemma 6.6. Let B be a nontrivial subgroup of U . Then (CV (B) + M)/M has dimension at most 1. If B ∩ X = 1 or |B| > q, then CV (B) ≤ M. In particular, if A is a subgroup of P and A ∩ U = 1, then dim CV (A)/CM (A) ≤ 1. Proof. Note that if 1 = x ∈ X, then CV (x) = CV (X) = M and the result is clear. Note also that all elements in U \ X are conjugate in P . So we can choose any such element x. Choose a basis for the natural module for G of the form e, f, v1 , . . . , vn−2 where e generates the space fixed by P and the vi are in e⊥ . Now choose x ∈ U with xf = f + vn−2 . Let z ∈ CV (B) \ M. So z = f ∧ v + m where m ∈ M. Thus, z = xz = xf ∧ v + m + e ∧ v , where v ∈ e⊥ . Thus, z = (f + vn−2 ) ∧ v + m. This implies that vn−2 ∧ v = 0. Thus, v is a multiple of vn−2 (modulo e) and the result follows. If |B| > q, the same argument shows that CV (B) ≤ M. This proves the first statement. The second statement follows for CV (A)/CM (A) ∼ = (CV (A) + M)/M ⊆ (CV (A ∩ U ) + M)/M. Lemma 6.7. Let A be an elementary abelian subgroup of U . If |AX| ≥ q d+1 , then CM (A)/W has dimension at most c(c − 1)/2 where c = 2n − 2 − d. Proof. Fix y ∈ U \ X. Let W1 be the fixed hyperplane of y on W . Fix w ∈ W not in W1 with yw = w + e. Identifying M/W with the exterior square of W , we see that

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y − 1 maps w ∧ w to e ∧ w. Thus, the image of y − 1 is precisely e ∧ W1 and has dimension 2n − 3. Moreover, we see that the fixed space of y on M/W is the exterior square of W1 . So we see that if y1 , . . . , yd are linearly independent elements in U/X, then the centralizer of the subgroup generated by them (modulo W ) is isomorphic to the exterior square of a c-dimensional subspace of W , whence the result. Lemma 6.8. Let A be an elementary abelian unipotent subgroup of G := Sp2n (q). Assume that n ≥ 3 and |A| ≤ q n(n−1)/2 . Then f (A) < |V |. Proof. The fundamental inequality from (2.1) and Proposition 3.12 for an offender A on V with centralizer codimension c on the natural module reads

c+1 c(2n − c − 1) ≤ 2 , 2 so c ≥ n − 1 and |A| ≥ q n(n−1)/2 . In particular A contains non-trivial elements with different centralizers, and the stronger inequality from Lemma 2.10(c) forces |A| > q n(n−1)/2 . Lemma 6.9. Let G = Sp6 (q). If V is the 14 dimensional irreducible (indecomposable in characteristic 3) with highest weight λ2 , then (1) V is not an F -module; and (2) f (A) ≤ q 15 for any elementary abelian unipotent subgroup 1 = A of G. Proof. By the previous result, f (A) < |V | if |A| ≤ q 3 . Note that f (Q) = q 15 . So this inequality holds for any subgroup A of Q as well (since either |A| ≤ q 3 or the weak closure of A in Q is Q and so f (A) ≤ f (Q)). Similarly any subgroup A of U satisfies f (A) ≤ f (U ) = q 14 . Indeed, if q is odd, then |A| ≤ q 3 and so f (A) < q 14 . Moreover, we can replace A by the Fq -subspace it generates (this will not change the centralizer in V ) and so f (A) ≤ q 13 for q odd. The same argument applies if q is even and |A| ≤ q 3 . Let A be an abelian unipotent subgroup of G with f (A) maximal. So |A| > q 3 . Set B = A ∩ U . If B = 1, then A embeds in L and so |A| ≤ q 3 , a contradiction. So B = 1. By Lemma 6.6 this implies that we may assume that X ≤ B (for either A ∩ X = 1 and CV (A)/CV (AX) has dimension at most 1 or CV (A) = CV (AX)). Similarly, we may assume that B is an Fq -subspace of U . If |B| > q 3 , then q is even and CL (B) has order at most q (the subspace generated by B will contain a nondegenerate space). Thus, f (A) ≤ q 2 f (B). Now CW (A) has dimension at most 3 and CW (B) = W , so in fact f (A) ≤ qf (B) ≤ q 15 . The same argument applies unless B/X is totally isotropic (which is always the case for q odd). So B/X is totally isotropic. In particular, f (B) ≤ q 13 . If |A| ≤ q, this implies that f (A) ≤ q 2 f (B) ≤ q 15 . So AU/U centralizes at most a 2-dimensional space on W . So f (A) ≤ |A/B|2 f (B)/q 2 ≤ q 15 unless |A/B| > q 2 . In the latter case, we see that the Fq -subspace generated by A/B in L is the unipotent radical of the

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stabilizer of a totally singular 2-space (in the natural module of L). It follows that A/B centralizes only a 1-dimensional space in the orthogonal module M/W . So CV (A) has dimension at most 3, whence f (A) ≤ q 12 q 3 = q 15 . This proves the second statement. Since f1 (A) = f (A)/|A|, the only possibility for f1 (A) ≥ |V | is that |A| ≤ q and centralizes a hyperplane. Since G contains no transvections on V , this cannot occur. Proposition 6.10. Let A be an elementary abelian unipotent subgroup of G := Sp2n (q). Assume that n ≥ 4. Then f (A) < |V | unless possibly f (A) = |V | when n = 4 and q is even. Proof. Choose A with f (A) maximal and assume the contrary. We keep notation as above. In particular, W and W are the spaces centralized by U and U opp and M = CV (X). Set V = M/W and as we have remarked V is either the exterior square or the irreducible constituent of the exterior square of the natural module for L. We induct on n. We may assume that A ≤ P , and, by Lemma 6.8, that |A| > q n(n−1)/2 . Set B = A ∩ U . Suppose that A centralizes some nondegenerate subspace of the natural module or more generally A ∩ U = 1. There is no loss of generality in assuming that A ≤ L. Thus, f (A) ≤ |A|2 |CV (A)||W |2 /q 2 . This is less than |V | by induction if n > 4 and by the previous lemma if n = 4. Suppose that A ≤ U . If A ≤ X, then f (A) ≤ f (X) < |V |, a contradiction. Otherwise, f (A) ≤ f (U ) = |U |2 q 2n−2 = q 6n−4 < |V |, also a contradiction. If |B| ≤ q, then CV (A)/CM (A) is at most 1-dimensional by Lemma 6.6. So |AU/U |2 |CV (A)| ≤ q|AU/U |2 |CV (AU/U )||W |/q and so f (A) ≤ |AU/U |2 |CV (AU/U )||W |q 2 < |V | (we are using induction if n > 4 and the previous lemma if n = 4). So assume that |B| > q. In particular, CV (A) = CM (A). If |B| > q n or more generally the space generated by BX/X contains a nondegenerate subspace (of dimension 2), then |A/B| ≤ q (n−2)(n−3)/2 . Thus, |A| ≤ q n+1 q (n−2)(n−3)/2 ≤ q n(n−1)/2 , a contradiction (since n ≥ 4). So BX/X is totally singular. On the other hand, |A| > q n(n−1)/2 , whence |A/B| > n(n−3)/2 . It follows that the fixed space of A/B on W is totally singular and so has q dimension at most n. Plugging this into our formula above yields f (A) ≤ |AU/U |2 |CV (AU/U )|q|B|2 q n−1 < |V |, unless |B|2 ≥ q 3n−4 . However, |B| ≤ q n , a contradiction unless n = 4. If q is even, this still gives the desired conclusion that f (A) ≤ |V |. The remaining case is n = 4, |B| = q 4 and q is odd. So we see that B/X is a maximal totally isotropic subspace of U/X. Thus, A ≤ CP (B) is contained in a conjugate of Q. Since f (Q) < |V |, the result follows.

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161

6.3. 2F -modules for Sp2n (q) Theorem 6.11. The absolutely irreducible 2F -modules in defining characteristic for Sp2n (q), q = a , n ≥ 3, are as given in Table 9 (see p. 163). Proof. Let P1 = U1 .Sp2n−2 (q)(q −1) and P2 = U2 .GLn (q) denote the two end-node n+1 parabolic subgroups, with unipotent radicals of order |U1 | = q 2n−1 and |U2 | = q ( 2 ) .

1. First assume that there exists an offender A ≤ U1 . If A lies in the invariant 1-dimensional Sp2n−2 (q)-subspace U0 of U1 (that is, in the center of U1 if is odd), then by Lemma 2.1 we may assume that A = U0 . This is a full long root subgroup. But 2n + 1 conjugates of it generate Sp2n (q), so the 2F -condition for a module V of dimension d defined over Ff reads df/(2n + 1) ≤ 2a, that is, d ≤ 2(2n + 1)a/f . Otherwise, if A is not contained in U0 , then again by Lemma 2.1 we may assume A = U1 . Now two conjugates of A generate Sp2n (q), so we arrive at the condition df/2 ≤ 2(2n − 1)a. Hence in any case d ≤ 4(2n − 1)a/f . If V = V1 ⊗ · · · ⊗ Vr then using that di = dim(Vi ) ≥ 2n and f ≥ a/r we get (2n)r ≤ 4(2n − 1)r.

This is only satisfied when either r = 2, n = 4, or r = 1. In the first case, we necessarily have f = a/r, so V1 , V2 are Galois conjugate, and d = 2n. Inspection shows that f (U0 ) < |V | and f (U1 ) < |V |. If r = 1 then we saw that d ≤ 4(2n−1), so V is a twist of one of the modules in Table 8. The ones not occurring in the conclusion are the spin module for n = 6, which was already ruled out in Proposition 5.9, and the modules with highest weights 2λn or λ1 for n = 3, > 2. Explicit computations show that U0 is not a centralizer. Similarly, computing the centralizer of U1 via Theorem 3.1 shows that U1 is not an offender on M(2λn ). On the other hand, the non-abelian group U1 is an offender on M(λ1 ) for n = 3; this case was excluded in 6.3. 2. From now on we may assume that there is no offender in U1 , hence by Proposition 2.4 the centralizer CV (U1 ) is a 2F -module for Sp2n−2 (q). These are known by induction starting with the case of Sp4 (q) in Proposition 6.2. In particular, the restriction of the highest weight of V to the first n − 1 nodes is known. If V is basic, then it occurs in Table 8. The symmetric square does not lead to an example, as can be seen by using Lemma 2.10(b), as in the proof of the previous proposition. The other modules are either excluded in Proposition 5.9, Propositions 6.3-6.5 or in Proposition 6.10 or occur in the conclusion. The case of tensor products V = V1 ⊗ · · · ⊗ Vr , r ≥ 2, of twists Vi of basic modules of dimension di leads to the inequality (3.4) d2 . . . dr ≤ rn(n + 1). Since d ≥ 2n this forces r ≤ 3. The case r = 3 can only hold when n = 3, d = 2n = 6, and the three modules form a Galois orbit. Moreover, A necessarily has d − 1-dimensional centralizer on Vi by Lemma 2.10. But then the rank of A is too small by Proposition 3.12.

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3. Thus r = 2. First assume that V1 , V2 are Galois conjugate, hence of the same dimension d. If Vi are not twists of the natural module, then any non-trivial unipotent element has rank at least 2 on Vi , so we obtain the sharper inequality d ≤ n(n + 1). This is only satisfied when = 2, n ≤ 4, and Vi are spin modules of dimension d = 2n . If n = 3 then (2.1) together with Lemma 5.1 shows that short and long root elements have rank at least 24 on V1 ⊗ V2 , so an offender has size |A| ≥ q 6 , maximal possible. So A is the stabilizer of a maximal isotropic subspace, with centralizer of dimension 1, too small. The case n = 4 is ruled out by induction. Thus V1 , V2 are Galois conjugate twists of the natural module. By Proposition 3.12 and Corollary 2.11(a) we obtain the inequality

c+1 max{2nc, 2c(2n − c)} ≤ 4 2 for the codimension c of the centralizer on the natural module of a possible offender 2 (a) A on V := Y2n ⊗ Y2n . Thus c ≥ (2n − 1)/2 > n − 1 and |A| ≥ q n /2 . Moreover, n+1 if c > n then |A| = q ( 2 ) is maximal, so A equals the unipotent radical U2 of P2 .

Similarly, if c = n then Proposition 3.12 shows that CY2n (A) must be maximally isotropic and again A ≤ U2 . But the centralizer of U2 on V is only of dimension n2 . 4. So we are left with the case that V is the tensor product of r = 2 factors which are not conjugate. Then d2 ≤ n(n + 1), so either d1 = d2 = 2n, or = 2, n ≤ 4 and d2 = 2n . The first case is certainly not an example, since even in the Galois conjugate case it wasn’t. In the cases involving a spin module, dim V is larger than the bound (3.3).

Proposition 6.12. Let V be an absolutely irreducible 2F -module in characteristic for a group G with F ∗ (G) = Sp2n (q), q = a , n ≥ 3. Then all composition factors of the restriction V |F ∗ (G) are 2F -modules for F ∗ (G). Proof. Field automorphisms are the only outer automorphisms of Sp2n (q), n ≥ 3, so assume that G = Sp2n (a ). involves a field automorphism. If V has reducible restriction to F ∗ (G) then the standard argument shows that the claim holds. So now let V be a module with irreducible restriction to Sp2n (a ). Since basic representations are not invariant this implies that V is a product of r ≥ 2 factors. Then (3.4) yields that r = = 2 and d1 = d2 = 2n. Let A be a candidate subgroup with centralizer of dimension 2n − c on the natural module. For these, by Lemma 2.10 and Proposition 3.12 we obtain n = c = a = 2, which is excluded here. 6.4. The examples for Sp2n (q). The conventions for the following table are as for Table 7. Case 6.1: The symplectic group Sp2n (q) acts as a transvection group on its natural module Y2n over Fq , thus we get an example with |A| = q. Case 6.2: The irreducible representation with highest weight λn−1 of Sp2n (a ) is the heart of the exterior square of the natural representation. Consider the maximal

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Classification of 2F -modules, II Table 9. 2F -modules for symplectic groups in defining characteristic

G Sp2n (a ) Sp2n (a )

d 2n 2n 2 − 1 − δ,n

Sp2n (2a ) Sp4 (2a )

2n 16

V f Y2n a 2 ∧˜ (Y2n ) a Zn (a) Y4 ⊗ Y4

a a

n = 2, 3 or = 2, n = 4 n = 3, 4, 5

log |A| a n+1 2 a 10a (2n − 1)a 3a

parabolic subgroup of type GLn . The centralizer of its unipotent radical U has weight λn−2 , hence dimension n2 . The unipotent radical U is elementary abelian of -rank a n+1 2 . Thus we find an example when n ≤ 3. In the case n = 2 we have an alternative interpretation: The 4-dimensional symplectic group is isomorphic to the 5-dimensional orthogonal group. The latter is a bi-transvection group on its natural module, so we get an example for Sp4 (q) in dimension 5, when = 2. When = 2 then this representation is reducible, and its irreducible constituent of dimension 4 yields an example. Case 6.3: The spin module Zn for Sp2n (2a ) has highest weight λ1 . Consider the maximal parabolic subgroup of type Sp2n−2 . The centralizer of its unipotent radical U is again the spin module for Sp2n−2 (2a ), hence has dimension 2n−1 . But U is elementary abelian of rank (2n − 1)a. Thus U is an offender on Zn if 2n − 2n−1 ≤ 2(2n − 1), so for n ≤ 5. Case 6.4: The tensor product of the natural module Y4 for Sp4 (2a ) with its twist (a) Y4 is defined over Fa . A maximal elementary abelian -subgroup A of order 3a has (a) 2-dimensional centralizer on Y4 , hence at least 4-dimensional centralizer on Y4 ⊗Y4 . Thus A is an offender.

7. Orthogonal groups (±)

In this section we consider orthogonal groups Spinn (q) of dimension n ≥ 7, defined over fields of odd order if n is odd. By the results on linear, unitary and symplectic groups in the previous sections and well-known isomorphisms this suffices to cover all finite quasi-simple orthogonal groups. 7.1. The case Spin2n+1 (q). We start by recording the basic modules of small dimension from [11, Thm. 4.4].

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Lemma 7.1. Let n ≥ 3, > 2, and M(λ) be a basic highest weight module for Spin2n+1 (a ) of dimension dim M(λ) ≤ 4n2 + 2n + 1. Then M(λ) is one of the modules in Table 10. Table 10. The basic Spin2n+1 (a )-modules, n ≥ 3, > 2, of dimension at most 4n2 + 2n + 1

λ λn λn−1 2λn λ1 2λ1 λ1 + λ3

M(λ) Y2n+1 ∧2 (Y2n+1 ) ˜ 2 (Y2n+1 )

Zn

dim M(λ) 2n + 1 2n+1 2n+2 2

2

− 1 − δ,2n+1 2n 35 40

n≤8 n=3 n = 3, = 7

Theorem 7.2. The absolutely irreducible 2F -modules in defining characteristic > 2 for Spin2n+1 (q), q = a , n ≥ 3, are as given in Table 12 (see p. 167). Proof. Let V be a 2F -module for G = Spin2n+1 (q), with offender A. 1. Let P1 = U1 .L1 denote the end node parabolic subgroup of G with Levi factor L1 of type Spin2n−1 (q). Then U1 is abelian of order q 2n−1 and irreducible under L1 . Thus, if A ≤ U1 then by Lemma 2.1 we may assume that A = U1 . By Proposition 3.7, the 2F -condition for V implies dim V ≤ 4(2n − 1)a/f if V is defined over Ff . By Lemma 7.1 any basic representation of G has dimension at least 2n + 1. This shows that V cannot be a tensor product of at least two factors. But otherwise, V is (a Frobenius twist of) one of the modules in Table 10 of dimension at most 4(2n − 1), and all these occur in the conclusion. 2. Thus, by Proposition 2.4 we may assume that CV (U1 ) is a 2F -module for Spin2n−1 (q). First assume that V is a basic representation, hence its highest weight ˜ 2 (Y2n+1 ) the basic inequality from (2.1) and Propois contained in Table 10. For sition 3.14 gives a contradiction. Similarly for V = ∧2 (Y2n+1 ) it gives n = 3, and |A| ≥ q 4 . But then A contains non-trivial elements with at least two different centralizers and Lemma 2.10(c) yields a contradiction. The spin module Z6 for n = 6 is not an example by Proposition 5.9. By step 1 this also rules out the spin modules for n ≥ 7. The module with highest weight 2λ1 for n = 3 is not an example since the centralizer of U1 , with the same highest weight, is not an example for Spin5 (q) ∼ = Sp4 (q) by Proposition 6.2. Explicit construction of the 40-dimensional module with highest weight λ1 + λ3 as a submodule of the tensor product of the natural module with the spin module shows that long root elements have 18-dimensional commutator space, hence by Corollary 3.4 and Proposition 3.14 this is not an example.

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3. Next assume that V = V1 ⊗ · · · ⊗ Vr is a tensor product of r ≥ 2 factors of dimensions d = d1 ≤ d2 ≤ · · · ≤ dr . Since G is not a transvection group, the commutator of any non-trivial element in any non-trivial representation is at least 2-dimensional. Then (3.4) shows that 1 r(n2 − n + 4). 2 This forces r = 2, V1 , V2 are Galois conjugate, and either d1 = d2 = 2n + 1, or n = 3, d1 = d2 = 8. Both cases are eliminated by the inequalities from (2.1) together with Lemma 3.14. d r−1 ≤

Proposition 7.3. Let V be an absolutely irreducible 2F -module in characteristic > 2 for a group G with F ∗ (G) = Spin2n+1 (q), q = a , n ≥ 3. Then all composition factors of the restriction V |F ∗ (G) are 2F -modules for F ∗ (G). Proof. We only have to care about field automorphisms. For induced modules the assertion is immediate from Proposition 2.5(i) since = 2. If V has irreducible restriction to F ∗ (G), then its highest weight is invariant under the field automorphism, in particular it is not basic. So V is a tensor product of r ≥ 2 factors. The standard inequality (3.4) then shows that r = 2 and = 2, which is excluded here. 7.2. The case Spin+ 2n (q). Again we first collect the basic modules of small dimension from [11, Thm. 4.4]. a Lemma 7.4. Let n ≥ 4, let M(λ) be a basic highest weight module for Spin+ 2n ( ) of 2 dimension dim M(λ) ≤ 4n − 2n + 4. Then M(λ) is one of the modules in Table 11.

2 Table 11. The basic Spin+ 2n (q)-modules of dimension at most 4n − 2n + 4

λ λn λn−1 2λn λ1 , λ2 λ1 + λ2 , λ1 + λ4 , λ2 + λ4

M(λ) Y2n ∧2 (Y2n ) ˜ 2 (Y2n )

Zn−1

dim M(λ) 2n 2n − (n, 2)δ,2 2 2n+1 − 1 − δ,n 2 2n−1 56 − 8δ,2

>2 n≤9 n=4

Theorem 7.5. The absolutely irreducible 2F -modules in defining characteristic for a Spin+ 2n (q), q = , n ≥ 4, are as given in Table 12 (see p. 167).

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Proof. 1. Let us first handle some particular cases. The natural module Y2n has highest weight λn and gives Example 7.1. The highest weight λ = (1 + a/2 )λn belongs to (a/2) the tensor product Y2n ⊗ Y2n . The 2F -condition for an offender with centralizer of codimension c on Y2n then reads 2c(2n − c)/2 ≤ 2 2c by (2.1) and Lemma 3.14, which implies c > n, but then the upper bound for the rank in Lemma 3.14 is violated. ˜ 2 (Y2n ) with highest weight 2λn we similarly For the heart of the symmetricsquare

c obtain c(2n − c + 1) − 2 ≤ 2 2 , again not possible. The highest weight module for λ = λn−1 is the heart ofthe exterior square of Y2n . Lemma 3.14 and (2.1) show that c(2n − c − 1) − 2 ≤ 2 2c . This forces c ≥ n, A of maximal rank. But then A contains elements with different centralizers, and Lemma 2.10 (c) gives a contradiction. The half spin modules for n ≤ 6 with highest weight λ1 or λ2 are Example 7.3. The half-spin modules Z6 for Spin+ 14 (q) were excluded in Proposition 5.5. 2. Let P1 = U1 .L1 be a maximal parabolic subgroup of G corresponding to the last node of the Dynkin diagram, that is, with Levi factor of type Spin+ 2n−2 (q). Then U1 is the natural module for L1 , hence irreducible. If A is contained in U1 , we may take A = U1 . Two conjugates generate G, hence df/2 ≤ 2(2n − 2)a for a 2F -module V defined over Ff with offender A. If V is the tensor product of r factors Vi , of dimensions di = dim Vi ≥ 2n, then we get (2n)r ≤ 8(n − 1)r. This shows that r = 1, and thus d < 2n2 . Hence V is a twist of one of the basic modules in Table 11 all of which were treated in the first part. 3. If U1 does not contain an offender, we may assume by Proposition 2.4 that V is an example for the Levi factor L1 , hence for Spin+ 2n−2 (q). Let first n = 4, so + ∼ L1 contains the linear group Spin6 (q) = SL4 (q). Then we may repeat the above argument for all three end nodes of the Dynkin diagram. By Proposition 4.7 this only leaves the possibilities λ4 , λ3 , 2λ4 , (1 + a/2 )λ4 and their twists and images under the graph automorphisms for λ. These were investigated in the first part. 4. Now let n ≥ 5 and P2 = U2 .L2 a maximal parabolic subgroup with L2 = GLn (q), U2 the exterior square of the natural module for L2 . If A ≤ U2 , then d ≤ 4 n2 a/f by the standard argument. This forces V to be a twist of a basic module, of dimension d < 2n2 , hence one of the cases in step 1. Otherwise, V is an example for SLn (q). This, together with induction starting from step 3 shows that λ is one of tλn (t = 1, 2, 1 + a/2 ), λ1 (n ≤ 7), λn−1 up to twists and the graph automorphism of order 2. These were considered in the first part of the proof. 7.3. The case Spin− 2n (q) Theorem 7.6. The absolutely irreducible 2F -modules in defining characteristic for a Spin− 2n (q), q = , n ≥ 4, are as given in Table 12.

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(a/2) ˜ 2 (Y2n ) Proof. 1. The natural module is Example 7.1. The cases Y2n ⊗ Y2n and are excluded precisely as for orthogonal groups of plus-type. The spin modules for n ≤ 5 are Example 7.4. The half-spin modules for Spin− 2n (q), n ≥ 6, were excluded 2 ˜ is excluded using Proposition 3.14 in in Proposition 5.7. The adjoint module ∧ conjunction with (2.1). 2. Let V be a 2F -module for G = Spin− 2n (q) and A an offender on V . Let P1 = U1 .L1 be the end node parabolic with Levi complement of type Spin− 2n−2 (q). (q) we find that either V is one of the Then arguing as in step 2 of the proof for Spin+ 2n modules treated in step 1, or CV (U1 ) is an example for L1 . 3. Assume that V is a basic module M(λ). Then dim V ≤ 4n2 − 2n + 4 by (3.3), and dim V ≤ 2n2 − n + 2 if the highest weight λ is not invariant under the graph automorphism exchanging nodes 1 and 2. Thus

λ ∈ {λn , λn−1 , 2λn },

or

λ = λ1 , λ2 for n ≤ 7,

or

λ = λ1 + λ2 for n = 4

by Proposition 7.4. All except the last case were considered in step 1. The weight λ1 + λ2 does not lead to an example by step 2, since CV (U1 ) is not an example for ∼ Spin− 6 (q) = SU4 (q) by Theorem 4.10. 4. If V is a tensor product of r ≥ 2 factors, then (3.4) shows that r = 2, both factors have dimension 2n and are Galois conjugate. Exactly the same calculation as for Spin+ 2n (q) allows to rule out this case. 7.4. The examples for Spin(±) n (q). In the first line of the following table, is odd when n is odd. Table 12. 2F -modules for orthogonal groups in defining characteristic

G

(±) Spinn (a ) Spin2n+1 (a ) a Spin+ 2n ( ) − a Spin2n ( )

d

V

f

n 2n 2n−1 2n−1

Yn Zn

a a a 2a

Zn−1 Zn−1

(±)

log |A| n≥7 n = 3, 4, 5 n = 4, 5, 6 n = 4, 5

a (2n − 1)a (2n − 2)a (2n − 2)a

Case 7.1: The orthogonal group Spinn (q) acts as a bi-transvection group on its natural module, hence we obtain an example with offender of order q. Case 7.2: The spin module Zn for Spin2n+1 (a ) has highest weight λ1 . Consider the maximal parabolic subgroup of type Spin2n−1 . The centralizer of its unipotent radical U is again the spin module for Spin2n−1 (a ), hence has dimension 2n−1 . But U is elementary abelian of rank (2n − 1)a. Thus U is an offender on Zn if 2n − 2n−1 ≤ 2(2n − 1), so for n ≤ 5.

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Case 7.3: For the half-spin modules Zn−1 of dimension 2n−1 of G = Spin+ 2n (q) let A be the unipotent radical of the maximal parabolic subgroup with Levi complement of + type Spin+ 2n−2 (q). Then the centralizer CV (A) is a half-spin module for Spin2n−2 (q), of dimension 2n−2 , while A is elementary abelian of order q 2n−2 , so we find an example whenever n ≤ 6. Case 7.4: The half-spin modules Zn−1 of dimension 2n−1 of G = Spin− 2n (q) are defined over Fq 2 . Let A be the unipotent radical of the maximal parabolic subgroup with Levi complement of type Spin− 2n−2 (q). Then the centralizer CV (A) is a spin modn−2 , while A is elementary abelian of order q 2n−2 , (q), of dimension 2 ule for Spin− 2n−2 so we find an example whenever n ≤ 5.

8. Exceptional groups For exceptional groups we make use of the table of -ranks m in [5, Table 3.3.1], which we reproduce here as Table 13 for the convenience of the reader, and the determination of small basic modules in [11, Thm. 4.4]. Moreover, we employ the tables in Lawther [9] on Jordan shapes of unipotent elements of exceptional groups on small modules in conjunction with information on the closure relation on the set of unipotent classes as can be found in Spaltenstein’s book [16], for example. Table 13. -ranks of exceptional groups

G

m (G)

G

2B (22a+1 ) 2

2a + 1

F4 (a )

2(2a + 1) 3a = 3 4a = 3

(a )

2G (32a+1 ) 2

G2 (a ) 3D 2F

4

(a )

4 2a+1 (2 )

5a 5(2a + 1)

E6 2E

6 (

a)

(a )

E7 E8 (a )

9a

m (G) >2

11a

=2

12a

16a >2

13a

=2 27a 36a

Theorem 8.1. Let V be an absolutely irreducible 2F -modules in defining characteristic for a group G with F ∗ (G) an exceptional group of Lie type. Then the composition factors of V |F ∗ (G) are as given in Table 17 (see p. 175). The proof will be given in the following subsections. We denote by G a finite group such that F ∗ (G) is a quasi-simple exceptional group of Lie-type in characteristic , and let S be the unique non-abelian simple composition factor of G. By our general

Classification of 2F -modules, II

169

considerations we may assume that F ∗ (G) is the universal covering group of S, quotiented out by the -part of the Schur multiplier. Thus, F ∗ (G) is the group of fixed points under a Frobenius morphism of a simple algebraic group of simply-connected type. We indicate this by a subscript sc. The corresponding simple group is written without this subscript. First we rule out the adjoint modules: Proposition 8.2. Let G be one of G2 (q), F4 (q), E6 (q)sc , 2E6 (q)sc , E7 (q)sc , E8 (q). Then the adjoint representation of G does not give rise to a 2F -module for G. Proof. Consider first the case of G = E8 (q), q = a . Let V be the adjoint representation of G on its Lie algebra. By [9, Tab. 9] root elements have rank 58 on V , so by Corollary 3.4 any offender A has order at least q 29 . Moreover, if x = 1 is not a root element, then its rank is at least 92, too large compared to the -rank 36a. Thus A consists of long root elements only. But then Proposition 3.9 shows that A has size at most q 8 , too small. For G = E7 (q)sc , an offender A on the adjoint representation V can only contain elements from classes labeled A1 , 2A1 , 3A 1 by [9, Tab. 8]. (Using the table of the closure relation of unipotent classes in [16, p. 248] and the argument in the proof of Corollary 3.4 we only have to know the centralizer dimensions of classes 3A 1 and 4A1 to verify this.) First assume that A contains an element x from class 3A 1 , with rank at least 53 on V . Then A is contained in the centralizer of x, with semisimple part of type F4 , which lies in the maximal parabolic subgroup P = U.L of type E6 , with U elementary abelian of order 27. By Theorem 3.1 we don’t have A ≤ U , and since E6 (q)sc has -rank 16a by Table 13, we neither have U ∩ A = 1. But any unipotent element of F4 (q) has commutator of dimension at least 16 on the Lie algebra of F4 , so A has centralizer of codimension at least 69, which is much too large. The case of elements of type 2A1 can be excluded similarly. So finally assume that A only consists of root elements, of type A1 , with centralizer codimension 34 on the Lie algebra. Then Proposition 3.9 gives a contradiction. For G = E6 (q)sc , long root elements have rank 22 on the Lie algebra, by [9, Tab. 6], so |A| ≥ q 11 . Thus A cannot consist of long root elements only by Proposition 3.9. By [9, Tab. 6] the only other non-trivial unipotent elements possible in A are those of type 2A1 , with rank 32. Their centralizer has semisimple part of type B3 , so is contained in the maximal parabolic subgroup P = U.L of type D5 , with elementary abelian radical of order q 16 . As usually we see that A is not contained in U , but no non-trivial unipotent element from L centralizes the Lie algebra of B3 , thus the centralizer has codimension larger than 32, which is impossible by Table 13. For G = 2E6 (q)sc , essentially the same argument applies. Now let G = F4 (q). By [9, Tab. 4] root elements have rank at least 16 on the 52-dimensional adjoint module, thus an offender A has |A| ≥ q 8 . Let P = U L denote a maximal parabolic subgroup containing A with Levi complement L of type Sp6 (q). Then the composition factors of U as L-module have dimensions 1, 14 for 2 q, respectively 1, 6, 8 for q even by [5, p. 99]. By the previous estimate, A is

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not contained in a proper L-submodule of U . Thus A ≤ U by Lemma 2.1 and Theorem 3.1. Since L has -rank 6a we can’t have A ∩ U = 1 either. Now adding up centralizer codimensions of a pair of elements 1 = x ∈ A ∩ U and y ∈ A \ U gives a contradiction to the 2F -condition. Finally consider the adjoint module V for G = G2 (a ), = 3. Let first = 2. Long root elements have rank 6 on V , and any other nontrivial unipotent element has rank at least 7, by [9, Tab. 2]. By Table 13, any elementary abelian subgroup of G has order at most q 3 , so it follows that either f (A) < |V | or A consists of long root elements. By Lemma 3.9, this implies that |A| ≤ q 2 and so f (A) ≤ q 12 < V . Suppose that = 2. As above, any nontrivial x ∈ A must have a fixed space of dimension 8. The only such elements are long or short root elements. This implies that the centralizer of x in G is [P , P ] where P is a maximal parabolic subgroup and the fixed space of x on V is the Lie algebra of [P , P ]. So if |A| > q, dim CV (A) < 8 and f (A) < |V |. If |A| ≤ q, then f (A) ≤ q 10 . 8.1. The Suzuki-groups Lemma 8.3. If F ∗ (G) = 2B2 (22a+1 ) then any 2F -module for G is as in the statement of Theorem 8.1. Proof. Let F ∗ (G) = 2B2 (q), q = 22a+1 . Since Suzuki groups have no outer automorphisms of even order or multiplier of odd order we in fact have G = 2B2 (22a+1 ). The basic representations have highest weights λ1 , λ2 , λ1 + λ2 . The first two are the natural representations, Examples 8.1. The Steinberg representation has dimension 16, larger than 10 ≥ 2 logq |G|, so cannot occur. For tensor products V of r ≥ 2 factors (of dimension at least 4) we reach the inequality 4r /r ≤ 10. Hence r = 2 and both factors have dimension 4. But then V is defined over Fq , q = 22a+1 , and the inequality is not satisfied. 8.2. Groups of type G2 . From [11, A.49] we find: Lemma 8.4. Let M(λ) be a basic highest weight module for G2 (q) of dimension d = dim M(λ) ≤ 29. Then M(λ) is one of the modules in Table 14. Lemma 8.5. If F ∗ (G) = 2G2 (32a+1 ) or F ∗ (G) = G2 (q) then any 2F -module for G is as in the statement of Theorem 8.1. Proof. Let first F ∗ (G) = 2G2 (q), q = 32a+1 and first consider the simple group. Here 2 logq |G| ≤ 14. The basic representations of dimension at most 14 are those with highest weights λ1 , λ2 , of dimension 7 both, by Table 14. The group G has two classes of elements of order 3, both of which have representatives already inside 2G (3) = Aut(L (8)). For the latter group, such elements have Jordan blocks of sizes 2 2 3, 2, 2 respectively 3, 3, 1 on the 7-dimensional module. In particular, the centralizer of any offender is at most 3-dimensional. But then A must have order q 2 , which is

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Table 14. The basic G2 (q)-modules of dimension at most 29

λ λ2 λ1 2λ2 2λ1

dim M(λ) 7 − δ,2 14 − 7δ,3 27 − δ,7 27

= 2 =3

maximal possible. In this case A intersects both conjugacy classes non-trivially. Now the sizes of Jordan blocks show that the centralizer has at most dimension 2, so we get no example. The case of tensor products of at least two factors is ruled out by the standard inequality (3.3). Now assume that 3 | 2a + 1 and G = 2G2 (32a+1 ).3 is the extension by the corresponding field automorphism. Let W be a basic representation. Since basic representations are not invariant, the induction V of W to G is irreducible. Outer elements have at most d/3-dimensional centralizer, where d = dim V . The standard inequality shows that this does not lead to examples. Similarly, tensor products can be ruled out. Let now F ∗ (G) = G2 (a ), a = 2 (the case a = 2 was treated in the first part [6, Prop. 4.3] as U3 (3)). First consider the simple group G = G2 (a ). The 6- and 7-dimensional basic modules are Examples 8.2 and 8.3. The adjoint module was handled in Proposition 8.2. The 26- and 27-dimensional representations are the heart of the symmetric square of the 7-dimensional representation(s) V0 . Since G2 (q) is not a transvection group, non-trivial unipotent elements have at most 5-dimensional centralizer on V0 , hence at ˜ 2 (V0 ) by (2.1). Thus they are not examples by most 15-dimensional centralizer on Table 13. The inequality (3.3) excludes non-trivial tensor products except for the case of two factors of dimension at most 7. Again, this does not lead to examples since non-trivial unipotent elements are not transvections. Now assume that G is G2 (a ) extended by a field automorphism of order , so | a. Let V be a 2F -module for G. The basic representations of G2 (a ) do not extend to G. Then (3.5) shows that dim V = 12 (and = a = 2), so V |F ∗ (G) indeed consists of 2F -modules. The case of tensor products is ruled out by (3.3). Finally note that graph-automorphisms (of order 2) only occur in characteristic 3, so need not be considered here. 8.3. The groups 3D4 (q) Lemma 8.6. If F ∗ (G) = 3D4 (q) then any 2F -module for G is as in the statement of Theorem 8.1. Proof. Let V be a 2F -module for G and A an elementary abelian unipotent subgroup of G with f (A) maximal.

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We note as usual that |V | ≤ q 56 . It follows by Table 11 that (up to quasiequivalence) V is either the natural 8-dimensional module Y8 (over Fq 3 ) or the adjoint ˜ 2 (Y8 ) for the 8-dimensional orthogonal group. The latter has dimension module ∧ (over Fq ) 28 if is odd and 26 if = 2. Consider the second case. If is odd, then dim CV (x) = 18 for x a long root element and dim CV (y) < 18 for 1 = y a unipotent element that is not a long root element by Table 3. If A consists of long root elements, then f (A) ≤ q 4 q 18 < |V | by Proposition 3.9. Otherwise f (A) ≤ q 10 q 17 < |V | by Table 13. If = 2, then dim CV (x) = 16 for x a long root element and dim CV (y) ≤ 14 for any involution y not a root element. The argument of the previous paragraph applies. Finally consider the case that V is 8-dimensional (over Fq 3 ). Let X be a long root subgroup centralizing A. Let P = NG (X) be a maximal parabolic subgroup with radical Q. Note dim CV (x) = 6 (over Fq 3 ) for x a long root element (and is no bigger for any other unipotent element). Thus, |A| ≥ q 3 . Note that the composition factors for the Levi complement L ≥ SL2 (q 3 ) have dimension 2, 2 and 4. The latter is a tensor product of two twists of the natural module. In particular, it follows that for any unipotent element y ∈ P \ Q, |CV (y)| ≤ q 12 . So if A is not contained in Q, then f (A) ≤ q 12 q 10 < |V |. Thus, we may assume that A ≤ Q. Note that AX is the normal closure of A in Q. Then f (AX) ≥ f (A) and since AX is abelian, there is no harm in assuming that X < A (and is proper). If CV (A) has dimension less than 5, then f (A) ≤ q 22 < |V |. Since A properly contains X and Q/X acts faithfully on W := CV (X) (a 6-dimensional space), it must be the case that CV (A) is 5-dimensional. Since L acts irreducibly on Q/X, Q is the normal closure of A in P and so f (A) ≤ f (Q) = |V |. So the only possibility is that f (A) = |V | and f (B) ≤ |V | for any subgroup B of Q. In particular, this implies that |A| = q 9/2 (and so q is a square). Choose g ∈ L with Ag = A. Then A ∩ Ag is trivial on W and so A ∩ Ag = X. Thus, AAg has order q 8 and fixes a 4 dimensional subspace of W . So f (AAg ) = q 28 > f (Q), a contradiction. Now assume that G is obtained by extending 3D4 (q) by a field automorphism of order . Basic invariant modules only exist when = 3, and the only one below the dimension bound is the adjoint module of dimension 28. But as before any inner element has rank at least 10. But the outer 3-rank is controlled by the centralizer G2 (a ), hence at most 4a/3 + 1. Thus we necessarily have a = 3, but then the centralizer of inner elements has dimension at most 16, and we obtain a contradiction. Induced basic modules can be ruled out by Proposition 2.5, tensor products can’t occur by (3.3). 8.4. Groups of type F4 . From [11, A.50] we find: Lemma 8.7. Let M(λ) be a basic highest weight module for F4 (q) of dimension d = dim M(λ) ≤ 105. Then M(λ) is one of the modules in Table 15.

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Table 15. The basic F4 (q)-modules of dimension at most 105

λ λ4 λ1 λ1

dim M(λ) 26 − δ,3 26 52

=2 = 2

Lemma 8.8. If F ∗ (G) = 2F4 (22a+1 ) or F ∗ (G) = F4 (q) then any 2F -module for G is as in the statement of Theorem 8.1. Proof. First assume F ∗ (G) = 2F4 (q) , q = 22a+1 . We only have to consider G = 2F (22a+1 ) since there is no odd multiplier and the outer automorphism of order 2 of 4 2 F (2) does not have representatives of order 2. We have 2 log |G| ≤ 52(2a + 1), 4 q so by Table 15 only the basic highest weight modules of dimension 26 might occur. The 2-rank of G is 5(2a + 1) by Table 13. Both classes of involutions 2A, 2B have representatives already inside 2 F4 (2) . Since these invert elements of order 5 respectively 13 we may conclude that their centralizers on the 26-dimensional modules are at most 16- respectively 14-dimensional. Hence a possible offender A must be 2A-pure of order 25(2a+1) . But the center of the centralizer of a 2A-element clearly has a smaller 2-rank. Thus we get a contradiction with Lemma 2.3. For a tensor product of r factors we need 26r ≤ 52r, which does not hold when r ≥ 2. Now let G = F4 (q), q = a . The small basic modules were collected in Table 15. The adjoint module was excluded in Proposition 8.2. We have to leave open the case of the 25- respectively 26-dimensional module. For tensor products, the inequality (3.3) leads to a contradiction. For groups G obtained from F4 (q) by extension with a field automorphism, the standard estimates rule out 2F -modules. The 2-rank of the centralizer of an exceptional graph automorphism of order 2 in characteristic = 2 is 5a + 1, and again the standard estimates rule out this case. 8.5. Groups of type En . From [11, A.51] we find: Lemma 8.9. Let M(λ) be a basic highest weight module for E6 (q)sc of dimension d = dim M(λ) ≤ 158. Then M(λ) is one of the modules in Table 16. Table 16. The basic E6 (q)sc -modules of dimension at most 158

λ λ1 , λ6 λ2

dim M(λ) 27 78 − δ,3

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Lemma 8.10. If F ∗ (G) = En (q)sc , n = 6, 7, 8, or F ∗ (G) = 2E6 (q)sc , then any 2F -module for G is as in the statement of Theorem 8.1. Proof. 1. First let G = E6 (q)sc . The unipotent radical U of a maximal parabolic subgroup of type Spin+ 10 (q) is a spin module for the Levi subgroup, hence irreducible. If A ≤ U is an offender on a 2F -module V defined over Ff then our usual argument gives df/2 ≤ 32a. In particular, V must be a twist of a basic module, of dimension at most 64. Hence by Table 16 it is one of the 27-dimensional modules. On the other hand, if U contains no offender, then CV (U ) is an example for Spin+ 10 (q). Applying this to both classes of D5 -parabolics and using the result for the orthogonal groups in Theorem 7.5 we see that up to twist V is one of the basic modules one Table 16. The 27-dimensional modules are Example 8.4, the adjoint modules were treated in Proposition 8.2. Now assume that G = E6 (q)sc . involves a field automorphism of order . If V is a tensor product, then 27r a/r ≤ 2 log |G|. This is only possible for r = 1. But basic modules are not invariant, so V is induced from a twist of a basic module W . Furthermore, a 2F -module for G is either already an example for the derived group, or every proper parabolic subgroup leads to an example. The results for SL6 and Spin10 show that this implies that W is (a twist of ) the 27-dimensional or the adjoint module. Using that outer elements have centralizer at most dim W on V and the -rank 8a + 1 we arrive at = 2, dim W = 27, which occurs in Table 17. Next assume that = 2 and G is obtained from E6 (q)sc by extension with the graph automorphism of order 2. The 27-dimensional module is not invariant, hence can be excluded by Proposition 2.5. Since the graph automorphism centralizes an F4 (q), outer involutions don’t act as bi-transvections, so B := A ∩ G = 1, and |A| ≤ 2q 11 by Table 13. The table in [9] shows that B has to consist of long root elements, but then |B| ≤ q 5 by Table 5, while long root elements have rank 22, contradiction. Non-trivial tensor products do not satisfy the basic inequality 3.4. If G involves a graph-field automorphism of order 2, then no basic representations is invariant, and no case arises. The same happens for the extension of E6 (2a )sc by the full group of graph and field automorphisms of order 2. 2. Now let G = 2E6 (q)sc . Arguing as above, only the basic representations of dimensions 27 and 78 − δ,3 need to be considered. The adjoint one is ruled out by Proposition 8.2. At present, we can’t decide whether the 27-dimensional module gives an example. The basic representations do not extend to the extension of 2E6 (q)sc by a field automorphism of order . Thus, as before, this case can be ruled out by the standard arguments. 3. Now let G = E7 (q)sc . By [11] the only small basic modules are the one of dimension 56 and the adjoint one of dimension 133 − δ,2 . The latter was ruled out in Proposition 8.2. We leave open the case of the 56-dimensional module. Tensor products do not lead to examples, using that G has -rank 27a (see Table 13). For extensions by field automorphisms of order the standard estimates allow to conclude. 4. Finally let G = E8 (q). Here [11] shows that the only possible basic module is the adjoint one of dimension 248. This was ruled out in Proposition 8.2. Tensor

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products do not give rise to examples by our basic inequality, using that the -rank of G is 36a (see Table 13). Similarly, the case of field automorphisms can be excluded by standard estimates. 8.6. The examples for exceptional groups. Case 8.1: Let A be the center of a Sylow 2-subgroup of the Suzuki group 2B2 (22a+1 ), of order 22a+1 . Then it is well known that A has 2-dimensional centralizer on the 4-dimensional modules, hence we obtain an example. Case 8.2: The group G2 (2a ) has 2-rank 3a, so we find an example on the 6dimensional module. Case 8.3: The group G2 (a ) has -rank at least 3a, so we find an example on the 7-dimensional module(s). Case 8.4: The Chevalley group F4 (2a ) contains the classical group SO9 (2a ) ∼ = Sp8 (2a ). The 26-dimensional module for F4 (2a ) restricts to the heart of the exterior ˜ 2 (Y8 ) for Sp8 (2a ), as can be seen on the extended square of the natural module ∧ Dynkin diagram. Since the latter is an example for the symplectic group by Table 9, it is a fortiori an example for F4 (2a ). Case 8.5: Let U be the abelian unipotent radical, of order q 16 , of a maximal parabolic subgroup of type D5 of E6 (q)sc . Clearly this gives an example on the 27-dimensional modules. Table 17. 2F -modules for exceptional groups in defining characteristic

G 2B (22a+1 ) 2 G2 (2a ) G2 (a ) F4 (2a ) E6 (a )sc F4 (a ) 2E (a ) 6 sc a E7 ( )sc

d 4 6 7 26 27 26 − δ,3 27 56

λ f λ1 , λ2 2a + 1 λ2 a λ2 a λ1 , λ4 a λ1 , λ6 a λ4 a λ1 , λ6 2a λ7 a

a>1 >2

|A| 2a + 1 3a 3a 10a 16a

>2

9. F -modules for quasi-simple groups In this section we obtain the list of all absolutely irreducible F -modules of finite quasisimple groups, as an easy application of our classification of 2F -modules in this paper and the previous part [6].

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Lemma 9.1. The 56-dimensional module for G := E7 (q)sc is not an F -module. Proof. Let P = U L where U is elementary abelian of order q 27 and L is of type E6 . Then V has an L-composition series F < W < M < V where F and V /M are 1-dimensional and W/F ∼ = U. = U ∗ and M/W ∼ Let A be a nontrivial elementary abelian unipotent subgroup of G with f1 (A) maximal and assume that f1 (A) ≥ |V |. Set B = A ∩ U . If B = 1, the result follows since the 27-dimensional module U for E6 (q) is not an F -module. If B does not consist solely of long root elements, then CV (B) has dimension at most 36 by [9, Table 7]. Since CW (y) is 27-dimensional (note that U acts trivially on W/F , whence [y, W ] = F for each nontrivial y ∈ U ), this implies that CV (A)/CW (A) has dimension at most 9. Thus, f1 (A) ≤ q 9 |A/B||CW (A/B)||B| < q 37 |B| since W/F is not an F -module for E6 (q). Thus, |B| > q 19 . By loc. cit. this implies that A/B consists of long root elements. Now we observe that if |A/B| > q, then |CB (A/B)| ≤ q 18 . (sketch: a long root element centralizes a space B0 of dimension 21 on B. Now work in the centralizer of this long root subgroup X in C := CE6 (X) = q 1+20 SL6 (q). The module B has three composition factors of dimensions 6, 15, 6 for SL6 (q). Suppose that y is in the unipotent radical Q of C outside X. Then y −1 y g is a nontrivial element of X for some g ∈ Q. Since an element of X has rank 6 on B0 , it follows that y has rank at least 3 on B0 , whence CB0 (y) = CB (y, x) has dimension at most 18. If y is not in U , then it fixes a 5-dimensional space at most on each of the 6-dimensional composition factors and a 13-dimensional space on the 15-dimensional composition factor). So in this case, |A| ≤ q|B|. Note that the action of U on W gives an identification of U with a subspace of W ∗ (y → y −1 and y −1 is a map from W to F ). In particular, any subspace R of U of dimension r centralizes a space of dimension 28 − r on W . In particular, |B||CW (B)| ≤ |W |. Thus, |CV (A)||A| ≤ q|W ||CV (A)/CW (A)| ≤ q 10 |W | < |V |. So B consists solely of long root elements. It follows from Table 5 that |B| ≤ q 7 . Thus, |A| ≤ q 7 |A/B|. Moreover, f1 (A) ≤ q 7 |A/B||CW (A)||CV (A)/CW (A)|. By the E6 -result, |A/B||CW (A)| < |W |. Even for y ∈ B a long root element, we have CV (y)/CW (y) of dimension at most 18, whence f1 (A) < q 53 , a contradiction. For the properties of M24 , Co1 and Co2 needed in the following proofs we refer to [5, Tab. 5.3e, 5.3k and 5.3l]. Lemma 9.2. Let V be an irreducible F2 -module of dimension 11 for G = M24 . Then V is not an F -module. Proof. We first note that the centralizer dimensions for involutions on V are 7 for a 2-central involution and 6 for the other class of involutions.

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Suppose that A is an offender with f1 (A) maximal. Let z be an involution of class 2A. We work with the subgroup A ≤ P with U = O2 (P ) elementary abelian of order 16 and P /U = A8 . Note that this contains D := CG (z) (after conjugating) where Q = O2 (D) is extraspecial of order 27 and D/Q = L3 (2). We claim that A must contain a conjugate of z. If not, then A ∩ U = 1 (since U intersects only one class of involutions). Then A embeds in P /U and so has order at most 16, but CV (A) has dimension at most 6, a contradiction. So we may assume that z ∈ A ≤ D. Set W = CV (z) of dimension 7 and M = [z, V ] of dimension 4. Note that W/M is an irreducible D-module. Suppose that A ∩ Q has order 2. Then |A| ≤ 8, a contradiction. So let z = y ∈ A ∩ Q be an involution. Since z = y −1 y g for some g ∈ Q, it follows that [y, W ] has dimension at least 2. Thus, CV (y) has dimension at most 5, whence exactly 5 and |A| = 26 . Thus, A is not contained in Q and so does not centralize M. Then CV (A) has dimension at most 4, a contradiction. Lemma 9.3. Let V be the 22-dimensional F2 -module for G = Co2 . Then V is not an F -module. Proof. Let A be an offender with f1 (A) maximal. Note that CV (x) has dimension at most 16 for any involution x. Thus, |A| ≥ 26 . Let z be a 2-central involution and set D = CG (z). Set U = O2 (D). Then D/U = Sp6 (2). Since |A| ≥ 64, A is not contained in U (the largest abelian subgroup of U has order 32). If A contains an involution not conjugate to z, then |CV (A)| ≤ 214 and so |A| ≥ 28 . We first show that A contains a conjugate of z. If not, then as we saw |A| ≥ 28 and B := A ∩ U = 1. Set W = CV (z) and M = [z, V ]. So dim W = 16 and dim M = 6. Now U centralizes M (since it is irreducible for D). If A/B has order 2, then f1 (A) ≤ f1 (B) < |V |. If A/B has order at least 4, then it follows that CV (A) has codimension at least 2 in CV (B) and so dim CV (A) ≤ 12, whence |A| = 210 . Since that is the maximal rank possible, it follows that A contains a 2-central involution. So we may assume that z ∈ A and B is a nontrivial proper subgroup of A. Note that CV (A) is properly contained in CV (B) as B = CM (A). If A/B has order 2, then f1 (A) ≤ f1 (B) < |V |. If A/B has order at least 4, it follows that CM (A) has dimension at most 4 and so CV (A) has codimension at least 2 in CV (B). If y ∈ B is an involution other than z, we see that CW (y) has codimension at least 3 in W . Thus, CV (B) has dimension at most 13, whence by the remark above, CV (A) has dimension at most 11 and f1 (A) < |V |. So B must have order 2 and |A| ≤ 27 . Then W = CV (z) is 16-dimensional and so CV (A) has dimension at most 14, whence f1 (A) ≤ 221 . Lemma 9.4. Let V be the 24-dimensional F2 -module for G = Co1 . Then V is not an F -module.

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Proof. Suppose that A is a nontrivial elementary abelian subgroup of G with f1 (A) ≥ |V | and maximal. Note that |A| ≤ 211 . Consider D = CG (z) with z an involution in class 2A. Set U = O2 (D) and recall that D/U = O+ 8 (2). Note that CV (A) has dimension at least 13, whence all involutions in A are conjugate to z (by centralizer dimensions). We assume that z ∈ A. Note that D has 3 composition factors on V , each of dimension 8. Set W = CV (z) (of dimension 16). Moreover, if x ∈ A \ U , then CW (x) has dimension at most 12 (since x does not induce a transvection on any composition factor). So if A is not contained in U , then f1 (A) ≤ |A|212 < |V |. If A is contained in U , then |A| ≤ 25 and so f1 (A) ≤ 221 . Theorem 9.5. The absolutely irreducible F -modules of finite quasi-simple groups are given in Tables 18 and 19. Proof. By definition, F -modules are in particular 2F -modules, so we just have to go through our list of 2F -modules and determine which of them satisfy the F-condition. We start with the groups of Lie type in their defining characteristic whose 2F -modules were classified in this paper. The symmetric square 2 (Yn ) of the natural module for SLn (q) is not an F -module by (2.1) in conjunction with Proposition 3.10. On the wedge cube V := ∧3 (Y6 ) of the natural module for SL6 (q), transvections have rank 6, so a possible offender A has |A| ≥ q 6 . Consider an end-node parabolic P = U L containing A. If A ≤ L = GL5 (q), then it is maximal possible, centralizing a 2- or 3-space on the natural module. But then the centralizer on V |GL5 (q) = ∧3 (Y5 ) ⊕ ∧2 (Y5 ) is just 4-dimensional. Thus A ∩ U = 1, and the centralizer of A is at most 11-dimensional, whence |A| = q 9 is maximal possible, the unipotent radical of a GL3 (q) × GL3 (q)-parabolic. This is ruled out by Theorem 3.1. (a) For the tensor product module V := Yn ⊗ Yn of SLn (2a ) we use induction. For n = 2 an offender A is contained in the unipotent radical of a Borel subgroup, but the latter violates the F-condition. For n > 2, induction and Theorem 2.4 show that a possible offender must lie inside the unipotent radical U of a maximal parabolic subgroup q n−1 GLn−1 (q). But U has centralizer dimension (n − 1)2 on V , thus we don’t get an example. For SU3 (q), the maximal size of -subgroups centralizing subspaces of the natural module of given dimension is bounded in Proposition 3.11. This shows that the Fcondition is not satisfied. ˜ 2 (Y2n ) of the natural module for Sp2n (q), n ≥ 2, The heart of the exterior square ∧ (a) is not an example by (2.1) in conjunction with Proposition 3.12. The case of Y4 ⊗ Y4 2a a for Sp4 ( ) is ruled out by the same argument. The spin modules for Sp2n (2 ), n ≥ 4, are not F -modules by Proposition 5.10. The spin module for Spin2n+1 (q), n ≥ 4, q = a odd, was excluded in Proposition 5.8. − The half-spin modules for Spin+ 2n (q), n ≥ 6, respectively for Spin2n (q), n ≥ 4, were excluded in Propositions 5.4 and 5.6.

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The Suzuki-group 2B2 (22a+1 ) has 2-rank 2a+1, but does not contain transvections, so the 4-dimensional module over F22a+1 is not an F -module. On the 7-dimensional module for G2 (a ), > 2, long root elements have rank 2, all other non-trivial unipotent elements have rank at least 4. Thus this is not an F module for > 3. For = 3 we obtain that the only possibility is that |A| = q 4 and CV (A) is 3-dimensional. Thus the corresponding tame abelian subalgebra I (A) has order q 4 , so must correspond to four abelian roots. The only possibility for this is the set of all non-simple roots. These roots are the ones of maximal weight in any reasonable ordering on the roots, so A = U (I (A)) is the subgroup generated by the root subgroups corresponding to the four positive nonsimple roots. Let X be any short root subgroup in A. Since CV (X) = CV (A), it follows that CV (A) is invariant under NG (X). Since there are two different short root subgroups in A and since NG (X) is a maximal parabolic subgroup of G, it follows that G leaves CV (A) invariant, a contradiction. For F4 (q) on the 25- or 26-dimensional module, long root elements have rank 6, all other non-trivial unipotent elements rank at least 10, by [9, Tab. 3]. So by Table 13 this is not an F -module when > 2. For = 2, the 26-dimensional module occurs as a direct summand in the restriction of the 27-dimensional module for E6 (q). Since the latter is not an F -module (see below), this case is excluded as well. Let P = U L be a maximal parabolic subgroup of E7 (q)sc with Levi factor L of type E6 . Then U is one of the 27-dimensional modules for E6 (q) ≤ L. Now U is the unique abelian subgroup of maximal order in a Sylow -subgroup of P by Lemma 3.15. Thus, by Lemma 2.6 the module U is not an F -module for E6 (q). The other 27-dimensional module for E6 (q) is the image under the graph automorphism of order 2, hence isn’t an F -module either. On the 27-dimensional modules for 2E6 (q) over Fq 2 long root elements have rank 6, all other non-trivial unipotent elements have rank at least 10, by [9, Tab. 5]. Thus Table 13 shows that the F-condition cannot be satisfied. In non-defining characteristic we just have to consider the 2F -modules determined in the first part [6] in Tables 6.3, 6.5, and 6.7. Subgroups of order k in Sn contain elements with at most n − k fixed points, so of rank ( − 1)k on the permutation module. Passing to the deleted permutation module we see that this is not an example when > 2. For = 2, subgroups of order 2k fixing n − 2k points necessarily contain odd elements. Thus if A ≤ An has size |A| = 2k then it contains elements of rank k + 1. If n is odd, the permutation module is a direct sum of the n − 1-dimensional irreducible module with the trivial one. Thus the deleted permutation module is not an F -module for An , n odd, in characteristic = 2. For the 8-dimensional representations of A9 over F2 not contained in the permutation module, explicit computation shows that elementary abelian subgroups of order 8 have at most 4-dimensional centralizer and those of order 16 have 2-dimensional centralizer. The 2-dimensional representation of 2.A5 over F9 does not give an example since the Sylow 3-subgroup has only order 3. On the 8-dimensional representation of

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2.A9 over F3 elements of order 3 have rank 4. Since the 3-rank is 3, we don’t get an example. The 6-dimensional F2 -representation of U3 (3) is the restriction from G2 (2). There long root elements have rank 2, so A has to contain other non-trivial unipotent elements by Table 5. But these have rank at least 3, while U3 (3) only has 2-rank 2, so there are no offenders. The 6-dimensional F3 -module for G := 2.L3 (4) is the restriction of the natural module for SO− 6 (3). Since G only has 3-rank 2, this is not an F -module for G by Proposition 3.14. The 7-dimensional F3 -module for S6 (2) is orthogonal. On restriction to the sub ∼ ∼ ∼ group SO− 6 (2) = SU4 (2) = S4 (3) = SO5 (3) of 3 -index we find a copy of the natural module for SO5 (3). This is the heart of the exterior square of the natural module for Sp4 (3), hence not an F -module by what we showed above. The 8-dimensional F3 -module for 2.S6 (2) is the restriction of the spin module for Spin7 (3). On restriction to the 3 -index subgroup SO5 (3) we obtain two copies of the spin module, the natural module for Sp4 (3). Now Proposition 3.12 shows that this is not an example. The 6-dimensional F4 -module V of 31 .U4 (3) is of orthogonal type, so involutions have rank at least 2. Hence |A| ≥ 24 , but then A centralizes at most a 3-dimensional subspace by Proposition 3.14. This contradicts the fact that 31 .U4 (3) has 2-rank 4. On the 8-dimensional F3 -module for 2.O+ 8 (2), elements of order 3 have rank 2 or 4 and subgroups of order 9 have centralizer dimension at most 4. So if there exists an offender, the maximal torus of order 34 is also an offender. In particular we may assume that A contains elements from all classes of 3-elements. Now the Weyl group of type E8 contains the direct product of two Weyl groups of type A4 (visible on the extended Dynkin diagram), as well as the product of four Weyl groups of type A2 (visible in E6 × A2 ). Thus there are 3-elements with Jordan type 32 12 and with Jordan type 24 . Clearly no subgroup containing these two types can have centralizer dimension 4. The 6-dimensional F4 -module V of 3.M22 is the restriction of the natural module for SU6 (2). Since 3.M22 does not contain transvections the centralizer of a 5dimensional subspace is trivial, and the centralizer of a 4-dimensional space has size less than 24 by Lemma 3.11. Since 3.M22 has 2-rank 4, this shows that V is not an F -module. The 6-dimensional F4 -module V of J2 is the restriction of the 6-dimensional module for G2 (4). By [9, Table 1] long root elements have rank 2, the other involutions have rank 3. Since J2 has 2-rank 4, an offender would have to consist of long root elements only. But such subgroups in G2 (4) have size at most 4 by Table 3.9. On the 10-dimensional F2 -module of M12 , involutions have rank 4, but the 2-rank equals 3. On the 10-dimensional F2 -module of M22 , involutions have rank 4, Klein four groups have centralizer codimension at least 5, but the 2-rank equals 4.

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On the 11-dimensional F2 -module of M23 , involutions have rank 4, Klein four groups have centralizer codimension at least 5, but the 2-rank equals 4. The 11-dimensional F2 -module of M24 , the 22-dimensional F2 -module of Co2 and the 24-dimensional F2 -module of Co1 were discussed in Lemmas 9.2–9.4. On the 5-dimensional F3 -module of M11 , elements of order 3 have rank 3, so this is no example. On the 6-dimensional F3 -module of M12 , elements of order 3 have rank 3 or 4, so this is not an example. 9.1. The examples for F -modules. The following two tables contain the F -modules V for finite quasi-simple groups. In the last column we give the size of one possible (±) offender on V . For the natural module Yn of Spinn (a ) in the fifth line, has to be odd when n is odd. Table 18. F -modules for groups of Lie type in defining characteristic

G SLn (a ) SLn (a ) SUn (a ) Sp2n (a ) (±) Spinn (a ) Spin7 (a ) a Spin+ 2n ( ) G2 (2a )

d n n 2

n 2n n 8 2n−1 6

V f Yn a 2 ∧ (Yn ) a Yn 2a Y2n a Yn a Z3 a Zn−1 a M(λ2 ) a

conditions n≥4 n≥4 n≥6 n = 4, 5

log |A| a (n − 1)a 4a a 3a 5a (2n − 2)a 3a

Case 9.1: On its natural module Yn , SLn (q) acts as a transvection group, so we get an example with |A| = q. Case 9.2: Thesubgroup of SLn (q) of order q n−1 in (4.1) has centralizer of dimension at least n−1 on ∧2 (Yn ), so this yields an F -module. 2 Case 9.3: For the special unitary group SUn (q), n ≥ 4, on its natural module Yn over Fq 2 Proposition 3.11 shows that there exists a subspace of codimension 2 whose centralizer contains an elementary abelian subgroup of order q 4 . Case 9.4: The symplectic group Sp2n (q) acts as a transvection group on its natural module Y2n , thus we get an example with |A| = q. (±) Case 9.5: For the orthogonal group Spinn (q), n ≥ 6, on its natural module Proposition 3.14 shows that there exists a subspace of codimension 3 whose centralizer contains an elementary abelian subgroup of order q 3 . Case 9.6: The spin module Z3 for Spin7 (q) has highest weight λ1 . Consider the maximal parabolic subgroup of type Spin5 (q). The centralizer of its unipotent radical

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U is again the spin module for Spin5 (q), hence has dimension 4. But U is elementary abelian of order q 5 . Thus U is an offender on Z3 . Case 9.7: For the half-spin modules V := Zn−1 of dimension 2n−1 of Spin+ 2n (q) let A be the unipotent radical of a maximal parabolic subgroup with Levi complement of + type Spin+ 2n−2 (q). Then the centralizer CV (A) is a half-spin module for Spin2n−2 (q), of dimension 2n−2 , while A is elementary abelian of order q 2n−2 , so we find an example whenever n ≤ 5. Case 9.8: The 6-dimensional module V embeds G2 (q), q = 2a , into Sp6 (q). Let B be a Borel subgroup of G2 (q). Then B fixes a totally singular 3-space W of V . This defines a homomorphism into GL3 (q), with kernel the centralizer A := CB (W ) of W . Thus |A| ≥ q 3 , and A is an offender. Table 19. F -modules in non-defining characteristic

G A2n 3.A6 A7

d f 2n − 2 1 2 3 2 2 4 1 2

|A| 2n−1 4 4

Case 9.9: Let V be the 2n − 2-dimensional heart of the permutation module of A2n in characteristic 2. Let A be the intersection with A2n of a maximal 2-subgroup of S2n generated by commuting transpositions. Then |A| = 2n−1 , A centralizes a subspace of dimension n on the permutation module, and by direct computation, still an n − 1-dimensional subspace of V . Thus A is an offender. (For A6 ∼ = Sp4 (2) this is ∼ the natural module, for A8 = SL4 (2) it is the exterior square of the natural module.) Case 9.10: The group 3.A6 on its 3-dimensional F4 -module gives an example with |A| = 4. Case 9.11: The group A7 on its 4-dimensional F2 -module (the natural module for SL4 (2) ∼ = A8 ) contains A6 ∼ = Sp4 (2) in its natural representation. The latter already is Example 9.9. Note added in proof: The open cases in Table 1 have now been settled in joint work with R. Lawther. The only case that is a 2F -module is F4 (3a ).

References [1]

M. Aschbacher, GF(2)-representations of finite groups. Amer. J. Math. 104 (1982), 683–771.

[2]

M. Aschbacher, S. Smith, The classification of quasithin groups, vol 1: Structure of strongly quasithin K-groups, vol 2: Main theorems: the classification of simple QTKE groups, Math. Surveys Monogr. 111, 112, Amer. Math. Soc., Providence, 2004.

Classification of 2F -modules, II

183

[3]

A. Chermak, A. Delgado, A measuring argument for finite groups, Proc. Amer. Math. Soc. 107 (1989), 907–914.

[4]

J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, R. A. Wilson, Atlas of finite groups, Clarendon Press, Oxford 1985.

[5]

D. Gorenstein, R. Lyons, R. Solomon, The classification of the finite simple groups, Number 3, Math. Surveys Monogr. 40.3, Amer. Math. Soc., Providence 1998.

[6]

R. M. Guralnick, G. Malle, Classification of 2F -modules, I, J. Algebra 257 (2002), 348–372.

[7]

I. Hughes, G. Kemper, Symmetric powers of modular representations, Hilbert series and degree bounds, Comm. Algebra 28 (2000), 2059–2088.

[8]

C. Jansen, K. Lux, R. Parker, R. Wilson, An atlas of Brauer characters, Clarendon Press, Oxford 1995.

[9]

R. Lawther, Jordan block sizes of unipotent elements in exceptional algebraic groups, Comm. Algebra 23 (1995), 4125–4156. Correction: ibid. 26 (1998), 2709.

[10] M. W. Liebeck, The affine permutation groups of rank three, Proc. Lond. Math. Soc. (3) 54 (1987), 477–516. [11] F. Lübeck, Small degree representations of finite Chevalley groups in defining characteristic, LMS J. Comput. Math. 4 (2001), 135–169. [12] U. Meierfrankenfeld, G. Stroth, On quadratic GF(2)-modules for Chevalley groups over fields of odd order. Arch. Math. (Basel) 55 (1990), 105–110. [13] U. Meierfrankenfeld, G. Stroth, Quadratic GF(2)-modules for sporadic simple groups and alternating groups. Comm. Algebra 18 (1990), 2099–2139. [14] Th. Meixner, Failure of factorization modules for Lie-type groups in odd characteristic, Comm. Algebra 19 (1991), 3193–3222. [15] A. Premet, Weights of infinitesimally irreducible representations of Chevalley groups over a field of prime characteristic, Math. USSS-Sb. 61 (1988), 167–183. [16] N. Spaltenstein, Classes unipotentes et sous-groupes de Borel, Lecture Notes in Math. 946, Springer-Verlag, Berlin–New York 1982. [17] G. Stroth, 2F -modules with quadratic offender for the finite simple groups, in: Groups and combinatorics—in memory of Michio Suzuki, Adv. Stud. Pure Math. 32, Math. Soc. Japan, Tokyo 2001, 391–400. Robert M. Guralnick, Department of Mathematics, University of Southern California, Los Angeles CA 90089-1113, U.S.A. E-mail: [email protected] Gunter Malle, FB Mathematik/Informatik, Universität Kassel, Heinrich-Plett-Str. 40, 34132 Kassel, Germany E-mail: [email protected]

On the real Schur indices associated with infinite Coxeter groups Allen Herman and Barry Monson∗

1. Introduction This article is motivated by an attempt to explain, using character theory, the three geometric formulae proved in [6] that give, for finite abstract regular polytopes, basic relationships between the arithmetic properties of their automorphism groups, their essential Wythoff space dimensions, and the dimensions of their realization cones. We will show that, while two of the three formulae are fairly easy consequences of character theory, the other cannot be explained using character theory as such. Indeed, this formula indicates a relationship between the geometric realizations of an abstract regular polytope and the realizability of the complex irreducible representations of its automorphism group over the field of real numbers.

2. Finite string C-groups A C-group is a group generated by a distinguished finite set S = {s0 , s1 , . . . , sn−1 } of involutions satisfying the intersection property with respect to S, i.e. for all subsets I , J ⊆ S, sk : k ∈ I ∩ sk : k ∈ J = sk : k ∈ I ∩ J . A string C-group is a C-group having the further property that (sj si )2 = 1, whenever |i − j | > 1. Thus is a rather special homomorphic image of some (possibly infinite) string Coxeter group, say W = ρ0 , ρ1 , . . . , ρn−1 , whose distinguished generators ρi satisfy the defining relations ρi 2 = 1, for all i ∈ {0, . . . , n − 1} (ρi−1 ρi )mi = 1 for all i ∈ {1, . . . , n − 1}, and (ρj ρi )2 = 1

if |i − j | > 1,

for m1 , . . . , mn−1 ∈ {∞, 2, 3, 4, . . . }. (The third set of relations implies that the Coxeter diagram for W has the shape of a string.) It is quite special that the homomor∗ The authors acknowledge the support of NSERC.

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phism induced by the mapping ρi → si should preserve the intersection condition, which does hold in any Coxeter group W [4, §5.5]. String C-groups are precisely the automorphism groups of abstract regular polytopes [7, §2B and §2E], which we discuss in § 4 below. In theory, string C-groups are quite plentiful. This is because any infinite Coxeter group W is a finitely generated linear group via the standard (faithful) reflection representation [4, §5.4]. Malcev’s theorem, concerning finitely generated linear groups, shows that W is residually finite; that is, for any finite subset {w1 , . . . , wm } of W \ 1, there exists a surjective homomorphism f : W → such that is finite and 1 ∈ {f (w1 ), . . . , f (wm )}. Using this property, one can (in a non-constructive way) prove the existence of several infinite families of finite abstract regular polytopes, each with its own finite string C-group of automorphisms [7, §4C]. −1 −1 For example, if m1 , m2 ≥ 3 are integers satisfying m−1 1 + m2 < 2 , then there are infinitely many finite regular maps of type {m1 , m2 }. (The latter notation indicates a regular map having only m1 -gonal faces, with m2 of them around each vertex.) On the other hand, while there are some explicit methods for producing plentiful examples of finite string C-groups, such methods typically involve very special constructions, such as the reduction of linear groups over a number field, modulo some prime ideal. Some other efforts to construct string C-groups involve taking the quotient of the Coxeter group W , say by prescribing the order of the Coxeter element ρ0 ρ1 . . . ρn , or by prescribing the orders of certain “holes” in the polytope. Several finite regular maps of these types, with group order up to 21 504, appear in Table 8 of [1].

3. Real Schur indices for finite Coxeter groups Let χ be an irreducible complex character of a finite group G, that is, the trace of an irreducible linear representation X : G → GLn (C) for some positive integer n. If K is a subfield of C, then χ is said to be realizable over K if there is a linear representation X which is similar to X such that X (G) ⊆ GLn (K). In order for χ to be realizable over K, it is necessary that K contain the field Q(χ ) := Q({χ (g) : g ∈ G}) generated by the character values over the rational field Q. In other words, it is necessary that K(χ ) = K. However, this condition is not sufficient for χ to be realizable over K. The Schur index of χ over the field K is the positive integer mK (χ) := min{[L : K(χ)] : χ is realizable over L}. There are three possibilities concerning realizability of complex irreducible characters over the field of real numbers R, which were classically explained by Frobenius

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and Schur using what is now known as the Frobenius–Schur indicator 1 ν2 (χ) := χ (g 2 ). |G| g∈G

The Frobenius–Schur indicator takes only the values 1, 0, and −1. It is 1 exactly when χ is realizable over R; it is 0 when χ is not realizable over R and R(χ ) = C; and it is −1 when χ is not realizable over R, R(χ ) = R, and χ has Schur index 2 over R. It has been known since the 70s that all irreducible representations of Weyl groups (i.e. finite Coxeter groups satisfying the crystallographic condition) are realizable over the field of rational numbers (see [3, §6.3]). In other words, all such representations have rational Schur index 1. On the other hand, there is one example of an irreducible representation of a finite Coxeter group with rational Schur index 2. (All others have rational Schur index 1.) The exception is an irreducible character of degree 48 of the√group of type H4 , which has rational character values but is only realizable over Q( 5) (see [3, §6.3]). Since this is a real subfield, it follows that every irreducible representation of a finite Coxeter group can be realized over the field of real numbers. For such groups then, all real irreducible representations are absolutely irreducible. On the other hand, there are examples of string C-groups that are known to possess irreducible characters with non-real fields of character values. For example, the groups PSL(2, 11) ∼ = [5, 5]5 := [5, 5]/(ρ0 ρ1 ρ2 )5 and PSL(2, 19) ∼ = [9, 9]3 := [9, 9]/(ρ0 ρ1 ρ2 )3 both possess a complex conjugate pair of irreducible characters of minimal nontrivial degree having imaginary quadratic fields of character values. (The value of the Frobenius–Schur indicator of such irreducible characters is 0.) Therefore, not all irreducible representations of finite string C-groups are realizable over the reals. However, no example is known of a finite string C-group that possesses an irreducible character with Frobenius–Schur indicator −1 (i.e. real Schur index 2).

4. Realizations of abstract regular polytopes An abstract regular n-polytope P is a partially ordered set having certain key properties of the face lattice of a classical convex regular n-polytope. (We refer to [7] for details.) The elements of P are called faces. P is equipped with a strictly monotone rank function on these faces, with range {−1, . . . , n}. Faces of rank 0 are vertices; those of rank 1 are edges; and there are unique minimal and maximal (improper) faces of ranks −1 and n, respectively.

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It is instructive to think of the vertices, edges and square facets of an ordinary cube P , along with ∅ and P itself, all ordered by inclusion. Note that the Euclidean symmetry group of P is the Coxeter group W = B3 generated by the ordinary reflections ρ0 , ρ1 , ρ2 . If abcd is a particular square facet of P , then we may suppose that ρ0 interchanges vertices a and b; that ρ1 interchanges edges ab, ad (while fixing a); and that ρ2 interchanges the two square facets which share edge ab. Note that the stabilizer in W of vertex a is ρ1 , ρ2 . Returning now to P , suppose that Fj is the face of rank j in some maximal chain F0 , . . . , Fn−1 of proper faces. Such a chain is called a flag; the polytope P is said to be regular if its automorphism group is transitive on the set of all flags. We may therefore fix F0 , . . . , Fn−1 as base flag. It is now a consequence of the axioms that for each j , 0 ≤ j ≤ n − 1, there exists a unique involutory automorphism sj moving only element Fj in the base flag. Furthermore,

= s0 , . . . , sn−1 is generated as a string C-group by these involutions; and Stab (F0 ) = 0 := s1 , . . . , sn−1 . Consequently, the vertices of P can be identified with the cosets of the standard subgroup 0 . We emphasize that, in general, P need not be a lattice, nor have any sort of familiar geometric structure. Despite this essentially combinatorial nature, however, it is naturally interesting to attempt to construct P as a ‘real’ object. When P is finite, we may modify the definitions in [5] as follows. For a finite P , let be any orthogonal representation of , of degree d. The Wythoff space for this representation is the subspace of points in Rd fixed by 0 . A realization of P is any pair P = [, x], where the base point x lies in the Wythoff space for . Let P0 denote the set of vertices of P . It follows that the map β : P0 → Rd given by β(g(F0 )) := (g)(x), g ∈ , is well-defined. Moreover, each g induces an isometric permutation of the geometric vertex set V := {(g)(x) | g ∈ }. This is the beginning of an inductive construction, wherein the representation is used to transfer the combinatorial structure of P to some sort of object in Rd (see [7, 5A]). We omit details, noting only that it may be inevitable that some of the structure of P is lost in the realization. In the extreme case, it can happen that the Wythoff space is trivial, so that x = o. We then obtain the trivial realization of P , with V = {o}.

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5. Character-theoretic proofs of some formulae concerning the geometric realizations of regular polytopes In [5], three equations were given concerning relationships between the dimensions and other key parameters for the realizations of finite, abstract regular polytopes. These formulae did not properly account for the possibility that some of the real irreducible representations involved might be reducible over the complex field. Corrected equations were later given by McMullen and Monson in [6]. The proofs of these results, in both the original and revised form, rely heavily on interesting convexity arguments. Here we show that two of the formulae can be viewed as consequences of charactertheoretic information concerning real representations of the automorphism group of the abstract polytope. Let Irr R ( ) and Irr C ( ) denote the set of characters of irreducible representations of the finite group over the real and complex numbers, respectively. For all α ∈ Irr R ( ), there is a χ ∈ Irr C ( ) such that when we consider α as a character over C, we have that either (1) α = χ , (2) α = χ + χ, ¯ or (3) α = 2χ . For any characters ϕ, α of , let (ϕ, α) denote the usual inner product of class functions. If ϕ is a real character of , and α ∈ Irr R ( ), then define if α is of type (1), (ϕ, α) (ϕ, α)∗ = 21 (ϕ, α) if α is of type (2), 1 4 (ϕ, α) if α is of type (3). Lemma 1. Suppose X is a real orthogonal representation of a finite group , affording the complex character ϕ. Then the decomposition of ϕ into real irreducible characters is (ϕ, α)∗ α. ϕ= α∈Irr R ( )

Proof. The decomposition of ϕ into complex irreducibles is (ϕ, χ )χ . ϕ= χ∈Irr C ( )

If χ is involved in a real irreducible character of type (1), then (ϕ, χ )χ = (ϕ, α)α = (ϕ, α)∗ α since χ = α. If χ is involved in a real irreducible character α of type (2), then α = χ + χ¯ . Since ϕ is a real character we must have (ϕ, χ) = (ϕ, χ¯ ). Thus (ϕ, χ )χ + (ϕ, χ¯ )χ¯ = (ϕ, χ )(χ + χ¯ ) = 21 (ϕ, α)α.

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If χ is involved in a real irreducible character α of type (3), then α = 2χ, and (ϕ, χ )χ = 41 [(ϕ, 2χ)2χ ] = 41 (ϕ, α)α. Since each χ ∈ Irr C ( ) is involved in exactly one of these cases we have (ϕ, χ)χ = (ϕ, α)∗ α. ϕ= χ∈Irr C ( )

α∈Irr R ( )

Let us consider again the finite abstract regular polytope P with automorphism group = ρ0 , . . . , ρn−1 . Let be any orthogonal representation of and let ψ be the (possibly reducible) complex character afforded by . Thus has degree dψ = ψ(1). If 0 is a subgroup of , we denote the restriction of a character ψ of

to 0 by ψ 0 , and the induction of a character θ of 0 to by θ . (Restricted and induced characters are characters in a natural way. The restriction of a character is the character of the restriction of the corresponding representation to 0 . Also, if M is a left C 0 -module affording θ, then θ is by definition the character afforded by the left C -module C ⊗C 0 M.) If = 1 is irreducible, then dψ coincides with the affine dimension of the orbit of any point x = o in the Wythoff space for . However, for 1 such an orbit is a single point, with affine dimension 0, even though d1 0 = 1. Note that the dimension of the Wythoff space is wψ = (1 0 , ψ 0 ), which equals ((1 0 ) , ψ) by Frobenius reciprocity. The dimension of the essential Wythoff space of is defined to be wψ ∗ := ((1 0 ) , ψ)∗ . (We are here factoring out the effect of the centralizer of ( ) in the orthogonal group for Rd , since such centralizing isometries merely map one realization to a congruent realization. Thus, as explained in [6], wψ ∗ measures the extent to which one and the same representation can provide incongruent realizations.) Concerning the trivial realization, it is a useful convention to agree that w1∗ = 0, even though w1 = 1. Recall from § 4 that |P0 | = v = [ : 0 ]. By pairing the v vertices of P with the standard basis vectors of Rv , we obtain a v-dimensional orthogonal representation of

, as well as a realization of P . We conclude that this representation of is induced by the trivial representation on 0 . Therefore, the character of this representation is (1 0 ) . Now, the simplex realization of the polytope P is obtained by observing that the sum of all standard basis vectors of Rv will be fixed by the action of , and so there is a representation of on (1, . . . , 1)⊥ ∼ = Rv−1 which affords the (real-valued) character

σ = (1 0 ) − 1 . Applying the lemma results in the two consequences of interest to us. Theorem 1 ([6], Theorem 3 (a), (c)). Let σ = (1 0 ) − 1 denote the real-valued character associated simplex realization of P . Then with the ∗ wα dα = v − 1, and (a) σ (1) = α∈Irr R ( ) (b) wα ∗ wα = wσ . α∈Irr R ( )

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Proof. By the lemma, we have σ (1) =

(σ, α)∗ α(1).

α∈Irr R ( )

As long as α = 1 , we have dα = α(1); also (σ, α) = ((1 0 ) , α) − 0 = wα , so (σ, α)∗ = wα ∗ . On the other hand, (σ, 1 )∗ = (σ, 1 ) = ((1 0 ) , 1 ) − (1 , 1 ) = (1 0 , 1 0 ) − 1 = 0 = w1∗ . Since σ (1) = v − 1, part (a) follows immediately. Part (b) concerns the dimension of the Wythoff space for the simplex realization: wσ = ((1 0 ) , σ ) (σ, α)∗ ((1 0 ) , α) = α∈Irr R ( )

= (σ, 1 )∗ ((1 0 ) , 1 ) +

(σ, α)∗ ((1 0 ) , α)

α =1

= (σ, 1 )∗ ((1 0 ) , 1 ) + =

wα ∗ wα

α =1

∗

wα wα ,

α∈Irr R ( )

again since w1∗ = 0.

6. The realization cone and realizability of characters As described in [5], the realizations of an abstract regular polytope P can be naturally arranged into isometry classes, and the set P˜ of these classes becomes a closed convex cone under the operations of blending of realization classes and scaling by positive real numbers. (For definitions of these terms, see [5].) A third formula from [6] equates the dimension r of the realization cone P˜ to the sum of the dimensions of the realization cones P˜α , as α runs through Irr R ( ). Here P˜α is the cone of all realization classes for which the corresponding orthogonal representation X : → On (E) affords the real irreducible character α. Furthermore, for each real irreducible character α, the dimension of the realization cone P˜α was shown (by geometric methods) to be 1 ∗ ∗ 2 wα (wα + 1). The number r, which counts the number of non-trivial diagonal classes in P , can also be expressed group-theoretically as r = |{ 0 g 0 ∪ 0 g −1 0 : g ∈ \ 0 }|.

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Similarly, we also have from [6] that the total number of double cosets is wσ + 1. If we now let N be the number of self-inverse double cosets of 0 in , then r + 1 = |{ 0 g 0 ∪ 0 g −1 0 : g ∈ }| 1 = N + (wσ + 1 − N). 2 On the other hand, J. S. Frame in [2, Theorem A], established character-theoretic formulas for the number of self-inverse double cosets of a subgroup in any finite group. These formulas in this case are N=

1 (1 0 ) (g 2 ) = | | g∈

ν2 (χ)((1 0 ) , χ) =

χ∈Irr C ( )

ν2 (χ )wχ .

χ ∈Irr C ( )

Since σ = (1 0 ) − 1 , we have wσ + 1 = ((1 0 ) , (1 0 ) ) =

((1 0 ) , χ)2 =

χ∈Irr C ( )

wχ 2 .

χ ∈Irr C ( )

Therefore r +1=

χ∈Irr C ( )

=

χ∈Irr C ( )

1 ν2 (χ)wχ + (wχ 2 − ν2 (χ )wχ ) 2 1 wχ (wχ + ν2 (χ )). 2

Comparing this with the expression for r in [6], we see that there is a perfect match over the irreducible characters χ with ν2 (χ) = 1. As for those characters with ν2 (χ ) = 1, it is convenient to first express the general formula in terms of real irreducible characters. For those χ ∈ Irr C ( ) with χ + χ¯ = α ∈ Irr R ( ), we have wχ = wχ¯ = wα ∗ and ν2 (χ ) = ν2 (χ¯ ) = 0, so 1 1 wχ (wχ + ν2 (χ)) + wχ¯ (wχ¯ + ν2 (χ¯ )) = wχ 2 = (wα ∗ )2 . 2 2 For those χ ∈ Irr C ( ) with 2χ = α ∈ Irr R ( ), we have wχ = ν2 (χ ) = −1, so

1 2 wα

= 2wα ∗ and

1 wχ (wχ + ν2 (χ)) = wα ∗ (2wα ∗ − 1). 2 Now in this calculation we have not had to assume that is a string C-group. It therefore follows from Frame’s formula that |{ 0 g 0 ∪ 0 g −1 0 : g ∈ G}| can be expressed as below in terms of real irreducible characters.

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Corollary 1. For any finite group and any subgroup 0 , |{ 0 g 0 ∪ 0 g −1 0 : g ∈ G}| equals 1 wα ∗ (wα ∗ + 1) + (wα ∗ )2 + wα ∗ (2wα ∗ − 1), 2 α=χ α=χ+χ¯

α=2χ

where each sum runs over the real irreducible characters α of that have the indicated decompositions as complex characters. Comparing this to the formula for r + 1, we obtain an interesting consequence. Theorem 2. Let α be a real irreducible character of the finite string C-group , which is not irreducible as a complex character. Then wα∗ = 0 or 1. In particular, if wα = wα ∗ then the corresponding realization cone Pα is one dimensional. Proof. Since McMullen and Monson’s expression for r + 1 is equal to the expression in Corollary 1, we have 1 1 wα ∗ (wα ∗ +1)+ wα ∗ (wα ∗ +1). (wα ∗ )2 + wα ∗ (2wα ∗ −1) = 2 2 α=χ +χ¯

α=2χ

α=χ+χ¯

α=2χ

Therefore 1 1 [(wα ∗ )2 − wα ∗ (wα ∗ + 1)] = [ wα ∗ (wα ∗ + 1) − wα ∗ (2wα ∗ − 1)]. (∗) 2 2 α=χ +χ¯

α=2χ

Since wα ∗ is a non-negative integer for all real irreducible characters α, we have 1 1 (wα ∗ )2 − wα ∗ (wα ∗ + 1) = [(wα ∗ )2 − wα ∗ ] ≥ 0 2 2 and 3 1 ∗ wα (wα ∗ + 1) − wα ∗ (2wα ∗ − 1) = [wα ∗ − (wα ∗ )2 ] ≤ 0. 2 2 Therefore, both sides of the equation (∗) must be 0. This implies wα ∗ = 0 or 1 for all real irreducible characters α which are not complex irreducible, and the theorem follows. Note that the group [5, 5]5 described in [6] admits a real irreducible character α for which the corresponding ν2 (χ) = 0. Indeed, α is the unique real irreducible character with wα ∗ = wα , and it is true that wα ∗ = 1. The next corollary is immediate from the theorem. Corollary 2. If dim P˜α > 1 for some α ∈ Irr R ( ), then α is necessarily the character of a complex irreducible representation that is realizable over the field of real numbers. Note that the above result really says nothing about whether there actually exists an irreducible character of a finite string C-group with Schur index 2, so the existence of such characters remains a mystery.

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Allen Herman and Barry Monson

References [1] H. S. M. Coxeter and W. O. Moser, Generators and Relations for Discrete Groups, SpringerVerlag, Berlin, Heidelberg 1980. [2] J. S. Frame, The double cosets of a finite group, Bull. Amer. Math. Soc. 47 (1941), 458–467. [3] M. Geck and G. Pfeiffer, Characters of Finite Coxeter Groups and Iwahori–Hecke Algebras, Oxford University Press, Oxford 2000. [4] J. E. Humphries, Reflection Groups and Coxeter Groups, Cambridge University Press, Cambridge 1990. [5] P. McMullen, Realizations of regular polytopes , Aequationes Math. 37(1989), 38–56. [6] P. McMullen and B. Monson, Realizations of regular polytopes II, Aequationes Math. 65 (2003), 102–112. [7] P. McMullen and E. Schulte, Abstract Regular Polytopes, Encyclopedia Math. Appl. 92, Cambridge University Press, Cambridge 2002. Allen Herman, Department of Mathematics and Statistics, University of Regina, Regina, SK, Canada S4S 0A2 E-mail: [email protected] Barry Monson, Department of Mathematics and Statistics, University of New Brunswick, Fredericton, NB, Canada E3B 5A3 E-mail: [email protected]

Some observations on products of characters of finite classical groups Gerhard Hiss and Frank Lübeck

Abstract. This is an extended version of the talk given by the first author at the conference. We report on the outcome of some experiments with the decomposition of products of (generalized) characters of some classical groups. In particular some results of the PhD-thesis of Dirk Mattig on the products of unipotent characters of general linear groups are presented and commented. In order to obtain similar patterns for other classical groups it seems necessary to replace unipotent characters by unipotent almost characters.

1. Introduction The unipotent characters of the general linear group GLn (q) are parametrized by partitions of n, independently of q. Let us fix n and consider three partitions λ, µ, and ν of n. For each prime power q, there are three associated unipotent characters q q q q χλ , χµ , and χν of GLn (q). We may consider the multiplicity of χλ in the product q q χµ · χν , as a function in q. Following Mattig, we show that this multiplicity function is a polynomial function over the rationals. Using CHEVIE, Mattig has computed the corresponding polynomials for all 1 ≤ n ≤ 8 (and all relevant triples of partitions). He found that in fact all these polynomials have non-negative integral coefficients. After a discussion of Mattig’s results, we present similar computations for the general unitary groups GUn (q). We also consider the problem of extending our observations to other series of classical groups. We suggest to replace unipotent characters by unipotent almost characters. Finally, we sketch some special results. For instance, we show that the constant q coefficient of the polynomial expressing the multiplicity of χν in the square of the Steinberg character, is equal to the degree of the irreducible character of the symmetric group Sn corresponding to ν. This fact has been observed by Lux and Malle on an example presented in the talk of the first author. Let us close this introduction by introducing two notational conventions. Let X be a subset of C and f : X → C, x → f (x) a function, and let S ⊆ C. We say that f is a polynomial in x over S, if there is a polynomial p ∈ C[X], with coefficients in S, such that f (x) = p(x) for all x ∈ X.

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If G is a finite group and if ϕ, ψ are class functions on G, we write 1 (ϕ, ψ)G = ϕ(g)ψ(g −1 ) |G|

(1)

g∈G

for the usual scalar product on the set of class functions of G.

2. Tensor product polynomials Let n be a positive integer, q a prime power and let G denote one of the groups GLn (q) or GUn (q). The unipotent characters of G form a distinguished subset of the set of absolutely irreducible ordinary characters of G (over the complex numbers). In case G = GLn (q) the unipotent characters of G are the irreducible constituents of the permutation character 1B G , where B denotes the Borel subgroup of upper triangular matrices of G. There is no elementary way to introduce the unipotent characters in case G = GUn (q). Here, they are constructed as characters on adic cohomology groups on Deligne–Lusztig varieties on which G acts (see, e.g., [2, Chapter 12]). There is, however, a common parametrization of the unipotent characters in both cases. Namely, there is a bijection between the unipotent characters of G and the partitions of n. In particular, the unipotent characters are parametrized independently q of q. Let us write χλ for the unipotent character of G labelled by the partition λ of n. q q In this parametrization χ(n) denotes the trivial character 1G and χ(1n ) the Steinberg character St G of G. If we fix n and three partitions λ, µ, and ν of n, we may consider the scalar products q

tλ,µ,ν (q) = (χλ , χµq · χνq )GLn (q)

and

q t¯λ,µ,ν (q) = (χλ , χµq · χνq )GUn (q)

(2)

as functions in q. Using the terminology introduced above, we can now state the first result. Proposition 2.1 (Mattig, [12, Propositon 3.1.6]). Let n be a positive integer and let λ, µ, ν be partitions of n. Then tλ,µ,ν (q) and t¯λ,µ,ν (q), as functions on the set of all prime powers q, are polynomials in q over Q. Let us sketch a proof of this result. First of all, the conjugacy classes of G can be grouped together into class types, and the class types can be classified independently of q. Let us write gs and gu for the semisimple and unipotent part of an element g ∈ G, respectively. Two elements g and h of G belong to the same class type, if CG (gs ) is conjugate to CG (hs ) and if gu is conjugate in CG (gs ) to a conjugate of hu lying in CG (gs ). (For a definition and more details on class types for general series of groups of Lie type see [6, Section 4].) Secondly, the unipotent characters of G are constant on class types, and thirdly, the numbers of conjugacy classes inside a class type, the lengths of conjugacy classes,

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and the values of the unipotent characters, viewed as functions of q, are polynomials in q over the rationals. Finally, the order of G is a polynomial in q over the integers, and hence, by (1), the scalar products (2) are rational functions in q. Since these take integer values for all prime powers q, they are polynomials in q over Q. Table 1 collects all the information that is needed to compute the tensor product polynomials for n = 2. The first column of that table gives the number of conjugacy classes of GL2 (q) in each of the four class types, the second column gives the length of each conjugacy class in a class type. The third and fourth columns give the values of the two unipotent characters of GL2 (q). Table 1. The generic unipotent character table of GL2 (q)

Nr. of Classes in Type

Length

χ(2)

χ(12 )

q −1 q −1 1 2 (q − 1)(q − 2) 1 2 (q − 1)q

1 q2 − 1 q(q + 1) q(q − 1)

1 1 1 1

q 0 1 −1

We call such a table a generic unipotent character table. The CHEVIE-system [6] contains the generic unipotent character tables of GLn (q) and GUn (q) for 2 ≤ n ≤ + 8, of CSp4 (q), CSp6 (q), SO− 8 (q), and Spin8 (q). Furthermore, CHEVIE includes programs to compute with these character tables. As examples we have computed the polynomials t(15 ),(15 ),ν (q) and t¯(15 ),(15 ),ν (q), where ν runs through the partitions of 5. The result is given in Table 2. Dirk Mattig has used CHEVIE to compute the polynomials tλ,µ,ν (q) for all triples of partitions of numbers up to 8. Let us write N for the set of non-negative integers. Observation 2.2. (Mattig, [12].) Let λ, µ, ν be partitions of the integer n with 1 ≤ n ≤ 8. Then the function tλ,µ,ν (q) is a polynomial in q over N for all prime powers q. The same statement does not hold for the polynomials t¯λ,µ,ν (q) but the calculations with CHEVIE lead to the following observation. Observation 2.3. Let λ, µ, ν be partitions of n, where n is an integer with 1 ≤ n ≤ 8. Then we have: (a) The functions t¯λ,µ,ν (q) are polynomials in q over the integers. (b) The difference functions tλ,µ,ν (q) − t¯λ,µ,ν (q) are polynomials in q over 2N, that is the coefficients are even non-negative integers. Remarks. There are formulae for the values of unipotent characters for GLn (q) and GUn (q) which are closely related by an operation called Ennola duality. Roughly speaking one has to substitute in the formulae for GL the parameter q by −q and to adjust some signs.

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But this does not lead to a simple relation between tλ,µ,ν (q) and t¯λ,µ,ν (q) (see the example in Table 2), since there is no such simple correspondence between the number of classes in class types (see the example in Table 3) and since the mentioned adjustment of signs depends of the class type. Note also that the polynomials do not specialize to the corresponding scalar products of characters of the symmetric group Sn at q = 1. For example, if n = 2 we have (St, St · St)GL2 (q) = 1. The Steinberg character of GL2 (q) corresponds to the alternating character of S2 , whose square equals the trivial character. Table 2. Some tensor products in GL5 (q) and in GU5 (q)

ν

(StG , StG ·χν ) for G = GL5 (q)

(5) 1 (4, 1) 4 (3, 2) q2 + 2 q + 5 (3, 12 ) q3 + 2 q2 + 4 q + 6 2 4 (2 , 1) q + q3 + 3 q2 + 5 q + 5 3 5 (2, 1 ) q + q 4 + 3 q 3 + 3 q 2 + 6 q + 4 q6 + q4 + 2 q3 + q2 + 3 q + 1 (15 )

(StG , StG ·χν ) for G = GU5 (q) 1 0 q2 + 1 q3 + 2 q + 2 4 q − q3 + q2 + q + 1 q5 + q3 − q4 − q2 6 q + q4 + q2 + q + 1

Table 3. Numbers of conjugacy classes in class types

GL3 (q)

GU3 (q)

q −1 q −1 q −1 (q − 1)(q − 2) (q − 1)(q − 2) 1 6 (q − 1)(q − 2)(q − 3) 1 2 2 q(q − 1) 1 3 q(q − 1)(q + 1)

q +1 q +1 q +1 (q + 1)q (q + 1)q 1 6 q(q − 1)(q + 1) 1 2 2 (q − 2)(q + 1) 1 3 q(q − 1)(q + 1)

3. Tensor product polynomials for other groups Now let {G(q)| q a prime power} be an arbitrary series of groups of Lie type. For example {Sp4 (q)}, the symplectic groups on a 4-dimensional vector space over Fq ,

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or {2 E6 (q)sc } the twisted groups of type E6 in a simply-connected algebraic group of this type. For a precise definition see [6, Section 4.1]. Again, each group G(q) has a distinguished set of unipotent characters which are parametrized for all groups in the series independently of q (see, e.g., [2, Sections 13.8,13.9]). We write for a set of parameters of the unipotent characters and q χλ for the character of G(q) corresponding to λ ∈ . The exact values of the unipotent characters are not yet known in all cases. But in those cases where they are known a certain generalization of Proposition 2.1 can be proved with the same arguments. Unlike the case {GLn (q)} it is in general no longer true that just one generic unipotent character table describes the values of the unipotent characters of G(q) for all prime powers q. The smallest example for this is the series {SL2 (q)}. For odd q these groups have three unipotent conjugacy classes while for even q there are only two. It turns out that for each series {G(q)} there is a number m ∈ N such that for any fixed 0 ≤ i ≤ m − 1 and all prime powers q ≡ i (mod m) the class types can be parametrized independently of q and the number of classes in a class type is given by a polynomial in q. Furthermore, the class length for all classes in a fixed class type is described by a polynomial in q. It is not clear a priori that the unipotent characters are constant on class types. But this is true in all known cases. Moreover, in the known cases all values of unipotent characters are given by polynomials in a square root q 1/2 of q with coefficients in some finite extension of Q which is independent of q within a congruence class of q modulo m as described above. Thus, fixing an integer 0 ≤ i ≤ m − 1 such that the generic unipotent character table of {G(q)| q ≡ i (mod m)} is known, we see as in Proposition 2.1 that the multiplicities q

tλ,µ,ν,i (q) = (χλ , χµq · χνq )G(q) for all prime powers q with q ≡ i (mod m) are polynomials in q over C. If we compute these polynomials in the case {Sp4 (q) | q even} (the character values were first computed by Enomoto [5]), we quickly find polynomials with non-integer coefficients. This shows that Observations 2.2 and 2.3 do not generalize directly to other series of groups. Almost Characters. To find such a generalization we have to look at the unipotent almost characters instead. These are certain C-linear combinations, with coefficients independent of q, of the unipotent characters. Almost characters were introduced by Lusztig when he described the parametrizations of unipotent characters. The transforming matrices from irreducible characters to almost characters are called Fourier transform matrices. For linear groups (like GLn (q) and SLn (q)) the unipotent characters coincide with the unipotent almost characters. For unitary groups (like GUn (q) and SUn (q)) they coincide up to sign. But for all other types of groups this is no longer true.

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It is conjectured by Lusztig and proved in many cases by Lusztig and Shoji that the almost characters are class functions associated to certain geometric objects called character sheaves, see [10], [11] and [14]. ˜ be a set of parameters for the unipotent almost characters of the groups Let ˜ we write χ˜ q for the corresponding almost character of G(q). For G(q); for λ ∈ λ q q q m as above and 0 ≤ i ≤ m − 1 set t˜λ,µ,ν,i (q) = (χ˜ λ , χ˜ µ · χ˜ ν )G(q) , viewed as a function on all prime powers q ≡ i (mod m). Then, in those cases where we know that the tλ,µ,ν,i (q) are polynomials in q the t˜λ,µ,ν,i (q) are also polynomials in q over the complex numbers. We have computed the polynomials t˜λ,µ,ν,i (q) for all 0 ≤ i ≤ m − 1 in the following cases.

Series

m

Series

m

{GLn (q)}, n ≤ 8 {SLn (q)}, n ≤ 8 {GUn (q)}, n ≤ 8 {SUn (q)}, n ≤ 8 {COn (q)}, n = 7, 9

1 n 1 n 2

{CSpn (q)}, n = 4, 6, 8 {Spn (q)}, n = 4, 6, 8 {Spin7 (q)} {Spin+ 8 (q)} {3D 4 (q)}

2 2 4 2 2

Note, that the case SLn (q) cannot easily be derived from the case GLn (q) although the sets of unipotent characters are in bijection via restriction. To see why, let us denote by ρ a linear character of GLn (q) generating the cyclic group GLn (q)/ SLn (q) of order q q − 1. The unipotent characters of SLn (q) are exactly the charactersχλ ↓SLn (q) , where λ runs through the partitions of n. Using Frobenius reciprocity and Clifford’s theorem we find q

(χλ ↓SLn (q) , χµq ↓SL

n (q)

q

· χνq ↓SLn (q) )SLn (q) = ([χλ ↓SLn (q) ]GLn (q) , χµq · χνq )GLn (q) =

q−2 q (χλ · ρ i , χµq · χνq )GLn (q) i=0

= tλ,µ,ν (q) +

q−2 q (χλ · ρ i , χµq · χνq )GLn (q) . i=1

q

q

q

If the center Z(GLn (q)) is in the kernel of ρ i , then (χλ · ρ i , χµ · χν )GLn (q) = 0 in general, and thus contributes to the tensor product polynomial for SLn (q). To give a specific example, t(13 ),(13 ),(13 ) = q +1, whereas in case 3 | q −1 the tensor product polynomial for SL3 (q) corresponding to ((13 ), (13 ), (13 )) equals 3q +1. This q q q reflects the fact that (χ(13 ) ·ρ i , χ(13 ) ·χ(13 ) )GL3 (q) = q for i = (q −1)/3 and 2(q −1)/3.

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Some of these tables of unipotent characters were available in CHEVIE and computed in [16], [5], [13], [4], [7], and [9]. The other non-linear group cases were computed by the second author using the results from [15]. Observation 3.1. In all cases mentioned above, the functions t˜λ,µ,ν,i (q) are polynomials in q over Z. Here, some negative coefficients occur in the cases GUn (q) with 5 ≤ n ≤ 8, SUn (q) with 3 ≤ n ≤ 8 (for some i), 3D 4 (q) and Spin7 (q) with q ≡ 3 (mod 4). This suggests that, if our observations are actually true for all series of groups of Lie type, one has to consider character sheaves for an explanation. Comparing the cases GL and GU it is interesting to know that the character sheaves leading to the values of unipotent characters are the same geometric objects for GL and GU. Maybe there is an interpretation of the coefficients of the tensor product polynomials using the same non-negative numbers which have to be added up with different signs. There does not seem to be an obvious notion of tensor products of character sheaves which correspond to the tensor products of the associated class functions.

4. Special results on the tensor product polynomials When the first author gave the talk on these results in Gainesville, Klaus Lux and Gunter Malle observed that the constant coefficient of the polynomial t(15 ),(15 ),ν in the first half of Table 2 equals ζ ν (1), where ζ ν is the irreducible character of the symmetric group S5 labelled by the partition ν. Gunter Malle observed furthermore, that the constant coefficient of the polynomial t¯(15 ),(15 ),ν in the second half of Table 2 equals 0, if χν does not lie in the principal series, and equals the degree of the corresponding character of the Weyl group of GU5 (q), otherwise. Here, the Weyl group is a dihedral group of order 8. During the conference a sketch of a proof for these facts emerged through discussions with Gunter Malle and Meinolf Geck. We give the details here. Proposition 4.1. Let n be a positive integer and let ν be a partition of n. Write ζ ν for the irreducible character of the symmetric group Sn labelled by ν. Put m := n/2. (a) The constant coefficient of t(1n ),(1n ),ν equals ζ ν (1). (b) The constant coefficient of t¯(1n ),(1n ),ν equals |ζ ν (σ )|, where σ ∈ Sn is an involution that is a product of m disjoint transpositions. The value |ζ ν (σ )| also has the following interpretation. Let κ and (α, β) denote the 2-core and the 2-quotient of ν, respectively. Then |ζ ν (σ )| equals 0, if |κ| > 1, and ζ (α,β) (1), otherwise. Here, ζ (α,β) denotes the irreducible character of the Weyl group of type Bm labelled by the bi-partition (α, β) of m. Proof. For 0 = f ∈ Q[X], f = Xl g with l ≥ 0 and X g, we write fX := Xl and fX := g = f/fX .

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The proof is given simultaneously for the general linear and unitary groups. Let ε be a parameter taking the values ±1. Let Gε denote the generic group giving rise to the series GLn (q), if ε = 1, and to GUn (q), if ε = −1. We write |Gε | for the order polynomial of Gε , i.e., the value of |Gε | at q is the order of the finite group Gε := Gε (q). Letting N := n(n − 1)/2, we have n (Xi − ε i ). |Gε | = X N

i=1

˜ label the class types ˜ be a set of labels for the class types of Gε and let ⊆ Let ˜ we of semisimple elements. Such sets can be chosen independently of ε. For ω ∈ write Stε (ω) and χν,ε (ω) for the value polynomials of the Steinberg character and the unipotent character χν , respectively. We also write nε (ω) and cε (ω) for the numbers of conjugacy classes of class type ω and the numbers of elements of a conjugacy class of class type ω, respectively. Finally, t ε denotes t, if ε = 1, and t¯, if ε = −1. We then have ε nε (ω)cε (ω) Stε (ω)2 χν,ε (ω). t(1 n ),(1n ),ν |Gε | = ˜ ω∈

This follows from Formula (1) with the above settings and the fact that χν,ε (ω) = χν,ε (ω ), where ω denotes the class type containing the inverses of the elements of class type ω. ˜ \ , then St ε (ω) = 0, and if ω ∈ , then St ε (ω) = ±(|Gε |/cε (ω))X (see If ω ∈ [2, Theorem 6.5.9]). Note that cε (ω) divides |Gε |, and that |Gε |/cε (ω) is the centralizer order polynomial corresponding to the class type ω. Let us write (|Gε |/cε (ω))X =: X mω,ε . Then, dividing both sides of the equation above by XN , we have ε t(1 n ),(1n ),ν |Gε |X =

nε (ω)cε (ω)X X mω,ε χν,ε (ω).

ω∈

Now mω,ε = 0 for ω ∈ , if and only if ω is a class type of regular semisimple elements, i.e., the generic centralizer is a torus. So, only regular semisimple classes ε contribute to the constant term of the polynomial t(1 n ),(1n ),ν . Finally, we have to consider the polynomials nε (ω) for class types of regular semisimple elements. Such class types are parametrized by the partitions of n as follows. Given a regular semisimple element in Gε , it has n pairwise different eigenvalues and the set of eigenvalues is permuted by raising these numbers to the εq-th power. All such sets of eigenvalues occur and each set of eigenvalues occurs for exactly one conjugacy class. The class type of the element is parametrized by the cycle type of this permutation on the eigenvalues. Using induction one can see that the number of sets of eigenvalues as above for a fixed cycle type is a polynomial in q which is divisible by q if there is: – any cycle of length greater than one, if ε = 1.

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203

– more than one cycle of length one or a cycle of length greater than two, if ε = −1. This shows that in each case there is only one class type, namely the one of the regular elements in a maximally split torus, which contributes to the constant term we are interested in. Let ω0 be the label of this class type. Then ε t(1 n ),(1n ),ν |Gε |X ≡ nε (ω0 )cε (ω0 )X χν,ε (ω0 )

(mod X).

Now, for ε = 1, i.e., for GLn (q), we find from the combinatorial interpretation of n1 (ω0 ) given above that n 1 (X − i). n1 (ω0 ) = n! i=1

Since c1 (ω0 )X =

n

i=1 (X

i

− 1) |Gε |X = , n (X − 1) (X − 1)n

it follows that the constant coefficient of t(1n ),(1n ),ν equals χν,1 (ω0 ). Now let ε = −1. In this case we have m 1 n−2m (X2 − X − 2i), n−1 (ω0 ) = m (X + 1) 2 m! i=1

and c−1 (ω0 )X =

n

(X

i i i=1 (X − (−1) ) . m + 1)n−2m i=1 (X 2 − 1)

It follows that the constant coefficient of t¯(1n ),(1n ),ν equals χν,−1 (ω0 ). To determine the value of χν,ε (ω0 ) we use the character formula for the values of Deligne–Lusztig generalized characters on semisimple elements (see [2, Proposition 7.5.3]), and the connection of unipotent characters with unipotent almost characters as presented in [1, p. 45]. Let ψν,ε denote the unipotent almost character of Gε corresponding to ζ ν . Then ψν,ε = χν,ε , if Gε = GLn (q), and ψν,ε = ±χν,ε , otherwise. Moreover, ψν,1 (ω0 ) = ζ ν (1), so we are done in case ε = 1. Let ε = −1. Then ψν,−1 (ω0 ) = ζ ν (σ ). Also, by [8, Corollary 2.7.33], |ζ ν (σ )| = 0, if |κ| > 1, and |ζ ν (σ )| = ζ (α,β) (1), if |κ| ≤ 1. It remains to show that the sign of ζ ν (σ ) equals the sign of ψν,−1 . By the definition of the almost characters, the degree of ψν,−1 is obtained from the degree of ψν,1 by replacing q by −q. Thus ψν,−1 = χν,−1 , if the generic degree polynomial has even degree, and ψν,−1 = −χν,−1 , otherwise. A formula for the generic degree polynomial can be found in [2, p. 466]. The sign of ζ ν (σ ) is determined in [8, pp. 80–82]. Using these descriptions and induction on the 2-weight of ν, it follows that ψν,−1 = −χν,−1 if and only if ζ ν (σ ) is negative.

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Remark. Note that the property that all but one of the numbers of classes in the class types of regular semisimple elements is divisible by q is a special property of the series GLn (q) and GUn (q). Thus there is no generalization of Theorem 4.1 to other series of groups. Let us now fix the positive integer n and put G = GLn (q) for the remainder of this section. The following result is also contained in Mattig’s PhD-thesis. Proposition 4.2 (Mattig, [12, Theorem 6.4.8]). For all partitions µ, ν of n, the polynomial t(n−1,1),µ,ν is constant (and hence a non-negative integer). In order to prove this, Mattig considers the permutation characters 1Pλ G on parabolic subgroups Pλ . Here, λ again is a partition of n and Pλ denotes the corresponding parabolic subgroup of G (whose Levi subgroup is the group of block diagonal matrices with blocks of sizes given by the parts of λ). Let us write Pn for the set of partitions of n. Given λ ∈ Pn , there are non-negative integers aλ,µ , independent of q, such that aλ,µ χµq (3) 1Pλ G = µ∈Pn

(see, e.g., [3, Theorem (70.24)]). Moreover, by [8, Section 2.2], the matrix of coefficients (aλ,µ )λ,µ∈Pn is invertible over the integers. In particular, the unipotent charq acters χλ can be expressed as Z-linear combinations of the permutation characters q 1Pµ G . For example, χ(n−1,1) = 1P(n−1,1) G − 1P(n) G = 1P(n−1,1) G − 1G . It follows from Proposition 2.1 and Equation (3) that there are polynomials rλ,µ,ν ∈ Q[X] such that rλ,µ,ν (q) = 1Pλ G , 1Pµ G · 1Pν G GL (q) n

for all prime powers q. Another interpretation of these polynomials is provided by G 1Pλ , 1Pµ G · 1Pν G GL (q) = |Pλ \G/Pµ ∩ xP ν |, (4) n

x∈Dµ,ν

where Dµ,ν denotes the set of distinguished double coset representatives corresponding to the parabolic subgroups Pµ and Pν . Note that Dµ,ν can be chosen, independently of q, as a subset of distinguished double coset representatives of the Weyl group Sn of G. Using (4), Mattig shows that r(n−1,1),µ,ν is constant for all partitions µ, ν of n. The remarks following (3) now prove Proposition 4.2. Of course, rλ,µ,ν ∈ N[X] if tλ,µ,ν ∈ N[X]. Let us consider a further special case, namely P(1n ) = B. We have q ζ λ (1) χλ . (5) 1B G = λ∈Pn

Moreover, by [2, Propositions 7.4.4 and 7.5.4] we have 1B G · St G = 1T G ,

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where T denotes the maximally split torus of G consisting of the diagonal matrices. Using Proposition 2.1, and Equation (5), we see that there is a polynomial sn ∈ Q[X] such that sn (q) = |B\G/T | = 1B G , 1T G G = 1B G , 1B G · St G G = ζ λ (1) ζ µ (1) tλ,µ,(1n ) (q). λ,µ∈Pn

Theorem 4.3 (Mattig, [12, Chapters 4 and 8]). The polynomials sn have integer coefficients for all n ≥ 1. The degree of sn equals (n − 1)(n − 2)/2, and the coefficient at X i is positive for all 0 ≤ i ≤ (n − 1)(n − 2)/2. Moreover, sn is monic except for n = 2. As examples, we give the polynomials sn for 1 ≤ n ≤ 6. s1 = 1 s2 = 3 s3 = X + 19 s4 = X3 + 6 X2 + 36 X + 211 s5 = X6 + 8 X5 + 35 X4 + 136 X3 + 410 X2 + 1253 X + 3651 s6 = X10 + 10 X9 + 54 X8 + 209 X7 + 685 X6 + 1969 X5 + 4951 X4 + 11592 X3 + 24415 X2 + 50547 X + 90921 Let us sketch the main arguments in the proof of Theorem 4.3. Let U denote the set of upper unitriangular matrices of G so that B = U T . Let W denote the Weyl group of G and w0 its longest element. For any w ∈ W , put Uw := U ∩ U w0 w . Then Uw is invariant under conjugation by T , and R := {wu | w ∈ W, u ∈ Uw } is a set of coset representatives of the right cosets of B in G. Moreover, for wu, w u ∈ R, we have BwuT = Bw u T if and only if w = w and u and u are in the same orbit of T on Uw , where T acts on Uw by conjugation. This shows that it suffices to determine, for each w ∈ W , the number of orbits of T on Uw as a function of q. Fix w ∈ W and write V := Uw . Let + denote the set of pairs {(i, j ) | 1 ≤ i < j ≤ n}, which corresponds to the set of positive roots of G. Then there is a subset ⊆ + such that | aii = 1 for 1 ≤ i ≤ n, and aij = 0 for i = j and (i, j ) ∈ / } V = {(aij ) ∈ Fn×n q (see [2, Proposition 2.5.16]). Choose an injective mapping ν : → N. A subset ⊆ can then be viewed as a weighted undirected graph, by associating ν(i, j ) to the edge (i, j ) ∈ . Since the weight function is injective, has a unique minimal spanning forest. Let X denote

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Gerhard Hiss and Frank Lübeck

the set of subsets of which arise as minimal spanning forests in this way. Then X contains a unique maximal element max . For a subset of let V∗ := {(aij ) ∈ V | aij = 0 if and only if i = j or (i, j ) ∈ }. For ∈ X, let V denote the union of the sets V∗ , where runs through the subsets of which have as their minimal spanning forest. Then, obviously, V is invariant under conjugation by T (since every V∗ is T -invariant), and V , V = ∈X

a disjoint union. The proof concludes by showing that the number of T -orbits on V equals q n , where n = |max | − ||.

References [1]

M. Broué, G. Malle, and J. Michel, Generic blocks of finite reductive groups, in Représentations unipotentes génériques et blocs des groupes réductifs finis, Astérisque 212 (1993), 7–92.

[2]

R. W. Carter, Finite Groups of Lie Type: Conjugacy Classes and Complex Characters, Pure Appl. Math., Wiley-Interscience , Chichester 1985.

[3]

C. W. Curtis and I. Reiner, Methods of Representation Theory I, Wiley, 1981; Methods of Representation Theory II, Wiley, New York 1987.

[4]

D. I. Deriziotis and G. O. Michler, Character tables and blocks of finite simple triality groups 3D 4 (q 3 ), Trans. Amer. Math. Soc. 303 (1987), 39–70.

[5]

H. Enomoto, The characters of the finite symplectic group Sp(4, q), q = 2f , Osaka J. Math. 9 (1972), 75–94.

[6]

M. Geck, G. Hiss, F. Lübeck, G. Malle, and G. Pfeiffer, CHEVIE—A system for computing and processing generic character tables, AAECC 7 (1996), 175–210.

[7]

M. Geck and G. Pfeiffer, Unipotent characters of the Chevalley groups D4 (q), q odd, Manuscripta Math. 76 (1992), 281–304.

[8]

G. D. James and A. Kerber, The Representation Theory of the Symmetric Group, Encyclopedia Math. Appl. 16, Addison-Wesley, Reading, MA 1981.

[9]

F. Lübeck, Charaktertafeln für die Gruppen CSp6 (q) mit ungeradem q und Sp6 (q) mit geradem q, Dissertation, Universität Heidelberg 1993.

[10] G. Lusztig, Character sheaves I to V, Adv. Math. 56, 57, 59, 61, 1985/1986. [11] G. Lusztig, Green functions and character sheaves, Ann. Math. 131 (1990), 355–408.

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[12] D. Mattig, Tensor Products of Unipotent Characters of GL(n, q), Dissertation, Universität Heidelberg 1999. [13] K. Shinoda, The characters of the finite conformal symplectic group CSp(4,q), Comm. Algebra 10 (1982), 1369–1419. [14] T. Shoji, Character sheaves and almost characters of reductive groups I and II, Adv. Math. 111 (1995), 244–313, 314–354. [15] T. Shoji, Unipotent characters of finite classical groups, in Finite reductive groups: related structures and representations (Luminy, 1994), Prog. Math. 141, Birkhäuser, Boston 1997, 373–413. [16] B. Srinivasan, The characters of the finite symplectic group Sp(4, q), Trans. Amer. Math. Soc. 131 (1968), 488–525. Gerhard Hiss, Lehrstuhl D für Mathematik, RWTH Aachen, 52056 Aachen, Germany E-mail: [email protected] Frank Lübeck, Lehrstuhl D für Mathematik, RWTH Aachen, 52056 Aachen, Germany E-mail: [email protected]

The number of finite p-groups with bounded number of generators Andrei Jaikin-Zapirain ∗

Abstract. We establish new bounds for the number of finite p-groups generated by a fixed number of elements. 2000 Mathematics Subject Classification: Primary 20D15; Secondary 20E18

1. Introduction The investigation of the number of finite p-groups is a classical topic in group theory. Denoting by f (n) the number of groups of order n, we have the following result of G. Higman and C. C. Sims (see [3, 8]): Theorem 1.1 (Higman, Sims). 2

f (pk ) = p 27 k

3 +o(k 3 )

,

where the term o depends only on k and not on p. In this paper we consider the function f (n, d), which denotes the number of groups of order n which can be generated by d elements. By results of A. McIver, P. M. Neumann and A. Mann (see [4, 5]), we have Theorem 1.2 (McIver, Neumann and Mann). pc1 (d)k ≤ f (pk , d) ≤ pc2 (d)k , 2

2

for some positive constants c1 (d) and c2 (d). Thus the following question seems reasonable: Problem 1.3 (A. Mann, [6, Question 6]). Is there a constant c(d) such that 2 2 f (p k , d) = pc(d)k +o(k ) , and what is the value of this constant? We have not been able to find an exact answer to this question, but we give rather reasonable estimates for c(d): ∗ This work has been partially supported by the MCYT Grants BFM2001-0201, BFM2001-0180, FEDER and the Ramón y Cajal Program.

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Theorem 1.4. We have that p

d−1 2 2 4 k +o(k )

≤ f (pk , d) ≤ p

d−1 2 2 2 k +o(k )

.

We explain now the general strategy of the proof of these two bounds. The detailed proof is given in the following sections. Let Fd be a free pro-p group on d generators. Then every finite p-group G generated by d elements is a quotient of Fd : G ∼ = Fd /N for some N Fd . First of all we want to know how many such subgroups N there are. Of course, their number is less than or equal to the order of the group of automorphisms of G. On the other hand | Aut(G)| ≤ |G|d . Thus we obtain: Proposition 1.5. Let Npk (Fd ) be the number of normal subgroups of index pk in Fd . Then f (p k , d) ≤ Npk (Fd ) ≤ pkd f (pk , d). As we have seen in Theorem 1.2, f (p k , d) is like pk , and so it is much bigger than p kd when k is big. Hence the main problem is to estimate Npk (Fd ) (this function is called the normal subgroup growth of Fd ). Now, fix a normal subgroup N of index p t in Fd . We want to know how many normal subgroups of Fd of index p in N there are. Define this set by MN : 2

MN = {K | K Fd , K ≤ N, |N : K| = p}. We have an easy characterization of the set MN : if |N : K| = p then K ∈ MN if and only if [N, Fd ]N p ≤ K. Thus, we have to estimate the rank of N/[N, Fd ]N p . The following characterization of the rank of N/[N, Fd ]N p is well-known (see, for example, [7, Proposition 7.8.2] and [1, Lemma D6]): d(N/[N, Fd ]N p ) = rp (Fd /N ) + d − d(Fd /N ),

(1)

where d(G) and rp (G) denotes the minimum number of generators and the minimum number of defining pro-p relations respectively of a finite p-group G. We set g(p t , d) = max{d(N/[N, Fd ]N p ) | N Fd , |Fd /N| = p t }. g(p t ,d)

t

Then we have |MN | ≤ p p−1 −1 < pg(p ,d) . Now, we come back to the function Npt (Fd ). This number is clearly less than the number of chains Fd = N0 > N1 > N2 > · · · > Nt , where Ni is a normal t−1 s subgroup of Fd and |Ni : Ni+1 | = p, which is less than p s=0 g(p ,d) by the previous observation. In this way we have an upper bound for f (pt , d). We will see now that in the lower bound of f (p t , d) the function g(pt , d) also plays a crucial role. Let N be a normal subgroup of Fd of index ps such that d(N/[N, Fd ]N p ) = g(p s , d). We have that every subgroup K, satisfying [N, Fd ]N p ≤ K ≤ N, is normal in Fd . Hence the number of normal subgroups of Fd of index ps+k is at least the number of subspaces in N/[N, Fd ]N p of codimension k, which is greater than s p k(g(p ,d)−k) . Playing with k and s such that k + s = t, we obtain the lower bound for f (pt , d). In Section 3 we will modify this method in order to obtain a better bound.

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211

All this leads us to study the function g(pt , d). In view of (1), we should estimate the following function: h(pt , d) = max{rp (G) | d(G) = d, |G| = pt }. Our result in this direction is the following: Theorem 1.6. If d is fixed, then (d − 1)t (1 + o(1)) ≤ h(pt , d) ≤ (d − 1)t − as t goes to infinity.

d(d−3) 2

The proof of the upper bound is based on the following remark. Let G be a finite p-group. We can associate with G a Lie algebra L = L(G) = i γi (G)/γi+1 (G), where γi (G) is the ith term of the lower central series of G. Then it is well-known that the minimum number of generators of L (we denote it by d(L)) is equal to d(G) and that rp (G) is less than or equal to the minimum number of relations needed to define L. Let F be a free Lie Zp -algebra on d(G) generators and H an ideal of F such that L ∼ = F /H . Then the minimum number of relations needed to define L is equal to dF (H ) = logp |H /([H, F ] + pH )|. Hence the upper bound in the previous theorem follows from its analogue for Lie rings: Theorem 1.7. Let F be a free Lie Zp -algebra on d generators, H ≤ [F, F ] + pF an ideal of F and L = F /H a nilpotent Lie ring of order pt . Then dF (H ) ≤ (d − 1)t − d(d−3) 2 . Remark 1.8. If we remove the condition H ≤ [F, F ] + pF in the theorem then we have that the minimum number of relations needed to define L is equal to dF (H ) − d + d(L) and so the theorem gives d(L)(d(L) − 3) + d − d(L). 2 We prove this theorem in Section 2. The lower bound in Theorem 1.6 is obtained by constructing an explicit family of p-groups {Gt } such that |Gt | = p t and rp (Gt ) ≥ (d − 1)t + o(t). We present this family in Section 3. The notation is standard. If L is a Lie Zp -algebra and M is a right L-module then dL (M) = logp |M/(ML + pM)|. dF (H ) ≤ (d(L) − 1)t −

2. The upper bound in Theorem 1.4 In order to prove Theorem 1.7 we need to do some auxiliary work. First we consider the case when pF ≤ H . We need the following lemma. Lemma 2.1. Let F be a free Lie Zp -algebra on d generators, H an ideal of F , pF ≤ H and L = F /H a nilpotent Lie ring of order pt > 1. Suppose Z = T /H ≤ [L, L] is a minimal ideal of L. Then dF (H /pF ) ≤ dF (T /pF ) + (d − 1).

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Andrei Jaikin-Zapirain

Proof. Note that dF (H /pF ) = logp |H /([H, F ] + pF )| and dF (T /pF ) = logp |T /([T , F ] + pF )|. We put F¯ = F /([H, F ] + pF ), T¯ = T /([H, F ] + pF ), H¯ = H /([H, F ] + pF ) and K¯ = ([T , F ] + pF )/([H, F ] + pF ). Choose z ∈ T¯ \ H¯ . Then K¯ = [z, F¯ ]. Since [z, [F¯ , F¯ ]] = 0, we obtain that ¯ ¯ d−1 . |K| ≤ pd . Hence |H¯ | ≤ |T¯ /K|p Lemma 2.2. Let F be a free Lie Zp -algebra on d generators, pF ≤ H ≤ [F, F ]+pF an ideal of F and L = F /H a nilpotent Lie ring of order pt . Then dF (H /pF ) ≤ t (d − 1) − d(d−1) 2 . Proof. When L is abelian, the result is trivial. The general case can be proved by induction on the order of L using the previous lemma. We will need the following auxiliary result. For the reader’s convenience we include the proof. Lemma 2.3. Let L be a Lie Zp -Lie algebra and M a right L-module. Then, if N is a L-submodule of M, dL (M) ≤ dL (M/N) + dL (N ). Proof. Since N L + pN ≤ (ML + pM) ∩ N , then |M : (ML + pM)| = |M : (ML + pM + N )||(ML + pM + N) : (ML + pM)| = |M : (ML + pM + N )||N : (N ∩ (ML + pM)| ≤ |M : (ML + pM + N )||N : (N L + pN)|. Proof of Theorem 1.7. We prove Theorem 1.7 by induction on the order of L. When L is abelian, the result is trivial. By Lemma 2.3, we have that dF (H ) ≤ dF (H ∩pF )+dF (H /(H ∩pF )) ≤ dF (H ∩pF )+dF ((H +pF )/pF ). (2) Consider first dF ((H + pF )/pF ). Applying Lemma 2.2, we obtain that d(d − 1) . 2 Now, consider dF (H ∩ pF ). We can divide by p in pF . Hence dF ((H + pF )/pF ) ≤ (d − 1) logp (|F /(H + pF )|) − dF (H ∩ pF ) = dF (p−1 (H ∩ pF )). Note that |F : p−1 (H ∩ pF )| = |pF /(H ∩ pF )| = |(pF + H )/H |. Put M = F /p −1 (H ∩ pF ).

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213

Then, by induction, using Remark 1.8, we have that dF (H ∩ pF ) ≤ (d(M) − 1) logp (|(pF + H )/H |) −

d(M)(d(M) − 3) + d − d(M) 2

≤ (d − 1) logp (|(pF + H )/H |) + d. (The equality occurs only when M = 0.) dF (H )(d − 1) logp |F /H | − d(d−3) 2 .

Therefore from (2) it follows that

Proof of the upper bound in Theorem 1.4. As we have seen before, Theorem 1.7 implies the upper bound in Theorem 1.6. Hence we have g(pt , d) ≤ (d − 1)t + d. Now, as was shown in Introduction f (p t , d) ≤ p t

g(pi , d) ≤

i=1

t−1 i=o

(3)

g(p i ,d)

. From (3) we get

d −1 (t − 1)t + td. 2

3. The lower bound in Theorem 1.4 Now we prove the second part of Theorem 1.4. First we need the following technical lemma: Lemma 3.1. Let Md (n) = n1 k|n µ(k)d n/k , where µ(m) is the Möbius function which is defined for positive integers by the rules µ(1) = 1, and for n = p1e1 . . . pses ; p1 , . . . , ps being distinct primes, µ(n) = 0 if ei > 1, and µ(p1 . . . ps ) = (−1)s . Then when d is fixed and n goes to infinity the following holds: n (1) Md (n) = dn (1 + o(1)); d n+1 (1 + o(1)). (2) sd (n) = ni=1 Md (i) = n(d−1) n−1 n+1 d (3) i=1 sd (i) = n(d−1)2 (1 + o(1)). Proof. The first statement is well-known approximation of Md (n) (see [2, Chapter VIII]). The second and third follow from the next result. Suppose that an+1 − an = k. lim an = lim bn = ∞ and lim n→∞ n→∞ n→∞ bn+1 − bn Then limn→∞

an bn

= k.

Let F be a free pro-p group on d generators x1 , . . . , xd . The lower p-series Pi = Pi (F ) of F is defined inductively by P1 = F

and

p

Pi+1 = [Pi , F ]Pi .

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Andrei Jaikin-Zapirain

We know that logp |Pi /Pi+1 | = sd (i). Put Bn = F /Pn . Now, we calculate the number of defining relations of Bn . Lemma 3.2. We have that rp (Bn ) = (d − 1) logp |Bn |(1 + o(1)) as n goes to infinity. p

Proof. Note that [Pn , F ]Pn = Pn+1 . Hence the result follows from (1) and Lemma 3.1. Now, we want to construct a series {Pn,k } of normal subgroups of F , such that P1 = P1,d > P1,d−1 > · · · > Pn−1,1 > Pn = Pn,sd (n) > Pn,sd (n)−1 > · · · , the index of Pn,k in Pn,k+1 is p and rp (F /Pn,k ) = (d − 1) logp |F /Pn,k |(1 + o(1)) as n goes to infinity. Clearly this will give the lower bound in Theorem 1.6. Define inductively a basis Wi of Pi /Pi+1 in the following way: W1 = {w1,1 P2 , . . . , w1,d P2 },

w1,i = xi .

Now suppose we have a basis Wn = {wn,i Pn+1 |1 ≤ i ≤ sd (n)} of Pn /Pn+1 . Put p wn+1,i = wn,i , when 1 ≤ i ≤ sd (n). We know that wn+1,1 Pn+2 , . . . , wn+1,sd (n) Pn+2 are independent in Pn+1 /Pn+2 and we can complete this set to a basis Wn+1 of Pn+1 /Pn+2 by adding some vectors wn+1,sd (n)+1 Pn+2 , . . . , wn+1,sd (n+1) Pn+2 , where wn+1,k (when k ≥ sd (n) + 1) is of type [xi1 , . . . , xin+1 ]. Let Pn,k be the normal subgroup of F generated by Pn+1 and wn,i with i ≤ k. Lemma 3.3. Suppose that n ≥ 3 and k ≥ p wn,k+1

Proof. Since k ≥

d(d+1) 2 ,

d(d+1) 2 .

Then p

∈ [Pn,k , F ]Pn,k .

we have

wn,k+1 = [[xi1 , . . . , xis−1 ], xis ]p

n−s

for some 1 ≤ i1 , . . . , is ≤ d and s ≥ 3 which can be calculated explicitly from n−s+1 p , xis ]Pn+2 . Now, note that k and n. Hence wn,k+1 Pn+2 = [[xi1 , . . . , xis−1 ]p [xi1 , . . . , xis−1 ]p

n−s+1

p

∈ Pn,k and Pn+2 ≤ [Pn,k , F ]Pn,k .

Put Bn,k = F /Pn,k . In the next lemma we relate the number of defining relations of Bn,k+1 and Bn,k : Lemma 3.4. Suppose n ≥ 3. We have that rp (Bn,k ) ≤ rp (Bn,k+1 )+d−1 if k ≥ and rp (Bn,k ) ≤ rp (Bn,k+1 ) + d in general.

d(d+1) 2

The number of finite p-groups with bounded number of generators

215

Proof. Use a similar argument as in Lemma 2.1 bearing in mind Lemma 3.3. Corollary 3.5. We have that rp (Bn+1 ) ≤ rp (Bn,k ) + (d − 1)(logp |Bn+1 | − logp |Bn,k |) +

d(d + 1) . 2

Theorem 3.6. We have that rp (Bn,k ) ≥ (d − 1) logp |Bn,k |(1 + o(1)) as n goes to infinity. Proof. Suppose that there exist and an infinite number of pairs (n, k) with 1 ≤ k ≤ sd (n) such that rp (Bn,k ) ≤ (d − 1 − ) logp |Bn,k |. Hence, by Corollary 3.5, d(d + 1) . 2 Bearing in mind that logp |Bn,k | ≥ (logp |Bn+1 |)/d, we obtain that rp (Bn+1 ) ≤ (d − 1) logp |Bn+1 | − log |Bn,k | +

rp (Bn+1 ) ≤ (d − 1 − /d) logp |Bn+1 | +

d(d + 1) , 2

which contradicts Lemma 3.2. Corollary 3.7. If d is fixed, then h(pt , d) ≥ (d − 1)t (1 + o(1)) as t goes to infinity. Proof of the lower bound in Theorem 1.4. In order to prove the lower bound in Theorem 1.4 we have to show that for any α < 1 there exists T , such that f (pt , d) ≥ α(d−1) 2 p 4 t for any t ≥ T . a = β > α, and put Choose a natural number a such that pp(d−1)−1 a (d−1) a e = p (d − 1) + 1. Let F be a free pro-p group on d generators x1 , . . . , xd and pa H a normal subgroup of F generated by x1 , x2 , . . . , xd . Then F /H is a cyclic group of order p a . The group H is a free pro-p group on e generators pa

y1 = x1 ,

xk

y(i−2)pa +k+2 = xi 1 ,

2 ≤ i ≤ d, 0 ≤ k ≤ pa − 1.

Put Pi = Pi (H ). Note that x1 acts on P1 /P2 as an automorphism of order pa permuting the elements {yi P2 }, whence x1 also acts on Pi /Pi+1 as an automorphism of order p a . As before we can construct a series {Pn,k } of normal subgroups of H , such that P1 = P1,e > P1,e−1 > · · · > Pn−1,1 > Pn = Pn,se (n) > Pn,se (n)−1 > · · · , the index of Pn,k in Pn,k+1 is p and rp (H /Pn,k ) = (e − 1) logp |H /Pn,k |(1 + o(1))

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Andrei Jaikin-Zapirain

as n goes to infinity. Moreover, without loss of generality we can assume that Pn,k are x1 -invariant. Our purpose is to calculate the number of x1 -invariant subgroups of p Mn,k = Pn,k /[Pn,k , H ]Pn,k . This will give an estimate of the normal subgroup growth of F . For this we need to understand the structure of Mn,k as Fp [x1 ]-module. Note pa that x1 acts trivially on Mn,k , whence we can consider Mn,k as Fp [Cpa ]-module. Claim 1. The Fp [Cpa ]-module Mn,k is isomorphic to a direct sum An,k ⊕ Dn,k , where An,k is a maximal free Fp [Cpa ]-submodule and Dn,k is some complement. Moreover, the Fp [Cpa ]-rank of An,k is (d − 1) logp |F /Pn,k |(1 + o(1)) as n goes to infinity. Proof. Let An,k be a maximal free Fp [Cpa ]-submodule of Mn,k . It is clear that there exists a Fp [Cpa ]-submodule Dn,k such that Mn,k = An,k ⊕ Dn,k . Now, the Fp [Cpa ]p rank of Mn,k is equal to d(Pn,k /[Pn,k , F ]Pn,k ) and so, by Theorem 1.6, it is less than (d − 1) logp |F /Pn,k | + 1. Hence if r1 is the Fp [Cpa ]-rank of An,k and r2 is the Fp [Cpa ]-rank of Dn,k , then r1 + r2 ≤ (d − 1) logp |F /Pn,k | + 1.

(4)

By the construction of Pn,k , the Fp -rank of Mn,k is equal to (e − 1) logp |F /Pn,k |(1 + o(1)) when n goes to infinity. On the other hand, it is less than or equal to pa r1 + (p a − 1)r2 . Hence we obtain that pa r1 + (p a − 1)r2 ≥ pa (d − 1) logp |F /Pn,k |(1 + o(1))

(5)

Comparing (4) and (5) we obtain r1 = (d − 1) logp |F /Pn,k |(1 + o(1)). Claim 2. Let M be a free Fp [Cpa ]-module of rank m. Then there are at least submodules N of M such that M/N is a free Fp [Cpa ]-module of rank s.

a p p s(m−s)

Proof. The proof goes in the same lines as the count of the number of subspaces of codimension r in an m-dimensional Fp -space. We have that there are p (p

a −1)m(m−s)

(pm − 1)(p m − p) . . . (p m − p m−s+1 )

(m − s)-tuples, generating free Fp [Cpa ]-submodules of rank (m − s). On the other hand, every free Fp [Cpa ]-module of rank (m − s) has p (p

a −1)(m−s)2

(pm−s − 1)(p m − p) . . . (p m−s − p m−s−1 )

generating (m − s)-tuples. Hence the number of Fp [Cpa ]-submodules of rank (m − s) is equal to p (p

a −1)(m−s)s

This proves the claim.

ps+1 − 1 p m − 1 p m−1 − 1 . . . . p m−s − 1 p m−s−1 − 1 p−1

The number of finite p-groups with bounded number of generators

217

Hence, using Claim 1 and Claim 2, we obtain that the group Mn,k has at least pp

a s((d−1) log p

|F /Pn,k |(1+o(1))−s)

a

x1 -invariant subgroups of index pp s . Let s be the integer part of t/2pa . Consider L = Pn,k such that the order of a F /L is p t−sp . By the previous observation, the number of x1 -invariant subgroups of a L/[L, H ]Lp of index psp is at least p sp

a ((d−1)(t−sp a )−s)+o(t 2 )

=p

β(d−1) 2 t +o(t 2 ) 4

.

Since every x1 -invariant normal subgroup of H is a normal subgroup of F , the normal β(d−1) 2 2 subgroup growth of F is at least p 4 t +o(t ) . This implies that there exists T , such α(d−1) 2 that f (pt , d) ≥ p 4 t for any t ≥ T . Since α is an arbitrary number less than 1, (d−1) 2 we obtain that f (p t , d) ≥ p 4 t (1+o(1)) . Acknowledgments. I thank A. Mann for helpful conversations and D. Segal for reading a previous version of this work. A part of this work was done while I was visiting the Hebrew University of Jerusalem. I would like to take this opportunity to thank the Institute of Mathematics and personally A. Lubotzky for his hospitality.

References [1] J. D. Dixon, M. P. F. du Sautoy, A. Mann and D. Segal, Analytic pro-p groups, 2nd edition, Cambridge Stud. Adv. Math. 61, Cambridge University Press, Cambridge 1999. [2] B. Huppert and N. Blackburn, Finite groups II, Springer-Verlag, Berlin, Heidelberg 1982. [3] G. Higman, Enumerating p-groups I. Inequalities., Proc. London Math. Soc. 10 (1960), 24–30. [4] A. McIver and P. M. Neumann, Enumerating finite groups, Quart. J. Math. Oxford Ser. (2) 38 (1987), 473–488. [5] A. Mann, Enumerating finite p-groups and their defining relations, J. Group Theory 1 (1998), 59–64. [6] A. Mann, Some questions about p-groups, J. Austral. Math. Soc. 67 (1999), 356–379. [7] L. Ribes and P. Zalesskii, Profinite groups, Springer-Verlag, Berlin, Heidelberg 2000. [8] C. C. Sims, Enumerating p-groups, Proc. London Math. Soc. 15 (1965), 151–166. Andrei Jaikin-Zapirain, Departamento de Matemáticas, Facultad de Ciencias, Universidad Autónoma de Madrid, Cantoblanco, Ciudad Universitaria, 28049 Madrid, Spain E-mail: [email protected]

Enumerating highly non-soluble groups Benjamin Klopsch

Abstract. Let G be a finite group and 1 = G0 ≤ G1 ≤ · · · ≤ Gr = G any composition series of G. The Jordan–Hölder theorem shows that, if I denotes the set of all indices i such that Gi /Gi−1 is Abelian, then α(G) := i∈I |Gi /Gi−1 | is an invariant of G. Let A ∈ N. For n ∈ N let gA (n) denote the number of (isomorphism classes of) groups G such that α(G) ≤ A and |G| ≤ n. We determine the asymptotic behaviour of gA (n) as n → ∞; this provides a satisfactory answer to a question raised by Thompson [13]. 2000 Mathematics Subject Classification: 20D60, 20F69.

1. Introduction For every n ∈ N let g(n) denote the number of (isomorphism classes of) groups of 2 2 order at most n. It is known that g(n) = n( 27 +o(1))(log2 n) as n → ∞; see Higman [5] and Pyber [12]. There are also estimates for the number of groups with some given restrictions; e.g. see Holt [7] or Mann [10]. Here we are interested in the enumeration of groups which are far from soluble in the following precise sense. Let G be a finite group and 1 = G0 ≤ G1 ≤ · · · ≤ Gr = G any composition series of G. The Jordan– Hölder theorem shows that, if I denotes the set of all indices i such that Gi /Gi−1 is Abelian, then α(G) := i∈I |Gi /Gi−1 | is an invariant of G. For every A ∈ N and n ∈ N let gA (n) denote the number of (isomorphism classes of) groups G such that α(G) ≤ A and |G| ≤ n. A result of Camina, Everest, and Gagen [1], originally conjectured by Thompson [13], states that for every A ∈ N we have gA (n) = o(g(n)) as n → ∞. This can be interpreted as a measure of rarity (“most finite groups consist largely of solvable chunks”) and agrees well with our intuition. Indeed, Camina et al. get away with some fairly crude estimates. The purpose of this note is to pin down the precise growth rate of gA (n) as n → ∞, turning “rare” into “really rare”. The special case A = 1 corresponds to enumerating groups without any Abelian composition factors and was successfully completed in [8]; here we prove more generally Theorem A. For every A ∈ N there exist b, c ∈ R>0 such that for all n ∈ N≥60 , nb log2 log2 n ≤ gA (n) ≤ nc log2 log2 n .

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Of course, the lower bound is a direct consequence of the special case A = 1 and requires no further comment. To obtain the upper bound we follow closely the strategy laid out in [8]. A new feature, however, is the appearance of non-trivial second cohomology groups; in this context we employ ideas of Holt [7]. As in [8] the classification of finite simple groups (CFSG) is used in a mild sense. Notation. For every x ∈ R>0 we write log x := log2 x. In addition we define a modified logarithm log∗ : R≥0 → R by setting log∗ x := 0 for 0 ≤ x ≤ 1 and log∗ x := log x otherwise. Let G be a finite group. We denote by com(G) the composition length of G and we write comAb (G), respectively comnA (G), for the number of Abelian, respectively non-Abelian, composition factors of G counted with repetitions. The socle of G is denoted by soc(G) and we write d(G) for the minimal number of generators for G.

2. Generators and permutation actions In this section we derive estimates for the minimal number of generators and the number of (equivalence classes of) permutation actions of a finite group. Proposition 2.1. Let G be a group acting faithfully on a finite set . Then we have com(G) ≤ || and comnA (G) ≤ ||/5. Proof. Put n := ||. First suppose that G acts intransitively on . Then splits into G-orbits 1 , 2 , . . . , r where r ≥ 2 and ri=1 |i | = n. For each i ∈ {1, 2, . . . , r} let Gi ≤ Sym(i ) denote the permutation group induced by G on i . So induction yields com(G) ≤ Then r G is a subdirect r product of G1 , G2 , . . . , Gr . nA com(G ) ≤ | | = n and similarly com (G) ≤ n/5. i i i=1 i=1 Next suppose that G acts transitively, but imprimitively on . Choose a non-trivial, proper G-congruence ≈ on and denote by K the kernel of the induced action of G on /≈. Write d := |/≈| and note that K acts faithfully on each ≈-class. So induction gives com(G) ≤ com(G/K) + com(K) ≤ d + n/d ≤ n and similarly comnA (G) ≤ n/5. Finally suppose that G acts primitively on . Then M := soc(G) is isomorphic to T k , the k-th direct power of some simple group T . Put t := |T |. By the O’Nan–Scott Theorem (as stated in [9]) we have to consider the following cases. I: Affine type. In this case T ∼ = Cp is Abelian, n = p k , and G is a subgroup of the affine linear group AGL(k, p); the group M consists of all translations. First we prove that com(G) ≤ p k . Let ν(k, p) denote the number of prime factors of |GL(k, p)| counted with multiplicities. Certainly we have com(G) = com(G/M)+ com(M) ≤ ν(k, p) + k. If (k, p) ∈ {(2, 2), (3, 2), (4, 2)}, then ν(k, p) is equal to 2,5,10 respectively, and hence ν(k, p) + k ≤ pk . Otherwise the estimate ν(k, p) ≤ log|GL(k, p)| ≤ k 2 log p suffices to obtain ν(k, p) + k ≤ pk .

Enumerating highly non-soluble groups

221

Now we show that comnA (G) ≤ pk /5. If k = 1 then G is soluble and nothing remains to be proved. Otherwise we simply dispose of the obvious Abelian composition factors and obtain comnA (G) ≤ log60 |SL(k, p)| ≤ (k 2 − 1) log60 p ≤ pk /5. II: Almost simple type. In this case T is non-Abelian, k = 1, and G is an extension of T by some subgroup of Out(T ). Since T embeds into Sym(n), the inequalities n ≥ 5 and t ≤ nn hold. It is a consequence of CFSG that Out(T ) is soluble and |Out(T )| ≤ log t; see [2]. This gives com(G) ≤ 1 + log log t ≤ 1 + log n + log log n ≤ n and comnA (G) = 1 ≤ n/5. III. In this case T is non-Abelian and k ≥ 2. There are three subcases. III.1: Simple diagonal action. Here n = t k−1 and G is an extension of M = T k by some subgroup of Out(T ) × Sym(k). By CFSG, Out(T ) is soluble and |Out(T )| ≤ log t. Since t ≥ 60, induction yields com(G) = com(M) + com(G/M) ≤ 2k + log log t ≤ t k−1 and comnA (G) ≤ 6k/5 ≤ t k−1 /5. III.2: Product action. Here n = d l where d ≥ 5 and l ≥ 2. The group G is a subgroup of Sym(d) Sym(l), hence an extension of a permutation group of degree ld by a permutation group of degree l. By induction we find com(G) ≤ l(d + 1) ≤ d l and comnA (G) ≤ l(d + 1)/5 ≤ d l /5. III.3: Twisted wreath action. Here n = t k and G is a twisted wreath product of T and a permutation group of degree k. Since t ≥ 60, induction yields com(G) ≤ 2k ≤ t k and comnA (G) ≤ 6k/5 ≤ t k /5. The bounds in Proposition 2.1 are not unreasonable; this can be seen from Example 2.2. Let k ∈ N. Then Sylow 2-subgroups of Sym(2k ) are k-fold iterated wreath product of groups of order two and thus have composition length 2k − 1. Corollary 2.3. Let G be a finite group whose soluble radical is trivial. Then we have com(soc(G)) ≥ 56 comnA (G). Proof. The socle M := soc(G) is isomorphic to a product ri=1 Siki of non-Abelian r simple groups Si with multiplicities ki . Note that k := i=1 ki = com(M) and r Aut(S ) Sym(k ). Since M has trivial centre, the group Aut(M) ∼ = i i i=1 G acts faithfully by conjugation on M and we obtain a homomorphism r ϕ : G → Aut(M) → i=1 Sym(ki ) → Sym(k). The kernel K of ϕ contains M, and K/M → ri=1 Out(Si )ki is soluble by the validity of Schreier’s Conjecture. Proposition 2.1 yields comnA (G) − com(M) = comnA (G/K) ≤ k/5 and the claim follows. Proposition 2.4. Every finite group G satisfies d(G) ≤ comAb (G) +

13 5

log∗ comnA (G) + ε(G)

where ε(G) := 0 if G is soluble and ε(G) := 2 otherwise.

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Proof. Let G be a finite group and put C := 2/ log(12/7) ≤ 13/5. We prove the slightly stronger assertion d(G) ≤ comAb (G) + C log∗ comnA (G) + ε(G). First suppose that G possesses a non-trivial Abelian normal subgroup A. Then d(G) ≤ d(A) + d(G/A) ≤ comAb (A) + d(G/A) and induction yields the desired bound. Now suppose that soc(G) has trivial centre. Then an easy modification of [8, Proof of Proposition 3.1], calling upon Corollary 2.3 in place of [8, Lemma 2.2], proves our claim. Lemma 2.5. Let G be a finite group and k ∈ N0 . Then #{N G | com

nA

(N) = k} ≤ 2

(log α(G))2 comAb (G)

comnA (G) . k

Proof. If G contains a non-Abelian minimal normal subgroup, we can argue as in [8, Proof of Lemma 4.1]. Now suppose that M := soc(G) is Abelian. Put α := α(G), c1 := comAb (G), c2 := comnA (G), and N := {N G | comnA (N ) = k}. Note that M has at most α c1 = 2(log α)c1 different subgroups and that every non-trivial N ∈ N intersects M non-trivially. Induction shows that for every non-trivial subgroup 2 M0 ≤ M the number of N ∈ N with N ∩ M = M0 is bounded by 2(log α−1) c1 ck2 . 2 Thus |N | ≤ 2(log α) c1 ck2 , as claimed. Proposition 2.6. Let n ∈ N. Then every finite group G admits at most 2(log α(G))

2 com Ab (G) log n

· 27n(com

Ab (G)+3 log∗ com nA (G)+log n)

transitive permutation representations of degree n (up to equivalence). Proof. We indicate how the proof of [8, Proposition 5.1] can be modified to yield the present, more general claim. Let G be a finite group. By Proposition 2.1 and Lemma 2.5 there are at most

n/5 nA

2 Ab ∗ nA (log α(G))2 comAb (G) com (G) 2 ≤ 2(log α(G)) com (G) · 2n log com (G) k k=0

normal subgroups K G such that G/K → Sym(n). This effectively reduces the problem to counting faithful transitive actions. Indeed, it suffices to verify: Every finite group G admits (up to equivalence) at most 2F (G,n) faithful transitive permutation representations of degree n ≥ 2, where F (G, n) := (log α(G))2 comAb (G)(log n − 1) + 7n(comAb (G) +

20 7

log∗ comnA (G) + log n).

Let G be a finite group. The special cases com(G) ≤ 1 and n ≤ 7 are easily dealt with, because d(G) ≤ 7 implies that there are no more than n7n homomorphisms from G to Sym(n). Now suppose that com(G) ≥ 2 and n ≥ 8. Choose a minimal

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normal subgroup N of G and put H := G/N . By Proposition 2.4 we have d(H ) ≤ ∗ nA comAb (H ) + 13 5 log com (H ) + 2. Let Tintr (G, n), respectively Ttr (G, n), denote the number of (equivalence classes of) faithful transitive permutation representations G → Sym(n) with N acting intransitively, respectively transitively. We claim that both, Tintr (G, n) and Ttr (G, n) are bounded by 21 · 2F (G,n) ; this will finish the overall proof. Arguing by induction, similarly as in [8, Proof of Proposition 5.1]1 , we obtain Tintr (G, n) ≤ n · max{2f (n,r) | 2 ≤ r ≤ n/2} where ∗ nA 13 5 log com (H ) + 2) + (log α(N ))2 comAb (N) log nr + 7 nr (comAb (N ) + 3 log∗ comnA (N ) + log nr ) 2 Ab Ab ∗ nA

f (n, r) = r log nr (comAb (H ) + + (log α(H )) com

(H ) log r + 7r(com

(H ) + 3 log com

(H ) + log r).

(Here r corresponds to the number of N -orbits; for details see [8, Proof of Proposition 5.1].) Rearranging terms yields Tintr (G, n) ≤ 2f (n) where f (n) = (log α(G))2 comAb (G)(log n − 1) + max log n + 7( nr log nr + r log r) + 2r log nr 2≤r≤n/2

+ (comAb (G) + 3 log∗ comnA (G)) (7( nr + r) + r log nr ) .

It is a matter of routine to verify that for 2 ≤ r ≤ n/2 the inequalities log n + 7( nr log nr + r log r) + 2r log nr ≤ 7n log n − 1, 3 · (7( nr + r) + r log nr ) ≤

20 7

· 7n

hold. So we obtain Tintr (G, n) ≤ 21 · 2F (G,n) as wanted. It remains to bound Ttr (G, n), i.e. the number of faithful transitive representations G → Sym(n) with N acting transitively. If N is Abelian, then N acts regularly and its action is known. If N is non-Abelian, then at least it is characteristically simple, and the same trick as in [8, Proof of Proposition 5.1] shows that there are 20 ∗ nA at most 26n( 7 log com (N)+log n) possibilities for a faithful transitive representation N → Sym(n). Once the action of N is fixed, any extension to an action of G is determined by the images of a single reference point under a set of generators of G modulo N . So by Proposition 2.4 we find that Ttr (G, n) ≤ 2f (n) where ∗ nA Ab ∗ nA 13 f (n) = 6n( 20 7 log com (N)+log n)+(log n)(com (H )+ 5 log com (H )+2).

This yields Ttr (G, n) ≤

1 2

· 2F (G,n) , as wanted.

1 Incidentally, there is a small mistake in [8, Proof of Proposition 5.1]: the induced action of N on a single N -orbit is wrongly taken to be faithful. But this only affects the various constants which appear in the estimates.

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Corollary 2.7. Let n ∈ N. Then every finite group G admits at most 2n(2+(log α(G))

3 )/2

· 27n(log α(G)+3 log

∗ com nA (G)+log n)

permutation representations of degree n up to equivalence. Proof. Let G be a finite group. Then a choice of orbits for a representation G → Sym(n) corresponds to an additive partition of n and there are less than 2n−1 such partitions. Now the claim follows from Proposition 2.6.

3. Cohomology groups and proof of Theorem A In this section we establish an expedient bound for the size of second cohomology groups and prove our main theorem. Proposition 3.1. Let G be a finite group and M a finite G-module. Then |H 2 (G, M)| ≤ |G|(2+com

Ab (G)) log |M|

.

Proof. We follow closely the beginning of [7, Proof of Proposition 3.1]. For any submodule L ≤ M we have |H 2 (G, M)| ≤ |H 2 (G, L)||H 2 (G, M/L)|, so we may assume that M is an irreducible G-module. If G acts faithfully on M, then |H 2 (G, M)| ≤ |G|2 log |M| by [6, Theorem 1]. So now suppose that CG (M) = 1 and argue by induction on com(G). If G is simple, then the universal coefficient theorem yields H 2 (G, M) ∼ = Hom(M(G), M), where M(G) denotes the Schur multiplier of G, and a list of multipliers of the finite simple groups, such as in [4] or [2], gives |H 2 (G, M)| ≤ |G|log |M| . Now suppose that com(G) ≥ 2. Choose a minimal normal subgroup N of G which lies inside CG (M). We use the inequality |H 2 (G, M)| ≤ |H 2 (N, M)G/N | · |H 1 (G/N, H 1 (N, M))| · |H 2 (G/N, M N )|. (3.1) This can be regarded as a consequence of the famous Hochschild–Serre spectral sep,q quence E2 := H p (G/N, H q (N, M)) ⇒ E p+q = H p+q (G, M). Indeed, the p,2−p 0,2 1,1 2,0 , E∞ , E∞ and each E∞ is limit term E 2 = H 2 (G, M) is composed of E∞ p,2−p a subquotient of the initial term E2 ; see [11, Chapter II] for a comprehensible account. If N is non-Abelian, then |H 1 (N, M)| = |Hom(N, M)| = 1, and hence (3.1) and induction yield |H 2 (G, M)| ≤ |N|2 log |M| · 1 · |G/N|(2+com

Ab (G/N )) log |M|

≤ |G|(2+com

Ab (G)) log |M|

.

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Now suppose that N is Abelian. Then |H 1 (N, M)| = |Hom(N, M)| ≤ |M|com(N ) , and so (3.1), induction, and the trivial estimate d(G/N ) ≤ log |G/N| give |H 2 (G, M)| ≤ |N|(2+com(N)) log |M| · |M|com(N )d(G/N ) · |G/N|(2+com ≤ |G/N |(2+com ≤ |G|(2+com

Ab (G/N)+com(N)) log |M|

Ab (G)) log |M|

Ab (G/N )) log |M|

· |N|(2+com(N )) log |M|

.

We are now ready to prove the upper bound in Theorem A; recall that the lower bound was already established in [8]. Let A, n ∈ N. Then we claim that gA (n) ≤ n(log A)

3 /2+8 log A+5

· nmax{1+3 log A,32} log log n .

(3.2)

Observe that every group G with α(G) ≤ A and |G| ≤ n satisfies comnA (G) ≤ log60 n ≤ 15 log n, d(G) ≤ log A + 3 log∗ log n (by Proposition 2.4).

(3.3)

Next we introduce some ad-hoc terminology. A (composition) type is just a tuple of non-trivial finite groups each of which is either a direct product of non-Abelian simple groups or elementary Abelian. Two types are said to be isomorphic if they have the same length and corresponding components are isomorphic. Let M = (M1 , . . . , Mr ) be a type. We define α(M) := ri=1 α(Mi ) and |M| := ri=1 |Mi |. A group G realizes M, if it possesses a normal series 1 = G0 ≤ G1 ≤ · · · ≤ Gr = G such that for all i ∈ {1, 2, . . . , r} we have Gi /Gi−1 ∼ = Mi and Gi /Gi−1 = soc(G/Gi−1 ) if Mi is non-Abelian, Gi /Gi−1 is a minimal normal subgroup of G/Gi−1 if Mi is Abelian. Note that every finite group realizes at least one type. Returning to the proof of (3.2), we divide the argument into two steps. Step 1. The number of (isomorphism classes of) types M with |M| ≤ n is at most n4+log log n . Step 2. Any particular type M with α(M) ≤ A and |M| ≤ n is realized by at most nF (A)+a log log n groups where F (A) := (log A)3 /2 + 8 log A + 1 and a := max{3 log A, 31}. Step 1. There are n choices for the destined order m ≤ n of our type M. The number m has, counted with repetitions, at most log m prime factors and these can be rearranged and grouped in at most (log m)! · 2log m ≤ n1+log log n ways to obtain an ordered factorisation m = m1 · · · · · mr of m. How many types M = (M1 , . . . , Mr ) are there with |Mi | = mi for all indices i? Let i ∈ {1, 2, . . . , r}. If mi is a prime power, then Mi has to be elementary Abelian of order mi . Otherwise, according to [12, Proof of Lemma 2.3] the number of groups of order mi which are direct products of non-Abelian simple groups is less than or equal to m2i . So altogether there are at most m2 ≤ n2 choices for the components Mi of M.

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Benjamin Klopsch

Step 2. Fix a type M = (M1 , . . . , Mr ) with α(M) ≤ A and |M| ≤ n. If r ≤ 1, only one group realizes M. So let us assume that r ≥ 2, and write m := |M1 |. Suppose that G is a group realizing M. Then G has a normal subgroup M ∼ = M1 such that H := G/M realizes the type (M2 , . . . , Mr ). We can view G as an extension of M by H , and conjugation in G induces a coupling homomorphism χ : H → Out(M). The extension 1 → M → G → H → 1 is determined up to equivalence by the prescribed isomorphism class of M, i.e. M ∼ = M1 , and (1) the isomorphism class of H , (2) the coupling homomorphism χ : H → Out(M), (3) a suitable element of H 2 (G, Z(M)) where the centre Z(M) of M is regarded as a G-module via χ. Of course, the same data then determine the group G up to isomorphism. So we can bound the number of groups realizing M by estimating the number of possible choices in (1), (2), and (3). Note that (1) can be dealt with by induction. First suppose that M is a direct product of non-Abelian simple groups. Then Z(M) is trivial and so (3) becomes irrelevant. We estimate the number of coupling homomorphisms similarly as in [8, Section 6], with Corollary 2.7 and (3.3) replacing [8, Proposition 1.2 and (6.1)], and see that the number of groups realizing M is at most (n/m)F (A)+a log log(n/m) · 2(log m)(2+(log A)

3 )/2

· 27(log m)(log A+3 log log(n/m)+log log m) · mlog A+3 log log(n/m) 3 /2+8 log A+1+31 log log n

≤ (n/m)F (A)+a log log n · m(log A) ≤ nF (A)+a log log n .

Now suppose that M is elementary Abelian. Then |Out(M)| = |Aut(M)| ≤ mlog m and the number of choices in (2) can be estimated by (3.3). The size of the cohomology group in (3) is restricted by Proposition 3.1. Thus the number of groups realizing M is at most log(A/m)+3 log log(n/m) (n/m)F (A/m)+a log log(n/m) · mlog m · (n/m)(2+log(A/m)) log m = (n/m)F (A/m)+(2+log(A/m)) log m+a log log n · mlog(A/m) log m+3 log m log log n . It is a routine matter to verify that F (A/m) + (2 + log(A/m)) log m ≤ F (A); so the claim follows and the proof of (3.2) is complete. Acknowledgements. I thank Larry and Lorien Wilson for their hospitality during the conference week; it was at their home that I convinced myself of some of the allegedly routine matters.

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227

References [1]

A. R. Camina, G.R. Everest, and T. M. Gagen, Enumerating non-soluble groups – a conjecture of John G. Thompson, Bull. Lond. Math. Soc. 18 (1986), 265–268.

[2]

J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson, Atlas of finite groups, Claredon Press, Oxford 1985.

[3]

W. Gaschütz, Zu einem von B. H. und H. Neumann gestellten Problem, Math. Nachr. 14 (1956), 249–252.

[4]

R. L. Griess Jr., Schur multipliers of the known finite simple groups. II, in The Santa Cruz Conference on Finite groups, Santa Cruz, 1979, Proc. Symp. Pure Math. 37, Amer. Math. Soc., Providence 1980, 279–282.

[5]

G. Higman, Enumerating p-groups. I: Inequalities, Proc. Lond. Math. Soc. 10 (1960), 24–30.

[6]

D. F. Holt, On the second cohomology group of a finite group, Proc. Lond. Math. Soc. 55 (1987), 22–36.

[7]

D. F. Holt, Enumerating perfect groups, J. Lond. Math. Soc. 39 (1989), 67–78.

[8]

B. Klopsch, Enumerating finite groups without abelian composition factors, Israel J. Math. 137 (2003), 265–284.

[9]

M. W. Liebeck, C. E. Praeger, and J. Saxl, On the O’Nan-Scott Theorem for finite primitive permutation groups, J. Austral. Math. Soc. (Ser. A) 44 (1988), 389–396.

[10] A. Mann, Enumerating finite groups and their defining relations, J. Group Theory 1 (1998), 59–64. [11] J. Neukirch, A. Schmidt, and K. Wingberg, Cohomology of number fields, Grundlehren Math. Wiss. 323, Springer-Verlag, Berlin 2000. [12] L. Pyber, Enumerating finite groups of given order, Ann. Math. 137 (1993), 203–220. [13] J. G. Thompson, Finite Non-Solvable Groups, in Group Theory: essays for Philip Hall (eds. K. W. Gruenberg and J. E. Roseblade), Academic Press, London 1984, 1–12. Benjamin Klopsch, Mathematisches Institut, Heinrich-Heine-Universität, 40225 Düsseldorf, Germany E-mail: [email protected]

3-Signalizers in almost simple groups Inna Korchagina

The term q-signalizer (q being a prime) was originally introduced by John Thompson to denote a q -subgroup of a group G normalized by a Sylow q-subgroup of G. With time the term came to mean a q -subgroup of G normalized by a “large” q-subgroup of G, where “large” has various meanings (e.g., of q-rank at least 3, or containing the full centralizer of some q-element, etc.). Control of the size and embedding of such signalizers has been a central theme of the Classification of Finite Simple Groups. The goal of the q-signalizer analysis is to obtain information about the q-local structure of G. In this work we study certain 3-signalizers of G. Definition 0.1. Let G be a finite group and t ∈ Aut(G) of order 3. Take p to be a prime distinct from 3. Let X be a p-subgroup of G such that (1) [X, t] = X; and (2) CG (t) ≤ NG (X). Then X is called a CG (t)-signalizer. In this case we say that the triple (G, t, X) is of type (H ). As a result of study of such 3-signalizers in finite simple K-groups we obtain the following result. Theorem 0.2. Let G be a non-sporadic finite simple K-group. Suppose that G admits a nontrivial automorphism t of order 3 and X is a CG (t)-signalizer. Then either X = 1, or one of the following conclusions holds: (1) G ∈ Lie(r), r = 3 and there exists a parabolic subgroup P of G, such that X is contained in the unipotent radical of P and CG (t) is contained in its Levi complement; (2) G ∼ = PSp (r) with r ∈ {5, 7} and l ≥ 2, 2l

∼ Z3 × Sp ∼ t ∈ Inn(G) with CG (t) = 2l−2 (r) and X ≤ G with X = Q8 ; (3) G ∼ = L2 (r) with r ∈ {5, 7}, t ∈ Inn(G) and X ≤ G with X ∼ = E4 ; ∼ ∼ ∼ (4) G = A7 , t ∈ Inn(G) with CG (t) = E9 and X ≤ G with X = E4 ; or

(5) G ∼ = A8 , t ∈ Inn(G) with CG (t) ∼ = Z3 Z2 and X ≤ G with X ∼ = E24 . We anticipate, that this result may be of use in the revised proof of the Classification Theorem of Finite Simple Groups for dealing with the situation when G is a group of

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even type and e(G) = m2,3 (G) = 3 (a difficult subcase of the small even case). For that purpose we also formulate and prove the following theorem. Theorem 0.3. Let G be a finite sporadic simple K-group with e(G) ≤ 3. Suppose that G admits a nontrivial automorphism t of order 3 and X is a CG (t)-signalizer. Then either X = 1, or one of the following conclusions holds: (1) G ∼ = Z3 × A4 and X ∼ = E4 ; = M22 , t ∈ 3A with CG (t) ∼ (2) G ∼ = Z3 × A5 and X ≤ E16 ; = M23 , t ∈ 3A with CG (t) ∼ ∼ (3) G = M24

(a) t ∈ 3A with C(3A) ∼ = 3A6 and X ∼ = 26 ; (b) t ∈ 3B with C(3B) ∼ = 26 ; = Z3 × L3 (2) and X ∼ (4) G ∼ = Z3 × A4 and = J2 , t ∈ 3B with C(3B) ∼ 2+4 ∼ 2 .(Z3 × S3 ); XCG (t) ≤ N (J (S2 )) =

(5) G ∼ = 6M22 and XCG (t) ≤ Q68 (3M22 .2); = J4 , t ∈ 3A with C(3A) ∼ (6) G ∼ = E4 . = H e, t ∈ 3B with C(3B) ∼ = Z3 × L3 (2) and X ∼

We shall subdivide the proof of both theorems into a series of lemmas and propositions in Sections 2, 3 and 4. We begin with some important background results.

1. Preliminary lemmas and their corollaries Throughout this section r denotes a prime number distinct from 3. Let F¯ = F¯ r be an algebraic closure of the field Fr of r elements. We shall suppose that the algebraic ¯ groups we are dealing with are F-algebraic groups. We begin by proving the following result about the automorphisms of algebraic groups. ¯ be a simple algebraic group with fundamental root system Proposition 1.1. Let G = {α1 , . . . , αl } and corresponding Dynkin diagram G¯ . Suppose that t¯ is a ¯ of order 3. Denote C¯ = C ¯ (t¯). Then one of the nontrivial inner automorphism of G G following conditions holds: (E) t¯ is of equal-rank type (see definition in [GLS3]) and S¯ = C¯ 0 is a semisimple algebraic group, whose Dynkin diagram 0 can be obtained by erasing a node ¯ from the extended Dynkin diagram of G. (P) t¯ is of parabolic type (see definition in [GLS3]), S¯ = [C¯ 0 , C¯ 0 ] is a semisimple algebraic group and the Dynkin diagram 0 of S¯ can be obtained in one of the following ways: ¯ is one of the following groups: Bl , Cl , G2 , F4 , E7 , E8 , then we have (a) If G to erase precisely one node from G¯ ; ¯ is one of the following groups: Al , Dl , E6 , then we have to erase at (b) If G most two nodes from G¯ .

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Proof. By Theorem 4.1.9 [GLS3], t¯ is either of equal-rank type or of parabolic type. If t¯ is of equal-rank type, the result follows immediately from Definition 4.1.8 [GLS3]. Thus suppose that t¯ is of parabolic type. By Theorem 4.1.8 [GLS3], S¯ = [C¯ 0 , C¯ 0 ] is a semisimple algebraic group. Let 0 be its fundamental root system and 0 corresponding Dynkin diagram. Theorem 4.1.9 [GLS3] together with Definition 4.1.8 [GLS3] implies that we may suppose that 0 is a subset of . ¯ actually are, let us consider In order to determine what 0 , 0 (and therefore S) ¯ Since t¯ is a semisimple the action of t¯ on LG¯ , where LG¯ is the Lie algebra of G. ¯ So for every element of order 3, it must be an element of some maximal torus T¯ of G. α ∈ and the corresponding element eα of LG¯ , the following condition holds: t¯.eα = ωf (α) eα with f (α) ∈ {0, ±1},

(1)

where ω ∈ F¯ is the cube root of unity. Now that most of the notation has been established, let us see what actually happens. ¯ is one of the following groups: Bl , Cl , G2 , F4 , E7 or Lemma 1.2. Suppose that G E8 . Denote l = ||. Then |0 | = l − 1. Proof. Since as we remarked earlier 0 ⊆ , we must show that | − 0 | = 1. Assume the contrary. Since t¯ is of parabolic type, this means that | − 0 | ≥ 2. Then there exist two roots β1 , β2 such that βi ∈ − 0 for i = 1, 2 and without loss of generality β1 , β2 can be chosen in the following way : 1. β1 lies to the left of β2 on G¯ , where we use Dynkin diagrams as shown on p. 12 in [GLS3]; 2. If is the smallest connected subdiagram of G¯ containing both β1 and β2 , then all other nodes of are elements of 0 ; and 3. All the nodes to the right of β2 correspond to the elements of 0 . ˜ = {α˜ 1 , . . . , α˜ n } of , we may suppose Therefore for the fundamental system that α˜ 1 = β1 , α˜ n = β2 and α˜ i ∈ 0 for 1 < i < n. Since βi ∈ − 0 , we have that t¯.eβi = eβi for i = 1, 2 and t¯.eα = eα for every α ∈ 0 . So in terms of new notation we obtain t¯.eα˜ i = eα˜ i if i ∈ {1, n}, and t¯.eα˜ i = eα˜ i otherwise. Using formula (1), we discover that up to the choice of ω one of the following conditions holds: 1. t¯.eα˜ 1 = ωeα˜ 1 and t¯.eα˜ n = ω−1 eα˜ n ; or 2. t¯.eα˜ 1 = ωeα˜ 1 and t¯.eα˜ n = ωeα˜ n . The goal now is to reach a contradiction by producing a root vector which is fixed by t¯ and yet is linearly independent of 0 . Let us see how we can achieve it in each of the situations described above. Case 1. Since is a connected, not necessarily proper subdiagram of G¯ , and because of the minimal choice of , it can only be of one of the following types: An , ˜ e ˜ as listed in Table 1. Then t¯.eα˜ = ωf (α) Bn , Cn , G2 or F4 . Let α˜ be the root of () α˜

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with f (α) ˜ = ω · ω−1 = 1 and so t¯ fixes eα˜ . Since α˜ is linearly independent of 0 , eα˜ is not supposed to be fixed by t¯. This is a contradiction which proves the result. Table 1

Type of An Bn Cn G2 F4

α˜ α˜ 1 + α˜ 2 + · · · + α˜ n α˜ 1 + α˜ 2 + · · · + α˜ n α˜ 1 + 2α˜ 2 + · · · + 2α˜ n−1 + α˜ n α˜ 1 + α˜ 2 α˜ 1 + α˜ 2 + α˜ 3 + α˜ 4

Case 2. This time the situation becomes a bit more difficult. Because of the ¯ 1 , which is a subgroup of G ¯ defined by the choice of , we may have to consider G fundamental root system 1 , where 1 is a subset of which contains all the roots beginning with β1 through all the ones that lie to the right of β1 on G¯ . Denote ¯ 1. k = |1 |. Then n ≤ k ≤ l. Let 1 denote the corresponding Dynkin diagram of G ¯ is one of the following groups: Bn , Cn , G2 or F4 , let us immediately proceed If G ¯ is either E7 , or E8 , is of type An and from the complete root lists with Table 2. If G given in Tables 3 and 4 [AS], we may see that for every αi ∈ and every αj ∈ ¯ j with c + c¯ = 3. with i = j , there exists α˜ ∈ () such that α˜ = cαi + · · · + cα ¯ ¯ In both situations above, apply t to eα˜ . This time we obtain: t .eα˜ = ω·ω−1 eα˜ = eα˜ . Therefore t¯ fixes eα˜ , but α˜ is linearly independent of 0 , which is a contradiction. The result now follows. Table 2

Type of An if 1 ∼ Bk An if 1 ∼ Ck A2 if 1 ∼ F4 Bn Cn G2 F4

α˜ α˜ 1 + 2α˜ 2 + · · · + 2α˜ n + 2αk−n+1 + · · · + 2αk α˜ 1 + 2α˜ 2 + · · · + 2α˜ n + 2αk−n+1 + · · · + 2αk−1 + αk α1 + 2α2 + 2α3 + α4 α˜ 1 + 2α˜ 2 + · · · + 2α˜ n 2α˜ 1 + 2α˜ 2 + · · · + 2α˜ n−1 + α˜ n α˜ 1 + 2α˜ 2 α˜ 1 + 2α˜ 2 + 2α˜ 3 + 2α˜ 4

¯ = Al , Dl or E6 . Denote l = ||. Then |0 | ≥ l − 2. Lemma 1.3. Suppose that G Proof. Assume the contrary. Then there exists {β1 , β2 , β3 } ⊆ − 0 . Without loss of generality, we can make the following assumptions: βi = αji with αji ∈ , i = 1, 2, 3, and j1 < j2 < j3 under the “usual” enumeration of coming from G¯

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(see p. 12 [GLS3]). Since βi ∈ − 0 for i = 1, 2, 3, t¯.eβi = eβi for i = 1, 2, 3, while t¯.eα = eα for every α ∈ 0 . Using formula (1), we obtain that up to the choice of ω, one of the following conditions holds: 1. t¯.eβi = ωeβi for i = 1, 2, 3; or 2. t¯.eβi = ωεi eβi with εi ∈ {±1} and i εi = 1, i = 1, 2, 3. Let us see what happens in each of those situations: Case 1. Let be the smallest connected subdiagram of G¯ containing {β1 , β2 , β3 }. Let {α˜ 1 , . . . , α˜ n } be the fundamental system of . Since is a connected, not necessarily proper subdiagram of G¯ , it must be of one of the following types: An , Dn ˜ as listed in Table 3. Consider the action of t¯ on or E6 . Let α˜ be the root of (), f ( α) ˜ ¯ eα˜ with f (α) ˜ = ω · ω · ω = 1. Thus t¯ fixes eα˜ . Since α˜ is linearly eα˜ : t .eα˜ = ω independent of 0 , eα˜ should not be fixed by t¯. This is a contradiction, which proves the result. Table 3

Type of An Dn E6

α˜ α˜ 1 + α˜ 2 + · · · + α˜ n α˜ 1 + 2α˜ 2 + · · · + 2α˜ n−2 + α˜ n−1 + α˜ n α˜ 1 + α˜ 2 + α˜ 3 + α˜ 4 + α˜ 5 + α˜ 6

Case 2. Obviously there exist {i, j } ⊆ {1, 2, 3} such that εi = 1 and εj = −1. Choose i, j and ij so that εi + εj = 0, and ij is the smallest connected subdiagram of G¯ containing βi and βj , and βk ∈ ij for k ∈ {1, 2, 3} − {i, j }. Clearly ij is of the type An for some n. Let {α˜ 1 , . . . , α˜ n } be a fundamental root system of ij . Without loss of generality, we may suppose that α˜ 1 = βi and α˜ n = βj and α˜ s ∈ 0 ˜ α˜ = α˜ 1 + α˜ 2 + · · · + α˜ n . for s ∈ {1, n}. Take α˜ to be the following root of (): f ( α) ˜ −1 Then t¯.eα˜ = ω eα˜ with f (α) ˜ = ω · ω = 1. Therefore t¯ fixes eα˜ . Since α˜ is linearly independent of 0 , eα˜ cannot be fixed by t¯, which is a contradiction. The result follows. Combining Lemma 1.2 and Lemma 1.3, we obtain the desired result. Remark 1.4. The analogous result can be proven for all primes p. For p = 2 it is Theorem 4.3.3 in [GLS3]. For p ∈ {5, 7} the proof is similar to the proof of the proposition, but requires more calculations. For p ≥ 11, it is simply a linear algebra computation. Let us apply the result we just proved to a finite situation. Corollary 1.5. Let G ∈ Lie(r) with r = 3. Suppose that the untwisted rank of G is at least 4. Let t be a nontrivial inner diagonal automorphism of G of order 3. Then there exists Z ≤ CG (t) such that Z ∼ = Zr × Zr .

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¯ σ ) be a σ -setup of G satisfying the conclusion of Theorem 2.5.14 Proof. Let (G, ¯ which induces [GLS3]. By Lemma 4.1.1 [GLS3], there exists the unique t¯ ∈ Aut(G) the automorphism t on G and satisfies the hypotheses of Proposition 1.1. Let C = CG (t) and L = O r (C). By Theorem 4.2.2 (a) [GLS3], the following statement is true: L = L1 · · · Ls where s ≥ 0 and Li ∈ Lie(r), while part (e) of Theorem 4.2.2 [GLS3] asserts that Li ∼ = dii (q mi ). Since the ¯ untwisted rank of G is at least 4, Proposition 1.1 implies that the centralizer of t¯ in G has a nontrivial component and so s ≥ 1. If s ≥ 2, the result follows immediately. Suppose that s = 1. If L = L1 ∼ = A± 1 (r), mr (L) ≥ 2 by Theorem 3.3.3 [GLS3] and again the result follows. Assume that L = L1 ∼ = A± 1 (r). Then by part (e) of ¯ Theorem 4.2.2 [GLS3], S = A1 , which, together with the fact that the untwisted rank of G is at least 4, contradicts Proposition 1.1. The result now follows. Remark 1.6. Let us fix the notation: if (G, t, X) is a triple of type (H ), denote C := CG (t) and L := O r (C). Lemma 1.7. Let K = X t be a Frobenius group with kernel X and complement t where X is a 2-group and t 3 = 1 = t. Let R be an r-group where r is an odd prime distinct from 3. Suppose that K acts on R and (RK, t, X) is of type (H ). Then K does not act faithfully on R. Proof. Assume the contrary. Let R0 = [R, X] and V = [R0 / (R0 ), X]. Then K acts faithfully on V . By Lemma 9.12 [GLS2], CV (t) = 1 and so by Lemma 11.3 [GLS3], CR0 (t) = 1. Since CR0 (t) ≤ NRK (X), [CR0 (t), X] ≤ R ∩ X = 1, i.e., CR0 (t) centralizes X. Therefore CV (t) centralizes X and so CX (CV (t)) = X, which contradicts Lemma 9.12 [GLS2], proving the result. Lemma 1.8. Let p and r be distinct primes, and p = 3. Let X be a nontrivial pgroup, and t an automorphism of X of order 3 with [X, t] = X. Suppose that X t acts faithfully on an r-group R and G = RX t with (G, t, X) of type (H ). Then r = 3. Proof. Assume that r = 3. By Proposition 11.11 [GLS2], R contains a critical subgroup Q of class at most 2 and exponent 3. In fact, by Lemma [Be1], we may assume that Q is abelian. Since X t acts faithfully on Q and p = 3, Q = [Q, X] × CQ (X). Thus X acts fixed points free on [Q, X] and [Q, X] is t-invariant 3-group. But 1 = C[Q,X] (t) ≤ NG (X), hence 1 = C[Q,X] (t) ≤ C[Q,X] (X), which is a contradiction. Lemma 1.9. Let G be a group such that F ∗ (N) = Or (N ) for every r-local subgroup of G. Let p be an odd prime distinct from r. Let t ∈ Aut(G) and X ≤ G be such that (G, t, X) is of type (H ). Suppose that there exists a t-invariant r-local subgroup P of G. Then [X ∩ P , t] = 1.

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Proof. Let X0 = X ∩ P . Then X0 is t-invariant. Assume that [X0 , t] = 1. Then we may suppose that X0 = [X0 , t]. Since X0 acts non-trivially on Or (P ), we are in the conditions of Lemma 11.14 [GLS2], which asserts that COr (P ) (t) = 1. But COr (P ) (t) ≤ C ∩ P ≤ NG (X0 ) and so [COr (P ) (t), X0 ] ≤ Or (P ) ∩ X = 1, i.e., X0 centralizes COr (P ) (t). By Lemma 11.14 [GLS2], p = r, which is a contradiction. The result follows. This lemma has very important consequences. Corollary 1.10. Let G ∈ Lie(r) and p be a prime distinct from r. Let t ∈ Aut(G) and X ≤ G be such that (G, t, X) is of type (H ). If there exists Z ≤ C with Z ∼ = Zr × Zr , then p = 2 and X is a non-abelian 2-group. Proof. Since Z ≤ C, Z normalizes X. Since p = r, we are dealing with the coprime action. By 5.3.13 [G], we have that X = CX (u) | u ∈ Z # . Denote Xu = CX (u). Fix an arbitrary element u ∈ Z # with Xu = 1. Denote U = u and consider Pu = NG t (U ). By Corollary 3.1.4 [GLS3] to the Borel–Tits Theorem, Xu t acts faithfully on R = Or (Pu ). First let us show that X is a 2-group. Assume the contrary, i.e., suppose that p is odd. By Lemma 1.9, [X ∩ Pu , t] = 1. But Xu ≤ X ∩ Pu , and so [Xu , t] = 1. But u was an arbitrary element of Z # with the property that Xu = 1. Therefore X = Xu | u ∈ Z # commutes with t, which is a contradiction. So p = 2 and X is a 2-group. Assume that X is abelian. Since X = [X, t], Theorem 5.2.3 [G] implies that CX (t) = 1. Therefore X t is a Frobenius group and so is Xu t. This contradiction to Lemma 1.7 proves the result. Combining Corollaries 1.5 and 1.10, we obtain the following result. Lemma 1.11. Let G ∈ Lie(r) and p be a prime distinct from r. Let t ∈ Aut(G) and X ≤ G be such that (G, t, X) is of type (H ). Suppose that the untwisted rank of G is at least 4. Then p = 2 and X is a non-abelian 2-group.

2. Simple groups of Lie type We now begin the proof of Theorem 0.2 for the simple K-groups of Lie type. Proposition 2.1. Let G ∈ Lie(r). Let t ∈ Aut(G) and 1 = X ≤ G be such that (G, t, X) is of type (H ). Then r = 3. Proof. Assume that r = 3. If G is isomorphic to either A1 (33 ), or 2G2 (33 ) and t is a field automorphism, then X = 1 from the local structure of G. Otherwise, there exists U ≤ CG (t) with U ∼ = E9 . Since X is a CG (t)-signalizer, U acts on X and by Proposition 11.13 [GLS3], X = Xu = CX (u) | u ∈ U #

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Thus there exists a non-trivial element u ∈ U with Xu = 1. Therefore Xu t ≤ CG (u) t. Since G ∈ Lie(3), Borel–Tits Theorem implies that Xu t acts faithfully on F ∗ (CG (u)) = O3 (CG (u)). This contradiction with Lemma 1.8 proves the result. Proposition 2.2. Let G ∈ Lie(r), t ∈ Aut(G) and 1 = X ≤ G be such that (G, t, X) is of type (H ) and p = r. Then there exists a parabolic subgroup P of G, such that X is contained in the unipotent radical of P and CG (t) is contained in its Levi complement. Proof. The result follows immediately by an application of Borel–Tits Theorem 3.1.3 [GLS3] and Corollary 3.1.4 [GLS3]. Remark 2.3. For the remainder of this section, p = r = 3. Proposition 2.4. Let G ∈ Lie(r) and X be a p-subgroup of G. Suppose that t is a graph or a graph-field automorphism of G and (G, t, X) is of type (H ). Then X = 1. Proof. Since G admits a graph or a graph-field automorphism of order 3, either G ∼ = D4 (r a ), or G ∼ = 3D4 (r a ). By 4.7.3 and 4.9.1[GLS3], mr (C) ≥ 2. Hence by Corollary 1.10, p = 2 and X is a non-abelian 2-group. Thus by minimality, there exists an involution z ∈ X such that XCG (t) ≤ CG (X). Combining the information about the structure of 2- and 3-automorphisms of G from 4.5.1, 4.7.3 [GLS3], we obtain that X = 1. Proposition 2.5. Let G ∈ Lie(r) and X be a p-subgroup of G. Suppose that t is a field automorphism of G and (G, t, X) is of type (H ). Then X = 1. Proof. By Theorem 1 [BuGrLy1], if C ∈ {A1 (2), A1 (3),2 B2 (2)}, NG (C) is maximal in G. In particular, in this case X = 1. Thus if X = 1, G must be isomorphic to one of the following groups: A1 (8), A1 (27), 2B 2 (8). If G ∼ = S3 . Using the local structure of G, we get X = 1. = A1 (8), then CG (t) ∼ ∼ If G = A1 (27), then CG (t) ∼ = A1 (3). From the local structure of G, X = 1. If G∼ = F r20 which is a maximal subgroup of G. Hence X = 1. = 2B2 (8), then CG (t) ∼ The result now follows. Let us now deal with the inner diagonal automorphisms of groups of Lie type. We begin with the following definition. Definition 2.6. We say that a finite group K ∈ Li for i ∈ {1, 2, 3, 4}, if it is isomorphic to one of the following groups: (L1 ) A1 (2a ) with a ≥ 3 or L such that |L| = 2|A1 (r a )| where r is an odd prime, a ≥ 1 and L/Z(L) ∼ = PGL2 (r a ); a (L2 ) A± 2 (r ) where a ≥ 1 if r is odd and a ≥ 3 if r = 2;

(L3 ) B2 (2a ) with a ≥ 1 or L such that |L| = 2|B2 (r a )| where r is an odd prime, a ≥ 1 and L/Z(L) ∼ = PSO4 (r a );

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a a a (L4 ) A± 3 (r ), or G2 (r ), or B3 (r ) with a ≥ 1.

where we use the usual notation to mean all possible standard versions of L. Lemma 2.7. Suppose that G is isomorphic either to an element of L1 ∪ L2 , or a G∼ = A± 3 (r ) with a ≥ 1. Let t ∈ Aut(G) and X ≤ G be such that (G, t, X) is of type (H ). If X is non-abelian, then p = 2, i.e., X is a 2-group. Proof. Assume the contrary. Then X is a non-abelian p-group and p is odd. By 3.11 [Is], the degree of any faithful representation of X is at least p. Since p ≥ 5, G cannot be represented in less than 5 dimensions. In particular, the universal version of G cannot be represented in less than 5 dimensions. But all the universal versions of the groups we are dealing with have natural representations in at most 4 dimensions, which leads to a contradiction, proving the result. Lemma 2.8. Suppose that G ∈ L1 . Let t and X be such that (G, t, X) is of type (H ). Then X = 1. Proof. Since X is a p-group, there are two situations to consider: p is odd and p = 2. If p is odd, then by Lemma 2.7, X must be abelian. Since X = [X, t], Proposition 5.2.3 [G] asserts that CX (t) = 1. Thus X t is a Frobenius group. Since |Z(G)| ≤ 2, we may suppose that O 2 (G) = SL2 (r a ). This implies that a Frobenius group with complement of order 3 has a 2-dimensional faithful representation, which contradicts Theorem 5.16 [Is]. Therefore p = 2. Hence r is odd and G ∼ = Z2 , a ≥ 1 = A1 (r a ).T where T ∼ a ∼ and G/Z(G) = PGL2 (r ). Since t ∈ Inn(G), the situation depends upon the version of G: 1. G = Gu . Then X is a subgroup of the generalized quaternion group. Since X admits an automorphism of order 3, X ∼ = Q8 . 2. G = Ga . Then X is a subgroup of a dihedral group. Since X admits an automorphism of order 3, X ∼ = Z2 × Z2 . In both cases we may see that NG (X)/CG (X) is isomorphic to a subgroup of S3 . This means that C/CC (X) ∼ = Z3 and so 3 | (r a + ε), where ε ∈ {±1} depending on r. 1. Let G = Gu . Then C is an abelian group which contains a subgroup isomorphic to Zr a +ε . By 5.1.1 [G], |CG (X)| = 2 and so (r a + ε)/3 = 2. Thus r a ∈ {5, 7}. If ˜ = G/Z(G) ∼ ˜ = G/Z(G) ∼ r = 5, then G = PGL2 (7). In = S5 . If r = 7, then G ∼ both cases CG˜ (t˜) = Z3 × Z2 and so by 5.3.15 [G], 4 divides the order of C. Thus |CG (X)| = 4, which is a contradiction. 2. Suppose that G = Ga . Since C ∼ = Z(r a +ε)/2 is abelian and there exists an involution outside of X centralizing X, m2 (G) ≥ 3. On the other hand, by Theorem 4.10.5 (b) [GLS3], m2 (G) = m2 (PGL2 (r a )) = 2, which is a contradiction, proving the result. Lemma 2.9. Suppose that G ∈ L2 . Let t and X be such that (G, t, X) is of type (H ). Then X = 1.

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Proof. Assume the contrary. Combining Lemma 4.1.1 [GLS3] with Proposition 1.1 and Theorem 4.2.2 [GLS3], we obtain that either L = 1 or L ∼ = A1 (r a ). Suppose first that L = 1. Assume that X is abelian. Then X t is a Frobenius group. Assume that there exists a nontrivial element h ∈ CG (t) of order coprime to 3. If h acts fixed points freely on X, then X th is a Frobenius group with complement of order bigger than 3 represented in 3 dimensions, which is a contradiction. Therefore CX (h) = 1. Since CX (h) is t-invariant, CX (h) t is a Frobenius group which acts absolutely irreducibly on the natural module of G. Therefore by the Schur’s Lemma, h is a scalar matrix, i.e., h ∈ Z(G). Contradiction. If t is a genuine element of order 3, we may assume that G = Gu . If t is diagonalizable, then CG (t) ∼ = Zr a ε × Zr a ε where G ∼ = SLε3 (r a ), otherwise CG (t) ∼ = a a a GU1 (r ). In all the cases r ± 1 = 3, i.e., r ∈ {2, 4}, which is a contradiction, since ± neither A± 2 (2), nor A2 (4) are in L2 . Thus we may suppose that t is an actual element of G of order 9, which “cubes” into the center of G (i.e., |Z(G)| = 3). If 9 | (r a + ε), then there exists an element t1 of order 3 which acts as an outer automorphism on G and such that CG (t) = CG (t1 ). Thus we are reduced to the previous situation. Hence 9 does not divide r a + ε. This means, that t is in the Coxeter torus of G and so, by the Zigmondy’s Theorem, there exists element t1 of prime order s = 3 which centralizes t. Again, we are reduced to the original situation, leading to a contradiction. Therefore X is a non-abelian group. In particular, by Lemma 2.7, X is a 2-group. By minimality, there exists an involution z ∈ Z(X) such that XCG (t) ≤ CG (z). Invoking the results of 4.5.1, 4.5.2 [GLS3] and the previous lemma, we obtain that X = 1. Therefore L ∼ = A1 (r a ). Let u be a non-trivial r-element of L acting on X. Assume that X is abelian. If CX (u) = 1, consider CG (u). Since G is a group of r-type, F ∗ (CG (u)) = Or (CG (u)). If p is odd, then we obtain contradiction with Lemma 1.9, otherwise contradiction follows from an application of Lemma 1.7. Therefore X is non-abelian and thus by Lemma 2.7, is a 2-group. By minimality, we may suppose that t centralizes Z(X). Take an involution z ∈ Z(X) such that [C, z] = 1. Then XC ≤ CG (z). By Theorem 4.5.1 [GLS3], CG (z) ∈ L1 . Hence, by Lemma 2.8, X = 1. Lemma 2.10. Suppose that G ∈ L3 . Let t and X be such that (G, t, X) is of type (H ). Then X = 1. Proof. If G ∼ = B2 (2), the result follows immediately from the local structure of G. Suppose that G ∈ L3 − {B2 (2)}. Combining the results of Lemma 4.1.1 [GLS3], Theorem 4.2.2 (a,e) [GLS3] and Proposition 1.1, we obtain that L = L1 · · · Ls where Li ∈ Lie(r), s ∈ {1, 2}, and the following conditions hold: 1. If s = 1, then L = L1 ∼ = A1 (r a ); 2. If s = 2, then L = L1 L2 with Li ∼ = Ai (r a ) for i = 1, 2. In both cases there exists a nontrivial r-element u ∈ L acting on X. First assume that X is abelian. If CX (u) = 1, then Frobenius group CX (u) t acts faithfully on the r-group R = Or (CG (u)). Since this contradicts Lemma 1.9 if p is odd and Lemma 1.7 if p = 2, CX (u) = 1. So X u is a Frobenius group with a complement of order r

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represented in 4 dimensions. Therefore r = 2. Since a ≥ 2, there exists Z ∼ = Z2 × Z2 such that Z ≤ L and Z acts on X. Therefore X = CX (u) | u ∈ Z, u = 1, i.e., X = 1. Thus X is a non-abelian p-group. Since it is represented in 4 dimensions, p = 2. By minimality, we may suppose that, t centralizes Z(X). Take an involution z ∈ Z(X) such that [C, z] = 1. Then XC ≤ CG (z). By Theorem 4.5.1 [GLS3], CG (z) is isomorphic to one of the following groups: A1 (r a ).Z2 , (A1 (r a ) A1 (r a )).E4 , A1 (r 2a ).E4 and so we may reduce to the situation when X t is a subgroup of an element of L1 . By Lemma 2.8, X = 1. Lemma 2.11. Suppose that G ∈ L4 . Let t and X be such that (G, t, X) is of type (H ). Then X is a non-abelian 2-group. Proof. Combining Lemma 4.1.1 [GLS3], Theorem 4.2.2 (a,e) [GLS3] and Proposition 1.1, we obtain that L = L1 · · · Ls where Li ∈ Lie(r) and s ∈ {1, 2}. Moreover one of the following conditions must hold: a a 1. s = 1 and L = L1 ∈ {A1 (r a ), A± 2 (r ), B2 (r )}; or a 2. s = 2 and L = L1 L2 where Li ∼ = Ai (r ). If there exists Z ≤ L with Z ∼ = Zr × Zr , then the result follows immediately by Corollary 1.10. Suppose now that mr (L) = 1. Then L ∼ = A1 (r) and either G ∼ = A± 2 (r), or G ∼ = G2 (r). Assume that X is abelian. Since [X, t] = X, Proposition 5.2.3 [G] implies that CX (t) = 1. Let u ∈ L be a nontrivial r-element acting on X. If Xu = CX (u) = 1, then Xu t is a Frobenius group acting faithfully on an r-group R = Or (CG (u)), which contradicts Lemma 1.9 if p is odd and Lemma 1.7 if p = 2. Therefore CX (u) = 1. Assume first that G ∼ = A± 2 (r). Then X u is a Frobenius group with a complement of order r. Moreover, we may assume that X u is represented in 4 dimensions. ∼ ∼ Therefore r = 2, and so G ∼ = A± 3 (2). Since X = [X, t], if G = A3 (2) = A8 , the ∼ only way X = 1 would be if X = Z7 . There are two conjugacy classes of elements of order 3 in G. The centralizer of the first one, 31 , is isomorphic to Z3 × A5 , the centralizer of the other one, 32 , is isomorphic to Z3 × Z3 . Since neither of them is ∼ contained in a 7-local subgroup of G, X = 1. If G ∼ = A− 3 (2) = B2 (3), then X is a 2-group. But now we have a contradiction with the 2-local structure of G. Therefore X is non-abelian and by Lemma 2.7, X is a 2-group. Suppose now that G ∼ = A1 (r). Then X tu is a Frobenius = G2 (r) and L ∼ group with a complement of order 3r. Since G has a natural representation in 14 dimensions, 3r ≤ 14, i.e., r = 2. Therefore G ∼ = G2 (2), and so |X| = 7. In this case a contradiction follows from the 7-local structure of G. For K ∈ Lie(r), denote by r(K) the untwisted Lie rank of K. We are now ready to prove the following result. Proposition 2.12. Let G = G1 · · · Gl where l ≥ 1 and for all i ∈ {1, . . . , l} either r(Gi ) ≥ 4 and Gi ∈ Lie(r) − {Cn (r) | n ≥ 2}, or Gi ∈ L1 ∪ L2 ∪ L3 ∪ L4 . Suppose

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that t ∈ Inndiag(G) and X is a p-group with p = r and (G, t, X) of type (H ). Then X = 1. Proof. Since t ∈ Inndiag(G), we have that t|Gi = ti ∈ Inndiag(Gi ) for i = 1, . . . , l. If there exists an i with ti = 1, then since [X, t] = X, we shall have X|Gi = 1. Therefore we may suppose that ti = 1 for i = 1, . . . , l. Let us prove the result using the induction on n, where n = max{(r(Gi )) | i = 1, . . . , l}. For n ≤ 3 the result follows by Lemmas 2.8, 2.9 and 2.11. Suppose now that n ≥ 4. By Corollary 1.10 and Lemmas 2.9, 2.10 and 2.11, Xi = X|Gi is either trivial or a non-abelian 2-group for i = 1, . . . , l. Assume that there exists i with Xi = 1. Consider the corresponding triple (Gi , ti , Xi ). We may suppose that Gi is the adjoint group. By minimality, there exists a nontrivial involution z ∈ Z(Xi ) such that [z, ti ] = 1 and [z, CGi (ti )] = 1. So Xi CGi (ti ) ≤ CGi (z). By Theorems 4.5.1, 4.5.3 and 4.2.2 [GLS3], there exists C 0 ≤ CGi (z) such that CGi (z)/C 0 is a 2-group and C 0 = (L1 L2 )T satisfies the following conditions: 1. Either Li = 1, or Li satisfies the hypotheses of the proposition; and 2. T is an abelian group inducing the inner diagonal automorphisms on L1 L2 . Since Xi is non-abelian and ti ∈ Inndiag(Gi ), ti must induce an inner diagonal automorphism on L1 L2 . Moreover, Xi ≤ L1 L2 . Therefore we are reduced to the case when everything is happening in the group (L1 L2 )T . Since r(L1 L2 ) < r(Gi ), by induction, Xi = 1, proving the result. We have dealt with most of the groups of Lie type. But there are a few exceptions which we ignored. For example, what happens in the symplectic groups? Or, why a did we avoid talking about A± 2 (2 ) for a ≤ 2? Lemma 2.13. Suppose that G ∼ = A2 (2), p is an odd prime and t and X are such that (G, t, X) is of type (H ). Then X ∼ = Z3 . = Z7 and t ∈ G with CG (t) ∼ Proof. Since G ∼ = L2 (7), the local structure of G implies that if X = 1, then = A2 (2) ∼ ∼ X = Z7 . Moreover |G|3 = 3 and C ∼ = Z3 normalizes X, proving the result. ∼ A2 (4), p is an odd prime and (G, t, X) is of type (H ). Lemma 2.14. Suppose that G = Then the following conditions hold: (1) If G ∼ = SL3 (4), then X ∼ = Z7 and CG (t) ∼ = Z3 × Z3 ; (2) If G ∼ = L± 3 (4), then X = 1.

Proof. First suppose that G ∼ = L3 (4). Then |X| ∈ {5, 7}. Since X = [X, t], X ∈ Syl7 (G), and NG (X) is a Frobenius group of order 21. On the other hand, a centralizer of any inner automorphism of order 3 contains Z ∼ = Z3 × Z3 . Thus t ∈ Inn(G). Assume that t is a diagonal automorphism. There are two conjugacy classes of such automorphisms. The centralizer of the first one, 31 , in G is a group of order 60, which is not contained in any 7-local subgroup of G. The centralizer of the other one, 32 , is precisely the normalizer of X, which means that [X, t] = 1, which is a contradiction.

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If G ∼ = U3 (4), then using the p-local structure of G, we obtain that CG (t) is not contained in any p-local subgroup N of G with [Op (N ), t] = Op (N ). Therefore X = 1. Finally, suppose that G ∼ = SL3 (4). Things work the same way as far as the diagonal automorphisms are concerned. But if t is an inner automorphism, then CG (t) ∼ = Z3 × Z3 normalizes some Sylow 7-subgroup X of G. The result now follows. Finally, let us deal with the symplectic “situation”. Lemma 2.15. Suppose that G = Sp2 (r a ) with a ≥ 1, p a prime distinct from r and t and X are such that (G, t, X) is of type (H ). If X = 1, then a = 1, r ∈ {5, 7}, p = 2 and X ∼ = Q8 . Proof. Assume that X is abelian. Since [X, t] = X, it follows that X t is a Frobenius group with a complement of order 3 represented in 2 dimensions, which contradicts Theorem 5.6 [Is]. Therefore X must be non-abelian. By the local structure of G, X is a 2-group and since it must admit an automorphism of order 3, X ∼ = Q8 . Recall that ∼ ∼ C normalizes X. Since Aut(X) = S4 , we obtain that C/CC (X) = Z3 . We also know that |CG (X)| = 2. Since 3 | (r a + ε) (where ε ∈ {±1}), (r a + ε)/3 = 2. Thus a = 1 and r ∈ {5, 7}. The result now follows immediately from the 2-local structure of G. Lemma 2.16. Suppose that G = PSp2 (r a ) with a ≥ 1, p is a prime distinct from r, and t and X are such that (G, t, X) is of type (H ). If X = 1, then a = 1, r ∈ {5, 7}, p = 2 and X ∼ = Z2 × Z2 . Proof. If p is odd, then because of the coprimeness, we must have the same thing happening in G as in 2G. So, by Lemma 2.15, X = 1. Therefore p = 2 and r is odd. Since X must admit an automorphism of order 3, the 2-local structure of G implies that X ∼ = S4 , we have = Z2 × Z2 . Recall that C normalizes X. Since Aut(X) ∼ that C/CC (X) ∼ = Z3 . We also know that |CC (X)| = 2. Since 3 | (r a + ε) (where ε ∈ {±1}), (r a + ε)/3 = 2. Thus a = 1 and r ∈ {5, 7}. The result now follows. Lemma 2.17. Suppose that G = C2 (r a ) with a ≥ 1, p is a prime distinct from r, and t and X are such that (G, t, X) is of type (H ). Then a = 1, r ∈ {5, 7}, p = 2 and X ∼ = Z3 × Z2 × A1 (r) and = Q8 . Moreover, t ∈ G is an element with CG (t) ∼ ∼ A1 (r) = Sp2 (r). Proof. Assume first that p is odd. Because of the coprimeness, we may suppose that G = Sp4 (r a ). If X were non-abelian, we would need at least p dimensions to represent it faithfully. Since p ≥ 5 and G has a 4-dimensional module, X must be abelian. Since [X, t] = X, X t is a Frobenius group. Let V be a natural module for G. Then V |X t = V1 ⊕ V2 where V1 is 1-dimensional and V2 is of dimension 3. Reducing the situation to the 3-dimensional case and applying Lemma 2.9, we obtain X = 1.

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Therefore p = 2 and r is odd. If a ≥ 2, then Proposition 1.1 together with Corollary 1.10 imply that X is non-abelian. Therefore we may suppose that G = Ga ∼ = PSp4 (r a ). As usual, there exists an involution z ∈ Z(X) such that [z, t] = 1 and [z, C] = 1. Therefore X t ≤ O r (CG (z)). By Theorem 4.5.1 [GLS3], O r (CG (z)) is isomorphic either to A1 (r 2a ), or to A1 (r a ) A1 (r a ). In the former situation we are reduced to the case G ∼ = A1 (r 2a ) and by Lemma 2.15, X = 1. In the latter case, a ∼ C = Z3 ×Z2 ×A1 (r ). Therefore Lemma 2.15 implies that X ∼ = Q8 with r a = 6±1. The result follows immediately. We are now ready to prove the following result. Proposition 2.18. Let G = Cl (r a ) with a ≥ 1, l ≥ 3, p a prime distinct from r and t and X are such that (G, t, X) is of type (H ). Then a = 1, r ∈ {5, 7}, p = 2 and X∼ = Z3 × Z2 × Cl−1 (r). = Q8 . Moreover, t ∈ G with C ∼ ¯ σ ) be a σ -setup of G satisfying the conclusion of Theorem 2.3.14 Proof. Let (G, ¯ which in[GLS3]. By Lemma 4.1.1 [GLS3], there exists a unique element t¯ ∈ Aut(G) duces the automorphism t on G and satisfies Proposition 1.1. Now Theorem 4.2.2 (a,e) [GLS3] together with Proposition 1.1 give us that L = L1 · · · Ls where Li ∈ Lie(r), s ∈ {1, 2} and one of the following conditions hold: a a 1. s = 1 and L = L1 is isomorphic either to A± l−1 (r ), or to Cl−1 (r ); or a a ∼ 2. s = 2 and L = L1 L2 with L1 ∼ = A± n (r ) and L2 = Cl−n−1 (r ). ∼ In both cases C contains a subgroup Z = Zr ×Zr . By Corollary 1.10, p = 2 and X is a non-abelian 2-group. As usual, by minimality, there exists an involution z ∈ Z(X) such that [z, t] = 1 and [z, C] = 1. Therefore XC ≤ CG (z). Suppose that L ∼ = Cl−1 (r a ). Combining this with Theorem 4.5.1 [GLS3], we obtain that (C, CG (z)) a a ± a a a is one of the following pairs: (A± l−1 (r ), Cl−1 (r )), (An (r ) Cl−n−1 (r ), Cn (r )

a Cl−n−1 (r )). Therefore we are reduced to the smaller dimensional case, and by induction, X = 1. Finally suppose that L ∼ = Cl−1 (r a ). Then by Theorem 4.5.1 [GLS3], we obtain that (C, CG (z)) is (Cl−1 (r a ), A1 (r a ) Cl−1 (r a )). Since t acts trivially on the second component of CG (z) and [X, t] = X, X must project on it trivially. So by Lemma 2.17, X ∼ = Q8 and the result follows.

3. Alternating groups Proposition 3.1. Let G = An with n ≥ 5, t ∈ Aut(G) and X ≤ G be such that (G, t, X) is of type (H ). Then either X = 1, or p = 2 and one of the following holds: ∼ E4 ; ∼ A5 and X = (1) G = (2) G ∼ = E4 and t is an element of G with CG (t) ∼ = E9 ; or = A7 , X ∼ ∼ A8 , X ∼ (3) G = = E24 and t is an element of G with CG (t) ∼ = Z3 Z2 .

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Proof. The proof of this proposition is an induction on n. To start the induction, we shall have to go up to n = 14 “by hand”. By direct calculations, we obtain that for 5 ≤ n ≤ 14, either X = 1, or one of the following conditions hold: ∼ E4 ; ∼ A5 and X = (1) G = (2) G ∼ = A7 , X (3) G ∼ = A8 , X

∼ = E4 and t is an element of G with CG (t) ∼ = E9 ; ∼ = E 4 and t is an element of G with CG (t) ∼ = Z3 Z2 . 2

Suppose now that n ≥ 15. Since (| Out(G)|, 3) = 1, t ∈ Inn(G) and without loss of generality we may suppose that t = (123) . . . is the product of l 3-cycles for some l ≥ 1. Consider Z ∼ = Z3 × Z3 such that Z = (123), (456). Since Z ≤ CG (t), Z must act on X and because of coprimeness, X = Xu = CX (u) | u ∈ Z # . If u ∈ Z is a 3-cycle, then CG (u) ∼ = Z3 × An−3 , otherwise CG (u) ∼ = (Z3 × Z3 × An−6 ).Z2 . Since X = [X, t], in both cases considering t|CG (u) , we are restricted to the situation when [Xu , t] t is contained in an alternating group of degree m < n. Since m ≥ 9, induction finishes the proof.

4. Sporadic groups Finally, let us deal with the “small” and “medium” sporadic groups. Proposition 4.1. Let G be a sporadic simple K-group with e(G) ≤ 3. Suppose that t ∈ Aut(G) and X ≤ G are such that (G, t, X) is of type (H ). Then either X = 1, or one of the following conditions holds: (1) G ∼ = Z3 × A4 and X ∼ = E4 ; = M22 , t ∈ 3A with CG (t) ∼ (2) G ∼ = M23 , t ∈ 3A with CG (t) ∼ = Z3 × A5 and X ≤ E16 ; ∼ (3) G = M24

(a) t ∈ 3A with C(3A) ∼ = 3A6 and X ∼ = 26 ; (b) t ∈ 3B with C(3B) ∼ = 26 ; = Z3 × L3 (2) and X ∼ ∼ Z3 × A4 and ∼ J2 , t ∈ 3B with C(3B) = (4) G = XCG (t) ≤ N (J (S2 )) ∼ = 22+4 .(Z3 × S3 );

(5) G ∼ = J4 , t ∈ 3A with C(3A) ∼ = 6M22 and XCG (t) ≤ Q68 (3M22 .2); (6) G ∼ = H e, t ∈ 3B with C(3B) ∼ = Z3 × L3 (2). and X ∼ = E4 .

Proof. Throughout this proof, we shall assume the results of section 5.3 [GLS3]. First of all recall, that e(G) = max{m2,p (G) | p odd}. Thus G is isomorphic to one of the following groups: M11 , M12 , M22 , M23 , M24 , J1 , J2 , J3 , J4 , HS, Mc, Co3 , Co2 , Suz, He, Ly, Ru, O’N, F5 , F3 .

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Since (| Out(G)|, 3) = 1, t ∈ Inn(G). If m3 (G) = 1, then G ∼ = J1 , |G| = 23 · 3 · 5 · 7 · 11 · 19 and CG (t) ∼ = Z3 × D10 . If Pp ∈ Sylp (G) for some odd prime p, then NG (Pp ) does not contain a subgroup isomorphic to CG (t). Therefore p = 2, and since S2 ∼ = E8 while X = [X, t], X∼ E . Since X does not admit an automorphism of order 5, X ≤ CG (P5 ) for some = 4 P5 ∈ Syl5 (G). But m2 (P5 ) = 1, hence X = 1. Suppose that m3 (G) ≥ 2. Then there exists s ∈ CG (t) such that U = s, t ∼ = E9 and Xs = CX (s) = 1. Moreover, we may suppose that Xs = [Xs , t]. Thus we are reduced into the centralizer of some 3-element, and if “nothing happens” for all possible choices of s, we may conclude that X = 1. On the other hand, if something does happen, i.e., Xs = 1 for some choice of s, it determines the possible value of p. Thus we must first look inside the centralizers of the 3-elements of the groups listed above. In order to do so, we will use Tables 5.3a–5.3z [GLS3], p. 262–287. ∼ ∼ M11 : Since G contains a unique conjugacy class of 3-elements, CG (s) = G= Z3 × S3 . Hence X = 1. G∼ = M12 : Looking at the list of all maximal subgroups of G (p. 263 [GLS3]), we obtain X = 1. ∼ G = M22 : Since G contains a unique conjugacy class of 3-elements, CG (s) ∼ = CG (t) ∼ = Z3 × A4 . If X = 1, then p = 2. Looking at the list of all maximal subgroups (p. 264 [GLS3]), we get X ∼ = E4 . ∼ G = M23 : Since G contains a unique conjugacy class of 3-elements, CG (s) ∼ = CG (t) ∼ = Z3 × A5 . If X = 1, then p = 2. Looking at the list of all maximal subgroups (p. 265 [GLS3]), we get X ≤ E16 . ∼ G = M24 : G has two conjugacy classes of 3-elements: 3A whose centralizer is isomorphic to 3A6 , and 3B whose centralizer is isomorphic to Z3 × L3 (2). Hence if X = 1, p ∈ {2, 7}. But P7 ∼ = = Z7 and NG (P7 ) ∼ Z7 .Z3 ×S3 . Hence p = 2. Looking at the list of all maximal subgroups (p. 266 [GLS3]), we get that X ∼ = 26 and both cases are possible, i.e., t ∈ 3B and t ∈ 3B happen. ∼ G = J2 : G contains two conjugacy classes of 3-elements: 3A with C(3A) ∼ = 3A6 and 3B with C(3B) ∼ = Z3 × A4 . If X = 1, then p = 2. Looking at the list of all maximal subgroups (p. 268 [GLS3]), we get t ∈ 32 and XCG (t) ≤ N (J (P2 )) ∼ = 22+4 .(Z3 × S3 ). ∼ ∼ J3 : G contains two conjugacy classes of 3-elements: 3A with C(3A) = G= ∼ E9 31+2 .2. If X = 1, then p = 2. But Z3 × A6 and 3B with C(3B) = no 2-local of G has a 3-part of its order bigger than 9. Hence X = 1. G∼ = 6M22 . = J4 : G has a unique conjugacy class of 3-elements 3A with C(3A) ∼ If X = 1, then from the M22 -case, p = 2. Since C(2A) ≥ NG (3A), XCG (t) ≤ Q68 (3M22 .2). G∼ = Z3 ×S5 . = HS: Since G has a unique conjugacy class of 3-elements, CG (s) ∼ But Xs = 1, and so X = 1.

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G∼ = Mc: G contains two conjugacy classes of 3-elements, and whether s ∈ 3A, or s ∈ 3B, F ∗ (CG (s)) = O3 (CG (s)). If Xs = 1, then Xs t must act faithfully on a 3-group, which contradicts Lemma 1.8. Hence Xs is trivial, and so is X. G∼ = = Co3 : G contains three conjugacy classes of 3-elements: 3A with C(3A) ∼ E .(Z × S ) and 3C with 31+4 .((4 SL2 (9)).2), 3B with C(3B) ∼ = 35 2 5 C(3C) ∼ = Z3 × L2 (8).Z3 . Hence, if X = 1, then p = 2, and since e(G) = m2,3 (G) = 3, t ∈ 3C. Moreover, by Lemma 1.8, s ∈ 3C. Since Xs = 1, t acts as an outer automorphism on Ls ∼ = L2 (8) and Xs ∼ = S3 does not normalize X. Hence Xs = = E4 . But CLs (t) ∼ X = 1. G∼ = = Co2 : G contains two conjugacy classes of 3-elements: 3A with C(3A) ∼ 31+4 .((D8 Q8 )S5 ) and 3B with C(3B) ∼ = Z3 × Aut(PSp4 (3)). Applying Lemma 1.8 if s ∈ 3A, and Proposition 2.1 if s ∈ 3B, we obtain that Xs = 1. Hence X = 1. G∼ = He: G contains two conjugacy classes of 3-elements: 3A with C(3A) ∼ = 3A7 .2 and 3B with C(3B) ∼ = Z3 × L3 (2). Hence, p ∈ {2, 7}. If p = 7, there exists X ∼ = Z7 with NG (X) ∼ = Z7 .Z3 × L3 (2). But the 3-element which acts faithfully on X is of type 3A. Hence, p = 2. Then X ∼ = S4 × L3 (2) contains CG (t) for t ∈ 3B. = E4 and NG (X) ∼ G∼ = = Ru: Since G has the unique conjugacy class of 3-elements, NG (t) ∼ NG (s) ∼ = 3 Aut(A6 ). So, if X = 1, then p = 2. Since no involution of G has a centralizer which contains a subgroup isomorphic to CG (t), CX (CG (t)) = 1. Studying the 2-local structure of G, we get X = 1. G ∈ {Ly, O’N, Suz, F5 , F3 }: No matter what t is, m3 (CG (t)) ≥ 4. mp,3 (G) ≤ 3, X = 1.

Since

The result now follows. Combining Propositions 2.1, 2.2, 2.4, 2.5, 2.18, 3.1 and 4.1, we obtain the results of Theorems 0.2 and 0.3. Acknowledgements. I would like to thank the referee for his valuable and patient comments. I would like to thank Ronald Solomon, under whose kind and thorough direction most of this work was done as a part of my Ph.D. thesis, and Richard Lyons for his invaluable discussions, corrections and support, which made it possible to complete this paper. I would also like to thank Peter Sin for his valuable comments and corrections.

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References [AS]

M. Aschbacher, G. Seitz, Involutions in Chevalley Groups Over Fields of Even Order, Nagoya Math. J. 63 (1976), 1–91.

[Be1]

H. Bender, Über den grössten p -Normalteiler in p-auflösbaren Gruppen, Arch. Math. 18 (1967), 15–16.

[BuGrLy1]

N. Burgoyne, R. L. Griess, R. Lyons, Maximal subgroups and automorphisms of Chevalley groups, Pacific J. Math. 71 (1977), 365–403.

[G]

D. Gorenstein. Finite Groups, Second Edition, Chelsea, New York 1980.

[GLS]

D. Gorenstein, R. Lyons, R. Solomon, The Classification of the Finite Simple Groups, Math. Surveys Monogr. 40.1, Amer. Math. Soc., Providence, RI, 1994.

[GLS2]

D. Gorenstein, R. Lyons, R. Solomon, The Classification of the Finite Simple Groups, Number 2, Math. Surveys Monogr. 40.2, Amer. Math. Soc., Providence, RI, 1996.

[GLS3]

D. Gorenstein, R. Lyons, R. Solomon, The Classification of the Finite Simple Groups, Number 3, Math. Surveys Monogr. 40.3, Amer. Math. Soc., Providence, RI, 1998.

[Is]

I. Martin Isaacs, Character Theory of Finite Simple Groups, Academic Press, New York 1976.

Inna Korchagina, Department of Mathematics, Rutgers University, New Brunswick, NJ 08903, U.S.A. E-mail: [email protected]

Classifying character degree graphs with 5 vertices Mark L. Lewis

Abstract. We study the degree graphs of solvable groups that have 5 vertices. Using known necessary conditions on one hand, and making use of straightforward constructions on the other hand, one can decide if a graph with 5 vertices occurs as a character degree graph for a solvable group– up to four exceptions. In this paper, we study these four exceptions. In particular, we show that two graphs that satisfy the previously known necessary conditions with 5 vertices cannot occur as degree graphs for solvable groups. We give an example showing that a third graph with 5 vertices does occur as the degree graph of a solvable group. Finally, we leave undecided whether one graph with 5 vertices is the character degree graph for a solvable group. 2000 Mathematics Subject Classification: 20c15

1. Introduction Throughout this paper, G will be a finite solvable group, and cd(G) will be the set of (irreducible) character degrees of G. Let ρ(G) be the set of primes that divide degrees in cd(G). The degree graph of G, written (G), is the graph whose vertex set is ρ(G). Two vertices p and q in ρ(G) are adjacent if there is some degree a ∈ cd(G) where pq divides a. These graphs have been studied in several places. The best sources for general information about these graphs are Theorem 14 of [3], Chapter 30 of [4], and Sections 18 and 19 of [12]. In this paper, we wish to consider which graphs with five vertices occur when G is solvable. The following theorem of Pálfy puts a restriction on which graphs can occur as degree graphs of solvable groups. Theorem 1.1 (Pálfy, [13]). Let G be a solvable group, and let π be a subset of ρ(G). If |π| ≥ 3, then there exist primes p, q ∈ π and a degree a ∈ cd(G) so that pq divides a. In particular, any three primes in ρ(G) must have an edge in (G) that is incident to two of those primes. We will say that a graph satisfies Pálfy’s condition if it has the property that every three vertices have an edge incident to two of those vertices. It is easy to see that a disconnected graph that satisfies Pálfy’s condition will have two connected components, and each connected component will be a complete graph. Therefore,

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there are only two possible disconnected graphs with five vertices that meet Pálfy’s condition. Both of these graphs occur as (G) for some solvable group G. We will present examples of groups with these graphs in Section 2. We now turn to the case where (G) is connected. In this case, we look at the diameter of the graph. As usual, the distance between two vertices is the number of edges in the shortest path between the vertices. The diameter of a connected graph is the maximum distance between points in the graph. For disconnected graphs, the diameter of the graph is the diameter of the largest connected component. It is not difficult to see that graphs satisfying Pálfy’s condition have diameter at most 3. It had been conjectured for solvable groups that in fact the diameter of (G) was at most 2, but in [10] we constructed a solvable group G where (G) had diameter 3. The graph we constructed had 6 vertices, and in [11], we proved the following theorem: Theorem 1.2 ([11]). If G has a solvable group and (G) has 5 vertices, then the diameter of (G) is at most 2. As mentioned in [11], this eliminates from consideration two graphs with 5 vertices that satisfy Pálfy’s condition. In Section 2, we will show how to use direct products to construct groups with various degree graphs having 5 vertices by using groups whose degree graphs have 4 or fewer vertices. When these constructions are completed there will remain four graphs with 5 vertices that satisfy Pálfy’s condition. These graphs are shown in Figure 1. The purpose of this paper is to show that (1) and (2) in Figure 1 cannot occur as degree graphs, and to construct a group whose degree graph is (3). Unfortunately, we have not been able to determine whether or not (4) is the degree graph of a solvable group. We codify this in the following theorem: Main Theorem. The graphs with 5 vertices that arise as (G) for some solvable group G are precisely the graphs with diameter at most 2 that satisfy Pálfy’s condition except for graphs (1), (2), and possibly (4) of Figure 1. .

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Figure 1. Graphs under consideration

2. Constructions We first present two groups G that have the two possible disconnected graphs with 5 vertices as (G). Both of these groups are affine semi-linear groups. For the first example, take the Galois field of order 210 acted on by its full multiplication group and

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then its Galois group. It is not difficult to see that this group has character degree set {1, 2, 5, 10, 3 · 11 · 31}. This gives a graph with two connected components: {2, 5} and {3, 11, 31} which are both complete graphs. For the second example, take the Galois field of order 216 acted on by its full multiplication group and then its Galois group. In this case, the character degree set is {1, 2, 4, 8, 16, 3 · 5 · 17 · 257}, and the degree graph has connected components: {2} and {3, 5, 17, 257} where the second component is a complete graph. These examples are in no way unique. Similar constructions will yield examples of both graphs, and it is not difficult to find examples where the groups have odd orders. One way to build groups from smaller groups is by taking direct products. It is not difficult to see that if H and K are groups, then ρ(H × K) = ρ(H ) ∪ ρ(K). Furthermore, there is an edge between p and q in ρ(H × K) if one of the following occurs: (1) p, q ∈ ρ(H ) and there is an edge between p and q in (H ), (2) p, q ∈ ρ(K) and there is an edge between p and q in (K), (3) p ∈ ρ(H ) and q ∈ ρ(K), or (4) p ∈ ρ(K) and q ∈ ρ(H ). This follows from Theorem 4.21 of [5]. All graphs with 3 or fewer vertices that satisfy Pálfy’s condition are known to arise as (G) for some solvable group G. In fact, the only graph with 3 or fewer vertices that does not satisfy Pálfy’s condition is the graph with 3 isolated vertices. Graphs with 4 vertices that satisfy Pálfy’s condition do occur as degree graphs for some solvable group. The two graphs with 4 vertices that satisfy Pálfy’s condition and do not occur as degree graphs are shown in Figure 2. For Graph (1) of Figure 2, this is proved in [14], and for Graph (2), this is the content of [15]. So we may refer to this fact later, we state the following theorem. Theorem 2.1 ([14] and [15]). If G is a solvable group, then (G) is neither of the graphs shown in Figure 2. ·

·

· (1)

·

·

·

·

·

(2)

Figure 2. 4-vertex graphs that do not occur

Notice that Graph (2) of Figure 2 is the only graph with four or fewer vertices that has diameter 3. Therefore, combining Theorem 1.2 with Theorem 2.1, we see that if G is solvable group and |ρ(G)| ≤ 5, then the diameter of (G) is at most 2. On page 25 of [3], Huppert listed all the graphs with at most 4 vertices that can arise as (G) for some solvable group G. For the record, there are 14 graphs with 5 vertices that satisfy Pálfy’s condition. We have seen that two of these graphs are disconnected and do occur as degree graphs. Two of the graphs have diameter 3, and by Theorem 1.2, they do not occur as degree graphs. This leaves ten graphs to consider, and of these six graphs can be obtained by looking at groups of the form H × K where ρ(H ) and ρ(K) are disjoint, and |ρ(H ) ∪ ρ(K)| = 5. The five graphs with 4 vertices give five graphs arising in the form H × K where |ρ(H )| = 4 and |ρ(K)| = 1. Since

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there are two graphs with 2 vertices and three graphs with 3 vertices, it would seem that we should get six graphs arising from H × K where |ρ(H )| = 3 and ρ(K) = 2, but it turns out that one graph arises in two different fashions, so we get five graphs of this form. We should mention that it is necessary to show for each of the pairs of graphs above that there exist groups H and K with the given graphs where ρ(H ) and ρ(K) are disjoint. Since for every prime p there is a group K with ρ(K) = {p}, this is fine when |ρ(H )| = 4 and |ρ(K)| = 1. When |ρ(H )| = 3 and |ρ(K)| = 2, it is less trivial, but still not difficult to show that such H and K will exist. The two lists have four graphs in common, so we get six graphs in this fashion, and we are left with the four graphs found in Figure 1.

3. Graph (1) of Figure 1 Throughout this paper, we will make use of the following results without further reference. By Itô’s theorem (Corollary 12.34 of [5]), we know that a solvable group G has a normal abelian Sylow p-subgroup if and only if p ∈ ρ(G). Suppose N is a normal subgroup of G, and the character θ ∈ Irr(N ) extends to G. Using Gallagher’s theorem (Corollary 6.17 of [5]), we can show that cd(G|θ ) = {θ (1)a | a ∈ cd(G/N )}. If (|G : N |, |N |) = 1, then we know via Corollary 6.28 of [5] that every character in Irr(N ) will extend to its stabilizer in G. Let F be the Fitting subgroup of G. In Lemma 18.1 of [12], it is shown that if G ⊆ F , then |G : F | ∈ cd(G). Let F2 /F be the Fitting subgroup of G/F . On page 254 of [12], a character in Irr(G) is produced that is divisible by all the prime divisors of |F2 : F |. We will also make use of the detailed information collected in [7] regarding a solvable group G where (G) is disconnected. It is actually quite easy to show that (1) from Figure 1 does not occur. In fact, we prove a couple of much more general results which are still quite easy. The hypotheses of these lemmas will arise in an inductive argument. Lemma 3.1. Let G be a solvable group, and suppose p is a prime in ρ(G). For every proper normal subgroup H of G, suppose that (H ) is a proper subgraph of (G). Assume that no subgraph of (G) obtained by removing the vertex p or by removing one or more of the edges incident to p is (H ) for any normal subgroup H of G. Then G = O p (G). Proof. Let M = O p (G). If M = G, then we are done, so we assume that M < G. By hypothesis, (M) is a proper subgraph of (G). Since |G : M| is a power of p, the only vertex of (G) that could be missing is p, and any edges that are missing must be incident to p. Thus, (M) is a subgraph of (G) obtained by deleting p or by deleting one or more of the edges incident to p. Since no such graph can be the degree graph of a normal subgroup of G, this is a contradiction, so we must have G = M.

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Lemma 3.2. Let be a graph with n vertices. Suppose that no proper subgraph of with either n vertices or n − 1 vertices occurs as (G) for some solvable group G. Then is not the degree graph (G) for some solvable group G. Proof. Let G be a counterexample with |G| minimal. Since is not empty, G > 1, and since G is solvable, there must be a prime p so that O p (G) < G. On the other hand, the minimality of |G| will imply that (H ) is a proper subgraph of for every proper normal subgroup H . Furthermore, any subgraph of obtained by removing p or edges incident to p will have n or n − 1 vertices, and so any graph obtained in this fashion cannot be the graph of a (normal) subgroup of G. Therefore, we obtain a contradiction via Lemma 3.1 which said O p (G) = G. We can now apply Lemma 3.2 to eliminate Graph (1) of Figure 1. Corollary 3.3. The pentagon (Graph (1) of Figure 1) is not the degree graph of a solvable group. Proof. We claim that no proper subgraph of Graph (1) of Figure 1 is the degree graph of a solvable group. The only proper subgraphs with 4 or 5 vertices that are not excluded by Pálfy’s condition are the two graphs in Figure 2, and these graphs are excluded by Theorem 2.1. It follows that the pentagon satisfies the hypotheses of Lemma 3.2, and applying that lemma, the pentagon is not the degree graph of any solvable group.

4. A nonabelian normal Sylow subgroup In the next three sections, we work to prove that Graph (2) of Figure 1 cannot occur. The next two sections will consist of lemmas that can be applied to a more general setting than just the graph under consideration. In this section, we concentrate on groups having a normal nonabelian Sylow subgroup. With this in mind, we make the following hypothesis. Hypothesis 4.1. Let G be a solvable group with a normal nonabelian Sylow psubgroup P for some prime p. Let π be the set of primes in ρ(G) that are adjacent to p in (G), and let ρ be the set of primes in ρ(G) that are not adjacent to p. Thus, ρ(G) is the disjoint union: {p} ∪ π ∪ ρ. The next lemma deals with the case where the p-complement is acting faithfully on P and the Fitting subgroup of the p-complement is a π-group. We show that the primes involved in the degrees of the p-complement lie in ρ. Lemma 4.2. Assume Hypothesis 4.1. Suppose that for every prime q ∈ π there is a prime r ∈ ρ so that q and r are not adjacent in (G). Let H be a p-complement for G. Assume that H acts faithfully on P , that ρ ⊆ ρ(H ), and that the Fitting subgroup of H is a π-group. Then ρ(H ) = ρ.

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Proof. If π is empty, then the result is immediate, so we may assume that π is not empty. Also, we can find a nonlinear character θ ∈ Irr(P ). We know that p divides every degree in cd(G|θ), and so no prime in ρ divides any degree in cd(G|θ ). By Theorem 12.9 of [12], G/P has an abelian Hall ρ-subgroup. Let F be the Fitting subgroup of H , and label the primes in π by π = {q1 , q2 , . . . , qm }. For each i, let Fi be the Sylow qi -subgroup of F , and let Ci = CH ([P , Fi ]P /P ). Since F = F1 × F2 × · · · × Fn , we have the following product and intersection: [F, P ] = [F1 , P ][F2 , P ] . . . [Fn , P ] and m i=1 Ci = CH ([P , F ]P /P ). Because F acts faithfully on P , the action of F on P /P will be faithful. Applying Fitting’s Lemma, F will act faithfully on [F, P ]P /P . On the other hand, if CH ([P , F ]P /P ) > 1, then CH ([P , F ]P /P ) will contain a minimal normal subgroup of H which would be contained in F . As CF ([P , F ]P /P ) = 1, m we conclude that CH ([P , F ]P /P ) = 1 and i=1 Ci = 1. For each i = 1, . . . , m, we fix a nonprincipal character λ ∈ Irr([P , Fi ]P /P ). Let ri be the prime in ρ that is not adjacent in (G) to qi . Since |H : CH (λ)| divides every degree in Irr(G|λ) and qi divides |H : CH (λ)|, it follows that ri does not divide |H : CH (λ)|. We see that CH (λ) contains a Sylow ri -subgroup of H . Observe that λ extends to P CH (λ), and using Gallagher’s theorem, ri ∈ ρ(CH (λ)). As usual, CH (λ) has a normal abelian Sylow ri -subgroup. Note that Ci ⊆ CH (λ), so if Ci contains a Sylow ri -subgroup of H , then H would have a normal abelian Sylow ri -subgroup which is a contradiction since ri ∈ ρ ⊆ ρ(H ). We conclude that a Sylow ri -subgroup of H acts nontrivially on [P , Fi ]P /P . By Lemma 1 of [8], either H /Ci ∼ = SL2 (3) or GL2 (3) and |[P , Fi ]P /P | = 9 or H /Ci has a normal abelian subgroup that acts irreducibly on [P , Fi ]P /P . Since (|P |, |H |) = 1, it must be that H /Ci has an abelian normal subgroup that acts irreducibly on [P , Fi ]P /P . From Lemma 2.2 of [12], H Ci /Ci is nilpotent. Because m i=1 Ci = 1, this implies that H is nilpotent. In particular, H ⊆ F , and thus, |H : F | ∈ cd(H ) = cd(G/P ). Since ρ ⊆ ρ(H ) and no prime in ρ lies in ρ(F ), we conclude that |H : F | is divisible by all the primes in ρ. Since every prime in π is not adjacent to some prime in ρ, it follows that |H : F | is not divisible by any prime in π , and F is the Hall π-subgroup of H . Our next goal is to prove that F is abelian. This will imply that H has an abelian normal Hall π -subgroup, and thus, ρ(H ) ⊆ ρ. It is sufficient to prove that Fi is abelian for each i, and so, we assume Fi is not abelian for some i, and we obtain a contradiction. Let Ri be a Sylow ri -subgroup of H . We claim that P Fi Ri is a normal subgroup of G. Since H /F is abelian, we see that F Ri is normal in H . Let Ei and Hi be Hall qi -complements of F and H , so F = Fi × Ei . There is a character ν ∈ Irr(Fi ) that is not linear. We see that ν extends to Fi CHi (ν). Since qi divides every degree in cd(H |ν) ⊆ cd(G), it follows that ri does not divide |Hi : CHi (ν)| and ri divides no degree in cd(CHi (ν)). As usual, we may use Gallagher’s theorem and Itô’s theorem to see that CHi (ν) contains a unique Sylow ri -subgroup of H . By conjugating, we may assume that Ri ⊆ CHi (ν).

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For any character µ ∈ Irr(Ei ), the stabilizer CHi (ν × µ) = CHi (ν) ∩ CHi (µ) contains some Sylow ri -subgroup for H . Since CHi (ν × µ) ⊆ CHi (ν), the uniqueness of Ri forces Ri ⊆ CH (ν × µ) ⊆ CH (µ), and hence, Ri stabilizes every character in Irr(Ei ). This implies that Ri centralizes Ei (see Lemma 3.2 (b) of [6]). It follows that Fi Ri is a normal subgroup of F Ri . Since Fi Ri is a Hall subgroup of F Ri , it is characteristic, and we use the fact that F Ri is normal in H to see that Fi Ri is normal in H . Therefore, P Fi Ri is a normal subgroup of G. It is not difficult to see that (P Fi Ri ) has two connected components: {ri } and {qi , p}. Looking at the classification in [7], we see that the only solvable groups K where (K) has two connected components and the Fitting subgroup of K/F (K) is not abelian has ρ(K) = {2, 3}. As |ρ(P Fi Ri )| = 3 > 2, this is a contradiction, and Fi is abelian. The next lemma deals with the situation where the primes dividing the degrees of the p-complement of G are precisely the primes in ρ. The lemma finds a subgroup whose degree graph has two connected components. In this lemma, it will be useful to have some additional notation. Let N be a normal subgroup of G. Then Irr(G|N) = {χ ∈ Irr(G) | N ⊆ ker(χ)}, and we define cd(G|N) = {χ (1) | χ ∈ Irr(G|N)}. Given a character θ ∈ Irr(N), we write cd(G|θ ) = {χ (1) | χ ∈ Irr(G|θ )}. If n is a positive integer, we define π(n) to be the set of primes dividing n. Lemma 4.3. Assume Hypothesis 4.1. Let H be a p-complement for G. Suppose that P is minimal normal in G, and ρ(H ) = ρ is not empty. Then there is a nonempty subset π ∗ of π so that (P H ) has two connected components: ρ and π ∗ . Furthermore, if |ρ| = n, then |π ∗ | ≥ 2n − 1. (In fact, P H satisfies the hypotheses of Example 2.4 of [7].) Proof. Since P is minimal normal in G, it follows that P ⊆ Z(P ). For any nonprincipal character α ∈ Irr(P ), we see that P CH (α) is the stabilizer of α in G. Using the Glauberman correspondence (Theorem 13.1 of [5]), we can find a character θ ∈ Irr(P |α) so that θ is CH (α)-invariant. On the other hand, θP = θ (1)α, and CH (θ ) = CH (α). The stabilizer of θ in G is P CH (α), and θ extends to H CH (α). It follows that p|H : CH (α)| divides every degree in cd(G|θ ), and so the prime divisors of |H : CH (α)| must lie in π. Also, we may use Gallagher’s theorem to see that ρ(CH (α)) ⊆ π . It follows that CH (α) contains a full Hall ρ-subgroup of H as a normal abelian subgroup. Now, α extends to P CH (α), its stabilizer in P H , and the prime divisors of degrees in cd(P H |α) must lie in π . Letting π ∗ be the set of prime divisors of degrees in cd(P H |P ), we conclude that π ∗ ⊆ π . Because ρ(H ) = ρ is nonempty, we use Itô’s theorem to see that H does not have a normal abelian Hall ρ-subgroup. Taking α as before, CH (α) < H , and π ∗ is not empty. We have cd(P H ) = cd(P H /P ) ∪ cd(P H |P ). We know that ρ(P H /P ) = ρ(H ) = ρ. In the previous paragraph, we showed that the degrees in cd(P H |P ) were divisible only by primes in π ∗ which is a subset of π. It follows that (P H ) has two connected components: ρ and π ∗ . Observe that P is a noncentral, abelian, normal Sylow p-subgroup of P H , and using Lemmas 3.1–3.6 of [7], the solvable

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groups with two connected components having such a subgroup satisfy the hypotheses of Example 2.4 of [7]. Let F and E/F be the Fitting subgroups of P H and P H /F . We know that the two connected components of (P H ) are π(|P H : E|) and π(|E : F |). Since P ⊆ F , we have that π(|P H : E|) = ρ(P H /F ) ⊆ ρ(H ) = ρ. This yields by the proof of Theorem 3 of [14] that if |ρ| = n, then |π ∗ | ≥ 2n − 1. The remaining lemmas in this section deal with groups where the neighbors of p are partitioned in two sets with no adjacencies between the sets. To make this more clear, we state the following hypothesis. Hypothesis 4.4. Assume Hypothesis 4.1. Suppose that π is the disjoint union of nonempty sets π1 and π2 , and assume that no prime in π1 is adjacent in (G) to any prime in π2 . This next lemma shows that G/P must have a normal abelian Hall π -complement. Lemma 4.5. Suppose Hypothesis 4.4 holds. Then π(G/P ) ⊆ π which is equivalent to saying G/P has a normal abelian Hall π -complement. Proof. Note that the equivalence of the conditions in the conclusion is due to Itô’s theorem. Let H be a Hall p-complement of G. Suppose that there exist nonlinear characters θ1 , θ2 ∈ Irr(P ) so that p1 ∈ π1 divides |H : CH (θ1 )| and p2 ∈ π2 divides |H : CH (θ2 )|. It follows that ppi divides every degree in cd(G|θi ), and thus, |H : CH (θi )| is divisible only by primes in πi . In particular, CH (θi ) contains a Hall πi -complement of H . Furthermore, θi extends to P CH (θi ), and using Gallagher’s theorem ρ(CH (θi )) ⊆ πi . By Itô’s theorem, CH (θi ) has a normal abelian Hall πi complement. Now, both CH (θ1 ) and CH (θ2 ) will normalize a Hall π -complement of H . Conjugating if necessary, we may assume that they normalize the same Hall π-complement of H . Since |H : CH (θ1 )| and |H : CH (θ2 )| are relatively prime, we conclude H = CH (θ1 )CH (θ2 ), and H has a normal abelian Hall π-complement, and the lemma is proved in this case. We may assume that we do not have a pair a characters like θ1 and θ2 in the previous paragraph. Because π1 and π2 are nonempty, there exist characters χ1 , χ2 ∈ Irr(G) so that ppi divides χi (1) for i = 1, 2 and primes pi ∈ πi . For each i, let γi be an irreducible constituent of (χi )P . We see that p divides γi (1), so γi is not linear. Using the previous paragraph, we cannot have both CH (γ1 ) < H and CH (γ2 ) < H . This implies that H = CH (γi ) for either i = 1 or i = 2. We fix the value i so that H = CH (γi ), and we note that γi extends to G. Applying Gallagher’s theorem, ρ(G/P ) = ρ(H ) ⊆ π . In addition to assuming Hypothesis 4.4, we now assume that the only primes dividing degrees in cd(G/P ) all lie in π1 . We show that the product of a Hall π2 subgroup of G with P will be normal in G and have an abelian quotient. Lemma 4.6. Suppose Hypothesis 4.4 holds. Assume that ρ(G/P ) ⊆ π1 , and let P2 be a Hall π2 subgroup of G. Then G ⊆ P P2 .

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Proof. Since no prime in π2 lies in ρ(G/P ), we use Itô’s theorem to see that G/P has a normal abelian Hall π2 -subgroup, and so P P2 is normal in G. Choose the character ψ ∈ Irr(G) so that ψ(1) is divisible by pp2 for some prime p2 ∈ π2 . Note that ψ(1) will be divisible only by primes in {p} ∪ π2 , and |G : P P2 | is divisible by no primes in {p} ∪ π2 . It follows that ψ(1) is relatively prime to |G : P P2 |, and so, ψP P2 is irreducible by Corollary 11.29 of [5]. By Gallagher’s theorem, ψ(1)a ∈ cd(G) for every degree a ∈ cd(G/P P2 ). Since ρ(G/P P2 ) contains no primes in {p} ∪ π2 , we see that 1 is the only degree in cd(G/P P2 ), and so, G/P P2 is abelian. This remaining lemma looks at the case where both π1 and π2 have primes dividing degrees in cd(G/P ). Lemma 4.7. Suppose Hypothesis 4.4 holds. Assume that ρ(G/P ) intersects both π1 and π2 nontrivially. Then one of the following holds: (1) G/P has a central Hall ρ-subgroup. (2) There is a prime r ∈ ρ so that G/P has a noncentral Sylow r-subgroup, and every other prime q ∈ ρ is adjacent in (G) to every prime in ρ(G/P ). Proof. By Lemma 4.5, we know that ρ(G/P ) ⊆ π . Since (G/P ) is a subgraph of (G), we see that (G/P ) has two connected components: ρ(G/P ) ∩ π1 and ρ(G/P ) ∩ π2 . If G/P has a central Hall ρ-subgroup, then the result holds; so we assume that G/P does not have a central Hall ρ-subgroup. By Theorem 5.5 of [7], G/P has at most one noncentral normal Sylow subgroup, and we will let r be the prime so that G/P has a noncentral Sylow r-subgroup. Since G/P has a normal abelian Hall ρ-subgroup (by Lemma 4.5), we see that if r is not in ρ, then G/P would have a central Hall ρ-subgroup, and we assumed this did not happen. Thus, we have r ∈ ρ. If ρ = {r}, then we are done, so we assume that ρ contains primes other than r. Looking at Lemmas 3.1–3.6 of [7], we see that if K is a solvable group where (K) is disconnected and K has a noncentral Hall ρ(K)-complement, then K satisfies the hypotheses of Example 2.4 of [7]. Therefore, G/P satisfies the hypotheses of Example 2.4 of [7]. Let H be a p-complement for G, and note that H ∼ = G/P . The Fitting subgroup of H has the form V × Z where V is a minimal normal r-subgroup of H and Z = Z(H ) (see Lemma 3.4 of [7]). Anything centralizing V will centralize V Z, and as V Z is the Fitting subgroup of H , we conclude that CH (V Z) = V Z. We now consider a prime q ∈ ρ with q = r. We need to show that q is adjacent to every prime in ρ(G/P ). For the purpose of finding a contradiction we assume that q is not adjacent to the prime s ∈ ρ(G/P ). Since q ∈ ρ ⊆ ρ(G), we see that the Sylow q-subgroup Q of H is not central in G. Since Q is central in H , we see that Q must act nontrivially on P , and this implies that Q acts nontrivially on P /P . Let C = CH ([P , Q]), and observe that C = CH ([P , Q]P /P ). Consider a nonprincipal character λ ∈ Irr([P , Q]P /P ). As q divides |H : CH (λ)|, it follows that s does not divide |H : CH (λ)|. Also, λ must extend to P CH (λ), so by Gallagher’s theorem s ∈ ρ(CH (λ)). Using Itô’s theorem, CH (λ) has a normal abelian Sylow s-subgroup of H . Now, C ⊆ CH (λ).

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If C contains a Sylow s-subgroup of H , then H would have a normal abelian Sylow s-subgroup which is a contradiction since s ∈ ρ(H ). Thus, a Sylow s-subgroup of H acts nontrivially on [P , Q]P /P . We have shown that H satisfies the hypotheses of Lemma 1 of [8] with respect to the prime s. By that result, either H /C ∼ = SL2 (3) or GL2 (3) with |[P , Q]P : P | = 9 or H /C has a normal abelian subgroup that acts irreducibly on [P , Q]P /P . Since |P | and |H | are coprime, we do not have H /C ∼ = SL2 (3) or GL2 (3). Since H /C has an abelian normal subgroup that acts irreducibly on [P , Q]/P , the Fitting height of H /C is at most 2 (see Theorem 19.3 of [12]). If V ⊆ C, then V C = V × C since V is minimal normal in H . We obtain C ⊆ Z. It follows that H /C has Fitting height 3, and this is a contradiction, so V ⊆ C ⊆ CH (λ). Let S be the Sylow s-subgroup of H contained in CH (λ), and recall that S is a normal subgroup of CH (λ). Because V is also a normal subgroup of CH (λ), we obtain V S = V × S, and S ⊆ CH (V ) = V × Z. We conclude that S is central in H . This contradicts the fact that s ∈ ρ(H ), and the lemma is proved.

5. Disconnected degree graphs In this section, we further the study of disconnected degree graphs. We begin with a general lemma about groups where the connected components of the degree graph have specified sizes. This next lemma should be viewed as a refinement of Theorem 5.4 of [7]. Lemma 5.1. Let G be a solvable group, and suppose that (G) has two connected components π and ρ where |π| = n > 1 and |ρ| = 2n − 1. Let F and E/F be the Fitting subgroups of G and G/F , respectively, and let Z be the center of G. Then G satisfies the hypotheses of Example 2.4 of [7], π = π(|G : E|), ρ = π(|E : F |), and |G : E| is square-free. Furthermore, there exists a prime q so that F /Z is an elementary abelian q-group of order q |G:E| and E/F is cyclic of order (q |G:E| − 1)/(q − 1). Finally, if m is a divisor of |G : E|, then the number of prime divisors of (q m − 1)/(q − 1) equals the number of nontrivial divisors of m which equals 2|π(m)| − 1. Proof. Reading the proof of Theorem 5.4 in [7], we see that G satisfies the hypotheses of Example 2.4 of [7], and from that example, we see that G has Fitting height 3. In addition from Lemma 3.4 of [7], we have (|G : E|, |E : F |) = 1, π = π(|G : E|) and ρ = π(|E : F |), and |F : Z| = q |G:E| for some prime power q and (q |G:E| −1)/(q−1) divides |E : F | which divides q |G:E| − 1. We wish to apply the Zsigmondy prime theorem to |E : F |. To do this, we need to see that we do not have any of the exceptional cases of the Zsigmondy prime theorem. Suppose 2 divides |G : E| and q is odd. It follows that 2 divides q + 1 = (q 2 −1)/(q −1). On the other hand, (q 2 −1)/(q −1) divides (q |G:E| −1)/(q −1) which

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divides |E : F |. Thus, 2 divides |G : E| and |E : F |, and this is a contradiction. Either |G : E| is odd, or q is even. Also, if q = 2 and 6 divides |G : E|, then 3 = (22 − 1)/(2 − 1) is a divisor of (2|G:E| − 1)/(2 − 1), which divides |E : F |. This is also a contradiction. We have shown that the exceptions to the Zsigmondy prime theorem cannot occur in our situation. We may apply the Zsigmondy prime theorem to see that the number of prime divisors of (q |G:E| − 1)/(q − 1) that do not divide q − 1 is at least the number of nontrivial divisors of |G : E|. Since |E : F | has 2n − 1 prime divisors, none of whom divide |G : E|, we see that |G : E| has at most 2n − 1 nontrivial divisors. Because n primes divide |G : E|, |G : E| has at least 2n − 1 nontrivial divisors. If some prime divides |G : E| with multiplicity more than 1, then |G : E| would have more divisors, so |G : E| is square-free. Furthermore, we see that the Zsigmondy primes account for all of the prime divisors of |E : F |, and so each prime dividing |E : F | must be a Zsigmondy prime for (q, m) where m is a nontrivial divisor of |G : E|. Further, this implies that the number of prime divisors of (q m − 1)/(q − 1) equals the number of nontrivial divisors of m which is 2|π(m)| − 1. We conclude that no prime dividing q − 1 divides |E : F |, and thus, |E : F | = (q |G:E| − 1)/(q − 1). If q is not prime, then the Zsigmondy prime theorem could be applied to the divisors of q, to see that (q m − 1)/(q − 1) would have more prime divisors than the number of nontrivial divisors of m, and so, q must be prime. In the next lemma, we consider adjacencies in (G) when G has a normal subgroup M so that (G/M) is similar to the degree graph studied in Lemma 5.1. Lemma 5.2. Let G be a group, and suppose M is a normal subgroup of G so that G/M is a solvable group that satisfies the hypotheses of Example 2.4 of [7]. Let F /M and E/F be the Fitting subgroups of G/M and G/F , respectively, and let r be the prime power so that |F /Z| = r |G:E| , where Z/M = Z(G/M). Suppose that there exist primes p ∈ π and q ∈ ρ and a degree a ∈ cd(G) so that pq divides a. If s ∈ π is not adjacent to q in (G), then (r |G:E| − 1)/(r |G:E|/s − 1) divides a. Proof. Let S/F be a Sylow s-subgroup of G/F . From [7], we see that we may view F /Z as the additive group of the field F of order r |G:E| , E/F as a subgroup of the multiplicative group of F, and G/E as the Galois group of F. Also by [7], the order of E/F is divisible by (r |G:E| − 1)/(r − 1), and |E/F | divides (r |G:E| − 1). We know that the action of S/F on E/F is the action coming from the Galois automorphism of order s. In F, the automorphism of order s has a fixed field of order r |G:E|/s , and looking at the associated elements in the multiplicative group of F, we obtain |E : C| = (r |G:E| − 1)/(r |G:E|/s − 1), where C/F = CE/F (S/F ). On the other hand, there exists a character χ ∈ Irr(G) with χ (1) = a. Let λ be an irreducible constituent of χF . Write T for the stabilizer of λ in G. Since all the Sylow subgroups of G/F are cyclic, it follows that all the Sylow subgroup of T /F are cyclic and λ extends to λˆ ∈ Irr(T ). There exists a character τ ∈ Irr(T /F ) so that

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ˆ G = χ . Since G/F has a normal abelian Hall ρ-subgroup, q ∈ ρ(T /F ), and (τ λ) thus, q does not divide τ (1). We deduce that q divides |G : T | or q divides λ(1). Because s does not divide χ(1), it will not divide |G : T |. Thus, T /F contains a Sylow s-subgroup of G/F . By conjugating, we may assume that S ⊆ T . Now, q divides every degree in cd(G|λ) and qs divides no degree in cd(G). We conclude that s divides no degree in cd(T /F ). Hence, T /F has a normal abelian Sylow s-subgroup, and so S/F is normal in T /F , and in particular, E ∩ T ⊆ C. This implies that |E : C| divides |G : T |. Since |G : T | divides a and |E : C| = (r |G:E| − 1)/(r |G:E|/s − 1), we obtain the desired conclusion.

6. Graph (2) of Figure 1 We now prove that Graph (2) of Figure 1 does not arise as the degree graph of a solvable group. Theorem 6.1. There is no solvable group G where (G) is the graph found in (2) of Figure 1. p1 p3 HH p2

p4

p5

Figure 3. Labeled Graph of Theorem 6.1

Proof. Suppose that G is a counterexample with |G| minimal. Label the primes in ρ(G) as in Figure 3. Our first goal is to show that G does not have any normal nonabelian Sylow subgroups. If P is a normal Sylow p-subgroup for some prime p, we have shown in Lemma 3 of [11] that cd(G/P ) = cd(G) \ {p}. From Theorem 2.1, we know that the graphs in Figure 2 cannot be the degree graphs for subgroups of G, and the only disconnected graph with 4 vertices that satisfies Pálfy’s condition has an isolated vertex and the remaining three vertices form a complete subgraph. We conclude that none of the subgraphs of (G) with vertex sets of {p1 , p2 , p3 , p5 } or {p1 , p2 , p4 , p5 } can arise as (H ) for a solvable group H . If G had a normal Sylow p3 -subgroup or normal Sylow p4 -subgroup P , then (G/P ) would be a subgraph of (G) with one of these two sets as its vertex set. Thus, G cannot have a normal Sylow p3 -subgroup nor a normal Sylow p4 -subgroup. Claim 1. Suppose H is a group where ρ(H ) = ρ(G), (H ) is a subgraph of (G), and |H | < |G|. Then (H ) has two connected components: {p1 , p2 } and {p3 , p4 , p5 }. In particular, H has no nonabelian normal Sylow subgroups. Proof. By minimality in the choice of G, we know (H ) = (G). Applying Pálfy’s condition to p1 , p2 , and p5 in (H ), we see that p1 and p2 must be adjacent in (H ).

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If (H ) is disconnected, we have already seen that Pálfy’s condition forces each component to be a complete graph, and so the two connected components must be {p1 , p2 } and {p3 , p4 , p5 }. In Lemma 5.1, we saw that H must satisfy the hypotheses of Example 2.4 of [7], and so, every normal Sylow subgroup of H is abelian. We assume (H ) is connected, and we work for a contradiction. If p3 and p4 are not adjacent, then (H ) will be a subgraph of Graph (1) of Figure 1. The only connected proper subgraph of Graph (1) of Figure 1 has diameter 4 which contradicts Pálfy’s condition, and so, (H ) will be Graph (1) of Figure 1 which was disallowed by Corollary 3.3. Therefore, p3 and p4 must be adjacent in (H ). Applying Pálfy’s condition to the sets {p1 , p4 , p5 } and {p2 , p3 , p5 }, we see that there must be an edge between p3 and p5 and an edge between p4 and p5 in (H ). Finally, since (H ) is connected, it must have an edge between p1 and p3 or between p2 and p4 . In fact, (H ) must contain both of these edges, or else it has diameter 3 which is not allowed by Theorem 1.2. We conclude (H ) = (G) which is a contradiction. We now work to prove G = O p3 (G) = O p4 (G). We show that this is true for p3 , and a similar argument will work to show that it is true for p4 . By the minimality of G, if H is a proper normal subgroup of G, then (H ) is a proper subgraph of (G). If p3 is removed from (G), we get Graph (2) from Figure 2 which does not occur by Theorem 2.1. On the other hand, removing one or more edges incident to p3 from (G), we obtain a graph that violates Pálfy’s condition, has diameter 3, or is Graph (1) of Figure 1, and we know that none of these graphs occur by Theorem 1.1, Theorem 1.2, and Corollary 3.3. We then use Lemma 3.1 to see that O p3 (G) = G. We show that G does not have a normal Sylow p1 -subgroup. A similar argument shows that G does not have a normal Sylow p2 -subgroup. We assume P is a normal Sylow p1 -subgroup of G, and we look for a contradiction. By Lemma 4.5, we see that ρ(G/P ) ⊆ {p2 , p3 }. If G/P is abelian, then O p4 (G) < G which is a contradiction. If |ρ(G/P )| = 1, then by Lemma 4.6, we see that G ⊆ P Q where Q is either a Sylow p2 -subgroup or Sylow p3 -subgroup of G. In either case, we again get O p4 (G) < G which is our contradiction. The final possibility is that ρ(G/P ) = {p2 , p3 }. In this case, we can apply Lemma 4.7 where ρ = {p4 , p5 }. If G/P has a central Sylow p4 -subgroup, then O p4 (G) < G, a contradiction. Thus, we have conclusion (2) of Lemma 4.7 with r = p4 . This implies that p5 must be adjacent to p2 which is a contradiction. Thus, G does not have a normal Sylow p1 -subgroup. We have shown that if G has a nonabelian Sylow p-subgroup for some prime p, then p must be p5 . We now eliminate this possibility. Claim 2. G does not have a normal Sylow p5 -subgroup. Proof. Suppose that G has a normal Sylow p5 -subgroup P . Let H be a p5 -complement for G, and let C = CH (P ). Note that C is normal in G and P C = P × C. Thus, any prime in ρ(C) must be adjacent to p5 in (G). It follows that ρ(C) ⊆ {p3 , p4 }, and hence, p1 p2 divides |H : C|. Since H /C acts faithfully on P , the primes in π(|H : C|) must lie in ρ(G/C), and so, {p1 , p2 } ⊆ ρ(G/C). Looking at the subgraphs of (G)

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that are allowed to be the degree graphs of solvable groups, we conclude that either ρ(G/C) = {p1 , p2 , p5 } or ρ(G/C) = ρ(G). Suppose ρ(G/C) = {p1 , p2 , p5 }. As we stated in the last paragraph, p3 and p4 will not divide |H : C|. Fix a nonlinear character θ ∈ Irr(P ). Then θ extends to θ × 1C in P × C. Given a character χ ∈ Irr(G|θ × 1C ), we see that p5 divides χ (1) and χ (1)/θ(1) divides |G : P C| = |H : C|. Since p1 and p2 do not divide χ (1) and the only primes that possibly divide |H : C| are p1 and p2 , we conclude that χP C = θ × 1C . This implies that χP = θ. Using Gallagher’s theorem, we have χ (1)a ∈ Irr(G) for every degree a ∈ Irr(H ) = Irr(G/P ), and so, ρ(H ) ⊆ {p3 , p4 }. Then H has a normal abelian Hall {p1 , p2 }-subgroup, and either O p3 (G) < G or O p4 (G) < G. This is a contradiction, so ρ(G/C) = ρ(G). On the other hand, G/C has a normal nonabelian Sylow subgroup P C/C. In view of Claim 1, we conclude that C = 1. Therefore, H acts faithfully on P , and P is the Fitting subgroup of G. Consider a nonlinear character θ ∈ Irr(P ) and any character χ ∈ Irr(G|θ ). We know that p5 divides both θ(1) and χ(1). It follows that p1 and p2 do not divide χ (1) nor χ (1)/θ (1). As this is true for all characters χ ∈ Irr(G|θ ), we use Theorem 12.9 of [12] to see that G/P has an abelian Hall {p1 , p2 }-subgroup. Let F be the Fitting subgroup of H , and observe that F P /P = F (G/P ). There exists a degree in cd(G) that is divisible by all the prime divisors of |F |. Suppose that p1 divides |F |. We know that p4 does not divide |F |. If p3 does not divide |F |, then F is a {p1 , p2 }-subgroup. Since H has an abelian Hall {p1 , p2 }-subgroup, it follows that F is the Hall {p1 , p2 }-subgroup of H . We have either O p3 (G) < G or O p4 (G) < G, and this is a contradiction. Thus, p3 must divide |F |, so F is a {p1 , p3 }-subgroup. We can find a character γ ∈ Irr(P F ) so that p1 p3 divides γ (1). For any character ψ ∈ Irr(G|γ ), we see that p1 p3 divides ψ(1), so p2 and p4 do not divide ψ(1) or ψ(1)/γ (1). From Theorem 12.9 of [12], H /F has an abelian Hall {p2 , p4 }-subgroup. Let M = O{p1 ,p3 } (H ), and note that F ⊆ M. Set E/M to be the Fitting subgroup of H /M. Since H /M has an abelian Hall {p2 , p4 }-subgroup, E/M is the Hall {p2 , p4 }subgroup of H /M. It is not difficult to see that ρ(P E) = ρ(G), and by Claim 1, G = P E. Because E/M is abelian, O p4 (G) < G, and this is a contradiction. We conclude that p1 does not divide |F |, and a similar argument shows that p2 will not divide |F |. This implies that F is a {p3 , p4 }-group and p1 , p2 ∈ ρ(H ). We now apply Lemma 4.2 with π = {p3 , p4 } and ρ = {p1 , p2 } to conclude that ρ(H ) = {p1 , p2 }. Suppose that X is a normal subgroup of G that is contained in P so that P /X is a chief factor for G. We know that ρ(G/X) ⊇ ρ(G/P ) = ρ(G)\{p5 } by Lemma 3 of [11], and p5 ∈ ρ(G/X) since G/X has a nonabelian Sylow p5 -subgroup. It follows that ρ(G) = ρ(G/X). By Claim 1, |G/X| = |G|, and X = 1. This implies that P is minimal normal in G. Finally, Lemma 4.3 yields a contradiction since it implies that |π| ≥ 22 − 1 = 3, and we know |π| = 2. We have shown that G has no normal nonabelian Sylow subgroups. Let F be the Fitting subgroup of G. We see that ρ(G) = π(|G : F |), and thus, ρ(G) = ρ(G/(G)) where (G) is the Frattini subgroup of G. Our next goal is to show that (G) = 1.

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We make a refinement of Claim 1. We prove that if M is normal in G with ρ(G/M) = ρ(G), then M = 1. We suppose that M > 1 for the purpose of finding a contradiction. By Claim 1, (G/M) has two connected components: π = {p1 , p2 } and ρ = {p3 , p4 , p5 }. In light of Lemma 5.1, G/M satisfies the hypotheses of Example 2.4 of [7]. Letting F /M and E/F be the Fitting subgroups of G/M and G/F , respectively, we know that |G : E| = p1 p2 and there is a prime q so that |E : F | = (q p1 p2 − 1)/(q − 1). Also, by Lemma 5.1, we see that (q p1 − 1)/(q − 1) is a prime power, so (q p1 p2 − 1)/(q p1 − 1) is divisible by at least 2 primes. Notice that p1 is adjacent to p3 , and p2 is not adjacent to p3 , so we may use Lemma 5.2 to see that p1 must be adjacent to all the primes that divide (q p1 p2 − 1)/(q p1 − 1) which is a contradiction since p1 is adjacent to only p3 in ρ and at least 2 primes in ρ divide (q p1 p2 − 1)/(q p1 − 1). Since ρ(G/(G)) = ρ(G), this implies that (G) = 1. By Hilfssatz III.4.4 of [2], there is a subgroup H so that G = H F and H ∩ F = 1. We write E for the Fitting subgroup of H . Claim 3. F is a minimal normal subgroup of G. Proof. Suppose that there is a normal subgroup N of G so that 1 < N < F . By a result of Gaschütz (Satz III.4.5 of [2]), there is a normal subgroup M of G so that F = N ×M. Since N > 1 and M > 1, we have ρ(G/N) < ρ(G) and ρ(G/M) < ρ(G). For any prime p ∈ ρ(G) \ ρ(G/N), we know that G/N has a normal abelian Sylow p-subgroup. The class of finite groups with an abelian and normal Sylow p-subgroup is a formation, so p must lie in ρ(G/M). (For information regarding formations, we suggest consulting [1].) We conclude that ρ(G) = ρ(G/N) ∪ ρ(G/M). If p ∈ ρ(G) \ ρ(G/N), then p is not in ρ(G/F ) = ρ(H ); so E must then contain the Sylow p-subgroup of H . Since p ∈ ρ(G), it follows that p divides |H |, and so, p divides |E|. Similarly, any prime in ρ(G) \ ρ(G/M) will divide |E|. Recall that cd(G) contains a degree divisible by all the prime divisors of |EF : F | = |E|. We conclude that ρ(G) \ (ρ(G/M) ∩ ρ(G/N)) lies in a complete subgraph of (G). Therefore, ρ(G) \ (ρ(G/M) ∩ ρ(G/N)) lies in one the subsets: (1) {p1 , p2 }, (2) {p3 , p4 , p5 }, (3) {p1 , p3 }, or (4) {p2 , p4 }. We begin by supposing that (1) occurs. This implies that E contains a Hall {p1 , p2 }subgroup of H . Since cd(G) has a degree divisible by all the primes dividing |E|, we see that |E| is divisible by no other primes, and E is the Hall {p1 , p2 }-subgroup of H . We can find a character χ ∈ Irr(G) with p1 p2 dividing χ (1). Let θ be an irreducible constituent of χF E . Now, χ(1)/θ(1) divides |G : F E| and χ (1) is relatively prime to |G : F E|. We determine that χF E = θ. Since p1 and p2 divide θ (1) and the only possible prime divisors of a ∈ cd(G/F E) are p3 , p4 , or p5 , we conclude via Gallagher’s theorem that cd(G/F E) = {1} and G/F E is abelian. We now have that O p3 (G) < G, a contradiction. Thus, (1) cannot occur. Suppose (2) occurs. This implies that ρ(G/N) ∩ ρ(G/M) ⊇ {p1 , p2 }. Since ρ(G/N ) and ρ(G/M) are not ρ(G), {p1 , p2 , p3 , p5 }, nor {p1 , p2 , p4 , p5 }, we may assume without loss of generality that ρ(G/N) = {p1 , p2 , p5 } and ρ(G/M) =

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{p1 , p2 , p3 , p4 }. Now, (G/N) has two connected components: {p1 , p2 } and {p5 }. From Theorem 5.5 of [7], G/N has either a central Sylow p3 -subgroup or a central p4 -subgroup. This would imply O p3 (G) < G or O p4 (G) < G which is a contradiction. Thus, (2) cannot occur. Suppose (3) occurs. This implies that ρ(G/N) ∩ ρ(G/M) ⊇ {p2 , p4 , p5 }. We know that one of ρ(G/N) or ρ(G/M) must contain p1 , say ρ(G/N). Since ρ(G/N ) < ρ(G), this implies that ρ(G/N) = {p1 , p2 , p4 , p5 }. We have previously seen that this set cannot be the degree set for a subgroup of G, so this is a contradiction. Thus, (3) cannot occur. A similar proof shows that (4) cannot occur, and the claim is proved. Let p be a prime divisor of |E|. We know via Lemma 2.10 of [9] that every degree in cd(G|F ) is divisible by all the prime divisors of |E|. There is a prime q ∈ ρ(G) that is not adjacent to p in (G). Consider a nonprincipal character λ ∈ Irr(F ). We know that every degree in cd(G|λ) is divisible by p, so CH (λ) contains a Sylow qsubgroup of H as a normal subgroup. Using Lemma 1 of [8], either (1) H ∼ = SL2 (3) or GL2 (3) or (2) H as an normal abelian subgroup that acts irreducibly on F . We know that ρ(G) = π(|H |). If (1) occurs, then π(|H |) = {2, 3} which is not allowed since |ρ(G)| = 5. Thus, (2) must occur. In case (2), H /E is abelian, and |H : E| ∈ cd(G). There is a degree in cd(G) divisible by all the prime divisors of |E|. Any prime in π(|H : E|) ∩ π(|E|) would be adjacent in (G) to all the primes in ρ(G). Since (G) has no such vertex, |H : E| and |E| are relatively prime. If p3 divides |H : E|, then O p3 (H ) < H and O p3 (G) < G. Since this was excluded, p3 does not divide |H : E| and p3 divides |E|. We have that p2 divides |H : E|. We then see that p5 divides |E|. This yields π(|H : E|) = {p1 , p2 } and π(|E|) = {p3 , p4 , p5 }. For i = 3, 4, 5, write Ei for the Sylow pi -subgroup of E. Let A be a Hall p3 complement for H , so H = E3 A and E3 ∩ A = 1. We can find a character χ ∈ Irr(G) with p1 p3 dividing χ(1). We know that p4 does not divide χ (1), so χ ∈ Irr(G|F ). It follows that χ ∈ Irr(G/F ) and χH is irreducible. Let θ be an irreducible constituent of χE3 . Now, p3 divides θ(1), so E3 is not abelian. The stabilizer of θ in H is E3 CA (θ ). Observe that E4 ⊆ CA (θ). Using the usual arguments, p2 ∈ ρ(CA (θ )), and CA (θ ) contains a Sylow p2 -subgroup P2 of H as a normal abelian subgroup. We see that P2 centralizes E4 , and P2 E = E3 E5 P2 × E4 . Since H /E is abelian, P2 E is normal in H . Let K = E3 E5 P2 , and note that F K is normal in G. Furthermore, (F K) has two connected components: {p2 } and {p3 , p5 }. Thus, F K is one of the examples in [7], but this is a contradiction as the only examples where EF /F is not abelian are Examples 2.2 and 2.3 of [7] and in both of those cases ρ(G) = {2, 3}. This is the final contradiction, and the theorem is proved.

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7. An example of Graph (3) of Figure 1 Example 7.1. We now construct a solvable group G so that (G) is the graph in Figure 1 (3). The group we construct will be similar to those groups constructed in [6]. We begin with a prime p1 with the property that 2p1 − 1 = p2 , a Mersenne prime, and (2p1 + 1)/3 is a power of a third prime p3 where the primes p1 , p2 , and p3 are different from 2 and 3. For example, we can take p1 = 5, p2 = 31, and p3 = 11. We take F to be the field of size 2p1 and E to be the field extension of F whose size p is 22p1 . Observe that E has the field automorphism σ defined by σ (α) = α 2 1 for all 2 α ∈ E. We know that F is the fixed field under σ and σ = 1. We take R to be the skew polynomial ring with coefficients in E and indeterminate X. We define Xα = α σ X to be the relation for multiplication involving an element α ∈ E and the indeterminate X. We observe that X3 R = RX3 is an ideal of R, and we have the quotient R = R/RX 3 . We let x denote the image of X in R, and we note that the Jacobson radical of R is Rx. We know that Q = 1 + Rx = {1 + αx + βx 2 | α, β ∈ E} is a group of order 4p 2 1 . It is not difficult to see that commutator [1 + α1 x + β1 x 2 , 1 + α2 x + β2 x 2 ] = 1 + (α1 α2σ + α1σ α2 )x 2 for αi , βi ∈ E. Since (α1 α2σ + α1σ α2 )σ = α1σ α2 + α1 α2σ , it is not difficult to show that Q = {1 + δx 2 | δ ∈ F}. Let C be the multiplicative group of E, so C is cyclic of order 22p1 − 1. Take G to be the Galois group of E over its base field, so G is a cyclic group of 2p1 . One can p show that C acts on Q via (1 + αx + βx 2 )c = 1 + αcx + βc2 1 +1 x 2 and G acts on R via (1 + αx + βx 2 )τ = 1 + α τ x + β τ x 2 for τ ∈ G. Also, G acts on C in the natural manner. Let C1 be the subgroup of C of order 2p1 + 1 and C2 the subgroup of order 2p1 − 1 = p2 . We see that the action of C2 on Q is a Frobenius action. On the other hand, let Z be the centralizer in Q of C1 , and we see that Z = {1 + βx 2 | β ∈ E}. By Fitting’s lemma, Q/Q = [Q, C1 ]Q /Q × Z/Q . We take P = [Q, C1 ]Q . Observe that |Q : P | = |Z : Q | = 2p1 , so P has order 23p1 . It is not difficult to show that Z(P ) = P = Q and that C and G both normalize P . We define G to be the semi-direct product P CG. An easy computation shows cd(CG) = {1, 2, p1 , 2p1 }. Note that C acts transitively on the nonprincipal characters in Irr(P /P ). If δ is a nonprincipal character in Irr(P /P ), then δ lies in an orbit of size 22p1 − 1. We may assume that the stabilizer of δ in G is P G. Since G is cyclic, we see that Irr(P G|δ) consists of extensions of δ. We conclude that cd(G|δ) = {22p1 − 1}, and thus, cd(G/P ) = {1, 2, p1 , 2p1 , 22p1 − 1}. Suppose λ is a nonprincipal character in Irr(P ). We see that C2 acts transitively on the nonprincipal characters of Irr(P ), so λ lies in an orbit of size 2p1 −1 = p2 . Without loss of generality, we may assume that the stabilizer of λ in G is P C1 G. Observe that P /P is a chief factor in P C1 . Using Problem 6.12 of [5], we see that λ must be fully-ramified with respect to P /P . Let λˆ be the unique irreducible constituent of λP . It follows that the stabilizer of λˆ in G is P C1 G. Since all the Sylow subgroups of C1 G are cyclic, λˆ will extend to P C1 G. One can show that cd(C1 G) = {1, 2, 2p1 }. By

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ˆ = Gallagher’s theorem (Corollary 6.17 of [5]), we obtain cd(P C1 G|λ) = cd(P C1 G|λ) p p +1 p +1 p p +1 p +1 1 1 1 1 1 1 ,2 p1 }, and thus, cd(G|λ) = {2 p2 , 2 p2 , 2 p1 p2 }. Since all {2 , 2 of the nonprincipal characters of Irr(P ) are conjugate, we obtain cd(G) = {1, 2, p1 , 2p1 , 22p1 − 1, 2p1 p2 , 2p1 +1 p2 , 2p1 +1 p1 p2 }. Since 22p1 − 1 = p2 (2p1 + 1) and the only prime divisors of 2p1 + 1 are 3 and p3 , we obtain the graph in Figure 4. 2

p1

HH

p2

HH

p3

3

Figure 4. Graph of Example

References [1]

K. Doerk and T. Hawkes, Finite Soluble Groups, de Gruyter Exp. Math. 4, Walter de Gruyter, Berlin 1992.

[2]

B. Huppert, Endliche Gruppen I, Springer-Verlag, Berlin 1983.

[3]

B. Huppert, Research in representation theory at Mainz (1984–1990), in Representation Theory of Finite Groups and Finite-Dimensional Algebras, ed. by G. O. Michler and C. M. Ringel (Bielefeld, May 15–17, 1991), Progr. Math. 95, Basel 1991, 17–36.

[4]

B. Huppert, Character Theory of Finite Groups, de Gruyter Exp. Math. 25, Walter de Gruyter, Berlin 1998.

[5]

I. M. Isaacs, Character Theory of Finite Groups, Academic Press, New York 1976.

[6]

I. M. Isaacs, Coprime group actions fixing all nonlinear irreducible characters, Canad. J. Math. 41 (1989), 68–82.

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M. L. Lewis, Solvable groups whose degree graphs have two connected components, J. Group Theory 4 (2001), 255–275.

[8]

M. L. Lewis, Bounding Fitting heights of character degree graphs, J. Algebra 242 (2001), 810–818.

[9]

M. L. Lewis, Derived lengths of solvable groups having five irreducible character degrees I, Algebr. Represent. Theory 4 (2001), 469–489.

[10] M. L. Lewis, A solvable group whose character degree graph has diameter 3, Proc. Amer. Math. Soc. 130 (2002), 625–630. [11] M. L. Lewis, Solvable groups with character degree graphs having 5 vertices and diameter 3, Comm. Algebra 30 (2002), 5485–5503.

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[12] O. Manz and T. R. Wolf, Representations of Solvable Groups, Cambridge University Press, Cambridge 1993. [13] P. P. Pálfy, On the character degree graph of solvable groups I: three primes, Period. Math. Hungar. 36 (1998), 61–65. [14] P. P. Pálfy, On the character degree graph of solvable groups, II: disconnected graphs, Studia Sci. Math. Hungar. 38 (2001), 339–355. [15] J. Zhang, On a problem by Huppert,Acta Scientiarum Naturalium Universitatis Pekinensis 34 (1998), 143–150. Mark L. Lewis, Department of Mathematical Sciences, Kent State University, Kent, OH 44242, U.S.A. E-mail: [email protected]

Heights of characters and defect groups Alexander Moretó

1. Introduction An important result in ordinary character theory is the Ito–Michler theorem, which asserts that a prime p does not divide the degree of any irreducible character of a finite group G if and only if G has a normal abelian Sylow p-subgroup. The famous Brauer’s height zero conjecture can be thought as the block version of this result. Given a block B with defect d the height of a character χ ∈ Irr(B) is the integer h such that χ (1)p = pa−d+h , where |G|p = pa . The height zero conjecture asserts that the height of any character χ ∈ Irr(B) is zero if and only if the defect group is abelian. The goal of this note is to discuss some possible extensions of the Ito–Michler theorem and its “block version”, the height zero conjecture. P. Fong [3] proved that all characters in a block with abelian defect group have height zero in a p-solvable group. The converse was proved by D. Gluck and T. Wolf in [4]. (However, both parts of the height zero conjecture remain open for arbitrary finite groups.) In their work in [4] Gluck and Wolf prove in fact a stronger result, namely that if e is the largest height of the characters in B, then the derived length of the defect group D cannot exceed 2e + 1 (we will keep this notation throughout this note). This generalized a result of Isaacs [8] which asserts that the derived length of a Sylow p-subgroup of a p-solvable group does not exceed 2f + 1, where pf is the largest p-part of the degrees of the irreducible characters of G (we will also maintain this notation in the remaining of this paper). The following conjecture asserts that these results should hold for arbitrary finite groups. Conjecture A. Let G be a finite group. Then the derived length of a Sylow p-subgroup is bounded in terms of f , where p f is the largest p-part of the degrees of the irreducible characters of G. More precisely, if B is a p-block of G with defect group D, then the derived length of D is bounded in terms of the largest height of the irreducible characters in B. It is clear that the first statement of Conjecture A follows from the second statement for the principal block. Isaacs bound was improved to a logarithmic bound in the case

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of solvable groups in [15] and it seems reasonable to hope that a logarithmic bound should hold in both statements for arbitrary groups. The work in [15] began as an attempt to relate p-parts of character degrees not with the derived length of a Sylow p-subgroup P but with the largest degree of the irreducible characters of P . It was conjectured in [12] that if we write b(P ) to denote the largest degree of the irreducible characters of P then logp b(P ) is bounded by a function of f , where pf is the largest p-part of the degrees of the irreducible characters of G. Now, we restate this conjecture and present its block version. Conjecture B. Let P be a Sylow p-subgroup of a finite group G and b(P ) = pb . Then b is bounded in terms of f . More precisely, if B is a p-block of G with defect group D then logp b(D) is bounded in terms of e. Again, the first statement of the conjecture follows from the second one for the principal block. It is tempting to conjecture that the bound logp b(D) ≤ 2e holds. For p ≤ 3, there are examples due to Isaacs in [8] and [12] that show that this bound would be best possible. It is clear that Conjecture B implies Conjecture A and that this bound would imply the desired logarithmic bound in Conjecture A (using Theorem D of [13], for instance). Our final conjecture relates the derived length of a Sylow p-subgroup (or of a defect group) and the number of p-parts of character degrees (or the number of heights of the characters in the block). Conjecture C. The derived length of a Sylow p-subgroup of a finite group is bounded in terms of the number of different p-parts of the degrees of the irreducible characters of the group. More precisely, the derived length of the defect group of a block B is bounded in terms of the number of different heights of the characters in B. As before, the second statement for the principal block implies the first statement and it is clear that this conjecture implies Conjecture A. It also seems tempting to conjecture that a logarithmic bound should hold but, among other things, one would need to solve the hard problem that says that the derived length of a p-group is bounded by a logarithmic function of the number of character degrees. In this note we discuss these conjectures for several important classes of finite groups: in Section 2 we consider p-solvable groups, in Section 3 the general linear groups in the defining characteristic, in Section 4 the symmetric groups and in Section 5 the sporadic groups. We prove that most of them hold for these groups. Before proceeding to do this, we consider the possible existence of reversed inequalities. There are p-groups of derived length 2 with arbitrarily many character degrees, so there is no hope that we can find any class of reversed bound in the situation of Conjectures A and C. Also, as Frobenius groups whose complement is an abelian p-group show, there is no possible reversed inequality for the first part of Conjecture B. Hence, it is perhaps surprising that something can be said about the second part of Conjecture B. It has recently been proved in [14] that if G is p-solvable, then the

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height of any character in a p-block with defect group D does not exceed 2 logp b(D) if p is odd and 5 logp b(D) if p = 2. However, as shown by an example due to G. Malle (see Section 4 of [14]) there is no hope to obtain any bound for arbitrary groups.

2. p-solvable groups As commented in the introduction, Conjecture A was proved for p-solvable groups in [8] and [4]. A logarithmic bound for the first statement for solvable groups was obtained in [15] For solvable groups, the first statement of Conjecture B was recently proved in Corollary B of [15] and that of Conjecture C in Theorem A of [13]. In the remainder of this section we discuss the block forms of these conjectures. Our first result reduces the p-solvable case of the modular form of Conjecture B to a problem in ordinary character theory. Theorem 2.1. The second statement of Conjecture B for p-solvable groups holds if the following is true: if Z is a cyclic central p -subgroup of a p-solvable group G, P ∈ Sylp (G), λ ∈ Irr(Z) and for any χ ∈ Irr(G|λ), χ (1)p ≤ p n , then logp b(P ) is bounded in terms of n. At first sight this statement looks very similar to that of the first part of Conjecture B, which was proved for solvable groups in [15]. However, it doesn’t seem possible to prove it using the methods of [15] Proof of Theorem 2.1. We begin working toward a proof of the second statement of Conjecture B for p-solvable groups. Using an argument due to Fong (see Theorems 9.14 and 10.20 of [16] or the first paragraph of the proof of Theorem A of [14]) we may assume that the defect group of B is a Sylow p-subgroup of G and that Irr(B) = Irr(G|θ) where θ ∈ Irr(Op (G)) is G-invariant. Put N = Op (G). Let (G∗ , N ∗ , θ ∗ ) be a character triple isomorphic to (G, N, θ) (see Definition 11.23 of [10]). By Theorem 5.2 of [9] and Theorem 11.28 of [10], we may assume that N ∗ ≤ Z(G∗ ) is a cyclic p -group. Furthermore, by Lemma 11.24 of [10] we know that the sets of p-parts of the degrees of the characters of G∗ that lie over θ ∗ and the set of p-parts of the degrees of G that lie over θ coincide. Also, by the definition of character triple, the Sylow p-subgroups of G/N and G∗ /N ∗ are isomorphic, and the result follows. We remark that the same proof would allow to obtain the corresponding restatement of the block form of Conjecture C for p-solvable groups. Using the results of [15] it is possible to reduce the block form of Conjecture B to a somewhat more restricted situation, but we do not think that it is worth including it here.

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3. General linear groups The goal of this section is to prove Conjectures A, B and C for GL(n, q) where q = pu . We prove the following result. Lemma 3.1. Suppose that G = GL(n, q), where q = p u and p is a prime. Then the set of p-heights of the ordinary irreducible characters of G lying in any of the p-blocks of defect bigger than 0 of G contains {ut (t − 1)/2 | t = 1, . . . , n − 1}. Proof. By [2], G has q − 1 blocks of defect zero and q − 1 blocks of full defect. Let B be a block of G of full defect whose characters lie over a character ρ ∈ Irr(Z(G)). By Proposition 3.1 of [18], the number of characters in B whose p-part of the degree is p ui is bigger than 0 whenever there is a partition µ of n such that n (µ) = i, where if µ = (a1l1 , . . . , aδlδ ) then

n (µ) =

δ aj

j =1

2

lj .

Now, it suffices to consider the partitions µt = (t, 1n−t ) for t = 1, . . . , n − 1. The result follows. Now, we can prove Conjectures A, B and C for the general linear group. Note that the bounds we obtain are good. By the structure of the blocks of GL(n, q) we may assume that B has full defect. Theorem 3.2. Let G = GL(n, q), where q is a power of p. Suppose that B is a block of full defect of G. Then the derived length of a Sylow p-subgroup P of G is bounded logarithmically in terms of the number of heights in B and also in terms of the largest height of the characters in B. Furthermore, logp b(P ) ≤ 2e. Proof. By Satz III.16.3 of [6], the derived length of P is of the order of log n. We have just proved that the number of heights is at least n − 1 and that the largest height is at least (n − 1)(n − 2)/2, so the first claim follows. In order to prove the second claim it suffices to use, for instance, the information on the character degrees of P that appears in [7].

4. Symmetric groups In this section we will find the set of heights of the characters in the blocks of the symmetric groups when p ≥ 5. As a consequence, we will see that Conjectures A, B and C hold for these groups. We need to recall some terminology. It is well-known that the irreducible characters of the symmetric group Sn are labelled by the partitions of n. If λ is a partition of n and [λ] is the Young diagram associated to λ a p-hook of

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[λ] is the part of the diagram associated to a hook of length p. The diagram (or the partition) obtained by successively removing all the rim p-hooks successively is called the p-core of λ (it is known that this does not depend on the order). The p-weight of a partition λ is the number of rim p-hooks that have to be removed before obtaining the p-core. By Nakayama’s conjecture (which was proved independently by Brauer [1] and de Robinson [19]), we know that two characters of Sn belong to the same block if and only if the associated partitions have the same p-core. Hence, it makes sense to define the weight of a block as the weight of the partition associated to any character of the block. For more details on all these concepts see [11]. Next, we need to recall some notation from [17]. We write c(p, n) to denote the number of p-core partitions of n and Fp (x) is the formal power series Fp (x) =

∞

c(p, n)x n .

n=0

We also put Fp (x)s =

∞

Cp (s, n)x n .

n=0

Now, we can prove the key result of this section. Theorem 4.1. If p ≥ 5, then the set of heights of the characters in any p-block B of Sn is the set of integers {0, 1, 2, . . . , (w − a0 − a1 − · · · − ar )/(p − 1)}, where w is the weight of B and w = a0 + a1 p + · · · + ar p r is the p-adic decomposition of w. Proof. It was proved in Corollary 3.8 of [17] that the maximal possible height of characters in B is (w − a0 − a1 − · · · − ar )/(p − 1) and that the number of characters of such height is Cp (p, w). It was proved in [5] that for every integer n there exists a p-core partition on n when p ≥ 5. Hence c(p, n) > 0. This means that for any s all the coefficients of Fp (x)s are non-zero positive integers. Hence it follows from Proposition 3.5 of [17] that the set of heights is the one asserted in the statement of the theorem (here we are using the fact that the set Ea (p, w) defined in [17] is not empty for all a with 0 ≤ a ≤ (w − a0 − a1 − · · · − ar )/(p − 1)). Corollary 4.2. Let B be a p-block of Sn for p ≥ 5 and D the defect group of B. Then logp b(D) is the maximum height of the characters in B. Proof. Write b(D) = p b . We have to show that b = (w − a0 − a1 − · · · − ar )/(p − 1), where w is the weight of B and w = a0 +a1 p+· · ·+ar p r is the p-adic decomposition of w. It is well-known that D is isomorphic to a Sylow p-subgroup of Spw . We have that pw = a0 p + a1 p2 + · · · + ar p r+1 , so D is isomorphic to the direct product of a0 copies of a Sylow p-subgroup of Sp , a1 copies of a Sylow p-subgroup of Sp2 ,…, ar copies of a Sylow p-subgroup of Spr+1 . It was proved in [7] that the largest character

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degree of a Sylow p-subgroup of Spn (for n ≥ 2) is p1+p+···+p

n−2

. We deduce that

b = a1 + a2 (1 + p) + a3 (1 + p + p 2 ) + · · · + ar (1 + p + · · · + p r−1 ) =

r

ai (p i − 1)/(p − 1) = (w − a0 − a1 − · · · − ar )/(p − 1),

i=1

as desired. Hence, we have proved that Conjecture B holds for symmetric groups when p ≥ 5. It is also clear from Theorem 4.1 that Conjectures A and C also hold in this case. Similarly, one could prove that all these conjectures also hold when p ≤ 3 for symmetric groups, but this would require some more tedious calculations using the structure of the 2-cores and 3-cores and the results of [17].

5. Sporadic groups Using the information provided by the GAP character table library, we have checked that the bound logp b(D) ≤ 2e holds for the sporadic groups. Actually, in order to check this bound we do not need to know the structure of the defect groups; it suffices to know their order. Acknowledgements. I thank G. Navarro for helpful conversations. This research was partially supported by the Spanish Ministerio de Ciencia y Tecnología, grant BFM2001-0180, the FEDER and the Basque Government.

References [1]

R. Brauer, On a conjecture by Nakayama, Trans. Roy. Soc. Canada Sect III 41 (1947), 11–19.

[2]

S. W. Dagger, On the blocks of the Chevalley groups, J. London Math. Soc. 3 (1971), 21–29.

[3]

P. Fong, On the characters of p-solvable groups, Trans. Amer. Math. Soc. 98 (1961), 263–284.

[4]

D. Gluck, T. R. Wolf, Brauer’s height conjecture for p-solvable groups, Trans. Amer. Math. Soc. 282 (1984), 137–152.

[5]

A. Granville, K. Ono, Defect zero p-blocks for finite simple groups, Trans. Amer Math. Soc. 348 (1996), 331–347.

[6]

B. Huppert, Endliche Gruppen I, Springer-Verlag, Berlin 1967.

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[7]

B. Huppert, A remark on the character-degrees of some p-groups, Arch. Math. 59 (1992), 313–318.

[8]

I. M. Isaacs, The p-parts of character degrees in p-solvable groups, Pacific J. Math 36 (1971), 677–691.

[9]

I. M. Isaacs, Partial characters of π -separable groups, in Representation Theory of Finite Groups and Finite-Dimensional Algebras, ed. by G. O. Michler and C. M. Ringel (Bielefeld, May 15–17, 1991), Progr. Math. 95, Birkhäuser, Basel 1991, 273–287.

[10] I. M. Isaacs, Character Theory of Finite Groups, Academic Press, New York 1976; reprinted by Dover, New York 1994. [11] G. James, A. Kerber, The Representation Theory of the Symmetric Group, Encyclopedia Math. Appl. 16, Addison-Wesley, Reading, MA 1981. [12] A. Moretó, Characters of p-groups and Sylow p-subgroups, in Groups St Andrews 2001 in Oxford, vol. II, London Math. Soc. Lecture Note Ser. 305, Cambridge Univ. Press, Cambridge 2003, 412–421. [13] A. Moretó, Derived length and character degrees of solvable groups, Proc. Amer. Math. Soc. 132 (2004), 1599–1604. [14] A. Moretó, G. Navarro, Heights of characters in blocks of p-solvable groups, to appear in Bull. London Math. Soc. [15] A. Moretó, T. R. Wolf, Orbit sizes, character degrees and Sylow subgroups, Adv. Math. 184 (2004), 18–36; Erratum ibid. p. 409. [16] G. Navarro, Characters and Blocks of Finite Groups, London Math. Soc. Lecture Note Ser. 250, Cambridge University Press, Cambridge 1998. [17] J. Olsson, McKay numbers and heights of characters, Math. Scand. 38 (1976), 25–42. [18] J. Olsson, K. Uno, Dade’s conjecture for general linear groups in the defining characteristic, Proc. London Math. Soc. 72 (1996), 359–384. [19] G. de B. Robinson, On a conjecture by Nakayama, Trans. Roy. Soc. Canada Sect III 41 (1947), 20–25. Alexander Moretó, Departament d’Àlgebra, Universitat de València, 46100 Burjassot, València, Spain E-mail: [email protected]

Problems on characters and Sylow subgroups Gabriel Navarro∗

I think that this is a very good opportunity to publicize some problems on characters and Sylow subgroups which, in my opinion, deserve attention. It is annoying that even for p-solvable groups, I am unable to solve most of these. Of course, it is very well possible that some of them have negative answers, but even if this is the case, that information will still be very valuable. There are hardly any new results on this note, but many questions. In fact, it all comes back to Problem 12 in the famous paper of R. Brauer [1]. If G is a finite group and P is a Sylow p-subgroup of G, Brauer asked how much the character table ct(G) of G determines the structure of P . In particular, he was interested in whether or not ct(G) determines if P is abelian. As this is answered in [7], a very natural question is the following. Question 1. Does ct(G) determine |P : P |? The invariant |P : P | is the p-part of the size of the important section NG (P )/P of G. Of course, the McKay conjecture asserts that there is a bijection Irrp (G) → Irr(NG (P )/P ), where Irrp (G) is the set of complex irreducible characters χ ∈ Irr(G) of degree not divisible by p. But this does not give a great deal of information unless, of course, the group NG (P )/P is abelian. We have a strong form of the McKay conjecture ([5], [12]) which is somehow relevant for the purpose of this talk. Conjecture 2. Let G be a finite group of order n and let p be a prime. Let e be a nonnegative integer and let σ ∈ Gal(Qn /Q) be any Galois automorphism sending e every p -root of unity ξ to ξ p . Then σ fixes the same number of characters in Irrp (G) as in Irrp (NG (P )). Using Conjecture 2, we were able to compute the exponent of P /P from ct(G) in [5]. As the reader interested in p-solvable groups will easily notice, Problem 1 reduces (in that case) to the following: Suppose that G has a normal elementary abelian p-subgroup N . Does ct(G) determine |N : N ∩ P |? ∗ Research partially supported by the Ministerio de Ciencia y Tecnología, Grant BFM2001-1667-C03-02.

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Of course, there is another variation to Brauer’s question. Question 3. Does ct(G) determine |Z(P )|? Next, we show how to find the exponent of Z(P ) from the character table of G. This is elementary. If G is a finite group and g ∈ G, let us denote by Fg = Q(χ (g)|χ ∈ Irr(G)). Also, Qn = Q(ξ ) is the cyclotomic field, where ξ ∈ C has order n. Notice, then, that Fg ⊆ Qo(g) . Surely the following must be well-known. Theorem 4. Suppose that G is a finite group and let g ∈ G. Then |NG ( g ) : CG (g)| = |Qo(g) : Fg |. Proof. Write o(g) = n and let ξ be a primitive n-th root of unity. Given an integer k coprime with n, there is a unique αk ∈ Gal(Qn ) sending ξ → ξ k and, in fact, Gal(Qn ) = {αk | (k, n) = 1}. Also, if ψ is any character of G, then ψ(g)αk = ψ(g k ). Write H = NG ( g )/CG (g). Given h ∈ H , there is an integer k coprime to n, uniquely determined modulo nZ, such that g h = g k . Now, the map f : H → Gal(Qn /Q) given by f (h) = αk is a one to one group homomorphism. Hence, it suffices to show that f (H ) = Gal(Qn /Fg ). Now, given h ∈ H , we have that g h = g k . Therefore, if χ ∈ Irr(G), we have that χ (g) = χ (g h ) = χ (g k ) = χ (g)αk , so f (H ) ⊆ Gal(Qn /Fg ). Conversely, if αt ∈ Gal(Qn /Fg ), then we have that αt fixes all χ (g) for χ ∈ Irr(G). Therefore, g and g t are G-conjugate. Hence, there exists x ∈ G such that g t = g x . Then x ∈ NG ( g ) and f (xCG (g)) = αt . The following was pointed out to me by M. Isaacs. Corollary 5. Suppose that 1 = g is a p-element of G such that |CG (g)|p = |G|p . If Fg = Q, then o(g) = p. Suppose that Fg = Q, and let e ≥ 1 be the integer such that Fg ⊆ Qpe but Fg is not contained in Qpe−1 . Then o(g) = pe . Proof. Suppose that o(g) = pf . By hypothesis and Theorem 4, we know that |Qpf : Fg | is not divisible by p. Hence, if Fg = Q, then necessarily f = 1. Suppose now that Fg = Q. We have that Fg ⊆ Qpf so 1 ≤ e ≤ f . If e < f , then p = |Qpf : Qpf −1 | divides |Qpf : Fg |, and this is not possible. Notice that if a Sylow p-subgroup of G is abelian, then Corollary 5 proves that we can find out from ct(G) the orders of the p-elements of G. Corollary 6. Suppose that G is a finite group and let P ∈ Sylp (G). Then ct(G) determines the exponent of Z(P ). Proof. We have that ct(G) determines the primes dividing the order of the elements of G (by Higman’s Theorem (8.21) of [4]) and the sizes of their centralizers. Now, notice that if g is a p-element of G, then |CG (g)|p = |G|p if and only if g ∈ Z(Q) for some Sylow p-subgroup Q of G. Now, Corollary 5 applies.

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We have already mentioned the groups P /P and NG (P )/P . So a natural problem is: Question 7. Does ct(G) determine |NG (P )|? It is a consequence of Conjecture 2, that ct(G) determines if NG (P ) = P ([12]). For solvable groups, we showed in [6], that ct(G) determines the set of primes dividing |NG (P )|, but of course, this is far from what we want. It is particulary disturbing in this problem that we do not know how to deal with what seems to be an easy case. Suppose that G has a normal p-complement K. Does ct(G) determine |CK (P )|? In this case, by using the Glauberman correspondence, it is very easy to determine the number of irreducible characters of CK (P ), but we cannot go much further than that. Perhaps, the problem is that we are very short of examples. Given a p-group P acting faithfully on p -groups K1 and K2 of the same size, when is it true that K1 P and K2 P have the same character table? Being unable to prove much, I decided to try Problem 7 in the case when we have the easiest possible Sylow subgroup: when P is cyclic. It follows from [7] that ct(G) determines if P is cyclic. But there is an elementary argument. Theorem 8. Suppose that P ∈ Sylp (G) has order p a , where a ≥ 2. Then P is cyclic if and only if there is a p-element g ∈ G such that Fg ⊆ Qpa but Fg is not contained in Qpa−1 . Proof. Suppose that P = g . Then Fg ⊆ Qpa . If Fg ⊆ Qpa−1 , then p = |Qpa : Qpa−1 | divides |Qpa : Fg |, and this is impossible by Theorem 4. Conversely, if g is a p-element such that Fg ⊆ Qpa but Fg is not contained in Qpa−1 , then necessarily g has order greater than or equal to pa . Now, using Theorems 4 and 8 and Corollary 5, we see that ct(G) determines |NG (P )| if P is cyclic. Since we know (if we believe Conjecture 2) that ct(G) determines when NG (P )/P is trivial, it is natural to ask the following. Question 9. Does ct(G) determine if NG (P )/P is abelian? About Problem 9 we can only say that we gave a positive answer in [11] for p-solvable groups. Unfortunately, our methods did not work for general groups. Question 10. Does ct(G) determine the character (1P )G ? This problem immediately leads us to study characters of groups which vanish off its p-elements. Could it be true that these characters are exactly those induced from characters of their Sylow subgroups? That would be quite an advantage toward a solution to 10. And, in fact, this is the case for p-solvable groups. (This is a not so well-known consequence of work by M. Isaacs.) Unfortunately, this is not true in full generality. If G = A6 and p = 3, then the character ψ = 2χ1 + χ2 + 3χ3 + 2χ6 + 4χ7

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vanishes off the 3-elements of G (where we are using the ATLAS notation), and ψ is not induced from any character of a Sylow 3-subgroup of G. We remark that in the case where P is cyclic, we have that the character (1P )G evaluated on the p-element x ∈ G is |NG ( x ) : P |. Therefore ct(G) determines (1P )G by Theorems 4 and 8 and Corollary 5. Notice that, using Frobenius reciprocity, Problem 10 would be solved if the following has a positive answer. Question 11. If K is a conjugacy class of p-elements of G, does ct(G) determine |K ∩ P |? We showed in [9] that this is the case, again, for p-solvable groups. Finally, we leave the character table of G, and we concentrate on the complex group algebra CG. Question 12. Does CG determine if P G? This sounds as a very elementary problem, but it seems hard to solve. At least, work in [2] and [3] show that the answer to 12 is positive when |G| is divisible by two primes only. Now, what kind of information do we have on the degrees of the irreducible characters of a group G with a normal Sylow p-subgroup P ? Given a nonnegative integer e, the following numbers seem relevant: χ (1)2 . se (G) = χ∈Irr(G),χ(1)p =p e

Lemma 13. Suppose that G/M is a p -group. Then se (G) = |G : M|se (M). Proof. Temporarily, let us write Irrpe (G) for the set of those χ ∈ Irr(G) such that χ (1)p = pe . Now, let be a complete set of representatives of the G-orbits of the action of G on Irrpe (M). It is then clear that Irrpe (G) = Irr(G|θ ) θ∈

is a disjoint union (where Irr(G|θ) is the set of irreducible characters of G lying over θ ). Now, if Tθ is the stabilizer of θ in G, by the Clifford Correspondence we have that |G : Tθ |2 ψ(1)2 se (G) = θ∈

=

ψ∈Irr(Tθ |θ )

|G : Tθ | |Tθ : M|θ (1)2

θ∈

= |G : M|

2

θ∈

|G : Tθ |θ (1)2

= |G : M|se (M), as desired.

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We see that if G has a normal Sylow p-subgroup P , then |G|p divides se (G) for all e. Although this is a very strong condition, M. Isaacs has found, with some effort, an example of a group satisfying this and not having a normal Sylow p-subgroup. On the other hand, notice that by Lemma 13, we have that s0 (G)p = |G|p if G has a normal Sylow p-subgroup. We do not know of any example where this holds and G has not a normal Sylow p-subgroup. (By work in [2], it is not possible to find counterexamples in groups of order divisible by two primes only.) However, it is very much possible that the condition s0 (G)p = |G|p is a rare numerical accident with no further significance. Having talked about s0 (G)p , we cannot resist mentioning a few facts on s0 (G)p , which come back to Problem 1. First, by using the fact that χ (1)2 , |G| = χ∈Irr(G)

notice that |G| ≡ s0 (G)

mod p2 .

In particular, we have that p divides s0 (G) if p divides |G|. (Also, s0 (G)p = p if |G|p = p.) Lemma 14. Suppose that Op pp (G) = G. Then s0 (G)p = |P : P |. Proof. By Lemma 13, we may assume that G has a normal p-complement K. Let Irr P (K) be the set of P -invariant irreducible characters of K. Then, it is easy to check that ˆ | λ ∈ Irr(G/G K)}, {θλ Irrp (G) = θ∈Irr P (K)

is a disjoint union, where θˆ ∈ Irr(G) is an extension of θ to G. Hence s0 (G) = |P : P | θ (1)2 . θ∈Irr P (K)

Now, if is a P -orbit on Irr(K) − Irr P (K), notice that p divides Hence, θ(1)2 mod p, |K| ≡ and we deduce that

2 ψ∈ ψ(1) .

θ∈Irr P (K) θ∈Irr P (K) θ(1)

2

is not divisible by p.

Unfortunately, there are many examples where the conclusion of Lemma 14 fails for general groups. Even more, A. Moretó has found an example where s0 (G)p = |G|p

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and P is not abelian (this is not so easy to find, though). However, we have the following. Theorem 15. Suppose that P ∈ Sylp (G) is abelian of order pa , and assume Brauer’s height zero conjecture. Then s0 (G) ≡ |NG (P )|

mod pa+1 .

In particular, s0 (G)p = |P |. The proof of Theorem 15, for which we require the height zero conjecture, follows from a fact on dimension of blocks (which was known to B. Külshammer). If B is a Brauer p-block of G, recall that the dimension of B is dim(B) = χ (1)2 . χ∈Irr(B)

Theorem 16. Suppose that B is a block of a finite group G with defect group D, and let b be the Brauer first main correspondent of B. Then dim(B)p ≡ (|G : NG (D)|2 )p dim(b)p

mod p.

Proof. This follows by using the formula in the third paragraph of Theorem (2.1) in [8]. Proof of Theorem 15. Suppose that b1 , . . . , bk are all the blocks of NG (P ). Hence, k

|NG (P )| =

dim(bj ).

j =1

Since P NG (P ), we have that all these blocks have defect group P . Now, let Bi = (bi )G , and by Brauer’s first main theorem, notice that {B1 , . . . , Bk } are exactly the blocks of maximal defect of G. By using the height zero conjecture, we have that Irrp (G) =

k

Irr(Bj )

j =1

is a disjoint union. Hence, s0 (G) =

k

dim(Bj ).

j =1

Now, by Theorem 16, we have that dim(Bj )p ≡ dim(bj )p

mod p

since |G : NG (P )| ≡ 1 mod p. Also, by Brauer’s Theorem (3.28) of [10], we have that dim(Bj )p = pa = dim(bj )p ,

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281

and therefore dim(Bj ) ≡ dim(bj )

mod pa+1 .

Now the proof of Theorem 15 easily follows. Acknowledgements. I thank Thomas Breuer, Marty Isaacs and Alexander Moretó for useful conversations on this paper.

References [1]

R. Brauer, Representations of Finite Groups, in Lectures on Modern Mathematics (ed. by T. L. Saaty), Vol. I, J. Wiley, New York 1963, 133–175; also in Richard Brauer, Collected Papers, Vol. II, MIT Press, Cambridge, MA 1980, 183–225.

[2]

J. Cossey, T. Hawkes, Computing the order of the nilpotent residual of a finite group from knowledge of its group algebra. Arch. Math. (Basel) 60 (1993), 115–120.

[3]

I. M. Isaacs, Recovering information about a group from its complex group algebra, Arch. Math. (Basel) 47 (1986), 293–295.

[4]

I. M. Isaacs, Character Theory of Finite Groups, Academic Press, New York 1976; reprinted by Dover, New York 1994.

[5]

I. M. Isaacs, G. Navarro, New refinements of the McKay conjecture for arbitrary finite groups, Ann. of Math. 156 (2002), 333–344.

[6]

I. M. Isaacs, G. Navarro, Character tables and Sylow normalizers, Arch. Math. (Basel) 788 (2002), 430–434.

[7]

W. Kimmerle, R. Sandling, Group theoretic and group ring theoretic determination of certain Sylow and Hall subgroups and the resolution of a question of R. Brauer, J. Algebra 171 (1995), 329–346.

[8]

G. Michler, Trace and defect of a block idempotent, J. Algebra 131 (1990), 496–501.

[9]

G. Navarro, Fusion in the character table, Proc. Amer. Math. Soc. 126 (1998), 165–166.

[10] G. Navarro, Characters of Blocks of Finite Groups, London Math. Soc. Lecture Note Ser. 250, Cambridge University Press, Cambridge 1998. [11] G. Navarro, Sylow normalizers and character tables II, Israel J. Math. 132 (2002), 277–283. [12] G. Navarro, The McKay conjecture and Galois automorphisms, to appear in Ann. of Math. Gabriel Navarro, Departament d’Àlgebra, Facultat de Matemàtiques, Universitat de València, 46100 Burjassot, València, Spain E-mail: [email protected]

Ovals in finite projective planes via the representation theory of the symmetric group Alan R. Prince

Abstract. We discuss the method, introduced by John Thompson, of studying ovals in finite projective planes via the representation theory of the symmetric group.

1. Introduction An oval configuration of involutions in the symmetric group Sn , where n is even, is a set of fixed-point-free involutions of Sn with the property that, given any product of two disjoint transpositions (a, b)(c, d) ∈ Sn , there is a unique involution in the set which contains (a, b)(c, d) in its expression as a product of disjoint transpositions. A counting argument shows that an oval configuration in Sn must contain (n − 1)(n − 3) involutions. An oval (sometimes called a hyperoval) in a projective plane of even order m is a set of m + 2 points, no three of which are collinear. If O is an oval in a projective plane of even order m, then by labelling the points of O by 1, 2, . . . , n, where n = m + 2, every point P ∈ O determines a fixed-point-free involution in Sn as follows: each line through P meeting O meets it in precisely two points i, j and the product of the disjoint transpositions (i, j ) corresponding to the lines through P meeting O is the required involution of Sn . The set of these involutions, as P ranges over the m2 − 1 points ∈ O, forms an oval configuration in Sn . Any oval configuration of involutions in Sn gives a start to constructing a projective plane of order m = n − 2 with an oval O but the lines not meeting O still need to be constructed and this may not be possible. All the known examples, except one, of oval configurations of involutions arise, via the construction described, from ovals in projective planes of even order. The exception is an oval configuration in S10 , discovered by Mathon [5]. It is easy to show that there are no oval configurations in S8 . Lam et al showed, by a long exhaustive computer search, that there are no oval configurations in S12 [3] and this formed part of the proof, by computer, of the non-existence of a projective plane of order 10 [4]. Since the known projective planes of even order all have order 2t , the known oval configurations of involutions in Sn occur only when n has the form 2t + 2 (there is

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always at least one example, if n is of this form, since the Desarguesian plane of order 2t contains an oval). The existence problem for n not of this form is open, for n ≥ 14. The author has shown that there are no oval configurations of involutions in S14 , which are invariant under conjugation by a Frobenius subgroup of order 39 [6]. The study of ovals in finite projective planes, via combinatorial properties of sets of involutions in the symmetric group, originated with Buekenhout [1] and the method can be applied also to ovals in projective planes of odd order n (which contain n + 1 points). For a survey of results on ovals in finite projective planes, see [2]. The idea of tackling the problem of the existence of oval configurations of involutions in Sn , n even, by using the representation theory of Sn was introduced by Thompson [8]. In Thompson’s theory, a certain upper-triangular matrix and its inverse play a key role and the main result of this paper is the observation that this matrix can be described precisely: a criterion is given for precisely when an entry in the matrix is nonzero and, moreover, it is shown that the nonzero entries are ±1 (Theorem 3.1). This result makes the computation of the inverse more feasible. We give the result, in Section 4, of some computations in S8 , as an illustration of the general theory. Of course, the aim is to be able to apply the theory in a case such as S14 to construct an oval configuration of involutions (and possibly a projective plane of order 12)!

2. Thompson’s theory In this section, we summarise some of the theory contained in [8]. Let Ck denote the set of all products of k disjoint transpositions in Sn , n = 2m, and let Mk be the free Zmodule with basis consisting of the elements of Ck . Denote the norm element x∈C k x y, of Mk by ν(Mk ). Let M = Mm and N = M2 . For x ∈ Cm , define λ(x) = where the sum is over all y ∈ C2 with y involved in x. Extending by linearity gives a linear map λ : M → N.(π)Let χ be the character of the action of Sn on Cm . In [8], it is shown that χ = χ , where π ranges over P , the set of all partitions of n with even parts. If T is a standard tableau associated to a partition π ∈ P , let u(T ) ∈ Cm denote the involution interchanging the entries in cells (i, 2j − 1) and (i, 2j ) throughout T . If H is the subgroup of the column group of T , consisting of all permutations fixing elements in the odd columns of T , set sg(h)u(T )h γ (T ) = h∈H

A key result of [8] is that the elements γ (T ) form a Z-basis for M. We also associate with the standard tableau T another element δ(T ) ∈ M, defined to be the sum of all the elements of Cm in the row group of T . We view M as a lattice with orthonormal basis Cm , so that we have an inner product, denoted · , · , defined on M. Using the usual ordering of partitions and the lexicographic ordering of standard tableaux for

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285

each partition, the tableaux are linearly ordered and the matrix C of inner products γ (Ti ), δ(Tj ) is upper triangular with 1’s on the diagonal. A different ordering of the standard tableau is used in [8] but the conclusion remains the same, with the lexicographic ordering, since Ti > Tj implies that there exist i, j in the same column of Ti and the same row of Tj . The coordinates, with respect to the δ(T )-basis, of the dual basis to the γ (T )-basis are given by the rows of (C −1 )t . The element ν(M) satisfies λ(ν(M)) = cν(N), where c = (n−5)(n−7) · · · 5·3·1. Let Q be the set of partitions [n], [n − 2, 2], [n − 4, 4], [n − 4, 2, 2]. The elements γ (T ), where T is a standard tableau associated with a partition of P − Q, form a Z-basis of the kernel K of λ. If ν(M) = a(T )γ (T ), where the sum is over all standard tableaux, then the row vector of coefficients a(T ) is (. . . , ι(δ(T )), . . . )C −1 , where ι(δ(T )) = ν(M), δ(T ) . For the tableaux T corresponding to π ∈ Q, the coefficients a(T ) are divisible by c. Thus, a(T )γ (T )} ν0 (M) = c−1 {ν(M) − π ∈P −Q

satisfies λ(ν0 (M)) = ν(N), ν0 (M) ∈ M, showing that ν(N) ∈ im(λ). If O is any set of fixed-point-free involutions of Sn , then the condition for O to be an oval configuration is just λ( x∈O x) = ν(N). Thus, the problem of finding an oval configuration is equivalent to that of finding an element m ∈ M, with coefficients 0, 1 when expressed in terms of the Cm -basis or equivalently of norm (n − 1)(n − 3), such that λ(m) = ν(N).

3. The upper triangular matrix C In this section, our main result is proved, giving a precise description of the upper triangular matrix C. First, we introduce some notation and terminology. If standard tableaux T and T have the property that no two elements in the same row of T are in the same column of T we write T ∼ T . If T ∼ T , then we can construct a rectangular tableau S as follows: place k in the (i, j )-cell of S if k is in the ith row of T and the j th column of T . Since T ∼ T , each cell in S is either empty or is assigned a single entry. We may assume that the cells of S form a rectangular array with the number of rows being the number of rows of T , while the number of columns is the number of columns of T . If T and T have the same shape and we ignore empty cells, then S is an ordinary tableau, not necessarily standard, of the same shape (see, for example, [7]). In the present context, however, we are also interested in the construction if T and T do not have the same shape, when the non-empty cells need not form the shape of a Ferrers diagram of a partition. The following are two examples of the construction of S.

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Alan R. Prince

1 4 5

2 6 7

3

8

1 4 1 4

2 6 2 6

3 7 3

5 8 8

7

5

1 4 6

2 5 7

3

8

1 4 1 4

2 6 2

3 7 3

6

7

5 8 8 5

If T ∼ T and S is the associated rectangular tableau, constructed as described, then there is a unique column permutation σ of T which when applied to T reorders the elements in each column so that they are in the order they appear in the columns of S. Let θ (T , T ) = sg(σ ). If S is a rectangular tableau, then a domino of S is a pair of cells in positions (i, 2j − 1), (i, 2j ) for some i, j (i.e. consecutive cells in the ith row in positions 1,2 or 3,4 or …). We say that S satisfies the domino condition if the cells in any domino are both non-empty or both empty. In the above examples, the rectangular tableau S satisfies the domino condition in the first case but not the second. With this notation and terminology in place, we can now state our main theorem. Theorem 3.1. If T ∼ T and the associated rectangular tableau S satisfies the domino condition, then γ (T ), δ(T ) = θ (T , T ). Otherwise, γ (T ), δ(T ) = 0. Proof. Suppose that T ∼ T and the associated rectangular tableau S satisfies the domino condition. Let u(S) ∈ Cm be the involution interchanging the entries in each domino. Then, u(S) is the unique involution involved in both γ (T ) and δ(T ) and appears in γ (T ) with coefficient θ(T , T ). Thus, γ (T ), δ(T ) = θ (T , T ). Suppose that T ∼ T and the associated rectangular tableau S does not satisfy the domino condition. Then, no involution is involved in both γ (T ) and δ(T ) and hence γ (T ), δ(T ) = 0. Finally, if T ∼ T , then there exist elements i, j occurring in the same row of T and the same column of T . Thus, the transposition (i, j ) is in the row group of T and the column group of T and it follows that γ (T ), δ(T ) = 0, since involutions involved in both γ (T ) and δ(T ) can be paired off, with contributions to the inner product being +1 and −1 for each pair (see also [8], where this argument is used in proving C is upper triangular). The previous theorem gives a precise description of the upper triangular matrix C. It is a sparse matrix with 1’s down the diagonal and nonzero entries ±1, making the computation of its inverse feasible, at least for n relatively small. We have seen (Section 2) that, if ν(M) = a(T )γ (T ), then the row vector of coefficients is (. . . , ι(δ(T )), . . . )C −1 , where ι(δ(T )) = ν(M), δ(T ) . Thus, these coefficients can be calculated once we know C −1 . For a tableau T corresponding to π ∈ Q, the coefficient a(T ) is divisible by c, while for a tableau T corresponding to π ∈ P − Q, γ (T ) ∈ K, the kernel of λ. Replacing

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each of the coefficients a(T ), for the tableaux T corresponding to π ∈ P − Q, by any integer multiple a(T )∗ of c, leaving the other coefficients a(T ) unchanged, and then dividing by c, we obtain an element m0 ∈ M m0 =

1 a(T )∗ γ (T ) c

which maps to ν(N). The element ν0 (M) of Section 2 is just the case where we set a ∗ (T ) = 0, for each tableau T corresponding to π ∈ P − Q. Alternatively, we could take a(T )∗ to be the multiple of c closest to a(T ), or some other close multiple. It may be possible, computationally, to exploit this idea to find elements m0 ∈ M of norm close to (n − 1)(n − 3), which map to ν(N). We illustrate this idea in the following section where some computations are given for the case S8 .

4. The case S8 It is easy to show that there are no oval configurations in S8 , as we may assume that the involutions containing (7, 8) are: (1, 2)(3, 4)(5, 6)(7, 8) (1, 3)(2, 5)(4, 6)(7, 8) (1, 4)(2, 6)(3, 5)(7, 8) (1, 5)(2, 4)(3, 6)(7, 8) (1, 6)(2, 3)(4, 5)(7, 8) and then attempts to extend this lead quickly to a contradiction, essentially because there are only 6 1-factorisations of the complete graph K6 . However, it is still instructive to consider the general theory in this particular case and we give the results of some calculations relating to S8 in this section. The partitions of 8 with all parts even are 8, 6 + 2, 4 + 4, 4 + 2 + 2, 2 + 2 + 2 + 2 and the corresponding number of standard tableaux is 1, 20, 14, 56, 14 with respective ι(δ(T ) being 105, 15, 9, 3, 1. The 14 elements γ (T ) corresponding to the partition [24 ] are a basis for K. The upper triangular matrix C is given by Theorem 3.1. Writing C as I +N, N has 266 nonzero entries and is nilpotent of index 7. From its inverse I − N + N 2 − N 3 + · · · + N 6 , the row vector of coefficients for ν(M), expressed as a linear combination of the basis vectors γ (T ), can be computed. These coefficients (ordered as described in the general theory) are:

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Alan R. Prince

105 −90 −90

15 15

15 −75

24 −36

−6

54 −27 18 3 −30 15 78 −9 40

15 45

9 −36

15

15

15

15

15

84

66

24

−12

−6

−15

−6 −6 48 0

3 3 −51 −48

−6 −78 −48 −30

−8 −48

22

−8

24 −12 24 39 −66 78 78 −30

88 −20

15 −90 −45 75

3 0 39 24 39 −162 3 −162 45 −36 −33 12 102 −9 75 63 −92

58 −57

0

15

15

15

39

−78

9

−12 −6 9 −678 −15 0 183 93

3 3 −3 −9

−102

210 −117

In this case, c = 3. Note that all the coefficients, except those in the final group, corresponding to the kernel K, are divisible by 3, as the general theory predicts. If we denote the 14 standard tableaux corresponding to [24 ] by T1 , T2 , . . . , T14 (in lexicographic order), then subtracting γ (T1 ) + γ (T2 ) + γ (T3 ) + γ (T4 ) + γ (T6 ) + γ (T7 ) + γ (T8 ) + γ (T9 ) from ν(M) gives a vector in which the coefficients have been changed to the nearest multiple of 3 and results, after dividing the coefficients by 3, and expressing the resulting vector in terms of the original basis of C4 ’s, in a vector of norm 65. However, replacing the coefficient of γ (T1 ) by 42 (instead of 39) and the coefficient of γ (T2 ) by 90 (instead of 87), results in a (0, 1, −1) solution of norm 45 in M which maps to ν(N ) and this appears to be a solution of minimum possible norm. We list this solution in full: 12 · 34 · 56 · 78 + 12 · 34 · 57 · 68 − 12 · 34 · 58 · 67 + 12 · 35 · 48 · 67 + 12 · 36 · 47 · 58 + 12 · 37 · 46 · 58 + 12 · 38 · 45 · 67 + 13 · 24 · 58 · 67 + 13 · 25 · 48 · 67 + 13 · 26 · 45 · 78 + 13 · 27 · 45 · 68 − 13 · 28 · 45 · 67 + 13 · 28 · 46 · 57 + 13 · 28 · 47 · 56 + 14 · 23 · 58 · 67 + 14 · 25 · 36 · 78 + 14 · 25 · 37 · 68 − 14 · 25 · 38 · 67 + 14 · 26 · 38 · 57 + 14 · 27 · 38 · 56 + 14 · 28 · 35 · 67 + 15 · 23 · 46 · 78 + 15 · 23 · 47 · 68 − 15 · 23 · 48 · 67 + 15 · 24 · 38 · 67 + 15 · 26 · 37 · 48 + 15 · 27 · 36 · 48 + 15 · 28 · 34 · 67 + 16 · 23 · 48 · 57 + 16 · 24 · 35 · 78 + 16 · 25 · 38 · 47 + 16 · 27 · 34 · 58 + 16 · 28 · 37 · 45 + 17 · 23 · 48 · 56 + 17 · 24 · 35 · 68 + 17 · 25 · 38 · 46 + 17 · 26 · 34 · 58 + 17 · 28 · 36 · 45 + 18 · 23 · 45 · 67 − 18 · 24 · 35 · 67 + 18 · 24 · 36 · 57 + 18 · 24 · 37 · 56 + 18 · 25 · 34 · 67 + 18 · 26 · 35 · 47 + 18 · 27 · 35 · 46

There is a nice pattern here: the involutions that occur with coefficient −1 are: (1, 2)(3, 4)(5, 8)(6, 7) (1, 3)(2, 8)(4, 5)(6, 7) (1, 4)(2, 5)(3, 8)(6, 7) (1, 5)(2, 3)(4, 8)(6, 7) (1, 8)(2, 4)(3, 5)(6, 7)

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and each gives rise to the involutions with coefficient +1 in the appropriate group above, in the following manner (illustrated for the (1, 2) group only): pair (1, 2) with each of (3, 4), (5, 8) and (6, 7) and take the two C4 ’s that involve each pair of transpositions other than (1, 2)(3, 4)(5, 8)(6, 7). In the same manner, (3, 4)(5, 8) gives rise to the involutions (1, 6)(2, 7)(3, 4)(5, 8) and (1, 7)(2, 8)(3, 4)(5, 8), which occur in the (1, 6) and (1, 7) sets above. Thus, we could write down other similar solutions. For example, we could start with the 5 involutions involving (7, 8), listed at the start of this section, taking each with coefficient −1, and constructing the involutions with coefficients +1 from them, as described. We conclude with an observation about the lattice K. The inner product matrix γ (Ti ), γ (Tj ) for the basis T1 , T2 , . . . , T14 of K is: 24 6 6 4 6 6 4 4 6 4 6 4 6 4 6 24 4 6 −6 4 6 6 4 −4 4 6 −6 −4 6 4 24 6 −6 4 6 6 4 −4 −6 −4 4 6 4 6 6 24 6 6 4 4 6 4 −4 −6 −4 −6 6 −6 −6 6 24 4 −4 −4 4 6 4 −4 −6 6 6 4 4 6 4 24 6 6 4 6 −6 −4 −6 −4 4 6 6 4 −4 6 24 4 6 −6 −4 −6 6 4 4 6 6 4 −4 6 4 24 6 −6 6 4 −4 −6 6 4 4 6 4 4 6 6 24 6 4 6 4 6 4 −4 −4 4 6 6 −6 −6 6 24 −4 4 6 −6 6 4 −6 −4 4 −6 −4 6 4 −4 24 6 −6 6 4 6 −4 −6 −4 −4 −6 4 6 4 6 24 6 −6 6 −6 4 −4 −6 −6 6 −4 4 6 −6 6 24 6 4 −4 6 −6 6 −4 4 −6 6 −6 6 −6 6 24 (Thompson [8] shows that the non-existence of an oval configuration in S8 follows from the fact that K is a doubly even lattice.) The coefficients of ν(M), not divisible by 3, are precisely the coefficients of γ (T1 ), γ (T2 ), γ (T3 ), γ (T4 ), γ (T6 ), γ (T7 ), γ (T8 ), γ (T9 ). It is interesting that the inner product of any two of these vectors is positive.

References [1] F. Buekenhout, Etude intrinseque des ovales, Rend. Mat. Appl. (5) 125 (1986), 525–534. [2] G. Korchmaros, Old and new results on ovals in finite projective planes, in Surveys in combinatorics, 1991 (ed. by A. D. Keedwell), London Math. Soc. Lecture Note Ser. 166, Cambridge Univ. Press, Cambridge 1991, 41–72. [3] C. W. H. Lam, L. Thiel, S. Swiercz and J. McKay, The non-existence of ovals in a projective plane of order 10, Discrete Math. 45 (1983), 319–321.

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[4] C. W. H. Lam, L. Thiel, S. Swiercz, The non-existence of finite projective planes of order 10, Canad. J. Math. 41 (1989), 1117–1123. [5] R. Mathon, The partial geometries pg(5,7,3), Congr. Numer. 31 (1981), 129–139. [6] A. R. Prince, Oval configurations of involutions in symmetric groups, Discrete Math. 174 (1997), 277–282. [7] D. E. Rutherford, Substitutional Analysis, Edinburgh University Publications 1, University Press, Edinburgh 1948; Hafner Publishing, New York 1968. [8] J. G. Thompson, Fixed point free involutions and finite projective planes, in Finite Simple Groups II (ed. by M. J. Collins), Academic Press, London 1980, 321–337. Alan R. Prince, School of Mathematical and Computer Sciences, Heriot-Watt University, Edinburgh EH14 4AS, United Kingdom E-mail: [email protected]

Rank invariance and automorphisms of generalized Kac–Moody superalgebras Urmie Ray

Abstract. We show that all infinite dimensional generalized Kac–Moody superalgebras are characterized by a unique Cartan matrix (up to equivalence) and describe their automorphisms. 2000 Mathematics Subject Classification: 17B

1. Introduction Automorphisms of finite dimensional simple Lie algebras are products of inner automorphisms and diagram automorphisms [10]. Unlike the finite dimensional case, for a Kac–Moody algebra, given a Cartan decomposition, the Borel subalgebra generated by the positive root spaces and the one generated by the negative root spaces are not in general conjugate under the action of inner automorphisms (or in other words of the corresponding Kac–Moody group) ([17], [15], 5.9). Hence the Chevalley involution has to be considered. The aim of this paper is to give the decomposition of an arbitrary automorphism of an infinite dimensional generalized Kac–Moody superalgebra. In order to do so, we show that Cartan subalgebras of generalized Kac–Moody superalgebras are conjugate under the action of inner automorphisms. This well-known result of Kac–Peterson [17] for Kac–Moody algebras therefore remains valid in the more general context. Furthermore, we show that all infinite dimensional generalized Kac–Moody superalgebras G are characterized by a unique Cartan matrix. This matrix is of generalized Kac–Moody type, i.e. it satisfies the usual conditions (a) − (d) (see §1 below). In other words, there are two conjugacy classes of Borel subalgebras, one corresponding respectively to positive and negative root spaces of a giving Cartan decomposition. This also generalizes a classical result of Kac–Peterson for Kac–Moody algebras [17] and implies that not only the rank of a Cartan subalgebra of an infinite dimensional generalized Kac–Moody superalgebra, but also the rank of the Kac–Moody sub-superalgebra generated by all the simple finite type (real) root spaces, are well defined.

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These results suggest that the structure of infinite dimensional generalized Kac– Moody superalgebras is closer to that of Kac–Moody algebras than to finite dimensional Lie superalgebras. Finite dimensional simple Lie superalgebras were extensively studied by many. In particular, their automorphisms were described by Serganova [22] and it is well known that in general, there are more than two conjugacy classes of Borel sub-superalgebras. Hence for finite dimensional simple Lie superalgebras, the situation differs much from the Lie algebra one. However, we show that if the finite dimensional classical Lie superalgebra [13] G is a generalized Kac–Moody superalgebra, then except when the derived Lie superalgebra G = [GG] is isomorphic to the 8 dimensional classical Lie superalgebra A(1, 0), it has a unique Cartan matrix of generalized Kac–Moody type associated to it. In other words, except in that one case, there are at most two conjugacy classes of Borel subalgebras corresponding to a Cartan decomposition of generalized Kac–Moody type, as in the infinite dimensional case. The uniqueness of the Cartan matrix also has a consequence in the Moonshine context. It follows that the moonshine module constructed by Frenkel, Lepowsky and Meurman [9] cannot be constructed as a lattice vertex algebra. That the construction of the Monster vertex algebra could not be as simple as that of the fake monster vertex algebra is known but we give a different proof of this fact.

2. Preliminary results We first need to fix some notations and to remind the reader of a few basic facts about generalized Kac–Moody superalgebras which will be needed later on. For details, see ([19], §1, 2). As usual G0¯ and G1¯ will denote the even and odd part of the Lie superalgebra G. The base field F will be either R or C. Let I be a finite set {1, . . . , n} or a countable one identified with Z+ and S be a subset of I . The set S indexes the generators contained in G1¯ . Let H be a F-vector space with a non-degenerate symmetric bilinear form ( · , ·) containing non-trivial vectors hi indexed by the set I . Set aij := (hi , hj ) and assume that (a) aij = aj i ; (b) aij ≤ 0 if i = j ; (c) (d)

2aij aii ∈ Z if aii > 0; aij aii ∈ Z if aii > 0 and

i ∈ S.

˜ = G(A, ˜ The Lie superalgebra G S, H ) is generated by the vector space H considered as an even abelian subalgebra and elements ei , fi , i ∈ I , satisfying the defining relations:

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(i) [ei , fj ] = δij hi ; (ii) [h, ej ] = (h, hj )ej , h ∈ H ; (iii) [h, fj ] = −(h, hj )fj , h ∈ H ; (iv) deg ei = deg fi = 0¯ if i ∈ S; (v) deg ei = deg fi = 1¯ if i ∈ S. Following Moody’s definition [16], we define the generalized Kac–Moody superalgebra G = G(A, S, H ) by generators and relations. Definition 2.1. The generalized Kac–Moody superalgebra G = G(A, S, H ) with generalized symmetric Cartan matrix A = (aij )i,j ∈I and Cartan subalgebra H is the Lie superalgebra generated by H , ei , and fi , i ∈ I and with defining relations (i)-(v) and with the following extra ones: (vi) 1 (ad ei ) j ∈ I;

−2aij aii

+1

ej = 0 = (ad fi )

−2aij aii

+1

fj for all i ∈ I such that aii > 0 and

(vii) [ei , ej ] = 0 = [fi , fj ] if aij = 0. The cardinality of the indexing set I is called the rank of the Lie superalgebra G. We will show that this concept is well defined. We first need to make some remarks about Cartan matrices. Remarks 2.2. If the matrix A is indecomposable and D is a diagonal matrix with positive entries, then the generalized Kac–Moody superalgebras G(A, S, H ) and G(DA, S, H ) are isomorphic. The matrix DA need not be symmetric and satisfies conditions (b) − (d) and if it is not symmetric, condition (a) has to be replaced by aij = 0 if aj i = 0. Note that one can also take a matrix D with negative entries. To include this possibility, the definition of a Cartan matrix A needs to be extended and modified in an obvious manner. Multiplying by −1 does not change anything: as long as all non-diagonal entries are of the same sign, whether they are taken to be non-positive or non-negative is only a matter of convention. Hence we need the concept of equivalent matrices. Definition 2.3. Two matrices B and C of the same size are said to be equivalent, if there exist non-singular matrices P and D with D diagonal, such that C = DP BP −1 . The generalized Kac–Moody superalgebras corresponding to equivalent Cartan matrices are isomorphic, given the same Cartan subalgebra H and indexing set S. Let G be the generalized Kac Moody superalgebra. Let Q be the root lattice form given by the generalized Cartan matrix A: with basis αi , i ∈ I and bilinear (αi , αj ) = aij . Let Q+ = { i∈I ki αi | ki ∈ Z+ }. For α ∈ Q+ (resp. −α ∈ Q+ ), let Gα be the subspace of G generated by all elements [. . . [ei3 [ei2 , ei1 ]] . . . ] (resp. [. . . [fi3 [fi2 , fi1 ]] . . . ]), where α1 + α2 + α3 + · · · = α (resp. −α). let denote 1 Note that there is a mistake in the statement of condition (vi) in ([20], §5.1).

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the set of roots, i.e. the non-zero elements α ∈ Q for which Gα = 0, + the set of positive roots, 0 the set of even roots and + 0 the set of even positive roots. The Cartan decomposition holds for the generalized Kac–Moody superalgebra G ([15], §1): G = (⊕−α∈ + Gα ) ⊕ H ⊕ (⊕α∈ + Gα ) Let N + := ⊕α∈ + Gα and N0+ = N + ∩ G0¯ . We give a simpler definition for infinite type roots that the one given in ([19], Def. 2.3) for the earlier one excluded roots of norm 0. Definition 2.4. A root α ∈ is said to be of finite type if for any root β ∈ , nα + β is a root for only finitely many integers n. Otherwise it is said to be of infinite type. So we can reformulate ([19], Lemma 2.4 and Prop. 2.6) as follows: Proposition 2.5. A root α ∈ + is of infinite type if for any root β ∈ {α, 21 α, 2α} for which (α, β) < 0, nα + β are roots for all positive integers n, unless α and β are both positive or both negative, β is of finite type and norm 0 and α − β is a root. In the case of a generalized Kac–Moody algebra of rank at least two, the former are the real roots and the latter the imaginary ones. All roots of positive norm are of finite type as in the algebra case. Remark 2.6. By (vii) the support of a root α is always connected. Let αi be a simple root of positive norm, then ri will denote the corresponding reflection when αi is even, and the reflection corresponding to 2αi , when αi is odd. Set W1 to be the group generated by these reflections. Roots of negative norm might be of finite type. The full Weyl group W is the group generated by all reflections corresponding to non-zero norm roots of finite type. Note that when the derived Lie superalgebra [G, G] has infinite dimension, or equivalently when the set of roots is infinite, the group W1 is the full Weyl group W for then G does not contain roots of finite type having negative norm ([19], Cor. 2.5). When G contains roots of the latter type, W1 is a proper subgroup of W . Lemma 2.7. For any root α ∈ + , if α has positive norm, then there exits w ∈ W1 , such that w(α) or 21 w(α) is a simple root. If α is an odd root of norm 0 and finite type, then exits w ∈ W1 such that w(α) is a simple root. For the proof of this result, see ([19], Lemma 2.2). There is an omission in the statement in [19]: as W1 contains the reflection ri corresponding to 2αi when the simple root αi is an odd root of positive norm, a positive root α of positive norm could be conjugate to 2αi rather than αi . Since the generalized Kac–Moody algebra G has no non-trivial ideal intersecting the Cartan subalgebra H trivially ([19], Lemma 2.1), the following two results are immediate.

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Corollary 2.8. The centre of the generalized Kac–Moody superalgebra G is contained in the Cartan subalgebra H . Corollary 2.9. The even part G0¯ of the generalized Kac–Moody superalgebra G is a generalized Kac–Moody algebra. Proof. The invariant, consistent, super-symmetric bilinear form induced on G by the bilinear form on H ([15], 2.2) is non-degenerate since its non-degenerate on H and any ideal intersecting H trivially is trivial. The result then follows by ([19], Theorem 3.3). We fix some more notation before proceeding to the main part of the paper. Let J = I when I = 1 and a11 = 0 and J = {i ∈ I | aii > 0} otherwise. So J is the set indexing the simple roots of finite type with non-zero norm. Let GJ be the Kac–Moody sub-superalgebra of G generated by the elements ei and fi for i ∈ J . We will refer to the cardinal of the set J as the Kac–Moody rank of G. We will show that this is a well defined notion. Let G0 be the Lie subalgebra of G generated by the Cartan subalgebra H , the elements ei , fi belonging to the even part G0¯ of the Lie superalgebra G and by the elements [ei , ei ], [fi , fi ], where ei , fi belong to the odd part G1¯ and the simple root αi is of finite type with non-zero norm. This is clearly a generalized Kac–Moody (even) subalgebra of GJ . Define U + (resp. U − ) as the Lie sub-superalgebra generated by the root spaces Gα where there exits some index i in the support of the root α ∈ + (resp. −α ∈ + ) such that the simple αi is either of infinite type or of finite type with norm 0.

3. Conjugacy of Cartan subalgebras of the Lie superalgebra G We first state an obvious fact: Lemma 3.1. G = U + ⊕ (GJ + H ) ⊕ U − as a direct sum of vector spaces. Proposition 3.2. If the generalized symmetric Cartan matrix A is indecomposable and the derived BKM superalgebra [G, G] is infinite dimensional, then (1) every element of G which act ad-locally finitely on G is contained in the Lie sub-superalgebra GJ + H ; (2) all Cartan subalgebras of G are contained in the Lie sub-superalgebra GJ + H .

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Proof. Suppose that K is a Cartan subalgebra of G not contained in GJ + H . By definition, K is an even subalgebra of G. Since K is a Cartan subalgebra, it acts on G in a locally finite way, i.e. for any element x ∈ K and v ∈ G, there is a finite dimensional subspace of G containing (ad x)n v for all n ∈ Z+ . Let x be an element of K not contained in GJ + H . Write x = u+ + v + u− , where u+ ∈ U + , u− ∈ U − and v ∈ GJ + H . We first assume that u+ = 0. So there exists positive roots β1 , . . . , βr ∈

containing either infinite type simple roots or finite type simple roots with zero norm in their support and root vectors ui ∈ Gβi such that u = ri=1 ui . Suppose that there exists a simple root αi such that for some j = 1, . . . , r, (βj , αi ) < 0. Since K is an even subalgebra, all roots βj are even. So by Corollary 2.9, the roots βj , 1 ≤ j ≤ r are of infinite type as they have non-positive norm and are even and by assumption, [G, G] being is infinite dimensional so that there are no roots of negative norm and finite type ([19], Corollary 2.5). This implies that (ad uj )n ei = 0 for all n ∈ Z+ ([19], Lemma 2.4). We claim that there are infinitely many vectors (ad x)n (ei ), n ∈ Z+ which are linearly independent. Let k ∈ {1, . . . , r} be such that βk is of maximum height with the property that βk + αi is a root. Then for each positive integer n, (ad u+ )n (ei ) is the sum of root vectors, exactly one of which corresponds to the root nβk + αi . Hence there are infinitely many linearly independent vectors (ad u+ )n (ei ), n ∈ Z+ . Indeed for all n ∈ Z+ , the component of (ad u+ )n (ei ) in the nβk + αi root space cannot be a linear combination of components of (ad u+ )r (ei ), r ≤ n − 1 since all the components of u− belong to negative root spaces and those of v belong to root spaces generated by simple finite type root spaces and to the Cartan subalgebra H , it follows that the claim holds. This contradicts the fact that K acts in a locally finite way. So (βj , αi ) ≥ 0

for all j = 1, . . . , r, i ∈ I.

(1)

Let αi be a simple root either of infinite type or of finite type and norm 0 in the support of βj . Then condition (1) forces (βj , αi ) = 0 for all j = 1, . . . , r. Hence the simple root αi has norm 0 and is orthogonal to all the simple roots in the support of each root βj , j = 1, . . . , r. Since the support of a root is connected and βj is an even root, we can deduce that βj = αi is a simple root of infinite type. As βj is simple, (βj , αk ) ≤ 0 for any simple root αk of the generalized Kac–Moody superalgebra G. Hence by (1), the simple root βj is orthogonal to all the simple roots of G. As the Cartan matrix is assumed to be indecomposable, it follows that j = 1 and I = {β1 }. So G is a Heisenberg algebra of rank 1. Therefore the Cartan subalgebra H is unique and in particular K = H , contradicting assumptions. So u+ = 0.

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If u− = 0, then similar arguments lead to a contradiction. It follows that x ∈ GJ + H and this forces the Cartan subalgebra K to be contained in GJ + H , contradicting assumptions and proving the result. Proposition 2.2 implies the first main result of this paper. We remind the reader that the derived Lie group of the Lie algebra G0 consists of the inner automorphisms. Theorem 3.3. All Cartan subalgebras of the generalized Kac–Moody superalgebra G are conjugate under the action of the derived Lie group of G0 . Proof. We first suppose that the derived Lie superalgebra [G, G] is infinite dimensional. Let H1 be a Cartan subalgebra of G. Then H1 ≤ H + GJ by Proposition 2.2. Let K be a finite subset of the indexing set J and HK = H1 ∩ (H + GK ). Write Z for the centre of the Lie superalgebra H + GK . By Corollary 2.8 applied to the generalized Kac–Moody superalgebra H + GK , Z ≤ HK ∩ H . For i ∈ K, let Vi+ (resp. Vi− ) be the HK -submodule of H + GK generated by the vector ei (resp. fi ) and φi+ (resp. φi− ) be the Lie algebra homomorphism from HK to gl(Vi+ ) (resp. gl(Vi− )). Then Z = ( i∈K ker φi+ )∩( i∈K ker φi− ). As the action of HK on G is locally finite, the modules Vi+ and Vi− are finite dimensional. Hence the subspaces HK /(ker φi+ ) and HK /(ker φi− ) are finite dimensional. So the subspace Z has finite co-dimension in HK since the set K is finite. Let L be a complement of Z in H1 . As L is finite dimensional, with obvious modifications for the superalgebra case, the Conjugacy Theorems of Peterson–Kac for Kac–Moody algebras [17] can directly be extended to L + GK . Thus, the subalgebra L of H + GJ is conjugate to a subalgebra of H under the action of some element g of the derived Lie group of G0 . Since Z is central in H + GK and contained in H , it follows that HK is conjugate to a subalgebra of H under the action of this element g. A symmetric argument applied to H shows that HK is conjugate to H under the action of G0 . Similarly H ∩ (H1 + GK ) is conjugate to H1 under the action of the derived Lie group of G0 . Hence H and H1 are conjugate under this action, proving the result. When the derived Lie superalgebra [G, G] is finite dimensional, the argument above shows that we may assume G to be finite dimensional. The result is then a well known result in the Lie algebra case; and has been proved in [22] in the context of Lie superalgebras. The next result is a also a basic one needed for the uniqueness of the Cartan matrix associated to the generalized Kac–Moody superalgebra G. Theorem 3.4. If the set of roots is infinite, then all bases of are conjugate to the base or − under the action of the Weyl group W . Proof. Note that as there are infinitely many roots, the group W1 is the full Weyl group W . Let be the base of defined in section 1 and 2 be another set of simple roots in Q with respect to which the symmetric Cartan matrix is B = (bij ). We may assume that the Cartan matrix A (with respect to ) is indecomposable. It

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is then clear that the Cartan matrix B with respect to 2 is also indecomposable. If the odd root α is of infinite type or has positive norm, then 2α is an even root. By Corollary 2.9, the Lie algebra G0¯ is a generalized Kac–Moody algebra and the set ˜ of 0 . ( ∩ 0 ) ∪ {2α | α ∈ − 0 , (α, α) = 0} can be completed to get a base ˜ Similarly the base 2 gives a base 2 of 0 . By ([15], §5.9, 11.10), we may assume ˜ 2 = . ˜ that We show that 2 ≤ + or ≤ − . If the root α ∈ 2 is not odd of norm 0, then ˜ 2 , and so α ∈ + . α (if α is even) or 2α (if α is odd) is in So suppose that the root α ∈ 2 is odd of norm 0 and that −α ∈ + . As it belongs to a base 2 , it is of finite type as there are no non-trivial ideals intersecting the Cartan subalgebra H trivially ([19], Lemma 2.1). If the Lie superalgebra G has finite growth, then it is an affine Lie superalgebra [14] since there are infinitely many roots. Affine generalized Kac–Moody Lie superalgebras have no odd root of finite type having norm 0 ([19], §2) and so there is nothing to show. Hence we may assume that the growth of the Lie superalgebra G is not finite. Then by ([19], Thm 3.3), either the matrix B or −B satisfies conditions (a) − (d). Furthermore, by ([19], Lemma 3.17), there are roots of infinite type and non-zero norm or else the indexing set I is infinite and all roots are of finite type. Suppose first that there is a positive (with respect to 2 ) root β of infinite type with non-zero norm. So (β, β) < 0. It follows that the matrix B satisfies conditions (a) − (d) for otherwise by ([19], Lemma 2.2), infinite type roots would have nonnegative norm. Also, depending on whether the root β is even or odd, β or 2β ∈ ˜ and thus β ∈ + . As the matrix B is indecomposable, we may assume that ˜ 2 = , (α, β) = 0, and so (α, β) < 0 by condition (b) satisfied by B. On the other hand, as −α ∈ + , by ([19], Lemma 2.2), there exists w ∈ W1 such that w(−α) ∈ , and so (β, −α) = (w(β), w(−α)) ≤ 0 since w(β) ∈ + . This contradiction implies that α ∈ + . Suppose next that the indexing set I is infinite and all roots are of finite type. Then, there are roots of positive norm and finite type in 2 , and so the matrix B again satisfies conditions (a) − (d). Since there are no roots of negative norm, and positive roots of positive norm with respect to 2 are sums of roots of positive norm belonging 2 different from α), to the base 2 ([19], Lemma 2.2) (and so are sums of roots in n 2 = {α = β0 , β1 , . . . , }, where for all integers n ≥ 0, i=0 βi is an odd root of norm 0, and βn is a root of positive norm and so is in + . Hence there is a minimal j integer n > 0 such that ni=0 βi ∈ + and − i=0 βi ∈ + for all j < n. Let r be the reflection with respect to the root βn . However these are not both positive roots, contradicting ([19], Lemma 2.2). This again shows that α ∈ + . Therefore, the base 2 ≤ + and so 2 = , proving the result. We are now ready to state the result on Borel sub-superalgebras and on the invariance of the Cartan matrix as it is an immediate consequence. Theorem 3.5. Suppose that the derived Lie superalgebra [G, G] is infinite dimensional. Then

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(1) there is a unique (up to equivalence) Cartan matrix associated to the generalized Kac–Moody superalgebra G; (2) all Borel subalgebras are conjugate to B + or B − under the action of the Weyl group W ; (3) the rank and Kac–Moody rank are well defined. Remark 3.6. Suppose that the Lie superalgebra [G, G] has finite dimension. It is well known that in general a simple finite dimensional classical Lie superalgebra has Cartan decompositions giving non-equivalent symmetric Cartan matrices [13]. For example, both 0 −1 0 2 −1 0 −1 2 −1 and −1 0 1 0 −1 0 0 1 −2 are Cartan matrices of A(1, 1). However, from the list of finite dimensional generalized Kac–Moody superalgebras in ([19], §2) and the list of all their simple root systems in ([13], §2.5.4), it follows that G is characterized by a unique Cartan matrix satisfying conditions (a) − (d) except when [G, G] is isomorphic to A(1, 0) when, up to equivalence, there are two such matrices associated to G: 0 −1 −1 0 (with S = I ) and

0 −1 −1 2

(with S = {1}). Furthermore, the Borel sub-superalgebras B + and B − are not W conjugate unless G is a Lie algebra or [G, G] is isomorphic to B(0, n) or to B(1, 1). Though in the latter case, there still are two conjugacy classes of Borel subalgebras (one with Cartan matrix A and the other −A). If [G, G] is isomorphic to A(1, 0), then there are three conjugacy classes of Borel subalgebras. We next give two applications of Theorems 3.4 and 3.5.

3.1. The Monster vertex algebra The crucial first step towards the proof of the Moonshine Theorem was the construction of a Z-graded moonshine module V for the Monster simple group [9]. It has a rich algebraic structure. One of its most important characteristics is that it is a vertex algebra [9], [1]. It would be useful to have a different construction of the Monster vertex algebra for its present one is very complex. In particular this may help to find an adequate integral form on V for modular moonshine questions [21], [6], [4], [5].

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Unlike the Monster vertex algebra, the Fake Monster vertex algebra has a nice simple construction. It is the lattice vertex algebra associated to the even unimodular non-degenerate Lorentzian lattice II 25,1 of rank 26 [1], [2]. Now, an important property of the moonshine module V is that it is a vertex algebra on which the Virasoro algebra acts with central charge 24, and an important aspect of a lattice vertex algebra constructed from an even non-degenerate lattice is that the Virasoro algebra acts on it with central charge equal to the rank of the lattice. Therefore, as the Monster vertex algebra is closely connected to the Leech lattice (i.e. the unique, up to isomorphism, positive definite even lattice of rank 24 not containing vectors of norm 2), it is natural to ask whether it can be constructed as a lattice vertex algebra VL , where L is a Niemeier lattice [8] (i.e. a positive definite even lattice of rank 24) as the Virasoro algebra acts on the VL with central charge 24. We remind the reader that the Virasoro algebra is the unique (up to isomorphism) central extension by a 1 dimensional centre of the Lie algebra of all derivations of the algebra of Laurent polynomials C[t, t −1 ]. It is generated by a central element c and elements Ln , n ∈ Z, satisfying 1 (m3 − m)δm+n,0 c. 12 Furthermore, there is a natural non-degenerate bilinear form on the vertex algebra VL such that the adjoint of the operator Ln is L−n for n ∈ Z. The monster Lie algebra G is constructed as follows: consider the tensor product of the moonshine module V with the lattice vertex algebra VII 1,1 . Tensor products of vertex algebras are vertex algebras. The quotient of the subalgebra V1 = {v ∈ V ⊗ VII 1,1 | L0 v = v; Ln v = 0, n ≥ 1} of V ⊗ VII 1,1 by the radical of the restriction of the bilinear form on V ⊗VII 1,1 to V1 is the monster Lie algebra. This is a generalized Kac–Moody algebra [3]. The Fake Monster Lie algebra is the generalized Kac–Moody algebra constructed in a similar way from the lattice vertex algebra VII 25,1 . Theorem 3.5 gives a different proof of the only known result that the Moonshine module does not as simple a construction as a lattice vertex algebra VL . [Lm , Ln ] = (m − n)Lm+n +

Corollary 3.7. If L is a Niemeier lattice, then VL is not isomorphic to the Moonshine module V . Proof. Suppose that the moonshine module is isomorphic to the lattice vertex algebra VL , where L is a Niemeier lattice. The direct sum L ⊕ II 1,1 is an even unimodular Lorentzian lattice of rank 26. Up to isomorphism there is a unique lattice II 25,1 with this property. It is the root lattice of the Fake Monster Lie algebra [2]. Therefore the tensor product vertex algebra V = VL ⊗ VII 1,1 is isomorphic to the Fake Monster vertex algebra. It therefore follows that the monster Lie algebra G is isomorphic to the Fake Monster Lie algebra. Hence by Theorem 3.5, they both have the same Cartan matrix (up to reordering). As this is false, it shows that these generalized Kac–Moody algebras are not isomorphic and proves the result.

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3.2. Automorphisms of generalized Kac–Moody superalgebras We finally give the decomposition of an arbitrary automorphism of the generalized Kac–Moody superalgebra G. Let ω be the Chevalley automorphism of period 4 of G [14]: i.e. fi if i ∈ S ω(ei ) = −fi otherwise, ω(fi ) = −ei for all i ∈ I , and ω(h) = −h for all h ∈ H . It is the Chevalley involution if G is a generalized Kac–Moody superalgebra. We remind the reader that a diagram automorphism θ of G satisfies θ ei = eθ˙ (i) and θfi = fθ˙ (i) for all i ∈ I , where θ˙ is a bijection on the indexing set I keeping the Cartan matrix invariant, i.e. aθi, ˙ θj ˙ = ai,j and such that θ˙ (S) = S. Consider the natural map from the root lattice Q to the Cartan subalgebra H : αi → hi , i ∈ I . In general this map is not injective: Suppose hi = hj , i = j ∈ I . Then aki = akj for all k ∈ I , i.e. the i-th and j -th columns of the Cartan matrix are equal. In particular this implies that the simple roots αi and αj are of infinite type or are both odd roots of norm 0 and finite type. When this is the case, it may be that hi = hj . However as i = j , αi = αj . Note that this map is always injective when G is a Kac–Moody superalgebra since then all the simple roots have positive norm. Set Ghi = {x ∈ G : [h, x] = (hi , h)x},

G−hi = {x ∈ G | [h, x] = −(hi , h)x}

and Ii = {j ∈ I | aki = akj for all k ∈ I }. Gαj and G−hi = j ∈Ii G−αj can have dimension greater than 1

Then Ghi = j ∈Ii when i ∈ I − J . Note that the Weyl group W clearly acts on the subspace of H generated by the elements hi , i ∈ I . We choose a set of representatives from each class of indices Ii : Iˆ = {i ∈ I | i ≤ j, for all j ∈ Ii }. The vectors ej (resp. fj ), j ∈ Ii form a basis of Ghi (resp. G−hi ). For simplicity of notation later, we choose an ordering of Ii = {j1 , . . . , jni }, where (s) (s) ni = dim Ghi and write ei = ejs and fi = fjs . For each i ∈ Iˆ, for each bijective linear map µi of the vector space Ghi ⊕ G−hi satisfying µi (Ghi ) = Ghi , µi (G−hi ) = G−hi , and [µi (ej ), µi (fj )] = µi ([ej , fj ]) for all j ∈ Ii , let µˆi be the automorphism of G given by µi (ej ), if j ∈ Ii µi (fj ), if j ∈ Ii µˆi (fj ) = µˆi (ej ) = otherwise; otherwise. ej fj

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First, a preliminary result. We write Aut(G) for the group of automorphisms of the generalized Kac–Moody algebra G, and In(G) for the derived Lie group of G0 . Lemma 3.8. (i) For fixed i ∈ Iˆ, the set Ai = {µˆ i | µi ∈ GL(Ghi )} is a group isomorphic to the general linear group GL(ni ), where ni = dim Ghi . (ii) The set D of diagram automorphisms is a group. (iii) The group of inner automorphisms In(G) is normal in G. (iv) For any i, j ∈ Iˆ, [Ai , In(G)] = 1 = [Ai , Aj ], and Ai is normal in Ai .D. (v) The Chevalley automorphism ω commutes with every element in Ai , D and In(G). Proof. (i) is obvious and (ii) and (iii) are standard results (the proof of the semisimple finite dimensional case applies). (v) and the second equality of (iv) are easy to see. (s) (s) With the above notation, for j ∈ Iˆ, for any k ∈ J , µˆ i ([ek , ej ]) = [ek , µˆ i (ej )] since dim Ghk = 1. The same holds with ek replaced by fk . Therefore µˆ i commutes with every inner automorphism. Since θ −1 µˆ i θ = µˆ θ˙ −1 i , Ai is normal in Ai .D. We are now ready to give the decomposition of an arbitrary automorphism of a generalized Kac–Moody superalgebra. Theorem 3.9. Suppose that the set of roots is infinite. Let φ be an automorphism of the generalized Kac–Moody superalgebra G. Then there is a diagram automorphism φ1 , an inner automorphism φ2 of G, and for each i ∈ Iˆ − J , automorphisms µi of G such that φ= µi ωi φ1 φ2 for some integer 0 ≤ i ≤ 3. i∈=I ˆ −J

This decomposition (in the given order) is unique. Proof. Let φ be an automorphism of the generalized Kac–Moody superalgebra G. Then φ(H ) is a Cartan subalgebra of G. So by Theorem 3.3, there exists an inner automorphism (i.e. an element of the derived Lie group G0 ) τ1 such that ψ1 = φτ1−1 fixes H . Then for all elements h ∈ H , [ψ1 (h), ψ1 (ei )] = ψ1 ([h, ei ]) = (h, h1 )ψ1 (ei )

for all i ∈ I ;

Thus ψ1 (ei ) is a root vector. Furthermore ψ1 ([G, G]) = [G, G] as its a Lie algebra homomorphism. Hence ψ1 fixes the subspace of H generated by the elements hi as < hi : i ∈ I >= H ∩ [G, G]. Clearly the ψ1 (ei ), ψ1 (fi ) and H are generators of the Lie superalgebra G satisfying relations (i)–(vii) of Definition 1.1. Also ψ1 induces a linear bijection of the root lattice Q keeping the set of roots invariant and mapping the base to another base of . Thus Theorem 3.4 implies that this isomorphism of the root lattice is given by an element w ∈ W . Now, there is an inner automorphism τ2 of the generalized Kac–Moody superalgebra G inducing the Weyl group element w (the argument remains the same as in the case of finite dimensional semisimple Lie

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algebras: see [10]). Hence ψ2 = ψ1 τ2−1 is an automorphism of G mapping the Cartan subalgebra to itself and ψ2 (ei ) ∈ Ceψˆ 2 (αi ) ,

ψ2 (fi ) ∈ Ceψˆ 2 (−αi ) ,

ψ2 (hi ) = hψˆ 2 (αi )

for all i ∈ I.

Set φ1 = τ2 τ1 . Since ω(h) = −h for all h ∈ H , the above implies that there exists some 0 ≤ l ≤ 3, and a diagram automorphism φ2 of G such that φφ1−1 ωl φ2−1 (Gα ) = Gα

for all α ∈ .

The decomposition of φ follows. Since we only need In(G)is normal in Aut(G), to prove uniqueness,

to check that (ω Ai ).D ∩ In(G) = 1. Suppose that φ = ωl µθ, where µ ∈ Ai and θ ∈ D is an inner automorphism. As φ keeps the Cartan subalgebra H invariant, there is an element w ∈ W corresponding to φ. Either w() = or −. For any root α, the height of w(α) or of −w(α) is the same as that of α. Suppose first that there are roots of infinite type. By Corollary 2.9, the even part ˜ ⊇ . By ([15], 11.13.3), G0¯ is a generalized Kac–Moody algebra and has a base + any element α ∈ Q with connected support containing an even number of indices in S satisfying (α, αi ) ≤ 0 is a positive even root of infinite type provided that it is not a ˜ with (β, β) ≤ 0. So for all α ∈ K, w(α) > 0, and multiple mβ for m ≥ 2 and β ∈ so w() = . Thus (w(α), w(αi )) ≤ 0 for all i ∈ I implies that (w(α), αj ) ≤ 0 for all j ∈ I . Hence w(α) = α by ([15], 3.12.b). This implies that w = 1. Suppose next that all roots are of finite type and that the indexing set I is countably infinite. Let w = ri1 . . . , rim be a reduced expression for w. Then as any finite subset of the indexing set I generates a finite subset of , there is a simple root αj ∈ such that (αj , αik ) = 0 for all 1 ≤ k ≤ m. Thus w(αj ) = αj and so w() = . However, w(αim ) < 0 by ([15], 3.11.b). This contradiction forces w = 1. Hence in all cases, φ acts as the identity on G. From this it easily follows that l = 0 and µ = 1 = θ. Remarks 3.10. 1. In the above definition of diagram automorphisms, we could have included the maps µi by considering the entire indexing set I rather than Iˆ. However, as the following example of the Monster Lie algebra shows the above method is more convenient. Indeed in practice, the roots are more naturally considered as elements of the Cartan subalgebra H than of an abstract root lattice Q, where the roots are linearly independent. 2. When i ∈ J , i.e. the corresponding simple root is of finite type, we do not have to include the automorphism µi . Indeed, the map given by ei → cei for non-zero constants c is accounted for by inner automorphisms. As an immediate consequence of Lemma 3.8 and Theorem 3.9, we can deduce that: Theorem 3.11. When dim([G, G]) = ∞, the automorphism group of the generalized Kac–Moody superalgebra G is the direct product of a central subgroup and of a

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semidirect product:

Aut(G) = ω × D Ai × In(G) .

Example 3.12. To construct the Monster Lie algebra 0 M, we take the Cartan subalgebra −1 ˆ H = C2 , with bilinear form given by the matrix −1 0 . The set I (as defined above) is countably infinite (S = ∅). Let h0 = (1, −1), hi = (1, i) for any integer i > 0. Then, the element (m, n) of Z 2 is a root of M with multiplicity c(mn), where for each n ∈ Z, c(n) is the coefficient of q nin the q-expansion of the modular invariant j (q) = q −1 + c(1)q + c(2)q 2 + · · · = n c(n)q n . The vectors (1, n) are the simple roots and have multiplicity c(n). Let us denote the bilinear form on H by u.v, for u, v ∈ H . As (1, n).(1, n) = −2n, the only simple root of finite type is (1, −1). Therefore the Weyl group W is cyclic of order 2 and GJ is isomorphic to sl2 . Hence the group Inn(M) of inner automorphisms of M is isomorphic to SL2 . The group of diagram automorphisms is trivial. Indeed (1, n).(1, n) = −2n. Hence the identity map is the only bijection of the set Iˆ keeping the bilinear form invariant, since under such a map the norm of the image of (1, n) must be the same as the norm of (1, n). For i > 0, the groups Ai are isomorphic to the general linear group glc(i) and the Chevalley involution has order 2. It follows that the group of automorphisms of G is isomorphic to: GLc(i) × SL2 . Aut(M) ∼ = Z2 × i>0

are representations of the Monster group. Thus the Monster The root spaces G(1,n) group is a subgroup of i>0 Ai . Remark 3.13. The finite dimensional case is classical when G is a Lie algebra and has been treated in [22] when G is a simple finite dimensional Lie superalgebra. Hence it need not be included in this paper. Acknowledgements. The author is very grateful to Richard Borcherds for useful discussions, and thanks the Centre de Recerca Matèmatica, the Institut des Hautes Etudes Scientifiques, the Max-Planck Institute and the University of California, Santa Cruz, for their financial support and hospitality.

References [1]

R. E. Borcherds, Vertex algebras, Kac–Moody algebras, and the Monster, Proc. Natl. Acad. Sci. USA 83 (1986), 3068–3071.

[2]

R. E. Borcherds, The Monster Lie algebra, Adv. Math. 83 (1990), 30–47.

[3]

R. E. Borcherds, Monstrous Moonshine and monstrous Lie superalgebras, Invent. Math. 109 (1992), 405–444.

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[4]

R. E. Borcherds, Modular Moonshine III, Duke Math. J. 93 (1998), 129–154.

[5]

R. E. Borcherds, Problems in Moonshine, in First International Congress of Chinese Mathematicians (Beijing, 1998), ed. by Lo Yang and S.-T. Yau, AMS/IP Stud. Adv. Math. 20, Amer. Math. Soc., Providence, RI 2001, 3–10.

[6]

R. E. Borcherds, A. J. E. Ryba, Modular moonshine II, Duke Math. J. 83 (1996), 435–459.

[7]

N. Bourbaki, Lie groups and Lie algebras, Part 1, Hermann, Paris 1975.

[8]

J. H. Conway, N. J. A. Sloane, Sphere Packings, Lattices and Groups, 3rd ed., Grundlehren Math. Wiss. 290, Springer-Verlag, New York, Berlin 1999.

[9]

I. B. Frenkel, J. Lepowsky, J. Meurman, Vertex operator algebras and the Monster, Pure Appl. Math. 134, Academic Press, Boston 1988.

[10] W. Fulton, J. Harris, Representation Theory, A First Course, Grad. Texts in Math. 129, Springer-Verlag, New York 1991. [11] O. Gabber, V. G. Kac, On defining relations of certain infinite-dimensional Lie algebras, Bull. Amer. Math. Soc. 5 (1981), 185–189. [12] V. G. Kac, Simple Irreducible graded Lie algebras of finite growth, Math. USSR-Izv. 2 (1968), 1271–1311. [13] V. G. Kac, Lie Superalgebras, Adv. Math. 26 (1977), 8–96. [14] V. G. Kac, Infinite-Dimensional Algebras, Dedekind’s η-Function, Classical Möbius Function and the Very Strange Formula, Adv. Math. 30 (1978), 85–136. [15] V. G. Kac, Infinite dimensional Lie algebras, 3rd ed., Cambridge University Press, Cambridge 1990. [16] R. V. Moody, A new class of Lie algebras, J. Algebra 10 (1968), 211–230. [17] V. G. Kac, D. H. Peterson, Infinite flag varieties and conjugacy theorems, Proc. Natl. Acad. Sci. USA 80 (1983), 1778–1782. [18] V. G. Kac, S. P. Wang, On automorphisms of Kac–Moody algebras and groups, Adv. Math. 92 (1992), 129–129. [19] U. Ray, A Characterization Theorem for a certain class of graded Lie superalgebras, J. Algebra 229 (2000), 405–434. [20] U. Ray, Generalized Kac–Moody algebras and some related topics, Bull. Amer. Math. Soc. 38 (2001), 1–42. [21] A. J. E. Ryba, Modular Moonshine?, in Moonshine, the Monster, and Related Topics (South Hadley, 1994), ed. by Chongying Dong and Geoffrey Mason, Contemp. Math. 193, American Mathematical Society, Providence, RI 1996, 307–336. [22] V. V. Serganova, Automorphisms of simple Lie superalgebras, Izv. Akad. Nauk. SSSR Ser. Mat. 48 (3) (1984), 585–598; Math. USSR-Izv. 24 (1985), 539–551. Urmie Ray, CRM, Apartat 50, 08193 Bellaterra, Spain E-mail: [email protected]

Bounding numbers and heights of characters in p-constrained groups Geoffrey R. Robinson

Abstract. Positive confirmation of Brauer’s k(B)-problem (which is to prove that the number of ordinary irreducible characters in a p-block B is bounded above by the order of a defect group for B ) has recently been achieved in the case that B is a block of a p-solvable group, which represents the culmination of the work of many authors, beginning with Nagao [6] and Knörr [4]. By contrast, the k(B) problem for p-constrained groups remains open at present. In this article, we present a new character-theoretic conjecture for p-constrained groups, which is, for p-solvable groups, strictly stronger than the k(B)-problem. The new conjecture seems neither to imply, nor to be implied by, the k(B)-problem for the general p-constrained group. In some ways, though, it is more precise for such a group. A key observation is Theorem 1.3, which may be of independent interest, and which clears the way for the charactertheoretic methods successfully employed in the k(GV )-problem to be used in more general situations. In particular, we prove that the new conjecture holds in fairly general circumstances, to be described more precisely later. The generalized characters constructed in Theorem 3.1 may also be of independent interest. 2000 Mathematics Subject Classification: 20C20

1. The conjecture and proof in special cases We consider a finite group G which has a non-trivial Abelian normal (p)-subgroup V such that CG (V ) is a p-group. Let S be a Sylow p-subgroup of G, and set P = S/V . For each v ∈ CV (S), set H (v) = CG (v)/V and let H (v)p denote the set of p-regular elements of H (v). We abbreviate H (1) to H . We let kd (G) denote the number of irreducible characters, χ , of G which satisfy p d χ (1)p = |G|p . We make the following: Conjecture 1.1. Under the hypotheses above, we have ∞ |H (v)p | kd (G) : v ∈ CV (S) . ≤ max p 2d |V ||H (v)| d=0

In particular,

∞ kd (G) d=0

p2d

with equality only possible if V ∈ Sylp (G).

≤

1 , |V |

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Remark 1.2. If it were the case that |H (v)p | |Hp | max : v ∈ CV (S) = , |H (v)| |H | this would allow a cleaner statement of the conjecture, but A. Maróti has recently furnished us with examples where this equality fails. We will henceforth refer to the second version of the inequality as the “weak version” of the conjecture. Note that it asserts that a necessary condition for equality is that V ∈ Sylp (G), but does not assert that this is sufficient to yield equality. Another interpretation of the conjecture (in the weak form, for ease of exposition), is that |G|p2 . χ(1)p2 ≤ |V | χ∈Irr(G)

For purposes of comparison, the reader might like to note that (after reconciling notation), it is a consequence of Brauer–Feit [1] that ∞ kd (G) d=0

p 2d

≤ 1,

and (implicitly) that ∞ kd (G) d=0

p 2d

≤

|Gp | . |G|

While our conjectured bound may at first glance look only marginally sharper, the difference is significant. For example, if we additionally assume that G/V has a normal p-complement, then (the stronger version of) our conjectured bound (suitably re-written) translates to µ(1)p2 ≤ |G|p . µ∈Irr(G)

This serves to illustrate that our conjectured bound is sometimes sharper than the bound predicted in Brauer’s k(B)-problem, which in the last example would reduce to k(G) ≤ |G|p . It also shows that the new conjecture (even in its weak form) implies the k(GV )-conjecture, and hence Brauer’s k(B) problem for p-solvable groups. Another consequence of the weak form of our conjecture would be that G/V should have at most |V | p-blocks of defect 0, or, more precisely, that G should have at most |V | irreducible characters which are V -projective (an irreducible character χ is V -projective under the current hypotheses if and only if its degree is divisible by [G : V ]p ). Notice that the weaker version of our conjecture would be sure to hold for G if we knew that k(G) ≤ |V |, since the degree of any irreducible character of G divides [G : V ] by Ito’s Theorem. However, it is not generally true that k(G) ≤ |V | under our hypotheses.

Bounding numbers and heights of characters in p-constrained groups

309

We also note that the inequality ∞ kd (G) d=0

≤

p 2d

1 |G|p

does not hold in general when G is p-solvable with Op (G) = 1, as the example G = GL(2, 3) already illustrates (for p = 2). We finally remark that the case of p = 3 and G the Hessian group of order 34 23 (recall that G is the semi-direct product on an extra-special group of order 27 and exponent 3 with SL(2, 3), where the action is the one obtained naturally from the fact that SL(2, 3) = Sp(2, 3)) shows that the inequality ∞ kd (G) d=0

p2d

≤

1 |Op (G)|

can fail even for p-solvable G (with Op (G) = 1), when Op (G) is non-Abelian. Hence the assumption that V is Abelian is crucial. Notation. If a finite group X acts on a p-group U, we will say that the Brauer character of X afforded by U is that afforded by the action of X on the direct sum of the chief factors of XU contained within U, viewed as GF(p)X-modules. We will prove that the conjecture holds in the case that the Brauer character afforded by V is real-valued, or more generally, in the case that there is an element v ∈ CV (S) such that ResG CG (v) (V ) has a submodule U on which no non-identity p-regular element acts trivially, and which affords a real-valued Brauer character. These are similar to the circumstances under which the k(GV )-conjecture was shown to hold in [9] (where S = V ), though the proof is in some ways more delicate in this case. We also recall some general facts which are useful, and will be freely used in what follows: i) If X is a finite group with a Sylow p-subgroup Q, and φ is a Brauer character (for the prime p) of X, then the class function of X which vanishes on all p-singular elements, and agrees with |Q|φ on all p-regular elements, is a generalized character. ii) X is as above, and X has a normal p-subgroup W, then there is a natural bijection between generalized characters of X which vanish on all p-singular elements and generalized characters of X/W which vanish on all p-singular elements. In this bijection, the generalized character θ of X corresponds to the generalized character θ ∗ of X/W which satisfies θ ∗ (xW ) =

θ (x) |CW (x)|

for all p-regular x ∈ X. The first assertion above is well-known. The second is frequently used in [9], and is due to W. F. Reynolds. The next auxiliary result is true in complete generality.

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Theorem 1.3. Let X be a finite group, A be an Abelian normal subgroup of X, and let µ be a complex irreducible character of X. Then for every y ∈ X, [X : CX (y)]µ(y) [A : CA (y)]µ(1) is an algebraic integer. i Proof. Let ω = exp( 2π |X| ) and let R = Z[ω]ρ be the localization at some prime ideal ρ of Z[ω]. Since ρ is arbitrary, it suffices to prove that

[X : CX (y)]µ(y) ∈ R. [A : CA (y)]µ(1) Let λ be a linear constituent of ResX A (µ), and let H = IX (λ). We may write X µ = IndH (ζ ) for some irreducible character ζ of H which lies over λ. Let S + denote the sum of the elements of S in RX, whenever S is a subset of X. A routine calculation shows that µ((y X )+ ) ζ ((y X ∩ H )+ ) = , µ(1) ζ (1) so it suffices by induction to consider the case that H = X, since y X ∩ H is a union of conjugacy classes of H, and since [A : CA (y )] = [A : CA (y)] whenever y is an X-conjugate of y. Hence, we may, and do, suppose that λ is X-stable. Let V be an RX-module affording character µ, and let σ : X → EndR (V ) be the associated representation. We view E = EndR (V ) as an interior X-algebra in the usual fashion, employing σ . We note that A acts trivially on E, since elements of A act as scalars on V by the stability of λ. Hence we see that X Tr X CX (y) (yσ ) = [A : CA (y)] Tr ACX (y) (yσ ),

while Tr X ACX (y) (yσ ) is a scalar matrix with entries in R. From this, we conclude that [X : CX (y)]µ(y) ∈ R, [A : CA (y)]µ(1) as required. Now let us return to our particular situation. If p is an odd prime, we let ε be the sign for which p ≡ ε (mod 4), and we let R √ be the ring of algebraic integers of Q[ εp]. If p = 2, we let R = Z. A key result for the cases of the conjecture we can prove, which is related to Knörr’s notion in [4], of “containing a square” is: Lemma 1.4. Let K be a finite group which acts on a p-group U in such a way that U affords a real-valued Brauer character of K. Then there is an R-combination of Brauer characters, θ, of K such that:

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311

i) |θ (x)|2 = [U : CU (x)] for each p-regular x ∈ K. √ ii) θ is integer valued if p = 2, and for p odd, θ (x) has the form ε(x)( εp)m(x) for some sign ε(x) and some integer m(x) (for each x ∈ K). Proof. It suffices to consider the case that U is elementary Abelian and completely reducible as GF(p)K-module. In that case, by the results of [9], which rely on the results of Gow [3], we can find an integer-valued Brauer character ψ of K such that either: i) ψ(x)2 = [U : CU (x)] for all p-regular x ∈ K, or ii) ψ(x)2 = [U : CU (x)] for all p-regular x ∈ K which act with determinant 1 on U , and ψ(x)2 = [U :CpU (x)] for all p-regular x ∈ K which act with determinant −1 on U . In the second case (which only occurs when p is odd), let L be the normal subgroup of K consisting of those elements which acts with determinant 1 on U . Let√χ be the √ 1+ εp class function which takes the value 1 on L and εp on K\L. Then χ = 2 1K + √ 1− εp 2 λ,

where λ is the character afforded by the determinant on U . Hence χ is an R-combination of characters of K, so we may set θ (x) = χ (x)ψ(x) for all p-regular x ∈ K to obtain the desired conclusion. Now let us turn to the proof of the conjecture in some special cases. Suppose first that CG (V ) is a p-group, and that V contains a normal subgroup U of G such that CG (U ) is a p-group and such that U affords a real-valued Brauer character. We let R be as above. Let P be a Sylow p-subgroup of G/V . We have an Rcombination of Brauer characters of G, say θ, such that |θ (x)|2 = [U : CU (x)] for all p-regular x ∈ G. We may, and do, regard θ as an R-combination of Brauer characters of G/V . Then |P |θ is an R-combination of characters of projective indecomposable modules of G/V , and the class function,ψ, of G defined by ψ(x) = |CV (x)||P |θ (x) for all p-regular x ∈ G is an R-combination of characters of projective indecomposable modules of G. Let us compute ψ, µ for an irreducible character µ of G. We know from (x)]µ(x) is an algebraic integer multiple of [V : CV (x)] for Theorem 1.3 that [G:CGµ(1) −1

) is an algebraic integer multiple of each p-regular x ∈ G. Hence [G:CG (x)]µ(x)ψ(x µ(1) −1 |P ||V |θ (x ) for each p-regular x ∈ G. Now θ (1) = ±1, while θ (x −1 ) is divisible √ by εp (or by 2 in the case p = 2) for each non-identity p-regular element x of G. Since |G|p = |P ||V |, we deduce that ψ, µ is divisible (in R) by µ(1)p but not by √ εpµ(1)p (or by 2µ(1)2 in case p = 2). In the following calculation, the reader may find it helpful to recall that there is a bijection between the classes of p-regular elements of G and of G/V , and that if x is G (x)] a p-regular element of G, then the number of conjugates of xV in H is [G:C [V :CV (x)] . We have |G| ψ, ψ = [G : CG (x)][U : CU (x)]|CV (x)|2 |P |2 ≤ x∈Gp /G

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Geoffrey R. Robinson

≤

[G : CG (x)]|V ||CV (x)||P |2 ,

x∈Gp /G

so that |H | ψ, ψ ≤

x∈Gp /G

[G : CG (x)]|V ||P |2 [V : CV (x)]

and hence

ψ, ψ ≤ Since

ψ, ψ ≥

|V ||P |2 |Hp | . |H | µ∈Irr(G)

µ(1)p2 ,

dividing both sides by |P |2 |V |2 gives the result. To be more precise, when p is odd, we see that

ψ, ψ = ψ σ , ψ σ where σ is a Galois automorphism fixing p -roots of unity and sending negative), and we note that

√ εp to its

|α σ |2 + |α|2 ≥ 2 for any non-zero α ∈ R. We have now dealt with the case that V affords real-valued Brauer character, or more generally, when it has a subgroup U which affords real-valued Brauer character and for which CG (U ) is a p-group. Suppose now that there is some non-identity element v ∈ CV (S) such that ResG CG (v) (V ) has a subgroup U with the same property with respect to CG (v). We may define a class function δ analogously for CG (v) so that |δ(x)|2 = [U : CU (x)] for all p-regular x ∈ CG (v). We may, and do, regard δ as an R-combination of Brauer characters of H (v) = CG (v)/V . Then |P |δ is an R-combination of characters of projective indecomposable modules of H (v), and the class function,ψ, of CG (v) defined by ψ(x) = |CV (x)||P |δ(x) for all p-regular x ∈ CG (v) is an R-combination of characters of projective indecomposable modules of CG (v). Now (letting ζ denote a primitive complex |v|-th root of unity), we define the Z[ζ ]combination of characters β of CG (v) via β(vx) = ψ(x) if x is p-regular, β(x) = 0 otherwise. We then consider the induced class function ψ = IndG CG (v) (β), which vanishes outside the p-section of v in G, and which has the same norm as β does. Again, we know from Theorem 1.3 that whenever µ is an irreducible charac(xv)]µ(xv) is an algebraic integer multiple of [V : CV (xv)] for each ter of G, [G:CGµ(1)

Bounding numbers and heights of characters in p-constrained groups

313

−1

) is an algebraic integer multiple p-regular x ∈ CG (v). Hence [G:CG (xv)]µ(xv)ψ((xv) µ(1) −1 of |P ||V |δ(x ) for each p-regular x ∈ CG (v). Again, δ(1) = ±1, while δ(x −1 ) is √ divisible by εp (or by 2 in the case p = 2) for each non-identity p-regular element x (v)]µ(v) of G. Furthermore, [G:CGµ(1) lies in Z[ζ ]\(1 − ζ )Z[ζ ], since µ is in the principal (and only) p-block of G, and v ∈ Z(S). We deduce that ψ, µ is divisible (in Z[ζ ]) by µ(1)p but not by (1 − ζ )µ(1)p . The computation showing that

ψ, ψ ≤

|V ||P |2 |H (v)p | |H (v)|

follows as before, using the fact that

ψ, ψ = β, β. We now need to invoke a standard argument using algebraic conjugation (essentially dating back to Burnside, and exploiting the arithmetic- geometric mean inequality in the case of totally positive algebraic integers). For every τ ∈ Gal(Q[ζ ]/Q), we have

ψ τ , ψ τ ≤

|V ||P |2 |H (v)p | , |H (v)|

and (for µ as above), we know that

ψ τ , µ ∈ Z[ζ ]\(1 − ζ )Z[ζ ]. µ(1)p Hence

τ ∈Gal(Q[ζ ]/Q)

| ψ τ , µ|2 ≥ | Gal(Q[ζ ]/Q)|µ(1)p2 .

Hence we may still conclude that µ∈Irr(G)

µ(1)p2 ≤

|V ||P |2 |H (v)p | , |H (v)|

which completes the proof that the strong form of the conjecture holds under the present hypotheses. Remark 1.5. The interested reader may care to note that it follows from the proof of the conjecture in the cases above, that if H is a subgroup of Sn and G = Z2 H, then we have χ(1)22 ≤ 23n−2 . χ∈Irr(G)

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Geoffrey R. Robinson

This already follows from the weak form of the conjecture, using the fact that a Sylow 2-subgroup of Sn has order at most 2n−1 . Notice in particular, that this yields χ(1)22 ≤ 23n−2 . χ∈Irr(H )

2. Clifford-theoretic reduction and residual obstacles The Clifford-theoretic reductions of the paper [9] and [8] carry over without any real difficulty to the present situation. We have seen that a sufficient condition for the conjecture to hold is the existence of an element v ∈ V with [G : CG (v)] coprime to p such that ResG CG (v) (V ) has a subgroup U which affords a real Brauer character for CG (v) and no non-trivial p-regular element of CG (v) acts trivially on U . In the situation of [9], the condition that [G : CG (v)] is coprime to p is automatically satisfied, but in this new setting, we need to keep track of the condition. However, this is routine. This means that one critical configuration which remains to consider after Cliffordtheoretic reductions have been performed is the following: We have a finite group L which acts faithfully and absolutely irreducibly on a GF(p)L-module U . L has a Sylow p-subgroup T , and there is no u ∈ U T for which ResL CL (u) (U ) has a faithful submodule W which affords a real-valued Brauer character. Furthermore, we may suppose that either: i) F ∗ (L) = Z(L)Oq (L), where q = p is a prime, and Oq (L) has all characteristic Abelian subgroups central. Furthermore, Q also acts absolutely irreducibly on U, or: ii) F ∗ (L) = Z(L)E, where E is a quasi-simple subgroup acting absolutely irreducibly on U . However, if E is a group of Lie type in characteristic p, for example, it is extremely likely that there will indeed be no such element u, since U T is 1-dimensional, so it seems that a different approach will be needed if such sections are involved in our group G. Our next result is relevant to this “natural characteristic” situation, and is really concerned with the case when U as above is the natural module, which may be the most difficult case. Returning to our group G, let be a set of representatives for the orbits of Irr(V ) under the action of G. For a non-trivial λ ∈ , let Sλ = IS (λ), and assume (as we may) that Sλ ∈ Sylp (IG (λ)). Let Pλ = Sλ /V . Suppose that IG (λ)/V is still p-constrained (in the strong sense that its largest normal p-subgroup contains its own centralizer). Let us count the sum of the squares of the p-parts of the degrees of those irreducible characters of G which lie over (G-conjugates of) λ. This is [P : Pλ ]2 times the corresponding sum for IG (λ). Let Vλ be the kernel of λ. Then all irreducible characters of Hλ which lie over λ contain Vλ in their kernels and V /Vλ is cyclic and central in IG (λ)/Vλ . Furthermore, λ may be identified with an IG (λ)/Vλ -stable linear character of V /Vλ .

Bounding numbers and heights of characters in p-constrained groups

We wish to evaluate

χ∈Irr(IG (λ),λ)

315

χ (1)p2 .

We may replace IG (λ) by Hλ = IG (λ)/Vλ and set Zλ = V /Vλ . By hypothesis, Hλ has only one p-block. From [7], we know that χ∈Irr(Hλ )

χ(1)p2 ≤

|Hλ |p2 |Zλ |

.

We wish to count those characters which lie over a linear character of maximal order of Zλ . We note that this contribution is independent of the algebraic conjugate chosen, and that there are at most (p − 1)|Zλ |/p such algebraic conjugates. Hence χ∈Irr(IG (λ),λ)

χ(1)p2 ≤

p|Pλ |2 . p−1

We have thus proved: Theorem 2.1. Let G, V be as above and assume further that IG (λ)/V is p-constrained for each non-trivial linear character λ of V . Then χ∈Irr(G):V ≤ker(χ)

χ(1)p2 ≤

p|P |2 (|| − 1) , p−1

where is a set of representatives for the orbits of G on Irr(V ). Remark 2.2. We remind the reader that the weak form of the conjecture asserts in this notation that we should have χ(1)p2 ≤ |P |2 |V |, χ∈Irr(G)

so that under the hypotheses of the theorem above, we are some considerable way to achieving this conjectured inequality. In fact, if G/V is itself p-constrained, the inequality holds, using the Brauer–Feit bound. More generally, the inequality holds if G/V has at most |V |− p(||−1) p−1 p-blocks. In the case that G/V is a quasi-simple group of Lie type in characteristic p and V is its natural module, there are 1 + |Z(G/V )| 1 p-blocks of G/V , and we certainly have |Z(G/V )| < (|V | − 1) 2 , so the required inequality holds comfortably.

3. Some generalized characters In this section, we note the existence of some generalized characters which are relevant to the questions considered earlier, but also have some independent interest. The relevance of these generalized characters lies in the fact that if the finite group G acts

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faithfully on a GF(p)G-module U, we are interested in constructing a virtual Brauer character θ such that |θ(y)|2 ≤ [U : CU (y)] for all p-regular y ∈ G, θ (y) ∈ π for all p-regular y ∈ G# , and θ(1) ∈ π (where π is a suitable prime ideal containing the rational prime p). Theorem 3.1. Let X be a finite group, p be a prime, n be a positive integer relatively prime to p. Let U be a GF(p)X-module, and for each g ∈ X, let mg (n, U ) denote the number of primitive n-th roots of unity (counting multiplicities) among the eigenvalues of g on U ⊗GF(p) GF(p). Define a class function ψn,U of X via ψn,U (g) = pmg (n,U ) for all g ∈ X. Then ψn,U is a generalized character of X. Proof. We note that ψn,U is constant on p -sections, so it suffices to check that ResX E (ψn,U ) is a generalized character for each Brauer elementary p -subgroup, E, of X. Hence we may, and do, assume that X is a p -group. We note that the result is visibly true when n = 1, since ψ1,U is the permutation character afforded by the action of X on the vectors of U . Let χ be the (Brauer) character of X afforded by U . Suppose that n > 1, and that prime divisors of n. Let J = {1, 2, . . . , m}. For each p1 , p2 , . . . , pm are the distinct subset I ⊆ J, set pI = i∈I pi , where the empty product is defined as 1. It is easy to check that for every g ∈ X, we have mg (n, U ) = (−1)|I | ResX pn (χ ), 1,

g

I ⊆J

so that

|I |even |CU (g

ψn,U (g) =

|I |odd |CU (g

I

n pI

)|

n pI

)|

.

Using so-called Adams operations, we know that for each integer r, the class function for which g → |CU (g r )| for all g ∈ X is a generalized character. Hence ψn,U is an integer-valued ratio of generalized characters whose values are coprime to |X|, so is a generalized character. This last fact is a result of G. Mason [5], but we include a proof of a slightly stronger version of Mason’s result for the convenience of the reader: Lemma 3.2. Let X be a finite group, γ be an algebraic integer-valued class function of X with the property that γ (g n ) = γ (g)σn whenever n is an integer coprime to |X| and σn is a Galois automorphism sending |X|-th roots of unity to their n-th powers. Suppose that there is a generalized character β of X such that β.γ is a generalized character of X and such that g∈X:γ (g)=0 β(g) is coprime to |X| (this quantity is an integer under the current hypotheses). Then γ is a generalized character. Proof. We may multiply β by various algebraic conjugates if necessary, and assume that β is integer-valued. We may find a positive integer r so that β(g)r ≡ 1 (mod |X|)

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whenever γ (g) = 0. Hence we may replace β by β r if necessary, and assume that (g) is an β(g) ≡ 1 (mod |X|) whenever γ (g) = 0. We now have that γ (g)β(g)−γ |X| algebraic integer for all g ∈ X. It follows that β.γ − γ , χ is an algebraic integer whenever χ is an irreducible character of X. Since our hypotheses together imply that β.γ , χ ∈ Z and that γ , χ ∈ Q, we deduce that γ , χ ∈ Z. Hence γ is a generalized character, as χ was arbitrary. Remark 3.3. We note that n>1 ψn,U takes the value [U : CU (x)] for all p-regular x ∈ X. We also remark that it is not really essential that U should be a module over the prime field to construct the generalized characters above, but that this is no real loss of generality in any case (first, it is no loss of generality to assume that U is irreducible. In that case, the argument works for any finite field of characteristic p and size q, as long as we replace pmg (n,U ) by q mg (n,U ) . However, the same generalized character can clearly be constructed from an irreducible GF(p)X-module).

References [1] R. Brauer and W. Feit, On the number of irreducible characters of finite groups in a given block, Proc. Nat. Acad. Sci. U.S.A. 45 (1959), 361–365. [2] D. Gluck and T. Wolf, Brauer’s height conjecture for p-solvable groups, Trans. Amer. Math. Soc. 282 (1984), 137–152. [3] R. Gow, On the number of characters in a block and the k(GV )-problem for self-dual V , J. London Math. Soc. (2), 48 (199), 441–451. [4] R. Knörr, On the number of characters in a block of a p-solvable group, Illinois J. Math. 28 (1984), 181-210. [5] G. Mason, Some applications of quasi-invertible characters, J. London Math. Soc. (2) 33 (1986), 40–48. [6] H. Nagao, On a conjecture of Brauer for p-solvable groups, J. Math. Osaka City University 13 (1962), 35–38. [7] G. R. Robinson, On the number of characters in a block, J. Algebra 138 (1991), 515–521. [8] G. R. Robinson, Further reductions for the k(GV )-problem, J. Algebra 195 (1997), 141–150. [9] G. R. Robinson and J. G. Thompson, On Brauer’s k(B) Problem, J. Algebra 184 (1996), 1143–1160. Geoffrey R. Robinson, School of Mathematics and Statistics, University of Birmingham, Birmingham B15 2TT, England E-mail: [email protected]

Classification of buildings Mark A. Ronan∗

Introduction This paper discusses the classification of buildings of all types: spherical, affine, etc. The question of constructing these buildings, which goes hand in hand with their classification, can be dealt with in several ways. The spherical buildings of classical type are constructed using classical forms on vector spaces, and other spherical buildings can be constructed using algebraic groups. More generally buildings can be constructed from groups having a BN-pair, and this is how the existence of affine buildings is established. There is also a direct construction [RT1] that I shall explain in Sect. 5. It produces the groups as by-products. Some types of buildings do not admit a classification because wild examples can be constructed. I shall deal with this too. The structure of the paper is as follows. The first section sets the scene, introducing apartments and buildings. Then in Sections 2 and 3 we deal with spherical and affine buildings – spherical ones arise largely from algebraic groups, and affine ones arise when the base field for the group has a specified valuation into the integers. Section 4 deals with twin buildings. These arise largely from Kac–Moody groups in the same way that spherical buildings arise from algebraic groups. They are like spherical buildings, but with infinite Weyl groups. Section 5 then explains some constructions of buildings, mainly using blueprints, and this leads to a short Section 6 mentioning some questions and conjectures. The assumption throughout this paper is that the reader has some familiarity with a few basic ideas in the theory of buildings, particularly regarding spherical buildings, which are the best known ones. For more details I suggest Tits’ original book [T3] and my book [R2]. The purpose of the present article is as a guide to known classification results, and some open questions. This paper originates from my lecture at the Florida conference celebrating John Thompson’s 70th birthday. I would like to thank the organizers for inviting me and thank my audience for their enthusiastic response. ∗ Partially supported by the NSA

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1. Buildings and apartments An apartment is a Coxeter complex for some Coxeter group W . This is a group generated by involutions s1 , . . . , sn along with defining relations (si sj )mij = 1 (when mij is infinity si and sj generate an infinite dihedral group). The rank of W is n. All Coxeter groups are non-trivial [T2], and moreover if J is a subset of {1, . . . , n} then the group WJ generated by the set {sj }j ∈J (along with the relations for pairs of generators) embeds as a subgroup of W . If for each j in J and k in K, each sj commutes with each sk , then the direct product WJ × WK embeds as a subgroup of W . A Coxeter group having no direct factors WJ is called irreducible. A Coxeter complex for W is a tiling of some space, such as a sphere, Euclidean space, or hyperbolic space in which the maximal dimensional tiles, called chambers, are mutually isometric. A chamber has n faces of codimension 1, called panels (n being the rank of W ), and the group acts simple-transitively on the set of chambers. Each conjugate of an si is called a reflection, and its fixed point set is called a wall. The halves of the Coxeter complex on either side of a wall are called roots. Here are some examples. Spherical apartments: These are tilings of an (n − 1)-sphere, and each chamber is an (n − 1)-dimensional simplex. The walls are reflecting hyperplane sections and the roots are hemispheres. Two chambers in an apartment are said to be opposite if they are antipodal on the sphere. Affine apartments: These are tilings of Euclidean space, and when W is irreducible each chamber is an (n − 1)-simplex. Otherwise the tiling is a direct product of lower dimensional tilings, the simplest example being a tiling of the Euclidean plane by rectangles, for which W is the direct product of two infinite dihedral groups. The walls are reflecting hyperplanes and the roots are half-spaces. Hyperbolic apartments: These are tilings of hyperbolic space, and there are many of them. Even when one restricts to the hyperbolic plane there are infinitely many. For example any triangle having dihedral angles that divide 180◦ and add up to less than 180◦ can be used to tile the hyperbolic plane, and there is a rank 3 Coxeter group in each case. Another example is provided by tiling the hyperbolic plane with regular hyperbolic n-gons (n ≥ 5) having dihedral angles of 90◦ . The chambers are n-gons, and the Coxeter group has rank n; it is generated by involutions si and relations (si si+1 )2 = 1 as i ranges over the integers modulo n. Buildings. A building of type W is a union of apartments of that type. Any two chambers lie in a common apartment, and any two apartments intersect in a convex subset (see [R2] for a formal definition). A building is called spherical, affine, etc. if its apartments are. As with Coxeter complexes, the codimension 1 faces of a chamber

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are called panels, and the number of panels on a chamber is the rank of the building. When the chambers are simplexes, the rank is one greater than the dimension. This is the case for all spherical and all irreducible affine buildings. Buildings of rank zero are vacuous, those of rank 1 are discrete sets of points, and those of rank 2 are bipartite graphs. A spherical building of rank 2 is called a generalized polygon, and a non-spherical building of rank 2 is a tree. Buildings, like their apartments, can be reducible or irreducible. Every building is a direct product of irreducibles in a unique way (see e.g. [R2, 3.10]), so the classification of buildings reduces immediately to the irreducible case. We shall first consider spherical buildings, which is where the whole subject started.

2. Spherical buildings and algebraic groups Buildings were invented by J. Tits who, back in the early 1950s, wanted to generalize classical projective geometry to the exceptional simple Lie groups (those of types E6 , E7 , E8 , F4 and G2 ), and thereby construct such groups over arbitrary fields. The group E6 had been dealt with by Dickson [D] using a cubic form in 27 dimensions, but Tits wanted to deal with all the exceptional groups in a uniform manner. In 1955, Claude Chevalley published his famous paper [Ch] on “Chevalley groups” and this was soon followed by variations on his methods by Chevalley himself, Steinberg [St] (Variations on a theme of Chevalley), Tits and others. These “twisted” Chevalley groups are now embraced by the theory of algebraic groups, where they appear as groups of rational points over fields, such as finite fields, that are not algebraically closed. While the theory of algebraic groups was moving forward, Tits developed the theory of buildings. He created axioms that could be given combinatorially, and other axioms that could be given group-theoretically in terms of a BN-pair. This was presented in his book [T3] where he goes on to classify spherical buildings. As mentioned earlier it suffices to consider the irreducible ones (those having a connected diagram), and in rank at least 3 this means one of the types An (n ≥ 3), Bn = Cn (n ≥ 3), Dn (n ≥ 4), E6 , E7 , E8 , F4 . The other types of finite Coxeter groups W , namely H3 and H4 , were eliminated by Tits [T4] using the non-existence of Moufang generalized 5-gons (see 2.4). Tits’ approach was to show that an irreducible spherical building of rank ≥ 3 must be determined by its local data, and must admit automorphisms taking one local datum to another. The chambers on a codimension 2 face of a spherical building form a rank 2 building, referred to as a rank 2 residue of , and the term local means the union of the rank 2 residues on a given chamber. To state the relevant theorem, let c be a chamber, and let E1 (c), respectively E2 (c), denote the subcomplex formed by chambers having a face of codimension 1, respectively 2, in common with c. A building is thick if each panel lies on at least three chambers.

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Theorem 2.1 (Tits [T3, 4.1.1]). Let and be thick irreducible spherical buildings of rank at least 3. If (c, d) is a pair of opposite chambers in , then any isomorphism from to is uniquely determined by its action on E1 (c) ∪ {d}. Theorem 2.2 (Tits [T3, 4.16]). Let (c, d) and (c , d ) be pairs of opposite chambers in thick spherical buildings and that are irreducible and of rank at least 3. Then any isomorphism from E2 (c) ∪ {d} to E2 (c ) ∪ {d } extends to a unique isomorphism from to . In particular the local structure determines the global structure. When and are the same building, these theorems lead directly to the existence of root groups, showing that must be Moufang. Moreover the rank 2 residues must be Moufang (the term “Moufang” was taken by Tits from its use in the special case of projective planes, which are buildings of type A2 ). In particular all buildings belonging to A2 sub-diagrams are Moufang planes, and this is the starting point for the classification. The classification in rank at least 3. In all cases except B3 and F4 there are subdiagrams of type A3 , and this implies (by a classical theorem of projective geometry – see [T3, 6.7]) that the planes must be Desarguesian, rather than just Moufang. Desarguesian means they are coordinatized by a field or skew-field. The classification of all spherical buildings of rank ≥ 3 is as follows. Type An (n ≥ 3): These all arise from projective space of dimension n over a field (possibly non-commutative), and the result is a classical theorem of projective geometry. The finite case: same as above except that finite fields are commutative. Types Dn (n ≥ 4), E6 , E7 , E8 : All these diagrams have single bonds. The existence of A3 subdiagrams implies that the planes are coordinatized by a field or skew-field, and the D4 subdiagram implies it cannot be a skew-field (see [T3, 6.12]). Thus there is one building of each of these types for each commutative field. The finite case: no change. Type Bn = Cn (n ≥ 3): These are called polar spaces by Tits. When n ≥ 4, there is an A3 sub-diagram, so all the projective planes are coordinatized by a field or skewfield K. Tits then showed that the polar spaces are embeddable in a vector space over K, possibly infinite dimensional, and arise from algebraic forms (quadratic forms, sesquilinear forms, pseudo-quadratic forms). This is hard work! When n = 3 there is no A3 subdiagram and the projective planes could be nonDesarguesian, though they must be Moufang. A Moufang non-Desarguesian plane is coordinatized by an alternative division algebra, and its automorphism group arises from a rational form of E6 (see [T1]). A B3 building having such planes arises from a rational form of E7 . These, along with the embeddable polar spaces give a complete classification of all buildings of type Bn (n ≥ 3).

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The finite case: Rank 3 is no longer a special case. All polar spaces are embeddable in a vector space whose dimension lies between 2w and 2w + 2 where w is the Witt index of the form. Type F4 : There are five types. One arises from a split group of type F4 , and exists over any field. A second one, known as 2E6 , arises from a quasi-split form of E6 . A third involves quaternionic planes and is a form of E7 . A fourth has a B3 residue with non-Desarguesian planes (arising from a rational form of E7 – see the paragraph above), and comes from a rational form of E8 . The fifth arises from a “mixed group” of type F4 over two non-perfect fields of characteristic 2. For details see [T3, Ch.10] and cf. [R2, Ch.8]. The finite case: Only the first two occur. Theorem 2.3. The above list gives a classification of all irreducible spherical buildings of rank at least 3. Rank 2 spherical buildings. For buildings of rank 2, known as generalized polygons, there is no classification unless one assumes the existence of root group automorphisms, in which case they are called Moufang (see above). Proofs published in 1979 by Tits [T5] and Weiss [W] showed that for Moufang m-gons, m must be 3, 4, 6 or 8, assuming there are at least three chambers per panel. More than twenty years later Tits and Weiss [TW] completed a classification of Moufang buildings of rank 2. Theorem 2.4 (see [TW, Ch.17]). For a Moufang m-gon, m = 3, 4, 6 or 8, and there is a complete list in each case. The finite case: When a generalized m-gon is finite and irreducible, and has at least three edges per vertex, W. Feit and G. Higman [FH] proved that m = 3, 4, 6 or 8. This remarkable result has nothing to do with the Moufang condition; there is no assumption on the existence of isomorphisms. In the cases m = 6 or 8 the only known finite examples are those arising from algebraic groups, so there may well be a classification theorem in these cases, but little progress has been made. For non-classical finite generalized m-gons, see [VM].

3. Affine buildings and discrete valuations Affine buildings first emerged from work of Iwahori and Matsumoto [IM] for Chevalley groups over p-adic fields. The general theory was then developed by Bruhat and Tits [BT], and a classification was finally given by Tits [T7]. Affine buildings arise from semisimple groups over fields endowed with integer valuations; fields such as the rational numbers Q with the p-adic valuation

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vp (p n a/b) = n, where p is a prime not dividing a or b. The completion of Q with respect to this valuation is the field of p-adic numbers Qp . A quite different example is obtained from a field k(t) of rational functions over a coefficient field k. There are many different integer valuations of k(t), one of which is given by v( fg ) = deg g − deg f , where f and g are polynomials. The completion of k(t) with respect to this valuation is the field k((t)) of Laurent series. Each field with a discrete valuation has a residue field, obtained from the ring comprising those elements having valuation ≥ 0, and factoring out by the maximal ideal of elements having valuation ≥ 1. Completion with respect to the valuation has no effect on the residue field. For example Q (with valuation vp ) and Qp have residue field Fp (the field of p elements), and k(t) and k((t)) have residue field k. Given a field K having an integer valuation with residue field k, a semisimple group over K acts on an affine building with local structure over the field k. For example if the group is SLn (K) and if x is a vertex of the affine building, then St(x) (the residue of x formed from the chambers having x as a vertex) is the spherical building for SLn (k). One immediate consequence of this is that two affine buildings, for example those for SLn (Qp ) and SLn (Fp ((t))), can be locally isomorphic but not globally isomorphic. The fields Qp and Fp ((t)) have residue field Fp , as does any totally ramified extension of Qp , so there are infinitely many isomorphism classes of affine buildings of dimension n having the same local structure. In particular Theorem 2.2 on spherical buildings fails for affine buildings. The simplest example of an affine building is the tree for SL2 (K) where K has an integer valuation with residue field k (see [S] for details). The star of each vertex in this tree is a set of edges in bijective correspondence with the rational points of the projective line over k. For example if k is Fp there are p + 1 edges per vertex. Notice that the trees for SL2 (Qp ) and SL2 (Fp ((t))) are isomorphic because if all vertices of a tree have the same valency, then this valency determines the tree up to isomorphism. However as soon as we move to SL3 the affine buildings for Qp and Fp ((t)) are not isomorphic, and in fact they differ already at the second neighborhood of a vertex. The building at infinity. Since the local-to-global theorem (Theorem 2.2) for spherical buildings fails for affine buildings, a classification cannot be given by local data. However, an affine building of dimension n has a spherical building of dimension n−1 at infinity, and for n ≥ 3 it is possible to use the classification of spherical buildings. The simplest example of an affine building is a tree; its building at infinity is its set of ends. The building at infinity arose in the work of Bruhat and Tits [BT], and was then used by Tits [T7] in the following theorems. Theorem 3.1 ([T7, Sect. 14]). Every irreducible affine building of dimension at least 3 has at infinity a spherical building of dimension at least 2 whose base field is complete with respect to an integer valuation. This valuation uniquely determines the affine building.

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Corollary 3.2 ([T7, Sect. 15]). The locally finite affine buildings that are irreducible and of dimension d ≥ 3 are the affine buildings of simple algebraic groups of rank d over locally compact fields that are non-discrete and totally discontinuous. Affine buildings of dimension 2 (rank 3) do not admit a classification because of a free construction [R1] – see Sect. 5 for more details. Non-complete fields. In the theorem above the field is complete with respect to the discrete valuation, and it is reasonable to ask what happens in the case of fields K, such as Q and k(t), that are not complete. The difference occurs with the system of apartments. Each affine apartment has a spherical apartment at infinity, and all apartments of the spherical building at infinity arise from affine apartments in this way. In the complete case all affine apartments are used, but in the non-complete case a certain subset of apartments, called a system of apartments is used. Going to infinity only in these apartments gives the spherical building for the group over K rather than the complete spherical building for the group over the complete field K. For example the SL2 affine building is a tree. Any two ends of this tree span an apartment, and the complete building at infinity is the set of all ends – it is in bijective . For the lesser field correspondence with the projective line over the complete field K K one has the same tree but a lesser set of ends, in bijective correspondence to the points of the projective line over K. Any two of these ends span an apartment in the system of apartments for SL2 (K). Tits’ work [T7] deals extensively with systems of apartments, and the appropriate definitions can be found in Section 1 of that paper.

4. Twin buildings and Kac–Moody groups Kac–Moody Lie algebras generalize semisimple Lie algebras, allowing infinite Weyl groups W , and they lead to Kac–Moody groups in the same way that semi-simple Lie algebras lead to Chevalley groups. When W is a finite group (the spherical case) the algebra is finite dimensional (and semisimple); when W is infinite the algebra is infinite dimensional. The buildings have the same rank as W and are finite dimensional. The first method of attaching groups to these algebras was given by Moody and Teo [MT]. Years later, Tits wrote a detailed study of Kac–Moody groups [T8] generating them using root groups Uα where α ranges over the set of roots of the Coxeter complex for W (recall that a root is half of a Coxeter complex, on either side of a reflecting hyperplane). These root groups, each isomorphic to the additive group of the base field k, generate a group G. If N denotes the subgroup of G normalizing the set of root groups, and T denotes the subgroup normalizing each individual root group, then as in the case of Chevalley groups, N/T is isomorphic to the Coxeter group W . The subgroup T , called a torus, is isomorphic to a direct product of copies of the multiplicative group of k.

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If c denotes a chamber of the Coxeter complex, the positive roots are those containing c, and the negative roots are those not containing c. The positive and negative root groups generate groups U+ and U− , which along with the torus generate subgroups B+ and B− respectively. The pairs (B+ , N) and (B− , N) are BN-pairs for G. A very important difference between the spherical case and the non-spherical case is that B+ and B− are conjugate in the spherical case, but not otherwise. The reason is simple. The element conjugating B+ to B− in a Chevalley group is the longest word in W , but when W is infinite there is no longest word. The BN -pairs (B+ , N) and (B− , N) generate two buildings + and − whose rank is the rank of W . There is a W -valued “codistance” between chambers of + and − , providing (+ , − ) with the structure of a twin building – see [T9] for the twin building axioms. When W is finite (the spherical case) then + = − , and the codistance is the usual W -valued distance pre-multiplied by the longest word in W . A spherical building is twinned with itself. Two chambers, one in one building, one in the other, are said to be opposite one another when the codistance between them is the identity element of W (and two vertices are opposite if they belong to opposite chambers and have the same type). Each twin building (+ , − ) contains twin apartments, which are pairs (A+ , A− ) of apartments one in each building, such that each chamber of A+ is opposite a unique chamber of A− . In the non-spherical case a twin apartment can be thought of as having its two components on opposite sides of a sphere, so that a chamber in one component of a twin apartment is opposite its antipode in the other component. Example. The group SLn over a ring of Laurent polynomials k[t, t −1 ] is a Kac– Moody group of affine type and acts naturally on a twin building (+ , − ). Both + and − are buildings for SLn (k(t)) but using different valuations v+ and v− , where v+ (t) = 1 and v− (t −1 ) = 1. The stabilizers of opposite vertices x+ and x− are isomorphic to SLn (k[t]) and SLn (k[t −1 ]) respectively, the stabilizer of the pair (x+ , x− ) being SLn (k). When k is a finite field, this implies that SLn (k[t]) acts discretely on − , and similarly with SLn (k[t −1 ]) acting on + . This makes affine twin buildings useful in the study of arithmetic groups over a ring of Laurent polynomials (see [A]). 2-spherical twin buildings. When all rank 2 residues have bounded diameter, or in other words are spherical (this eliminates twin tree residues), the twin building is called 2-spherical. In this case Tits’ methods for spherical buildings can be used to prove that the rank 2 residues must be Moufang. A complete proof is given in [R3, Theorem 3]. Theorem 4.1 ([R3]). Let be a thick 2-spherical twin building with no direct factors of rank 2. Then every residue of type {i, j } for which mij ≥ 3 is Moufang. In particular mij is 2, 3, 4, 6 or 8 for all {i, j }.

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Moreover one can show that the local structure usually determines the global structure, as in 2.2. Recall that E1 (c), respectively E2 (c), means the union of the rank 1, respectively rank 2, residues containing the chamber c. The following theorems are generalizations of 2.1 and 2.2 for spherical buildings. An isometry is a map preserving distances and codistances, and a building is thick if each panel is on at least three chambers. Theorem 4.2 ([R3]). Let and be 2-spherical twin buildings of rank at least 3 that are thick (at least 3 chambers per panel). If (c, d) is a pair of opposite chambers in , then any isomorphism from to is uniquely determined by its action on E1 (c) ∪ {d}. Theorem 4.3 ([MR]). Let (c, d) and (c , d ) be pairs of opposite chambers in thick 2spherical twin buildings and respectively, and assume has no residues of type Sp4 (2), G2 (2), G2 (3) or 2F4 (2). Then any isometry from E2 (c) ∪ {d} to E2 (c ) ∪ {d } extends to a unique isometry from to . In particular the local structure of determines its global structure. The reason for excluding certain residues over the fields of 2 or 3 elements is that in these cases the set of chambers opposite a given chamber is not connected. In all other cases the set of chambers opposite a given chamber is connected, and the building is said to satisfy condition (co). Using this local-to-global theorem the classification of 2-spherical twin buildings has been achieved in the following way (see [M1] and [M2] for further details). Theorem 4.4 ([M2]). Let be an irreducible, 2-spherical twin building satisfying (co). Then one of the following holds: (1) is associated with a classical group over a division ring that is infinite dimensional over its center. In this case is of spherical or of affine type. (2) can be obtained from the class of twin buildings associated with Kac–Moody groups over algebraically closed fields by a sequence of constructions (CS), (FP), (L), (FP), (FP). (CS) refers to Convex Sub-buildings of the same type, obtained by descending to a suitable subfield. (L) refers to Limits, such as moving from finite dimensional to infinite dimensional forms of finite Witt index. (FP) refers to Fixed Points. All three procedures occur in the spherical case when descending from a building over an algebraically closed field, but are only needed in the sequence (CS), (FP), (L). In the non-spherical case there are added complications because the diagram can contain circuits, and the fixed point procedure can involve descent from a building of infinite rank to one of finite rank (see [M2] for more details).

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When a twin building fails to be 2-spherical it has rank 2 residues that are trees. These residues and their opposites form twin trees. A single tree is a simple structure, but twin trees are far more complicated. A single tree can be twinned with an isomorphic copy of itself in infinitely many different ways, as I shall now point out. Twin trees. The definition of a twin tree can be given in terms of an integer codistance between vertices (see [RT2]), rather than using a W -valued codistance between chambers as in the higher rank case [T9]. When this codistance is zero the vertices are said to be opposite, and two such vertices necessarily have the same valency. When the trees are thick (meaning that every vertex has valency at least three), then vertices in the same tree at even distance from one another, or vertices in opposite trees at even codistance from one another, have the same valency. Thus the two trees are isomorphic and each tree is semi-homogenous. Example. In the twin tree for the group SL2 over a ring of Laurent polynomials Fq [t, t −1 ], where Fq is the field having q elements, each vertex has valency q + 1. Moufang twin trees and Kac–Moody groups. Rank 2 Kac–Moody algebras Lie are constructed using a field k along with a generalized Cartan matrix b2 a2 , where a and b are negative integers. When ab = 1, 2 or 3 the algebras are finite dimensional, giving rise to Chevalley groups of types A2 , B2 = C2 , and G2 , respectively. Their buildings are generalized m-gons for m = 3, 4, and 6, respectively. When ab ≥ 4 the algebra is infinite dimensional and the buildings are twin trees. If the field k is finite with q elements, then the twin trees have valency q + 1, and the example above is the special case where a = b = −2. These “Kac–Moody” twin trees are Moufang in the sense that they admit root groups (see [RT2] for the definition of root groups for twin trees), but they are by no means the only Moufang examples. In the case of valency 3 (three edges per vertex) Tits [T10] has produced uncountably many Moufang twin trees. Constructing twin trees. In the Kac–Moody examples over a finite field, the number of edges per vertex is q + 1 where q is a prime power or an infinite cardinal number. An obvious question is whether examples can exist for any q. The answer is that there are uncountably many, as the next theorem shows. Theorem 4.5 ([RT3, 8.2]). A thick semi-homogeneous tree whose set of vertices has cardinality α admits 2α isomorphism classes of twinnings, and among these 2α have trivial automorphism group. The method of proof is to examine a partial twinning between a tree and a ball of radius 2 around a given vertex. This partial twinning is called a uniform 2-ball. In [RT3] we show two main things: (1) that a uniform 2-ball can have a wild structure, and (2) that the uniform 2-balls at different vertices of a tree twinned with T can be

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independent of one another. The point is that one can amalgamate uniform 2-balls in a random way to obtain a twin tree. We now return to single buildings, and find that wildness can occur there too. There is a difference, however. Wildness does not need the absence of rank 2 spherical residues, as it does with twin buildings, but the absence of rank 3 spherical residues.

5. Construction and existence of buildings of all types In [R1] I showed that a classification of affine buildings of rank 3 (dimension 2) was not possible, even in the locally finite case. The method used a free construction and applies to any building whose rank 3 residues are non-spherical. It works by starting at one chamber and working outwards. The rank 2 residues can be chosen independently, the only restriction being the obvious one that if two rank 2 residues have a rank 1 residue in common then they confer the same valency on this rank 1 residue. Assuming valency at least 3 this implies that two rank 1 residues of the same type have the same valency. In order to state a theorem, let us say that a building has parameters (qi ), i ∈ I , if every rank 1 residue of type i has qi + 1 chambers; we allow the possibility that qi may be infinite. This means that every rank 2 residue of type {i, j } is a generalized mij -gon with parameters (qi , qj ); if mij is infinity this means it is a tree. The fact that every rank 3 residue is non-spherical means that m1ij + m1j k + m1ki ≤ 1 for every set of three types i, j , k ∈ I . Define (qi ) to be an admissible set of parameters if for each i, j there exists a generalized mij -gon having parameters (qi , qj ). For example if mij = 3 then qi = qj and in all known finite examples qi is a prime power. Theorem 5.1 ([R1]). Let (qi ) be an admissible set of parameters, and W a Coxeter group having no rank 3 spherical subgroups. Then there exists a building of type W with these parameters, whose rank 2 residues of type {i, j } range through any desired set of generalized mij -gons having parameters (qi , qj ). This theorem, of course does not apply to irreducible, spherical buildings of rank at least 3, and for that we need a more sophisticated approach. This approach, developed by Tits and the author uses the concept of a “blueprint”. Blueprint constructions. If a building contains irreducible spherical rank 3 residues (namely of types A3 or B3 ) then things are very different. For example in an A3 or B3 residue two rank 2 sub-residues intersecting in a rank 1 residue must give compatible structures on that intersection. For example if this rank 1 residue is a projective line it must receive the same projective structure from all rank 2 residues containing it. This compatibility problem led the author and J. Tits [RT1] to a new method of construction

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that, although not as free as the earlier one in [R1], applies to buildings of all types. Here is how it works. Let I be the indexing set for the fundamental roots, or equivalently for the different types of panels in the building. Attach to each i in I a set Ui . The idea is to use the sets Ui , along with the Coxeter group, to parametrize the building, starting with one chamber b and working outwards. A gallery from b to another chamber x is a sequence of chambers b = x0 , x1 , . . . , xd = x with each one adjacent to the next; we say it has type (i1 , . . . , id ) if xj −1 is ij -adjacent to xj . Using the sets Ui we then specify x by a sequence, where uj ∈ Uj , and (i1 , . . . , id ) is the type of a minimal gallery from b to x. If b is joined to x by minimal galleries of different types then it is necessary to know when sequences of two different types yield the same chamber. Fortunately, if we fix the end chambers b and x, one minimal gallery can be transformed into another by a sequence of alterations (called elementary homotopies) in rank 2 residues (see [T6, (3.4)]). The datum consisting of the sets Ui along with these rank 2 homotopies, for each type of rank 2 residue, is called a blueprint. A building that can be parametrized in accordance with a blueprint is said to conform to that blueprint, and a blueprint is called realisable if there exists a building conforming to it. The secret in producing realisable blueprints is to let the Ui be root groups for the fundamental roots – see [RT1] for details. Clearly any realisable blueprint must be realisable in all rank 3 spherical residues, and these are the residues that formed an obstruction to the previous construction. What is surprising is that this is the only obstruction. Theorem 5.2 ([RT1, Theorem 1]). A blueprint is realisable if its restriction to each rank 3 spherical subdiagram is realisable. In this case there is a unique building which conforms to it. [Note: the statement in [RT1] inadvertently omits the word spherical]. The reason the rank 3 spherical restriction is sufficient can be explained in terms of galleries as follows. A sequence of rank 2 homotopies may yield a gallery having the same type as the initial gallery. But the initial sequence (ui1 , . . . , uid ) and the final sequence (vi1 , . . . , vid ) must yield the same end chamber, so uij = vij . To ensure this happens, it suffices to check such self-homotopies in rank 3 spherical residues because of a theorem for Coxeter complexes due to Tits [T6, Prop.4]. Foundations. In order to avoid specifying an explicit blueprint, the authors [RT1] define a foundation to be an amalgam of rank 2 residues containing a given chamber. It is said to support a building if there exists a building and a chamber c in that building such that E2 (c) – the union of the rank 2 residues containing c – is isomorphic to the foundation concerned. The point of using foundations is the following powerful existence theorem.

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Theorem 5.3 ([RT1, Theorem 2]). Let E be a foundation. If each rank 3 residue of type A3 or B3 supports a building, then E supports a building. The theorem is proved by producing a realisable blueprint, and hence a building conforming to it. It is now time to ask which buildings these are that conform to blueprints. In the spherical case all buildings conform to a blueprint. In particular the theorems above prove the existence of spherical buildings, such as those of type E8 , that were previously only known from the existence of the Chevalley groups. In the affine case the buildings conforming to a blueprint are those for groups over a power series field, rather than a p-adic field. The p-adic affine buildings do not conform to blueprints, apart from an exceptional case for SL3 (Q2 ). The affine buildings over power series fields are precisely those that arise from Kac–Moody groups. For example in the last section the Kac–Moody group SLn (k[t, t −1 ]) acts on an affine twin building, its two components being the affine building for SLn (k((t))).

6. Questions and conjectures The last paragraph brought out a connection between twin buildings, and buildings conforming to a blueprint. This leads to the following questions. 1. Do the buildings of a twin conform to a blueprint? The answer is yes in the 2-spherical case when condition (co) is satisfied because Theorem 4.3 shows that a twin building is Moufang, and this yields root groups that give a natural blueprint for the building. 2. Are buildings that conform to a blueprint twinnable? The immediate answer is no because there are examples conforming to some odd-ball blueprints that are not twinnable. For example there are A˜ 2 blueprints involving non-Moufang planes, but by Theorem 4.1 there are no twin buildings of such type. Of course this example does not involve rank 3 spherical residues, and begs the question of what happens when such residues exist. Here is a conjecture. Conjecture 6.1. If a building conforms to a blueprint and all its rank 3 residues are spherical, then it admits a twinning. Conforming to a blueprint is a strong condition, and this conjecture seems quite likely to be true. On the other hand we can forget about blueprints and just ask what happens if all rank 3 residues are spherical. This is true of every affine building having rank ≥ 4, and we already know such buildings that are not twinnable (for example the p-adic ones). But what about the non-affine case? Hyperbolic buildings of rank ≥ 4 in which all rank 3 residues are spherical are known only from Kac–Moody groups. Are there others, comparable to the p-adic affine buildings? No-one knows of suitable groups, so it is worth asking the following question.

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Question 6.2. Can one prove that a non-affine building whose rank 3 residues are spherical is necessarily twinnable? If this were true there would be a classification of such buildings, by using twin buildings, as in 4.4, at least when condition (co) is satisfied. To attack this question one might proceed in two steps: (i) twin the building with a single chamber, and (ii) extend outwards to neighboring chambers, obtaining a twinning with another building. The first step seems to be impossible in the p-adic affine case, and is probably where the obstruction lies, if there is one.

References [A]

P. Abramenko, Twin Buildings and Applications to S-Arithmetic Groups, Lecture Notes in Math. 1641, Springer-Verlag, Berlin 1996.

[BT]

F. Bruhat and J. Tits, Groupes réductifs sur un corps local, I. Données radicielles valuées, Inst. Hautes Études Sci. Publ. Math. 41 (1972), 5–252.

[Ch]

C. Chevalley, Sur certains groupes simples, Tohoku Math. J. (2) 7 (1955), 14–66.

[D]

L. Dickson, Linear Groups, originally published 1900, reprinted by Dover, New York 1958.

[FH]

W. Feit and G. Higman, The nonexistence of certain generalized polygons, J. Algebra 1 (1964), 114–131.

[IM]

N. Iwahori and H. Matsumoto, On some Bruhat decomposition and the structure of the Hecke ring of a p-adic Chevalley group, Inst. Hautes Études Sci. Publ. Math. 25 (1965), 5–48.

[MT]

R. Moody and K. Teo, Tits systems with crystallographic Weyl groups, J. Algebra 21 (1972), 178–190.

[M1]

B. Mühlherr, Locally split and locally finite twin buildings of 2-spherical type, J. Reine Angew. Math. 511 (1999), 119–143

[M2]

B. Mühlherr, Twin Buildings, in Tits Buildings and the Model Theory of Groups, ed. by K. Tent, London Math. Soc. Lecture Note Ser. 291, Cambridge University Press, Cambridge 2002, 103–117.

[MR]

B. Mühlherr and M. Ronan, Local to Global Structure in Twin Buildings, Invent. Math. 122 (1995), 71–81.

[R1]

M. Ronan, A Construction of Buildings with no Rank 3 Residues of Spherical Type, in Buildings and the Geometry of Diagrams (Como 1984), ed. by L. A. Rosati, Lecture Notes in Math. 1181, Springer-Verlag, Berlin 1986, 242–248.

[R2]

M. Ronan, Lectures on Buildings, Perspect. Math. 7, Academic Press, Boston 1989.

[R3]

M. Ronan, Local Isometries of Twin Buildings, Math. Z. 234 (2000), 435–455.

[RT1] M. Ronan and J. Tits, Building Buildings, Math. Ann. 278 (1987), 291–306.

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[RT2] M. Ronan and J. Tits, Twin Trees I, Invent. Math. 116 (1994), 463–479. [RT3] M. Ronan and J. Tits, Twin Trees II, Israel J. Math. 109 (1999), 349–377. [S]

J.-P. Serre, Arbres, Amalgames, SL2 ; Astérisque 46, 2nd corr. ed., Soc. Mathém. de France, Paris 1979; English translation: Trees, Springer-Verlag, Berlin 1980.

[St]

R. Steinberg, Variations on a theme of Chevalley, Pacific J. Math. 9 (1959), 875–891.

[T1]

J. Tits, Le plan projectif des octaves et les groupes de Lie exceptionelles, Acad. Roy. Belgique. Bull. Cl. Sci. (5) 39 (1953), 309–329.

[T2]

J. Tits, Le problème de mots dans les groupes de Coxeter. Ist. Naz. Alta Mat., Symposia Math. 1 (1968), 175–185.

[T3]

J. Tits, Buildings of Spherical Type and Finite BN-Pairs, Lecture Notes in Math. 386, Springer-Verlag, Berlin 1974.

[T4]

J. Tits, Endliche Spiegelungsgruppen, die als Weylgruppen auftreten, Invent. Math. 45 (1977), 283–295.

[T5]

J. Tits, Non-existence de certains polygones généralisés, I, II, Invent. Math. 36 (1976), 275–284; ibid. 51 (1979), 267–269.

[T6]

J. Tits, A Local Approach to Buildings, in The Geometric Vein – The Coxeter Festschrift, ed. by C. Davis, B. Grünbaum and F. A. Sherk, Springer-Verlag, New York 1982, 519–547.

[T7]

J. Tits, Immeubles de type affine, in: Buildings and the Geometry of Diagrams (Como 1984), ed. by L. A. Rosati, Lecture Notes in Math. 1181, Springer-Verlag, Berlin 1986, 159–190.

[T8]

J. Tits, Uniqueness and Presentation of Kac–Moody Groups over Fields, J. Algebra 105 (1987), 542–573.

[T9]

J. Tits, Twin Buildings and Groups of Kac–Moody Type, in Groups, Combinatorics and Geometry (Durham 1990), ed. by M. W. Liebeck and J. Saxl, , London Math. Soc. Lecture Note Ser. 165, Cambridge University Press, Cambridge 1992, 249–286.

[T10]

J. Tits, private communication

[TW]

J. Tits and R. M. Weiss, Moufang Polygons, Springer Monogr. Math., Springer-Verlag, Berlin 2002.

[VM]

H. Van Maldeghem, Generalized Polygons, Monogr. Math. 93, Birkhäuser Verlag, Basel 1998.

[W]

R. Weiss, The Nonexistence of certain Moufang Polygons, Invent. Math. 51 (1979), 261–266.

Mark A. Ronan, Mathematics Department, University of Illinois at Chicago, 851 S. Morgan, mc 249, Chicago, IL 60607, U.S.A. E-mail: [email protected]

Characters of algebra groups and unitriangular groups Josu Sangroniz∗

Abstract. In this work we study the irreducible complex characters of finite algebra groups, with a special interest in the groups of unitriangular matrices over finite fields. By an algebra group we mean a group of the form 1 + J , where J is the Jacobson radical of a finite associative algebra over a commutative ring R and the additive group of J is a p-group. We start with an exposition of Kirillov’s method adapted to our setting, which yields a description of the irreducible characters of the algebra group 1 + J when J p = 0. Although this method can be applied immediately to the unitriagular groups when the size of the matrices does not exceed the characteristic of the field, it fails in general to describe all the irreducible characters. However, we shall show that the method is still useful to describe some of the irreducible characters, namely, those whose degree is small or large enough.

1. Introduction This work is a contribution to the study of the (complex, irreducible) characters of the unitriangular groups (i. e., the groups formed by the n × n upper-triangular matrices with entries in a finite field and all of whose diagonal elements are 1). Isaacs found it useful in [6] to consider unitriangular groups only as part of a larger family of groups: the algebra groups. By an algebra group he means a group of the type G = 1 + J , where J is the Jacobson radical of a finite dimensional algebra over a finite field with identity element 1. This setting lends itself to inductive arguments which do not always work when confined to unitriangular groups. In the abovementioned work, Isaacs proved for instance that the degrees of the irreducible characters of algebra groups are always powers of q, the size of the underlying field, a result originally conjectured for unitriangular groups. Our goal here is twofold. First we explain Kirillov’s method of coadjoint orbits (see [11]) in the context of algebra groups. Our presentation, inspired by [10], also reflects some of the ideas of Isaacs’ in [7]. For an alternative approach, see [2]. We also change slightly the concept of algebra group by allowing the algebras to be ∗ This research has been supported by the Feder, the Spanish Ministerio de Ciencia y Tecnología, grant BFM2001-0180 and the University of the Basque Country, grant 1/UPV00127.310-E-13817/2001.

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defined over a commutative ring rather than a field. As we shall see, this seemingly unimportant modification has some noteworthy consequences. In the second part of this paper we focus our attention on unitriangular groups. As it was proved by Isaacs and Karagueuzian in [8], there are serious limitations in Kirillov’s method when we try to apply it to unitriangular groups if the size of the matrices exceeds the characteristic of the field. However, we shall show that even in this case the method still works to describe the irreducible characters provided the degree is small or large enough. These results can be seen as a generalization of [13] (where the irreducible characters of degree 1, q and q 2 are considered) and [1] (which includes a description of the characters of maximal degree). I would like to thank M. Isaacs and A. Jaikin for their helpful comments. Part of this work was done while I was visiting the Department of Mathematics of the University of Wisconsin-Madison in the fall 2002. I would like to express my gratitude for their kind hospitality.

2. Kirillov’s method and algebra groups In this section R will denote a commutative ring with unity and J will be a multiplicatively closed submodule inside an associative R-algebra with identity 1. We shall also assume that J n = 0 for some positive integer n and that the additive group of J is a finite p-group. Under these hypotheses, the set G = 1 + J is a finite p-group. A group like G will be called an algebra group. Some preliminary work is necessary before stating the key result of Kirillov’s method. We start by introducing a convenient notion of ‘multiplicative bilinearity’. A multiplicative bilinear alternating form on J will be for us a map B : J × J → C∗ satisfying the following conditions: (i) B(u + v, w) = B(u, w)B(v, w) and B(u, v + w) = B(u, v)B(u, w) for all u, v, w ∈ J . (ii) B(ru, u) = 1 for all u ∈ J and r ∈ R. Expanding the equality B(r(u + v), u + v) = 1 and using (ii), we obtain B(ru, v)B(rv, u) = 1. For r = 1 this yields B(u, v) = B(v, u)−1 for all u, v ∈ J , so B(ru, v) = B(rv, u)−1 = B(u, rv). The radical of B is the submodule defined by: rad B = {u ∈ J | B(u, v) = 1 for all v ∈ J } = {u ∈ J | B(v, u) = 1 for all v ∈ J }. If rad B happens to be trivial we shall say that B is non-degenerate. Suppose for a moment that this is the case. Then there is a canonical isomorphism φ between J and Jˆ, the set of irreducible (linear) characters of J , mapping u ∈ J to Bu , which sends v ∈ J to B(u, v) ∈ C∗ . For a submodule (or subgroup) H ≤ J we define H ⊥ to be the set of elements of Jˆ whose kernel contains H . It is clearly isomorphic, as

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a group, to J /H, so in particular |H ⊥ | = |J : H |. Now suppose that H is isotropic with respect to B (that is B(u, v) = 1 for all u, v ∈ H). Then φ maps H into H ⊥ , whence |H | ≤ |H ⊥ | = |J : H| and |H|2 ≤ |J |. We claim that the equality holds precisely when H is a maximal isotropic submodule of J . Indeed, if H is maximal isotropic and we are given ζ ∈ H ⊥ , we can represent ζ as ζ = Bu for some u ∈ J and the fact that ζ is in H ⊥ translates into the condition B(u, v) = 1 for all v ∈ H . Then K = Ru + H, the submodule generated by u and H , is also isotropic, whence K = H and u ∈ H. This means that φ is an isomorphism between H and H ⊥ , which is simply another way to formulate our claim. The degenerate case is reduced to the non-degenerate one by the usual procedure of considering B as a form on J / rad B. It follows that all maximal isotropic submodules of J have the same order, namely (|J || rad B|)1/2 . The algebra group G = 1 + J acts by conjugation on the additive group J (it is actually the action of a group on an R-module), so it also acts by duality on Jˆ. Given ζ ∈ Jˆ, we denote by ζ the orbit of ζ under this action (these are the coadjoint orbits in Kirillov’s terminology). There is a close relation between ζ and the multiplicative bilinear form Bζ defined by Bζ (u, v) = ζ ([u, v]), where [ , ] denotes the standard Lie bracket in J . It can be checked routinely that the correspondence u → 1 + u maps the radical of Bζ onto the stabilizer of ζ , S. In particular, |ζ | = |G : S| =

|J | = |H / rad Bζ |2 , | rad Bζ |

(2.1)

where H ≤ J is a maximal isotropic submodule of J . This shows that the lengths of the orbits are perfect squares. If R = Fq is a finite field we can identify Jˆ with the dual vector space J ∗ : simply fix a non-principal linear character ψ : Fq → C∗ of the additive group of the field and notice that the map λ → ψ λ gives a bijection from J ∗ onto Jˆ. If ζ = ψ λ, we can also identify Bζ with the ordinary bilinear map Bλ given by Bλ (u, v) = λ([u, v]) and both the radical of Bζ and the usual radical of Bλ coincide. The action of G on Jˆ translates into an action of G on J ∗ and, for λ ∈ J ∗ , the size of the orbit λ of λ is q rk Bλ . The next step is to prove that the submodule H in (2.1) can be chosen to be multiplicatively closed. The key observation here is that if B is a multiplicative bilinear form on J and U ≤ J is an isotropic submodule, then any maximal isotropic submodule W inside U⊥ = {v ∈ J | B(u, v) = 1 for all u ∈ U} is also maximal isotropic when viewed in the whole J (the reason for this is that any isotropic submodule of J containing W is actually contained in U⊥ ). Proposition 1. Let ζ ∈ Jˆ and B = Bζ . Then there exists a maximal isotropic submodule which is multiplicatively closed. In addition, this submodule can be taken to contain any particular isotropic ideal of J .

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Proof. We argue by induction on the order of J . The result is obvious if B is trivial. Otherwise, J = 0 and, by refining the series {J k }, we obtain a chain of ideals J = J1 ⊃ J2 ⊃ · · · ⊃ Jr+1 = 0 such that all the factor modules Ji /Ji+1 are cyclic. We can also suppose that this series contains any particular isotropic ideal A. Let Ji , i ≥ 1, be the largest non-isotropic ideal in the series. Since Ji /Ji+1 is cyclic, Ji+1 cannot be contained in the radical of ⊥ is properly contained in J and it is not difficult B, so its orthogonal submodule Ji+1 to see that is multiplicatively closed (the key is that the bilinear form B = Bζ has the ⊥ . By property B(uv, w) = B(u, vw)B(v, wu)). Notice also that A ⊆ Ji+1 ⊆ Ji+1 ⊥ has a maximal isotropic submodule multiplicatively closed containing induction Ji+1 A which, by the previous remark, gives the desired submodule. If ζ ∈ Jˆ and B = Bζ , we shall denote by Hζ the submodule whose existence we have just proved in the last proposition and we set Hζ = 1 + Hζ , which is an algebra subgroup of G = 1 + J . We insist that in the sequel this notation will be used without any further explanation. Since Hζ is isotropic, the subgroup Hζ fixes the restriction of ζ to Hζ and the function ϕζ defined by ϕζ (1 + u) = ζ (u) is a class function of Hζ . We compute now the induced function ϕζG . Theorem 2. Let ζ ∈ Jˆ and ζ = ϕζG . Then, for all u ∈ J , µ(u). ζ (1 + u) = |ζ |−1/2 µ∈ζ

Proof. By definition of ζ , ζ (1 + u) =

1 ◦ 1 ◦ g ϕ ζ (1 + ug ) = ζ (u ), |Hζ | |Hζ | g∈G

◦

g∈G

◦

where ϕ ζ (resp. ζ ) is the extension of ϕζ (resp. ζ ) which vanishes outside Hζ (resp. Hζ ). By the orthogonality relations on the group J /Hζ we have that |J : Hζ |, if v ∈ Hζ ; µ(v) = 0, otherwise, µ∈J /Hζ

hence, ζ (1 + u) =

1 1 ζ (ug )µ(ug ) = ν(ug ), |J | g∈G |J | g∈G µ∈J /Hζ

(2.2)

ν∈E

where E is the set of linear characters of J which agree with ζ on Hζ . Note that the subgroup Hζ acts on E (because Hζ fixes ζ|Hζ ) and contains the stabilizer S of ζ in

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G (because of the maximality of Hζ ), thus the length of the orbit of ζ under Hζ is /Hζ | = |J |/|Hζ |, so bearing |Hζ : S| = |Hζ |/| rad Bζ |. On the other hand, |E | = |J in mind the formula for the order of Hζ we conclude that |E | = |Hζ : S|, which means that Hζ acts transitively on E . Then if ν ∈ E , there exists h ∈ Hζ such that ν = ζ h and ν(ug ) = ζ h (ug ) = ζ (ug ). g∈G

g∈G

g∈G

Putting this into (2.2) we obtain ζ (1 + u) =

|E | g 1 g ζ (u) = ζ (u) |J | |Hζ | g∈G

g∈G

|Gζ | = µ(u) = |ζ |−1/2 µ(u). |Hζ | µ∈ζ

µ∈ζ

Corollary 3. Let G = 1 + J and, in addition to the usual assumptions, suppose that J p = 0. Then any irreducible character of G is induced by a linear character of an algebra subgroup of G, that is, of the form 1 + H for H a multiplicatively closed submodule of J . Moreover, if χ is an irreducible character of G, then there exists ζ ∈ Jˆ, unique up to the action by G, such that µ(u). χ(exp u) = |ζ |−1/2 µ∈ζ

Proof. The exponential is well-defined because of the condition J p = 0 and maps bijectively J onto G. For ζ ∈ Jˆ we set χζ and αζ to be, respectively, the functions on G and Hζ defined by χζ (exp u) = ζ (1 + u), u ∈ J ; αζ (exp u) = ϕζ (1 + u) = ζ (u), u ∈ Hζ . By the Hausdorff formula (and the isotropy of Hζ ) it is clear that αζ is a linear character of the subgroup Hζ and, by the last theorem, χζ = αζG , so χζ is a genuine character of G. Now, (χζ , χζ ) = 1 or 0 depending on whether ζ and ζ are in the same orbit by G or not (this an easy consequence of the orthogonality relations for the linear characters of J ), so we have produced as many irreducible characters of G as coadjoint orbits. Finally, we know, by Brauer’s permutation lemma, that the number of coadjoint orbits is the same as the number of orbits of the action of G on J , which is of course the number of classes of G, thus the characters χζ account for all the irreducible characters of G. Under the hypotheses of the last corollary, the technique we have used also allows us to describe explicitly the irreducible characters of the subgroups of 1+J associated to Lie subalgebras of J (we insist that J p = 0). To be more precise about the relation between Lie subalgebras and subgroups, note that if L is a Lie subalgebra of J then,

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by the Hausdorff formula, the set L = exp L = {exp u | u ∈ L} is a subgroup of G = 1 + J . This subgroup acts by conjugation on L since for u, v ∈ L, uexp v = u +

[u, v] [u, v, v] . . . , v] [u, v, p−2 + + ··· + 1! 2! (p − 2)!

(2.3)

is an element lying on L. The formula (exp u)g = exp(ug ) tells us that this action is permutation equivalent to the action of L on itself by conjugation, so the number of classes of L coincides with the number of orbits of L acting on L which, by Brauer’s ˆ For permutation lemma, is also the number of orbits of the dual action of L on L. ˆ ζ ∈ L we define the bilinear form Bζ as before. An element exp v ∈ L fixes ζ if and only if . . . , v] [u, v, p−2 [u, v] [u, v, v] + + ··· + ∈ Ker ζ 1! 2! (p − 2)!

(2.4)

for all u ∈ L, a condition that is certainly fulfilled if [v, L] ⊆ Ker ζ . Conversely, if . . . , v] (2.4) holds for all u ∈ L, by substituting u successively by [u, v], . . . , [u, v, p−3 we realize that [u, v] ∈ Ker ζ (for any u ∈ L), whence [v, L] ⊆ Ker ζ . Thus the exponential maps the radical of Bζ onto the stabilizer of ζ . In particular, ζ is fixed by L if and only if [L, L] ⊆ Ker ζ . ˆ Then Proposition 4. Let L be a finite nilpotent Lie algebra over a ring R and ζ ∈ L. there exists a Lie subalgebra Hζ such that Hζ is a maximal isotropic submodule with respect to Bζ . Proof. Mimic the proof of Proposition 1 (obviously, changing the ideals Ji by Lie ideals). Note also that if I is a Lie ideal of L, then the orthogonal submodule I ⊥ is a Lie subalgebra. The computations in the proof of Theorem 2 can be carried out almost word for word to the context of Lie algebras, so that the following analogous result to Corollary 3 can be obtained. Theorem 5. Let L = exp L, where L is a Lie subalgebra of J and J satisfies the conditions of Corollary 3. Then any irreducible character of L can be induced by a linear character of a subgroup exp H for some Lie subalgebra H ⊆ L. In addition, ˆ unique up to the action of L, if χ is an irreducible character of L, there exists ζ ∈ L, such that for all u ∈ L, χ(exp u) = |ζ |−1/2 µ(u). µ∈ζ

It is clear that under the hypotheses of the last theorem the nilpotency class of the group L is at most p − 1. It should be noted that actually any p-group of class less than p can be realized in this way. Indeed, suppose G is such a group and let pe be its exponent. We consider the group algebra RG, where R is the ring of integers

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modulo p e and as usual, denote by the augmentation ideal of RG. Then, by a e result of Moran [14], G ∩ (1 + p ) = Gp γp (G) = 1, so G can be embedded in the algebra group associated to J = /p , an algebra that satisfies our requirement J p = 0. Finally, the exponential induces a bijection between the Lie subrings (in this case there is no need to distinguish between subrings and subalgebras) of J and the subgroups of 1 + J (because of the Hausdorff formula and its inverses), so our group G is isomorphic to a group exp L for a Lie subalgebra L of J . Notice that L is the Lie ring corresponding to G by the Lazard correspondence. For some purposes it may be more useful to change the point of view and start with an abstract Lie ring L of class less than p whose underlying additive group is a p-group. Then we can pass to its group counterpart G and realize G as exp L for some Lie subalgebra L inside an associative algebra J with J p = 0. Since L is also the Lazard correspondent to G, it follows that L and L are isomorphic as Lie rings. We can use this fact for instance to obtain the degrees of the irreducible characters of G performing all computations in L instead of G: by Theorem 5 this ˆ If L is actually an set consists of the numbers |L : rad Bζ |1/2 for ζ running on L. 1 algebra over Fp , these numbers can be rewritten as p 2 rk Bλ with λ ∈ L∗ . We give as an application another construction of a p-group (for odd p) with prescribed character degrees, different from the ones proposed by Isaacs in [4] and Isaacs and Slattery in [9]. Theorem 6. Let p be an odd prime and X any set of powers of p containing 1. Then X is the set of degrees of the irreducible characters of some p-group of class 2. Proof. By the preceding discussion the statement can be reformulated in the following terms: for any set Y of non-negative integers containing 0, there exists a Lie algebra L over Fp of class 2 such that the set of numbers 21 rk Bλ | λ ∈ L∗ equals Y . Suppose then that the positive numbers in Y are n1 > · · · > nr and take a vector space L over Fp of dimension 2n1 + r with a basis x1 , . . . , x2n1 , y1 , . . . , yr . Set [xi , xj ] = yk and [xj , xi ] = −yk for i + j = 2n1 − k + 1, 1 ≤ i ≤ nk , 1 ≤ k ≤ r and define the rest of the Lie products among the basis elements to be zero. This clearly endows L with a Lie algebra structure of class 2. If λ is a linear functional on L, the rank of Bλ is the rank of the antisymmetric matrix (λ([xi , xj ])), and this number is twice the largest nk for which λ(yk ) = 0 (or 0 if λ annihilates all the yk ). Note that if X = {1, pn1 , . . . , pnr }, n1 > · · · > nr , the order of the group we have constructed is p 2n1 +r , compared to p2n1 +n2 +···+nr +r in Isaacs’ example.

3. Unitriangular groups For the rest of this work we shall be concerned mainly with unitriangular groups, so all the algebras will be supposed to be defined over (finite) fields. Recall that in this case Jˆ can be identified with the dual space J ∗ by means of a fixed non-principal

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character ψ of the additive group of the field. If the characteristic is p and J p = 0 the exponential map is no longer defined but a natural substitute for it is the truncated

p−1 exponential map Exp : J → G defined by Exp u = i=0 ui /i!. It is easy to see that Exp is a bijection between J and G (actually any polynomial map with constant term 1 is a bijection from J to G). Notice that we reserve the notation exp for the ordinary exponential map defined when J p = 0. There is also a ‘Hausdorff formula’ for Exp: Lemma 7. Let u, v ∈ J . Then there exists γ ∈ J p + [J, J ] such that Exp u Exp v = Exp(u + v + γ ). Proof. The result is clear if J p = 0 by the Hausdorff formula, so we can suppose J p = 0. Let n > p be such that J n−1 = 0 but J n = 0. Arguing inductively the formula can be assumed to be true for the algebra J /J n−1 . The algebra group associated to this factor algebra can be identified naturally with the factor group (1 + J )/(1 + J n−1 ), thus the relation for the truncated exponential on J /J n−1 gives rise to the equality in 1 + J : Exp u Exp v = Exp(u + v + γ )(1 + w), for some γ ∈ J p + [J, J ] and w ∈ J n−1 . Since wJ = J w = 0, it is clear that Exp(z + w) = (Exp z)(1 + w) for any z, thus Exp u Exp v = Exp(u + v + γ + w) and γ + w ∈ J p + [J, J ] (remember that J n−1 ⊆ J p ). If λ ∈ J ∗ we take a subalgebra Hλ which is a maximal isotropic subspace with respect to Bλ and define the class function αλ on Hλ = 1 + Hλ = Exp Hλ by αλ (Exp u) = ψ(λ(u)). For u ∈ J we also set χλ (Exp u) = αλG (Exp u) (notice that this notation is consistent with the notation in Section 2). Then, by Theorem 2, ψ(µ(u)). (3.1) χλ (Exp u) = |λ |−1/2 µ∈λ

By the definition, it is clear that χλ is a character of G whenever αλ is a character of Hλ . Moreover, in this case χλ is an irreducible character (because (χλ , χλ ) = 1). This p happens for instance when Hλ ⊆ Ker λ (by the last lemma). It should be noted at this point that there is no hope to describe all the irreducible characters of the unitriangular groups by means of a formula like (3.1). The reason is that these formulas always yield functions that take values in the field Q(ε) for ε a primitive pth root of unity, in particular, they are rational valued if p = 2. However, there exist matrices over the field with two elements which are not conjugate to their inverses (see examples in [8]), so these groups do have characters which are not rational valued (actually, for any prime p there exist unitriangular groups over Fp with characters whose values are not in Q(ε), see [15]). As an illustration of the ideas above, we determine the irreducible characters of the unitriangular groups of ‘small degree’. For matrices of size n × n over the finite

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field Fq this group will be denoted by Un (q) and un (q) will be the associated nilpotent algebra. Theorem 8. Let 0 ≤ k ≤ log2 p − 1. Then the correspondence λ → χλ induces a bijection between the irreducible characters of Un (q) of degree q k and the coadjoint orbits of size q 2k . Proof. Suppose first that λ is a linear functional on J = un (q) whose orbit has size q 2k for some k in the specified range. We know that the codimension of Hλ is precisely k. Then, there exists an ascending series of subalgebras Hi , 0 ≤ i ≤ k, beginning with Hλ and ending in J such that each one is an ideal of the next. We have that 2 ⊆ H , thus J 2k ⊆ H and, since p ≥ 2k+1 , Hi+1 i λ Jp ⊆ J2 p

k+1

k

k

= [J 2 , J 2 ] ⊆ [Hλ , Hλ ] ⊆ Ker λ, k+1

k

k

so Hλ ⊆ Ker λ (note that the equality J 2 = [J 2 , J 2 ] is a property of the algebra p un (q) and not a general one). We conclude that Hλ + [Hλ , Hλ ] ⊆ Ker λ. This implies by the previous lemma that αλ is a linear character of Hλ = 1 + Hλ and, as remarked before, this is all we need to ensure that χλ is a character of G = Un (q). Conversely suppose now that χ in an irreducible character of G of degree q k and 0 ≤ k ≤ log2 p − 1. We need to find λ ∈ J ∗ such that χ = χλ . By [5, Theorem 5.12] we have that G(k+1) ≤ Ker χ (it is crucial here to use the fact that the degrees of the irreducible characters of G are powers of q) and it is well-known k+1 that G(k+1) = 1 + J 2 , hence χ can be viewed as a character χ of the factor group k+1 k+1 G/(1 + J 2 ), which is isomorphic to the group associated to J = J /J 2 . Again, p as p ≥ 2k+1 , J = 0, thus χ is determined by a linear functional on J , which lifts to a linear functional λ on J . Now, the formula for χ , when viewed for G, is exactly the equality χ = χλ that we wanted to prove. Theorem 2 yields a description of the irreducible characters of Un (q) for n ≤ p. However this result can be pushed further if we make use of some specific properties of the algebra un (q). If r and s are two positive integers with r + s = n, then un (q) = Kr,s ⊕ Ar,s , where u 0 0 a Kr,s = | u ∈ ur (q), v ∈ us (q) and Ar,s = | a ∈ Mr×s (q) . 0 v 0 0 (3.2) Notice that K = Kr,s is an algebra and A = Ar,s is an ideal with A2 = 0. For algebras J admitting such a decomposition, the hypothesis J p = 0 in the previous section can be relaxed a bit. This will suffice to extend the result on the characters of Un (q) to the case n < 2p and, when no condition is imposed on n, to describe the irreducible characters with large degree. We begin work toward our goal with two rather technical lemmas. Lemma 9. Suppose J = K ⊕ A, where K is a subalgebra, A an abelian ideal and K i Ap−i = 0 for all 0 ≤ i ≤ p (for i = 0 and i = p, read this condition as Ap = 0

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and K p = 0, respectively). Then, if u ∈ K and a ∈ A, there exists an element γ ∈ [J, J ] such that exp u exp a = Exp(u + a + γ ). Proof. Note first that there is no harm in replacing K by the subalgebra generated by u inside J , so without loss of generality, we can suppose that K is a commutative algebra. We claim that [K, A], the subspace spanned by all the Lie products [u, a] for u ∈ K and a ∈ A, is an ideal of J . Indeed, if u, u ∈ K and a, a ∈ A, a [u, a] = a ua − a au = a ua − aa u = [a u, a] = 0 ∈ [K, A], where we have made use both of the commutativity of A and the fact that it is an ideal. Also u [u, a] = u ua − u au = uu a − u au = [u, u a] ∈ [K, A]. Something similar happens when multiplying on the right. Thus [K, A] is an ideal and, in the quotient algebra J = J /[K, A], the elements of K and A commute, so p

J = (K + A)p =

p

i

K A

p−i

= 0,

i=0

that is J p ⊆ [K, A]. In particular J p ⊆ [J, J ] and the result follows directly from Lemma 7. Lemma 10. Let J be as in Lemma 9 and suppose that λ ∈ J ∗ is invariant by the algebra group G = 1 + J (or equivalently, [J, J ] ⊆ Ker λ). Then, the function αλ is a linear character of G. Proof. We observe first that the group G is the semidirect product of 1 + K and 1 + A and, because K p = Ap = 0, the usual exponential is defined between K and A and these subgroups. Hence, any element g ∈ G can be written as g = exp u exp a for some u ∈ K, a ∈ A. Now, if another element g ∈ G is written similarly, we obtain by using the Hausdorff formula on K and the commutativity of A that

gg = exp u exp a exp u exp a = exp u exp u (exp a)exp u exp a

= exp(u + u + γ ) exp(a exp u + a ), for some γ ∈ [K, K]. By using the previous lemma we finally obtain

gg = Exp(u + u + a exp u + a + γ ), where γ lies in [J, J ]. Applying the definition of αλ and bearing in mind the invariance of λ, we conclude that

αλ (gg ) = ψ(λ(u) + λ(u ) + λ(a exp u ) + λ(a )) = ψ(λ(u) + λ(u ) + λ(a) + λ(a )).

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On the other hand, again by the last lemma, we have g = exp u exp a = Exp(u + a + γ1 ), with γ1 ∈ [J, J ], so αλ (g) = ψ(λ(u + a)) and similarly for g . It is now clear that αλ (gg ) = αλ (g)αλ (g ). Theorem 11. Let J be as in Lemma 9. Then the functions χλ are the irreducible characters of G = 1 + J . Proof. All we have to see is that for any λ ∈ J ∗ , the function αλ is a character of Hλ . We can suppose that A ⊆ Hλ , so that Hλ = (Hλ ∩K)⊕A and the hypotheses on J are inherited by Hλ . The result follows now from Lemma 10, because [Hλ , Hλ ] ⊆ Ker λ by the very definition of Hλ . Corollary 12. Suppose n < 2p and let λ1 , . . . , λr be the different coadjoint orbits of the unitriangular group G = Un (q). Then χλ1 , . . . , χλr are the different irreducible characters of G. Proof. Choose positive integers r and s with r < p, s ≤ p and n = r + s. Then if we define K = Kr,s and A = Ar,s as in (3.2), it is clear that K p = 0 = K i Ap−i for 0 ≤ i ≤ p − 2 (remember that A2 = 0) and it is immediate to check that also K p−1 A = 0, hence the last theorem can be applied. We turn now to the problem of determining the irreducible characters of Un (q) of large degree. ‘Large’ will mean here one of the [p(p − 1)/4] + 1 largest possible values. Some explicit computations must be carried out in the algebra un (q). For the sake of notation, K and A will be as in (3.2), which sometimes will be convenient to identify with ur (q) ⊕ us (q) and Mr×s (q), respectively. Given δ ∈ A∗ , there is a unique matrix m ∈ Ms×r (q) such that for all a ∈ Mr×s (q), δ(a) is the trace of am, δ(a) = tr(am). Now, if g = x0 y0 is an element in 1 + K, then the corresponding matrix for δ g is y −1 mx. By an adequate choice of x and y it is possible to obtain in this way a matrix, which we shall call quasi-monomial, in which at most one single element in every row and column is non-zero. Actually, one can prove that the set of quasi-monomial matrices is a set of representatives for the action of Ur (q) × Us (q) on Ms×r (q) given by m(x,y) = y −1 mx. Thus, there is no loss of generality if we assume from the outset that the matrix corresponding to δ is quasi-monomial. Let us calculate now the orthogonal subspace of A with respect of Bλ if λ ∈ J ∗ is an extension of δ. Since A2 = 0, A is isotropic, so A ⊆ A⊥ and A⊥ = K0 ⊕ A, where K0 = K ∩ A⊥ . The Lie product of an element (u, v) ∈ K and a ∈ A is the element ua − av in A, hence (u, v) ∈ K0 if and only if tr(ua − av)m = 0 for all a ∈ Mr×s (q) or, equivalently, tr(mu)a = tr a(vm). By specializing this condition for the matrices a = ei,j , whose (i, j ) entry is 1 and is zero elsewhere, the equality mu = vm follows. Hence K0 = {(u, v) ∈ ur (q) ⊕ us (q) | mu = vm}. The following definition is taken from [8].

(3.3)

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Definition. We say that a subalgebra of un (q) is a pattern algebra if it has a basis consisting entirely of matrices of the type ei,j . Of course, any pattern algebra determines and is determined by a set of pairs P ⊆ {(i, j ) | 1 ≤ i < j ≤ n} whose cardinality is the dimension of the algebra. Also, given an arbitrary set of pairs P , the condition for the subspace spanned by the matrices ei,j with (i, j ) ∈ P to be a pattern algebra is that whenever two pairs (i, j ) and (j, k) are in P , so is (i, k). Although it is not always the case that K0 is a pattern algebra, its structure is closely related to them, as shown by the next result. Recall that a subspace of a direct sum is called a subdirect sum if both projections are surjective. Lemma 13. Let m ∈ Ms×r (q) be a quasi-monomial matrix. Then the algebra K0 defined by (3.3) is a subdirect sum of two pattern algebras. Proof. Since m is quasi-monomial, if an element mi,j is not zero, then there is no k other than j with mi,k = 0. This defines an injective correspondence σ from {1, . . . , s} into {1, . . . , r}, but not necessarily defined on the whole domain {1, . . . , s}. Then, an element (u, v) ∈ ur (q) ⊕ us (q) is in K0 if and only if for all 1 ≤ i ≤ s and 1 ≤ j ≤ r, mi,σ (i) uσ (i),j = vi,σ −1 (j ) mσ −1 (j ),j ,

(3.4)

where we make the convention that, whenever σ (i) or σ −1 (j ) is not defined, the corresponding side of (3.4) is read as zero. Set P1 to be the set of pairs (k, j ) with 1 ≤ k < j ≤ r such that σ −1 (k) is not defined or both σ −1 (k) and σ −1 (j ) are defined and σ −1 (k) < σ −1 (j ). There is a pattern algebra K1 associated to P1 as one can easily check. We claim now that if (u, v) ∈ K0 , then u ∈ K1 . For, otherwise, the matrix u would have a nonzero element uk,j with (k, j ) ∈ P1 , which means that σ −1 (k) is defined and either σ −1 (j ) is not defined or else, σ −1 (k) ≥ σ −1 (j ). In any of the two cases (3.4) implies uk,j = 0, a contradiction. Similarly, one obtains that v ∈ K2 , where K2 is the pattern algebra given by the entries (i, l) with 1 ≤ i < l ≤ s for which σ (l) is not defined or both σ (i) and σ (l) are defined and σ (i) < σ (l). Thus, K0 is a subalgebra of K1 ⊕ K2 . It only remains to be proved that the projections on K1 and K2 are onto. We consider only the first one, the other being similar. Thus, it suffices to exhibit for each (k, j ) ∈ P1 a matrix v such that (ek,j , v) ∈ K0 . Take v = 0 if σ −1 (k) is not defined and v = m−1 m −1 e −1 , otherwise. −1 σ −1 (j ),j σ (k),k σ (k),σ (j ) This lemma has the following useful consequence for our purposes. Lemma 14. Let K0 be as in the previous lemma. If K0l = 0, then the dimension of K0 is at least l(l + 1)/2. Proof. Note first that it suffices to prove the lemma for pattern algebras. Indeed, K0 is a subdirect sum of pattern algebras and, since K0l = 0, the l-th power of one of these algebras cannot be zero. On the other hand their dimensions are at most that

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of K0 , so all we have to prove is that if K is a pattern algebra and K l = 0, then dim K ≥ l(l + 1)/2. Now, the hypothesis K l = 0 implies that there exist matrices ei1 ,j1 , . . . , eil ,jl in K whose product is not zero. This forces j1 = i2 , . . . , jl−1 = il . Put il+1 = jl . Then for any 1 ≤ r ≤ s ≤ l, eir ,is+1 = eir ,ir+1 . . . eis ,is+1 ∈ K and in particular, dim K ≥ l(l + 1)/2. We need one more result before giving our last theorem. Proposition 15. Let A be an abelian ideal of J with Ap = 0 and let χ be an irreducible character of G = 1 + J . Take an irreducible constituent τ of χ1+A and denote by δ the linear functional on A defined by τ (exp a) = ψ(δ(a)) for all a ∈ A. Suppose that the conclusion of Theorem 11 holds for the algebra group 1 + A , where A = {u ∈ J | [u, A] ⊆ Ker δ}. Then there exists λ ∈ J ∗ such that χ = χλ . Proof. The inertia group of τ in G is T = 1 + A , the stabilizer of δ, thus by Clifford’s correspondence, there exists an irreducible character of T η such that χ = ηG and η1+A = η(1)τ . By the hypothesis on the characters of T , η is given by Kirillov’s formula for some linear functional δ of A . This can also be restated in the following way: if H is a subalgebra which is a maximal isotropic subspace of A with respect to Bδ , then η = α T , where α is the class function defined on 1 + H by α(Exp h) = ψ(δ (h)). We can suppose that H contains A. Then α is an extension of τ , whence δ is an extension of δ. Now we take λ ∈ J ∗ , an extension of δ to J . Clearly A is A⊥ , the orthogonal subspace of A with respect to Bλ and A is isotropic, so H is also a maximal isotropic subspace in J with respect to Bλ (recall the remark just before Proposition 1). Hence, χλ = α G = (α T )G = ηG = χ and we are done. The set of character degrees of Un (q) is well-known (this set is determined in [3] and [12] under the additional assumption that only q-powers belong to it and, in fact, this hypothesis can be dropped by Isaacs’ result in [6]): it consists of all the powers q i with 0 ≤ i ≤ f (n), where n(n−2) if n is even; 4 f (n) = n−1 2 if n is odd. 2 Moreover, q f (n) is the index of the abelian subgroup 1 + Ar,s for r = s = n/2, if n is even and r = (n − 1)/2, s = (n + 1)/2, if n is odd. In the sequel, we simply write K and A instead of Kr,s and Ar,s for these particular values of r and s. On the other hand, the size of a coadjoint orbit cannot exceed q 2f (n) . This is so because, as we know, the number of elements in the orbit of λ is q 2m , where 2m is the rank of Bλ . The number m is also the codimension of a maximal isotropic subspace and A, which has codimension f (n), is isotropic, whence m ≤ f (n).

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Theorem 16. Let G = Un (q) and 0 ≤ k ≤ p(p − 1)/4. Then the map λ → χλ gives a bijection between the irreducible characters of G of degree q f (n)−k and the coadjoint orbits of size q 2(f (n)−k) . Proof. Suppose first that λ is a coadjoint orbit of size q 2(f (n)−k) with k in the specified range. We pick as usual a subalgebra H which is a maximal isotropic subspace of un (q) containing A. There is no loss of generality if we suppose that the restriction of λ to A is represented by a quasi-monomial matrix m so that A⊥ = K0 ⊕ A and K0 is as defined in (3.3). Since H is isotropic, H ⊆ A⊥ , so H = (H ∩ K0 ) ⊕ A. If we p−1 prove that K0 = 0, then of course (H ∩ K0 )p−1 = 0 and we can apply Lemma 10 to conclude that the function αλ is a character of 1 + H . But αλG = χλ , so this last function is also a character and the theorem will follow in one direction. Suppose p−1 then by way of contradiction that K0 = 0. By Lemma 14, dim K0 ≥ p(p − 1)/2. To estimate the dimension of H ∩ K0 note that this is a maximal isotropic subspace of K0 (otherwise, we could find an isotropic subspace containing H properly), so its dimension is at least half the dimension of K0 and actually, this inequality is strict because equality occurs only when the alternating form is non-degenerate, which is not our case since the center of K0 is inside the radical of the restriction of Bλ to K0 . In short, 1 p(p − 1) dim K0 ≥ ≥ k. (3.5) 2 4 But the dimension of H ∩ K0 can be computed as follows: write 2m for the rank of Bλ and r for the dimension of its radical. Then, the assumption on the size of λ is that m = f (n) − k, thus dim H ∩ K0 >

dim H ∩ K0 = dim H − dim A = dim H − dim un (q) + f (n) = f (n) − m = k, in contradiction with (3.5). In the reverse direction, suppose that χ is an irreducible character of Un (q) of degree q f (n)−k , again with 0 ≤ k ≤ p(p − 1)/4. We take τ , an irreducible constituent of χ1+A , and δ ∈ A∗ such that τ (1 + a) = ψ(δ(a)) (observe that A2 = 0, so exp a = 1 + a). At the expense of replacing τ by one of its conjugates, we can suppose that δ is represented by a quasi-monomial matrix m. Thus, A = {u ∈ un (q) | [u, A] ⊆ Ker δ} = K0 ⊕ A, where K0 is as in (3.3). Bearing in mind Proposition 15 and Theorem 11, it suffices p−1 to prove that K0 = 0. Suppose this is not true, so that dim K0 ≥ p(p − 1)/2. By Clifford’s correspondence, there exists an irreducible character η of the inertia group of τ , which is T = 1+A , lying over τ such that χ = ηG . But T is the semidirect product of 1 + K0 and 1 + A and the latter is a normal and abelian subgroup, so τ extends to T and, if we take a fixed extension τ , then by Gallagher’s Theorem [5, Corollary 6.17], η = βτ for some irreducible character β of 1 + K0 . Then η(1)2 = β(1)2 < |1 + K0 |. But χ (1) = |G : T |η(1), whence η(1) = |T : 1 + A|/q k = |1 + K0 |/q k and it

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follows that |1 + K0 | < q 2k , that is 2k > dim K0 ≥ p(p − 1)/2, a contradiction. This completes the proof.

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C. A. M. André, Basic sums of coadjoint orbits of the unitriangular group, J. Algebra 176 (1995), 959–1000.

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C. A. M. André, Irreducible characters of finite algebra groups, in Matrices and group representations, Universidade de Coimbra, Coimbra 1999.

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B. Huppert, A remark on the character-degrees of some p-groups, Arch. Math. 59 (1992), 313–318.

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I. M. Isaacs, Sets of p-powers as irreducible character degrees, Proc. Amer. Math. Soc. 96 (1986), 551–552.

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I. M. Isaacs, unpublished notes.

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I. M. Isaacs and M. C. Slattery, Character degree sets that do not bound the class of a p-group, Proc. Amer. Math. Soc. 130 (2002), 2553–2558.

[10] D. Kazhdan, Proof of Springer’s hypothesis, Israel J. Math. 28 (1977), 272–286. [11] A. A. Kirillov, Variations on the triangular theme, in Lie Groups and Lie Algebras: E. B. Dynkin’s Seminar, Amer. Math. Soc. Transl. (2) 169, Amer. Math. Soc., Providence, RI 1995, 43–73. [12] G. I. Lehrer, Discrete series and the unipotent subgroup, Compositio Math. 28 (1974), 9–19. [13] M. Marjoram, Irreducible characters of small degree of the unitriangular group, Irish Math. Soc. Bull. 42 (1999), 21–31. [14] S. Moran, Dimension subgroups modulo n, Proc. Camb. Phil. Soc. 68 (1970), 579–582. [15] A. Vera-López and J. M. Arregi, Computing in unitriangular matrices over finite fields, preprint. Josu Sangroniz, Departamento de Matemáticas, Facultad de Ciencias, Universidad del País Vasco, Apartado 644, 48080 Bilbao, Spain E-mail: [email protected]

The signalizer method Ronald Solomon∗

General remarks The fundamental problem of the simple group classification may be formulated vaguely as follows: Given certain (local) information about a finite group G, determine whether or not G has a proper normal subgroup. More precisely, we may say (at least, a posteriori) that the problem is: Given a finite group whose local structure differs in some way from the local structure of a known simple group, prove that G has a proper normal subgroup. By local subgroup, we mean the normalizer of a non-identity p-subgroup of G. Let π be a set of primes and a set of Sylow p-subgroups of G, one for each p ∈ π . By π-local information we will mean the structure and embedding of the subgroups NG (R) as R ranges over the non-identity subgroups of members of . The work of Frobenius and Burnside from the early 1890s until 1910 achieved significant progress on the following part of the problem: Given a finite group G. Ascertain whether G has a nontrivial abelian quotient. In particular Burnside introduced what we now call the transfer homomorphism, which was further refined in ensuing decades by Grün, Wielandt, Yoshida and others. Thus by 1950, it is more or less fair to say that one could safely assume that G had no nontrivial solvable quotients. The next question was: Does G have a nontrivial solvable normal subgroup? In the context of finite groups, all successful approaches to date have proceeded from the following point of view. The prime graph (G) has vertex set the prime divisors of G with {p, q} an edge if and only if G has an element of order pq. We must begin our p-local study of G at some prime p and, since the Odd Order Paper [FT] of Feit and Thompson, 2 has been the obvious choice. We can then move among the primes p ∈ π2 where π2 is the set of primes which are vertices of the connected component of (G) containing the vertex 2. Other primes may not be accessible. ∗ Partially supported by an NSF research grant.

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The following small example illustrates the difficulty. Let G∗ = V H , where V is elementary of order 112 and H ∼ = SL(2, 5) acting faithfully on the normal subgroup V . Now let G be a finite group with the same 2-local structure as G∗ . Then π2 = {2, 3, 5}. Question: Given a group G with the same π2 -local data as G∗ , can one show that ∼ G = G∗ ? Answer: No. H as the same π2 -local data as G∗ . Weaker question: Given a group G with the same π2 -local data as G∗ , can we show that G has a nontrivial solvable normal subgroup? Answer: Yes. Note that if G ∼ = H , then Z(G) is a nontrivial solvable normal subgroup of G. Thus we may assume that H < G. G is then a member of a class of groups called Frobenius groups and a partial answer is given by the following theorem of Frobenius from 1901. Frobenius’ Theorem ([GLS2, 9.10]). Let G be a finite group with a proper subgroup H such that H ∩ gH g −1 = 1 for all g ∈ G − H . Then G has a proper normal subgroup K with K ∩ H = 1 and G = KH . Indeed K is a solvable (in fact, nilpotent) normal subgroup of G. (This fact was first proved by Thompson in his PhD dissertation in 1958 [GLS2, 9.11 (i)].) Frobenius finds K as the kernel of a nontrivial character of G. This may be regarded as a subtle (complexified) counting argument, made possible by the fact that conjugates of H intersect trivially. The methodology does not seem to extend far beyond the trivial intersection situation, though remarkable tours de force in this vein were achieved in the 1950s and early 1960s by Brauer, Suzuki, Feit and others. In particular Brauer and Suzuki extended the Frobenius-type result to all groups with (generalized) quaternion Sylow 2-subgroups ([GLS2, 15.2]). New methods were needed to deal with finite groups having a rich subgroup lattice with many nontrivial intersections. This is where the story of signalizers begins.

Signalizers We start with a trivial observation: Suppose that G is a finite group with proper subgroups M and N such that G = M, N. If X M and X N , then X G. Normality in both M and N is an unlikely condition to achieve. Much more likely is subnormality. It is always useful to bear in mind the fundamental example: G∗ = V H , where V is an elementary abelian r-group for some odd prime r and H is a nonabelian simple group.

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Assume that G is a finite group with the same 2-local structure as G∗ . We would like to prove that G has a nontrivial normal r-subgroup. Since H is nonabelian simple, the Brauer–Suzuki Theorem implies that H contains a Klein 4-group U ∗ and so CV (u∗ ) = 1 for some involution u∗ ∈ U ∗ . Moreover CV (u∗ ) ∩ N V ∩ N N for every subgroup N of G. So we may hope to prove that there is a nontrivial rsubgroup of CG (u) which is subnormal in every 2-local subgroup of G which contains it. (Here u is the involution of G corresponding to u∗ ∈ G∗ .) Methods to attack such problems were provided by Feit and Thompson in the Odd Order Paper. For a subgroup A of G, they define IG (A) to be the set of all Ainvariant subgroups of G which intersect A trivially, and IG (A; π ) to be the set of all π-subgroups in IG (A), for π a set of primes. Typically A is a Sylow p-subgroup of G or, more generally, a p-subgroup of G which is Sylow in ACG (A). The following fundamental lemma treats the solvable situation. Thompson–Bender Lemma ([GLS2, 23.3]). Let X be a solvable group and let p be an odd prime. Let A be a p-subgroup of X which contains every element of order p in CX (A). Then Op (X) is the unique maximal member of IX (A, p ). Under some circumstances we can say more. Glauberman’s K-Theorem ([GG]). Let X be a solvable group of odd order and let r be a prime with F (X) = Or (X). Let R be any r-subgroup of X containing F (X). Then there exists a nontrivial characteristic subgroup of R, K(R), such that K(R) X. Indeed the hypothesis of solvability can be weakened to a p-stability hypothesis. In particular if A is a p-subgroup of X for some prime p = r, then we may choose R to be a maximal element of IX (A, r). We let I∗X (A, r) denote the set of maximal elements of IX (A, r).

Signalizers: moving around We have yet to confront the fundamental issue – comparing the structures of two different maximal subgroups X and Y of a finite simple group G, where X and Y have a large intersection. Let us suppose then that there is a prime r such that F ∗ (X) and F ∗ (Y ) are both r-groups. Suppose further that p is a prime with p = r and that A is a p-subgroup of X ∩ Y . If Glauberman’s K-Theorem applies to both X and Y , then X = NG (K(R)) and Y = NG (K(S)), where R ∈ I∗X (A, r) and S ∈ I∗Y (A, r). Moreover Or (X) ≤ R and Or (Y ) ≤ S. Thus if F ∗ (X) ∩ F ∗ (Y ) = 1, then likewise R ∩ S = 1. What we need is a Sylow-type theorem relating R and S. If A = 1, then R and S are Sylow r-subgroups of G and the ordinary Sylow Theorem will suffice. In general we need an A-equivariant Sylow Theorem. This is provided in many contexts by the Feit–Thompson Transitivity Theorem.

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Feit–Thompson Transitivity Theorem ([GLS2, 23.12]). Let A be an abelian psubgroup of the finite group G which contains every element of order p in CG (A). Let r be a prime different from p and suppose that every r-local subgroup of G is solvable. If R, S ∈ I∗G (A, r) with R ∩ S = 1, then S is CG (A)-conjugate to R. Moreover if every p-local subgroup of G is solvable and A has p-rank at least 3, then all members of I∗G (A, r) lie in the same CG (A)-orbit. Corollary (A Uniqueness Theorem). Let G be a finite group and let p and r be distinct primes. Let A be a abelian p-subgroup of G with m(A) ≥ 3, such that A contains every element of order p in CG (A). Suppose further that every r-local subgroup H of G is solvable with F ∗ (H )(= F (H )) = Or (H ), and every p-local subgroup of G is solvable. If CG (A) is contained in a maximal r-local subgroup M of G, then M is the unique maximal r-local subgroup of G containing A. (Note: If Or (G) = 1, then M = G.) Proof. Let M be as given and suppose that N is a maximal r-local subgroup of G containing A. Let R ∈ I∗M (A, r) and let S ∈ I∗N (A, r). By Glauberman’s KTheorem, M = NG (K(R)) and N = NG (K(S)). Thus R, S ∈ I∗G (A, r). Now by the Transitivity Theorem, S is CG (A)-conjugate to R. Thus N = M c for some c ∈ CG (A). But CG (A) ≤ M, whence N = M, as claimed. This is an example of a Uniqueness Theorem. Typically one hopes to establish eventually that M is a strongly p-embedded r-local subgroup of G, and then to derive a contradiction. Of course the hypothesis of solvability of r-local and p-local subgroups is unrealistic in the general case, and much of the work of Gorenstein and Walter and somewhat later, of Bender, was directed towards finding the appropriate generalizations. The hypothesis that F ∗ (H ) = Or (H ) for every r-local H is also unjustified in general, but the following theorem of Bender hints at the fact that this is still the most important case. Bender’s Maximal Subgroup Theorem ([GLS2, 19.5]). Let M and N be distinct maximal subgroups of the finite simple group G. Suppose that there exist X ≤ F (M) and Y ≤ F (N ) with NG (X) ≤ N and NG (Y ) ≤ M. Then there exists a prime p such that F ∗ (M) = Op (M) and F ∗ (N) = Op (N). Notice that the hypothesis X ≤ F (M) is equivalent to the assumption that X is a nilpotent subnormal subgroup of M. Typically the Maximal Subgroup Theorem is used in the following way. We have a maximal subgroup M of the finite simple group G and we wish to establish that NG (X) ≤ M for many of the subnormal nilpotent subgroups X of M. Suppose then that NG (X) ≤ N with N maximal in G. Let Y be a non-trivial subgroup of NF (M) (X) with Y M. (For example let Y = Z(F (M)).) Then Y ≤ N and NG (Y ) ≤ M. So, if we can just show that Y ≤ F (N), then the hypotheses of the Maximal Subgroup Theorem hold and we either have the desired conclusion that NG (X) ≤ M or we have that F ∗ (M) and F ∗ (N ) are p-groups for

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some prime p. In the latter case we may try to use Glauberman’s K-Theorem and transitivity arguments to get a contradiction. The main problem then is to force subgroups like Y to lie in F (N). We now impose a hypothesis, which plays a central role in the Signalizer Analysis. Signalizer Hypothesis. There exists a subgroup M and an element t ∈ M of prime order r with CG (t) ≤ M. There exists a t-invariant r -subgroup Z of M, and there exists a second subgroup N containing Z, t. Hence Z ∈ IN (CN (t), r ), i.e. Z is a CN (t)-invariant r -subgroup of N. The general philosophy is that such r-signalizers should “sink” into Op (N ), and indeed, if they are p-groups, into Op (N) ≤ F (N). We subdivide the discussion into two cases. Case 1: [Z, t] = 1. This is the A × B Lemma case. Lemma. If N is solvable, then Z ≤ Or (N). Proof. Let N = N/Or (N). The result follows from the A × B Lemma applied to the action of t × Z on Or (N ). An immediate application is the following result. Balance Theorem of Thompson. Let t1 and t2 be commuting r-elements of G and assume that CG (ti ) is solvable for i = 1, 2. Then Or (CG (ti )) ∩ CG (tj ) = CG (ti ) ∩ Or (CG (tj )). The equation is called the Balance Equation. An easy consideration of commuting semisimple elements in finite groups of Lie type (or of commuting r-elements with large (but distinct) supports in alternating groups) quickly reveals that the Balance Equation routinely fails in the absence of solvable centralizers. Gorenstein’s idea was to restrict attention to a proper subgroup of Or (CG (t)) for which the analogue of Thompson’s Balance Equation will hold. The subgroup {1} is an obvious choice, but this yields no useful information. Again we should bear in mind the fundamental example where G∗ = V H , V a vector space in characteristic p, H a nonabelian simple group. Consider the following specific example: G∗ = V H where V is elementary abelian of order 113 and H = SL(3, 11) acting faithfully on V . If t is an involution of H , then O2 (CG∗ (t)) = Vt Ht , where |Vt | = 11 and |Ht | = 5. Moreover the eigenspaces of t and of Ht on V coincide. Thus if we take t to be an involution of H commuting with t, then O2 (CG∗ (t)) ∩ CG∗ (t ) = Ht O2 (CG∗ (t )).

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Thus Balance fails. But the subgroup we wanted to identify in any case was V . So instead of O2 (CG∗ (t)) we should choose Vt and then of course Vt ∩ CG∗ (t ) = 1 ≤ Vt . The problem though is to name Vt in the hypothetical finite simple group G which has the same 2-local structure as G∗ . If we set A = t, t , then the following solution suggests itself. Define θ1 (a) = [O2 (CG (a)), A] for each a ∈ A# . Now we see that in our example θ1 (a) = CV (a) for each a ∈ A# . For technical reasons, it turns out to be preferable to define θ(a) = [O2 (CG (a)), A]A , where A =

O2 (CG (a)).

a∈A#

Notice that A = 1 in our example and so again θ (a) = CV (a), as desired. The function θ on A# is an example of a signalizer functor and it was introduced by Goldschmidt in [G]. It was later used in the treatment of 2-components of type 2An [S] and, with some refinements, by Aschbacher in his characterization of groups of Lie type in odd characteristic [A]. When extended to arbitrary primes p, it was dubbed the 23 -balanced functor by Gorenstein and Lyons [GLS2, 21.9] and became the first of a family of k + 21 -balanced functors they defined. Earlier, versions of had been used by Gorenstein and Walter to define k-balanced functors for k ≥ 2. But I have gotten ahead of myself and have not defined a signalizer functor. The following definition is due to Goldschmidt, refining an earlier definition by Gorenstein. Here we let P (G) denote the set of all subsets of G. Definition. If A is an elementary abelian r-subgroup of the finite group G, then an A-signalizer functor is a function θ : A# → P (G) satisfying for all a, a ∈ A# : (1) θ (a) is an A-invariant r -subgroup of CG (a); and (2) θ (a) ∩ CG (a ) = CG (a) ∩ θ(a ). The crucial theorem is the Signalizer Functor Theorem. A preliminary version was proved by Gorenstein. Then, for r = 2, the theorem was proved by Goldschmidt. Under the additional hypothesis that θ(a) is a solvable group for all a ∈ A# , the theorem was proved by Glauberman. Finally, under a weak assumption on proper simple sections of G (valid for all (known) simple groups), the full theorem was established by McBride. The Signalizer Functor Theorem ([GLS2, 21.3,21.4]). Let A be an elementary abelian r-subgroup of the finite group G with m(A) ≥ 3. Let θ be an A-signalizer functor on G such that either

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(1) θ (a) is solvable for all a ∈ A# ; or (2) G has the r-McBride property ([GLS2, 2.6]). Then WA = θ (a) : a ∈ A# is an r -subgroup of G. One then hopes to prove that NG (WA ) is a strongly r-embedded r -local subgroup of G; and then that WA G with θ(a) = CWA (a) for all a ∈ A# . If this is accomplished, then we will (in our example) have reconstructed the normal vector space V from the pieces visible inside r-local subgroups of G. As noted in the example above, a naive choice of θ will often lead to the failure of the Balance Equation. If the 23 -balanced functor is used, then often it suffices to establish the following: Xa := [θ(a) ∩ CG (a ), A] ≤ Or (CG (a )). Let C = CG (a )/Or (CG (a )). Then, as in the proof of Thompson’s Balance Theorem, the A × B Lemma shows that [Or (C), Xa ] = 1. It easily follows that an obstruction to balance must be an a, Xa -invariant quasisimple component K of C such that 1 = Xa ≤ [Or (C K,Xa ,a (a)), A]. Unfortunately such “locally unbalanced” configurations are ubiquitous in alternating groups and groups of Lie type in odd characteristic. Fortunately we are permitted to make a judicious choice of A as well. The prototypical good choice is the following: A ≤ Z ∗ (K) = Z ∗ (K1 K2 K3 ), where K/O2 (K) ∼ = SL2 (q) × SL2 (q) × SL2 (q) for some odd q. This is the choice made by Aschbacher in the “generic” case of his characterization of groups of odd Lie type, and it leads to an easy verification of local balance. Good generation of (most) Lie type groups (proved by Seitz and Lyons [GLS3, 7.3.3]) then leads easily to the dichotomy that either the Goldschmidt functor is trivial (as desired) or that G has a strongly embedded subgroup, contradicting the Suzuki-Bender Theorem ([GLS4, Theorem SE]). The triviality of the Goldschmidt functor then leads easily to a weak (but sufficient) version of Thompson’s B-Conjecture, namely that L2 (CG (a)) is semisimple for all a ∈ A# . A similar and almost equally easy argument may be found in [GLS5] to handle the “wide” Lie type groups of characteristic 2. As usual the “narrow” groups cause all the mischief. Case2: [Z, t] = 1. This case plays an important role in Bender’s proof of the 2-Signalizer Functor Theorem and also in the analysis of the Special Odd Case [GLS1, Part 2, 14.2]. In particular, since the Signalizer Functor Theorem requires that m(A) ≥ 3, this case plays a crucial role in the analysis of simple groups G with m2 (G) = 2. Though it has antecedents in the work of Thompson, the shape of this analysis was largely sculpted by Helmut Bender and it is often called the Bender Method. We assume in this case a slightly modified version of the Signalizer Hypothesis.

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Signalizer Hypothesis. N is a finite group and p and r are distinct primes with t ∈ N of order r and Z a CN (t)-invariant p-subgroup of N with [Z, t] = Z. We let N = N/Or (N) and we wish to prove that Z = 1, whence Z is a nilpotent subnormal subgroup of N. In the solvable case (or more generally, when E(N) = 1), we confront a Hall-Higman Configuration. Dropping the bars, we may consider H = QZ t, where Q is a Z t-invariant q-group for some prime q = p and Z = [Z, t] acts faithfully on V := [Q/(Q), Z]. Lemma. If Q, Z and t are as above with p odd (or r not a Fermat prime) and if Z is CH (t)-invariant, then q = r. Proof. A theorem of Hall-Higman type implies that t has a free summand in its action on V unless p = 2 and r is a Fermat prime. In particular, CV (t) = 1. If also r = q, then CV (t) = CQ (t)(Q)/(Q). But Z ∈ IN (CN (t), p), whence [CQ (t), Z] ≤ Q ∩ Z = 1, a contradiction. In particular, if p = 2 and Z, t ≤ N , then Z centralizes O2 (F (N)). Indeed the following is true. Lemma (Thompson [T, 5.17]). Z centralizes F (N) unless O2 (N) has nilpotence class 2. We cannot say much more because of examples such as the following: H = GL2 (3), Q = Q8 , |Z| = 3, |t| = 2; and H = QZ t ≤ L t, L ∼ = U3 (4), |Q| = 26 , |Z| = 5, with t an involutory graph automorphism of L. We conclude with a few remarks about the way this Case 2 analysis is used in the classification of simple groups G of Special Odd Type. We begin with a simple group G and a 2-central involution t of G. The Special Odd Type hypothesis quickly implies that F ∗ (CG (t)) is not a 2-group. If M = CG (t) is a maximal subgroup of G, then Bender’s Maximal Subgroup Theorem is effective and a Case I type analysis gives control of O2 (CG (t)). Suppose then that CG (t) is properly contained in a maximal subgroup M of G. With some exceptions (notably the dihedral Sylow 2-subgroup case), we easily conclude that F ∗ (M) = F (M) has odd order. The heart of the analysis is aimed at proving that NG (X) ≤ M for most t-invariant subgroups X of F (M). (After that, a short argument using the G-fusion of involutions yields a contradiction or leads to the parabolic structure of PSL(3, pn ).)

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Thus assume that CG (t) ≤ M with M a maximal subgroup of G and with F ∗ (M) of odd order. Let X be a non-trivial t-invariant subgroup of F (M) and embed NG (X) in a maximal subgroup N of G. Then half of the hypotheses of Bender’s Maximal Subgroup Theorem apply to the pair (M, N ). It remains to find Y ≤ F (N) with NG (Y ) ≤ M. We proceed by a bootstrap argument, incrementally forcing more and more subgroups Y of F (M) to satisfy NG (Y ) ≤ M. Typically we arrive at the following situation: Set Y = [Op (M) ∩ N, t] with Y = 1. (This must occur for some odd prime p by the A × B Lemma.) Suppose we know that NG (Y0 ) ≤ M for some Y0 ≤ Y . Then it will suffice to show that Y ≤ Op (N). Note that Y = [Y, t] and Y ∈ IN (CN (t), p). So the Case 2 analysis applies to yield that the obstruction to Y ≤ Op (N ) must be either a class 2 2-subgroup of N/Op (N) or a quasisimple component of N/Op (N ). Korchagina has done an exhaustive analysis of the quasisimple obstructions which can arise if t has order 2 or 3. These seem to be the crucial cases for the Classification proof. (In the Special Odd Case, t has order 2.) If Y can be shown to lie in F (N ), then Bender’s Maximal Subgroup Theorem yields that either M = N (the desired conclusion) or that both F ∗ (M) = Op (M) and F ∗ (N ) = Op (N). In the latter case Glauberman’s K-Theorem and transitivity arguments usually suffice to yield a contradiction. The existence of large 2-group and quasisimple obstructions to “pushing down” signalizers makes this method (at least in the naive form outlined here) impractical in large groups. Particularly serious is the possibility of a quasisimple component K of N/Op (N ) with K ∼ = PSL(n, pm ) and containing a maximal parabolic H with H = Y CK (t), i.e. Y projects onto the unipotent radical of a maximal parabolic subgroup H of K and t lies in the center of a Levi complement of H . Fortunately, the Signalizer Functor Method, as outlined in Case I, is quite effective in large groups. Thus the only case where it seems necessary to confront the situation described above is when G has semidihedral or wreathed Sylow 2-subgroups and K ∼ = PSL(3, pm ). Luckily in this situation we are rescued by a modular character argument of Brauer. (There is in fact an unpublished alternate local approach to this configuration, due to Aschbacher.)

Concluding remarks The Signalizer Method pioneered by Feit and Thompson in the Odd Order Paper has proved to be remarkably robust, extending from the original context of solvable locals of odd order to the general context of arbitrary local subgroups (with known composition factors). It has provided the mechanism to prove weak versions of the B-Theorem (whose full-strength version, stated below after a few definitions, is a corollary of the Classification Theorem):

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Definitions. Let H be a finite group and let p be a prime. A p-component L of H is a subnormal subgroup of H minimal subject to covering a quasisimple component L of H = H /Op (H ). The Gorenstein–Walter p-layer of H , Lp (H ) is the product of all p-components of H . Then Bp (H ) is the product of those p-components of H which are not quasisimple, while Ep (H ) is the product of all quasisimple p-components of H . Thus Lp (H ) = Bp (H )Ep (H ) with Bp (H ) ∩ Ep (H ) ≤ Z(Ep (H )). Note that a p-component must by definition have order divisible by p. Thus E2 (H ) = E(H ), but in general Ep (H ) may be a proper characteristic subgroup of E(H ). The B-Theorem. Let G be a finite group, p a prime and H a p-local subgroup of G. Then Bp (H ) ≤ Bp (G). In particular, Op (Bp (H )) ≤ Op (G). This theorem guarantees the semisimplicity of p-layers in p-local subgroups of a finite simple group G. We can then use an amalgam of quasisimple p-components to identify G. The weak version of the B-Theorem establishes the result only for certain p-locals of a K-proper simple group G, but enough p-locals to produce the desired amalgam of quasisimple p-components. Is the good the enemy of the better in this context? Perhaps. We can only hope that some new researchers will bring fresh skeptical eyes to this problem and find a beautiful new approach. We offer a few random comments: In the Killing–Cartan classification of semisimple Lie groups and Lie algebras the analogue of the Signalizer Problem is the Solvable Radical Problem, which is brilliantly resolved by the Killing form and Cartan’s Criterion. Is there an analogue – perhaps some trilinear map on a finite group G which detects the existence of a nontrivial solvable normal subgroup? Lyons has observed a suggestive analogy between the Signalizer Functor Theorem and the Curtis–Tits Theorem (characterizing the finite groups of Lie type from local data). Let A be an elementary abelian r-subgroup of G of rank 3 and let θ be a signalizer functor defined on A. For each hyperplane B of A define θ (b). θ(B) = b∈B #

The subgroups θ(B) play the role of the rank 1 subgroups in the Curtis–Tits set-up, and the subgroups θ(a) play the role of the rank 2 subgroups. The condition θ(a) = θ(B) ∩ θ(a) : B ≤ A, |A : B| = r is an immediate consequence of the Balance Condition. The Signalizer Functor Theorem then asserts that if the rank 1 and rank 2 subgroups are all r -subgroups, then the full group θ(1) := θ(a) : a ∈ A# is also an r -group.

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This suggests the possibility of bypassing the Signalizer Functor Theorem and proceeding directly to a characterization of finite groups generated by a family of rank 1 groups KB , labelled by the hyperplanes B of some fixed elementary abelian r-group A, with K/Or (K) ∈ Chev(p) of BN -rank 1, for some fixed distinct primes p and r, subject to a Balance Condition specifying the nature of the rank 2 amalgams of these rank 1 groups. In any event, Lyons’ remark highlights the crucial role played by the number 3 in both the Signalizer Functor Theorem and the Curtis–Tits Theorem. Knowledge of rank 1 and rank 2 groups is presupposed and only for rank at least 3 is new information obtained. Brauer proved that for any finite group H and any prime p, Op (H ) is the intersection of the kernels of the ordinary irreducible characters in the principal p-block of H . This certainly suggests a role for modular character theory in the analysis of signalizers, a fact which has not escaped the attention of Alperin, Broué, Puig, Geoff Robinson and others. Certain nagging doubts persist, however. For one thing, there is no nice relationship between Op (H ) and Op (G) for H a p-local subgroup of G in general. Even if G is of characteristic p-type, i.e. all p-locals of G are p-constrained (even solvable), problems arise when the p-rank of G is at most 2. The B-Theorem is a general fact, but no one has given a representation theoretic characterization of Bp (G). The fact that general statements about signalizers and strongly p-embedded subgroups only become true for p-rank at least 3 seems to argue against a role for character theory, unless p-modular results specific to the case of p-rank at least 3 are uncovered. Thus far, the contrary seems to be true, i.e. the strongest p-modular results are obtained for p-rank at most 2. Indeed this “miracle” saves the Classification proof. Borovic has extended Signalizer Analysis to the context of infinite groups of finite Morley rank. He discovered that Signalizer Analysis is more elegant in this setting because one can name the vector-space part of the signalizer effectively. This is due to the fact that one can easily distinguish the additive group of an algebraically closed field from the multiplicative group of an algebraically closed field. By contrast there is no way to distinguish the additive group of the field of order 2 from the multiplicative group of the field of order 3. Perhaps then suffering is inevitable in the finite case. While looking forward to new minds with new insights, we must also look back with gratitude for the creativity and perseverance of the heroes of the “first generation” signalizer analysis: Feit and Thompson, Gorenstein and Walter, Suzuki and Bender, Goldschmidt and Glauberman, Alperin and Brauer, Gorenstein and Harada, Aschbacher, Gorenstein and Lyons; but most of all: John G. Thompson.

References [A]

M. Aschbacher, A characterization of Chevalley groups over fields of odd order, Ann. of Math. 106 (1977), 353–468.

[FT]

W. Feit and J. G. Thompson, Solvability of groups of odd order, Pacific J. Math. 13 (1963), 775–1029.

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[G]

D. M. Goldschmidt, Weakly embedded 2-local subgroups of finite groups, J. Algebra 21 (1972), 137–148.

[GG]

G. Glauberman, Prime power factor groups of finite groups, II, Math. Z. 117 (1970) 46–56.

[GLS1] D. Gorenstein, R. Lyons, R. Solomon, The Classification of the Finite Simple Groups, Math. Surveys Monogr. 40.1, Amer. Math. Soc., Providence, RI 1994. [GLS2] D. Gorenstein, R. Lyons and R. Solomon, The Classification of the Finite Simple Groups, Number 2, Math. Surveys Monogr. 40.2, Amer. Math. Soc., Providence, RI 1996. [GLS3] D. Gorenstein, R. Lyons and R. Solomon, The Classification of the Finite Simple Groups, Number 3, Math. Surveys Monogr. 40.3, Amer. Math. Soc., Providence, RI 1998. [GLS4] D. Gorenstein, R. Lyons and R. Solomon, The Classification of the Finite Simple Groups, Number 4, Math. Surveys Monogr. 40.4, Amer. Math. Soc., Providence, RI 1999. [GLS5] D. Gorenstein, R. Lyons and R. Solomon, The Classification of the Finite Simple Groups, Number 5, Math. Surveys Monogr. 40.5, Amer. Math. Soc., Providence, RI 2002. [GLS6] D. Gorenstein, R. Lyons and R. Solomon, The Classification of the Finite Simple Groups, Number 6, Math. Surveys Monogr. 40.6, Amer. Math. Soc., Providence, RI, to appear. [K2]

I. Korchagina, 2-Signalizers in almost simple groups, to appear.

[K3]

I. Korchagina, 3-Signalizers in almost simple groups, in Finite Groups 2003 (C. Y. Ho, P. Sin, P. H. Tiep, A. Turull, eds.), Proceedings of the Gainesville Conference on Finite Groups, March 2003, Walter de Gruyter, Berlin 2004, 229–246. R. Solomon, Finite groups with intrinsic 2-components of type Aˆ n , J. Algebra 33 (1975), 498–522.

[S] [T]

J. G. Thompson, Nonsolvable finite groups all of whose local subgroups are solvable, I, Bull. Amer. Math. Soc. 74 (1968), 383–437.

Ronald Solomon, Department of Mathematics, The Ohio State University, Columbus, Ohio 43210, U.S.A. E-mail: [email protected]

Classical groups in dimension 5 which are Hurwitz M. Chiara Tamburini and Alexandre E. Zalesski

Abstract. In this paper we provide new examples of Hurwitz classical groups. 2000 Mathematics Subject Classification: 20D, 20F

1. Introduction We recall that a group G is said to be (2, 3, k)-generated if it can be generated by an involution and an element of order 3 whose product has order k. A finite (2, 3, 7)generated group is called a Hurwitz group. The relevance of such groups is due to the famous result of Hurwitz, that a finite group of order 84(g − 1) can act faithfully on a compact Riemann surface of genus g if and only if it is (2, 3, 7)-generated. Moreover 7 is the first value of k for which the triangle group T (2, 3, k) is infinite and non-soluble. The smallest Hurwitz group is PSL2 (7), already discovered by Klein. But the first significant examples are due to Macbeath who, in 1969 [12], classified the Hurwitz groups of type PSL2 (q). They are precisely PSL(2, p) if p ≡ 0, ±1 (mod 7), and PSL(2, p3 ) if p ≡ ±2, ±3 (mod 7). This result provided in particular many transitive permutational representations of T (2, 3, 7): they arise from the action of PSL2 (q) on the cosets of its subgroups. These representations, and a clever method of joining them via handles developed by Graham Higman, culminated in the famous theorem of Conder that for n > 167 the alternating groups An are Hurwitz [2]. He also proved the stronger result that for k ≥ 7 almost all of the alternating groups are (2, 3, k)generated, and an even more striking generalization of this was given in [6], where it is shown that any Fuchsian group surjects almost all of the alternating groups. There is now evidence that the property of being a Hurwitz group is quite common among finite simple groups. In fact Conder’s constructive permutational methods were later generalized to the linear context, and led to the discovery that most finite classical groups are Hurwitz (see [10] and [11]). On the other hand, using charactertheoretic methods, Malle determined the precise values of q, m and n for which the exceptional simple groups of Lie type G2 (q), 2 G2 (32m+1 ), 3 D4 (p n ) and 2 F4 (22n+1 ) are Hurwitz (see [13, 14] and, for Ree groups, see also [7] and [16]). Finally, by the contribution of several authors, it is known exactly which of the 26 sporadic simple groups are Hurwitz (see, for example, [3] and [25]). For more information in this very

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wide area we refer to the papers in the references and the literature quoted there, and to the recent survey [22]. Although the above mentioned results on classical groups are satisfactory if considered asymptotically with respect to the ranks of the groups under consideration, the lower bounds for their ranks are certainly much higher than necessary for the existence of Hurwitz generators. But a constructive search of Hurwitz generators for groups of Lie type of small rank cannot be done in a uniform way. The main difficulty consists in the fact that the property of being Hurwitz may depend also on the size of the field and its characteristic, and not only on the rank and type. Evidence of this is given by the above mentioned results of Macbeath and Malle, and by this paper. Further evidence is given in [4], where is made a systematic application of a well-known formula of Scott [17] to show that many classical groups of small rank are not Hurwitz. In this paper we concentrate on the small rank case. Let us first recall that the Hurwitz subgroups of PSL3 (F), where F is an algebraically closed field of positive characteristic, are isomorphic to those discovered by Macbeath, as proved in [1]. And this fact, which also applies to PSL4 (F) (see [22]), may have erroneously discouraged, for a long time, the search for (projective) linear Hurwitz groups. Actually the reason why PSL3 (F) and PSL4 (F) do not contain new Hurwitz groups is due to Scott’s formula which gives a very strong restriction for small dimensional matrix groups. However in dimension 5, we detect new examples. In fact the main result of this paper is the following: Theorem 1.1. Assume that k ≥ 7 is a prime number. If k = p, set q = p. Otherwise, let n be the order of p modulo k, and suppose n ≡ 0 (mod 4). Set q = pn if n is odd, n q = p 2 if n is even. Then the following groups are (2, 3, k)-generated: PSL5 (q), if p ≡ 1 (mod 5); PSU5 (q 2 ), if p ≡ −1 (mod 5); PSU5 (q 4 ), if p ≡ ±2 (mod 5). If k = 7, the only other irreducible Hurwitz subgroups of PSL5 (F) are isomorphic to PSL2 (q), with q as above, except PSL2 (8) and PSL2 (27). We note that our assumptions on n are certainly satisfied if k ≡ 3 (mod 4). For k = 7, in each characteristic p = 0, 5 we obtain one new example of a Hurwitz group. Namely, if p = 7, this is the group PSU5 (74 ). Otherwise, every p = 7 determines n (equal to 1 or 3), hence q, and depending on the residue of p modulo 5, one can pick a Hurwitz group among the three groups in the theorem list. We emphasize that SL5 (q) and SU5 (q 2 ) are not Hurwitz for any q according to [4]. Our proof, which uses the knowledge of the subgroups of SL5 (p m ), is based on results on product of matrices, Scott’s formula, Lang-Steinberg’s theorem and the notion of rigidity. The latter notion is widely explored in the theory of Riemann surfaces and their applications, in particular, to differential equations and Galois theory. Especially a version of the rigidity notion, introduced by John Thompson in [23], plays a prominent role in recent developments of inverse Galois theory.

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365

2. Preliminary results In this paper F denotes an algebraically closed field of characteristic p > 0. Definition 2.1. Let a1 , a2 , a3 ∈ GLn (F). The triple (a1 , a2 , a3 ) is called linearly g g g rigid if a1 a2 a3 = 1 and, for all elements gi in GLn (F) satisfying a1 1 a2 2 a3 3 = 1, gi g gi there exists g ∈ GLn (F) such that ai = ai for i = 1, 2, 3. (Here ai is meant to be gi−1 ai gi ). Given a group H and a representation f : H → GLn (F), let V be the vector space Fn , considered as an H -module, via f . For any subset A of H , define VA as the subspace of fixed points of f (A) and denote by dVA its dimension over F. Define dˆVA in the same way, with respect to the dual representation. Then a special case of Scott’s formula reads Theorem 2.2. Assume that H is generated by x and y. Then y

xy

dVx + dV + dV ≤ n + dVH + dˆVH .

(2.1)

a = According to the notation used in [4], for each a ∈ M = Matn (F), we set dM dim CM (a). Applying formula (2.1) to the action of f (H ) by conjugation on M, we have the following:

Corollary 2.3. Suppose that f : H → GLn (F) is irreducible, and H = x, y. Then y

xy

x + dM + dM ≤ n2 + 2. dM

(2.2)

The following lemma appears as Theorem 2.3 in [21]: Lemma 2.4. Let a, b ∈ GLn (F) and set (ab)−1 = c. Suppose that the group a, b is irreducible and that b c a dM + dM + dM = n2 + 2.

Then the triple (a, b, c) is linearly rigid. In the following, for each power q of p, σ will denote one of the Frobenius automorphisms φq or φq− of GLn (F), defined respectively by: q

(gij ) → (gij )

and

q

(gij ) → (gj i )−1 .

GLn (q) is the subgroup of the fixed points of φq , i.e. GLn (q) = CGLn (F) (φq ). Similarly Un (q 2 ) = CGLn (F) (φq− ). By a well known result of S. Lang and R. Steinberg (see [20]), the map GLn (F) → GLn (F) given by g → g −1 σ (g) is surjective. Corollary 2.5. [19] Let σ be as above and assume that A is a set permuted transitively by GLn (F). Assume further that τ : A → A is a map such that τ (g(a)) = σ (g) (τ (a)) for all g ∈ GLn (F) and all a ∈ A. Then there exists a ∈ A such that τ (a) = a.

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Proof. Let a ∈ A. As GLn (F) acts transitively on A, there exists g ∈ GLn (F) such that g (τ (a)) = a. By Lang-Steinberg’s theorem, there exists x ∈ GLn (F) such that x −1 σ (x) = g. Then x −1 (σ (x) (τ (a))) = a implies σ (x) (τ (a)) = x(a), i.e. τ (x(a)) = x(a). We conclude that a = x(a) is fixed by τ . The following theorem is known and it is used a few times in [15] but it is not stated explicitly. We provide a proof for the reader’s convenience. Theorem 2.6. Let (a1 , a2 , a3 ) be a linearly rigid triple, with ai ∈ GLn (F). For i = 1, 2, 3, let Ci be the conjugacy class of ai and suppose that Ci ∩CGLn (F) (σ ) is nong empty, where σ is as above. Then there exists g ∈ GLn (F) such that ai ∈ CGLn (F) (σ ), for all i = 1, 2, 3. Proof. Set W = GLn (F)3 and make GLn (F) act on W via g g g (w1 , w2 , w3 ) → w1 , w2 , w3 for all (w1 , w2 , w3 ) ∈ W and all g ∈ GLn (F). Define A = {(b1 , b2 , b3 ) ∈ C1 × C2 × C3 | b1 b2 b3 = 1} . Clearly A is invariant under this action and the rigidity of (a1 , a2 , a3 ) is equivalent to the fact that GLn (F) acts transitively on A. Let τ : W → W be the map defined by (w1 , w2 , w3 ) → (σ (w1 ), σ (w2 ), σ (w3 )) . From Ci ∩ CGLn (F) (σ ) = ∅, it follows that σ (Ci ) ⊆ Ci for i ≤ 3. Indeed, if g σ (g) ci ∈ Ci ∩ CGLn (F) (σ ) then σ ci = ci for all g ∈ GLn (F). Thus τ (C1 × C2 × C3 ) ⊆ C1 × C2 × C3 whence also τ (A) ⊆ A. Moreover g g g g g g τ (b1 , b2 , b3 ) = σ (b1 ), σ (b2 ), σ (b3 ) = σ (b1 )σ (g) , σ (b2 )σ (g) , σ (b3 )σ (g) , that is, τ satisfies the assumptions of Corollary 2.5. Hence there is in A a triple fixed by τ . This means that there exist bi ∈ Ci such that b1 b2 b3 = 1 and σ (bi ) = bi . The latter is equivalent to bi ∈ GLn (q) (respectively ∈ Un (q 2 )) for i ≤ 3.

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367

3. Hurwitz groups Lemma 3.1. Consider the matrices x and y in SL5 (F), of respective orders 2 and 3, defined as follows: 1 0 −1 1 −1 0 1 , 1 0 0 0 1 y = x = 0 1 1 0 0 1 0 0 1 0 Moreover, for each prime k ≥ 7, consider z in SL5 (F), of order k, defined as follows. If p = k, 1 0 0 0 0 1 1 0 0 0 z= 0 1 1 0 0 0 0 1 1 0 0 0 0 1 1 If p = k, let ε denote a primitive k-th root of 1 in F and set θj = j + −j for an integer j . For all integers = j such that 1 ≤ , j ≤ k−1 2 define 1 0 −1 1 θ z = z,j = 0 −1 1 θj Assume p = 5 and let 1 = η ∈ F be such that η5 = 1. Then there exist x conjugate to x and y conjugate y such that xy = ηz. Moreover G = x, y is an irreducible subgroup of SL5 (F). Proof. The existence of x and y with the required properties follows from Silva [18, Theorem 2] if p = 7, and from Kurtz [8, Theorem 1.1] if p = 2, 3. As to the irreducibility of G, we note that any abelian epimorphic image of a group generated by two elements whose orders divide 2 and 3 respectively, has order dividing 6. So, assume by contradiction that U is a proper G-invariant subspace of F5 . Then, for each g ∈ G, det(g|U )6 = 1. But ηI5 ∈ G and the restriction of ηI5 to U has determinant of order 5. We conclude that G is irreducible. Remark 3.2. It maybe worth noting that the previous Lemma actually holds also in characteristic 0. Theorem 3.3. Assume p = 5, k ≥ 7 a prime, and let G = x, y be as in Lemma 3.1. If k = p, set q = p. Otherwise, let n be the order of p modulo k, and suppose

368

M. Chiara Tamburini and Alexandre E. Zalesski n

n ≡ 0 (mod 4). Set q = p n if n is odd, q = p 2 if n is even. Then q is the minimum power of p such that: (1) G is conjugate to a subgroup of SL5 (q), if p ≡ 1 (mod 5); (2) G is conjugate to a subgroup of SU5 (q 2 ), if p ≡ −1 (mod 5); (3) G is conjugate to a subgroup of SU5 (q 4 ), if p ≡ ±2 (mod 5). Proof. Recall that G is irreducible. From y

ηz

x dM + dM + dM = 13 + 9 + 5 = 52 + 2 it follows that the triple x, y, (ηz)−1 is linearly rigid by Lemma 2.4. Let us denote by Cx , Cy , Cηz and Cz the conjugacy classes of x, y, ηz and z. From x , y ∈ SL5 (p) it follows that

Cx ∩ SL5 (p m ) = ∅,

Cy ∩ SL5 (p m ) = ∅

for all m ≥ 1.

Moreover, noting that any permutation matrix preserves the hermitian form defined by the identity matrix, and recalling that SL2 (pm ) SU2 (p2m ) preserves a nondegenerate hermitian form, it is easy to see that Cx ∩ SU5 (p2m ) = ∅,

Cy ∩ SU5 (p 2m )

for all m ≥ 1.

From xy = ηz, with o(η) = 5 and o(z) = k, it follows that ηI5 ∈ (xy)k . Thus ηI5 belongs to any conjugate of G. Thus, by Theorem 2.6, G is conjugate to a subgroup of SL5 (p ), for some , precisely when ηI5 ∈ SL5 (p ) and Cz ∩ SL5 (p ) = ∅. Similarly G is conjugate to a subgroup of SU5 (p 2 ), for some , precisely when ηI5 ∈ SU5 (p 2 ) and Cz ∩ SU5 (p 2 ) = ∅. By definition of q, we have GF(q) = GF(p) (θ ) = GF(p) θj . Hence z ∈ SL5 (q). Moreover z ∈ SU5 (q 2 ) ∩ SU5 (q 4 ). This fact, if p = k, is easily seen recalling that SL2 (pm ) SU2 (p2m ) preserves a non-degenerate hermitian form. And, if p = k, one may consider for example the matrix 1 0 0 0 0 −1 1 0 0 0 0 −1 1 0 0 0 −2−1 1 1 0 0 −2−1 1 1 1 conjugate to z, and preserving the hermitian form defined by J = antidiag(1, 1, 1, 1, 1). Our claims follow immediately noting that, as q is supposed to be a power of p with odd exponent, we have q ≡ p (mod 5). Finally q is the minimum power of p for which cases (1), (2) and (3) hold. This is clear if p = k. On the other hand, if p = k, it follows from the facts 2 2 that the characteristic polynomial of z is (t − 1)(t − θ t + 1)(t − θj t + 1) and GF(q) = GF(p) θ + θj , θ θj since we are assuming n ≡ 0 (mod 4).

Classical groups in dimension 5 which are Hurwitz

369

Theorem 3.4. Let k, n and q be defined as in Theorem 3.3. (1) The following groups are (2, 3, k)-generated: PSL5 (q),

if p ≡ 1

(mod 5);

PSU5 (q ), if p ≡ −1

(mod 5);

PSU5 (q ), if p ≡ ±2

(mod 5).

2 4

(2) If k = 7, the only other irreducible Hurwitz subgroups of PSL5 (F) are isomorphic to PSL2 (q) with q as above, except PSL2 (8) and PSL2 (27). Proof. (1) We claim that the group G = x, y of Theorem 3.3 coincides respectively with SL5 (q) in (1), SU5 (q 2 ) in (2), and SU5 (q 4 ) in (3). The centre of G has order 5. In particular G cannot be a simple group and cannot be an orthogonal group. Moreover G is irreducible, as shown in Lemma 3.1. Thus inspection of the subgroups of SL5 (pm ) (see [5] and [24]) gives that G coincides with SL5 (p ) with p ≡ 1 (mod 5), or SU5 (p2 ) with p ≡ −1 (mod 5) or SU5 (p4 ) with p ≡ ±2 (mod 5), for some . Since q is the minimum power of p for which cases (1), (2) and (3) of Theorem 3.3 hold, and q ≡ p (mod 5), we conclude p = q. (2) Let H be the linear preimage of an irreducible (2, 3, 7)-generated subgroup of PSL5 (F). Then, up to conjugation, we may assume H = x, y with x 2 = y 3 = I and either (i) (xy)7 = ηI5 where η ∈ F has order 5, or (ii) (xy)7 = 1.

y

xy

x ≥ 13, d In both cases dM M ≥ 9 and dM ≥ 5. On the other hand, Scott’s formula (2.2) and irreducibility force the condition y

xy

x + dM + dM ≤ 25 + 2. dM

Thus x must be conjugate to x and y must be conjugate to y . Moreover, if p = 7, z = (xy)5 must have five different eigenvalues and determinant 1. Thus z can only be conjugate to one of the matrices z,i , as in Lemma 3.1, with k = 7. And, if p = 7, z must be conjugate to the Jordan block J5 . In case (i), by the rigidity condition, H is conjugate to the group G of Theorem 3.3. In case (ii), by the rigidity condition, the group H can belong to at most 3 conjugacy classes. Let us recall that the groups PSL2 (p m ) have an embedding into SL5 (F), arising from the action on homogeneous polynomials of degree 4 in 2 variables. In this action, the three non-conjugate (2,3,7) generating triples of PSL2 (q), when Hurwitz according to Macbeath’s Theorem, remain non-conjugate in SL5 (F). This embedding is irreducible precisely when p ≥ 5. Thus, in case (ii) with p = 2, 3, H coincides with one of the groups in the statement. Finally, if p = 2, 3, H is finite by 2.6 and H preserves a symmetric bilinear form by [4]. It follows p = 3 as groups in characteristic 2 preserving symmetric bilinear forms are reducible. Thus it could only be an irreducible subgroup of 5 (3m ) PSp4 (3m ), for some m. Moreover, by [4], H cannot be SL5 (3a ) or SU5 (3a ), for any a ≤ m.

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Hence, inspection of the irreducible subgroups of 5 (3m ), H should be isomorphic to 5 (3a ) for some a. But these groups are not even (2, 3)-generated by [9].

References [1]

J. Cohen, On non-Hurwitz groups and non-congruence subgroups of the modular group, Glasgow Math. J. 22 (1981) 1–7.

[2]

M. Conder, Generators for alternating and symmetric groups, J. London Math. Soc. (2) 22 (1980), 75–86.

[3]

M. D. E. Conder, R. A. Wilson and A. J. Woldar, The symmetric genus of sporadic groups, Proc. Amer. Math. Soc. 116 (1992), 653–663.

[4]

L. Di Martino, C. Tamburini and A. Zalesskii, On Hurwitz groups of low rank, Comm. in Algebra 28 (2000), 5383–5404.

[5]

L. Di Martino and A. Wagner, The irreducible subgroups of PSL5 (q), where q is odd, Results Math. 2 (1979), 54–61.

[6]

B. Everitt, Alternating quotients of Fuchsian groups, J. Algebra 223 (2000), 457–476.

[7]

G. A. Jones, Ree groups and Riemann surfaces, J. Algebra 165 (1994), 41–62.

[8]

C. Kurtz, On the product of diagonal conjugacy classes, Comm. Algebra 29 (2001), 769–779.

[9]

M. W. Liebeck and A. Shalev, Classical groups, probabilistic methods, and the (2, 3)generation problem, Ann. of Math. 144 (1996), 77–125.

[10] A. Lucchini and M. C. Tamburini, Classical groups of large rank as Hurwitz groups, J. Algebra 219 (1999), 531–546. [11] A. Lucchini, M. C. Tamburini and J. S. Wilson, Hurwitz groups of large rank, J. London Math. Soc. (2) 61 (2000), 81–92. [12] A. M. Macbeath, Generators of the linear fractional groups, Proc. Symp. Pure Math. 12 (1969), 14–32. [13] G. Malle, Hurwitz groups and G2 (q), Canad. Math. Bull. 33 (1990), 349-357. [14] G. Malle, Small rank exceptional Hurwitz groups, Groups of Lie type and their geometries, in Groups of Lie Type and Their Geometries (Como, 1993), ed. by W. M. Kantor and L. Di Martino, London Math. Soc. Lecture Note Ser. 207, Cambridge University Press, Cambridge 1995, 173–183. [15] G. Malle and B. H. Matzat, Inverse Galois Theory, Springer Monogr. Math., SpringerVerlag, Berlin 1999. [16] C. H. Sah, Groups related to compact Riemann surfaces, Acta Math. 123 (1969), 13–42. [17] L. L. Scott, Matrices and cohomology, Ann. of Math. 105 (1977), 473–492. [18] F. C. Silva, The eigenvalues of the product of matrices with prescribed similarity classes, Linear and Multilinear Algebra 34 (1993), 269–277.

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[19] T. A. Springer and R. Steinberg, Conjugacy classes, in A. Borel, R. Carter, C. W. Curtis, N. Iwahori, Seminar on Algebraic Groups and Related Finite Groups, Lecture Notes in Math. 131, Springer-Verlag, Berlin 1970; corr. 2nd printing 1986. [20] R. Steinberg, Endomorphisms of linear algebraic groups, Mem. Amer. Math. Soc. 80, Amer. Math. Soc., Providence, RI 1968. [21] K. Strambach and H. Völklein, On linearly rigid tuples, J. Reine Angew. Math., 510 (1999), 57–62. [22] C. Tamburini and M. Vsemirnov, Hurwitz groups and Hurwitz generation, Survey article, to appear. [23] J. Thompson, Some finite groups which appear as Gal(L/K), where K ⊆ Q(µn ), J. Algebra 89 (1984), 437–499. [24] A. Wagner, The subgroups of PSL5 (2a ), Results Math. 1 (1978), 207–226. [25] R. A. Wilson, The Monster is a Hurwitz group, J. Group Theory 4 (2001), 367-374. M. C. Tamburini, Dipartimento di Matematica e Fisica, Università Cattolica del Sacro Cuore, Via Musei 41, 25121 Brescia, Italy E-mail: [email protected] A. E. Zalesski, School of Mathematics, University of East Anglia, Norwich NR4 7TJ, UK E-mail: [email protected]

A classification of Weyl groups as finite {3, 4}-transposition groups Franz G. Timmesfeld

1. Introduction A set D of involutions of the finite group G is called a set of {3, 4}-transpositions of G, if the following hold: (1) G = D and D g ⊆ D for all g ∈ G. (2) If d, e ∈ D then o(de) ≤ 4. Finite groups with a set D of {3, 4}-transpositions and containing no solvable normal subgroup have been classified under the additional condition If d, e ∈ D with o(de) = 4, then (de)2 ∈ D.

(+)

(See [2] and [3].) Without this condition a classification has never been tried, since there are many examples (for example the Baby Monster). But using the classification and known properties of involutions in the automorphism groups of the simple groups this should now be no big problem. In this note we go into an other direction, i.e., we look for conditions under which such a {3, 4} transposition group G is a Weyl group, which means we must deal with groups which are not nearly simple. We prove: Theorem 1.1. Let G be a finite group generated by a set D of {3, 4}-transpositions satisfying: (a) If a, b, c ∈ D and X = a, b, c, then X is an epimorphic image of a Weyl group and a, b and c are images of reflections. (b) If ai , x ∈ D, i = 1, . . . , 4 satisfying (i) o(ai x) = 3 for i = 1, . . . , 4 and (ii) o(ai aj ) ≤ 2 for i, j ≤ 4, then X = x, ai | i = 1, . . . , 4 is an epimorphic image of W (D4 ) and x and the ai are images of reflections. Then G is an epimorphic image of a Weyl group W and D is the image of the set of reflections of W .

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Franz G. Timmesfeld

Notice that the Weyl group W of the theorem does not need to be irreducible and that all Weyl groups, which do not have a direct factor isomorphic to W (G2 ), do occur. In fact the group G of the theorem can occur in different disguises as an image of a Weyl group. (For example D8 as W (C2 ), but also as an image of W (C2 ) × W (A1 ), in which case all involutions of D8 are in D.) The main reason for proving the above theorem is, that it will be used to give a presentation type classification of arbitrary ‘Lie-type groups’ (In the sense of [Ti2, II §5]) similar to theorem 2 of [5], in which such a {3, 4}-transposition group plays the role of a Weyl group. (For the proof of theorem 2 of [5] we had to give a classification of W (A ), W (D ) and W (E ) as finite 3-transposition groups satisfying (a) and (b), see [Ti3, (6.7)].) The reader will notice that for most parts of the proof we just rely on condition (a). Hence it should not be difficult to give a classification of {3, 4}-transposition groups G satisfying condition (a) with O2 (G) = 1. (Without condition (b) there exist additional cases in the conclusion of the theorem, for example the symplectic and orthogonal groups over GF(2).) But since for the application I have in mind we must allow O2 (G) = 1, I introduced condition (b).

2. Preliminary results Let in this section D be a set of {3, 4}-transpositions of the finite group G satisfying conditions (a) and (b) of the theorem. We will often use pictorial notation, i.e. we will write for d, e ∈ D: d

e

e

e

d

e

e

e

d

e

e

iff o(de) = 2 iff o(de) = 3 iff o(de) = 4

e

Let F (D) be the graph with vertex set D and edges pairs e, d ∈ D with o(ed) = 3. Lemma 2.1.

(1) There exist no a, b, c ∈ D satisfying

a

b e T e T e c

or

a

b e TT TT e c e

A classification of Weyl groups as finite {3, 4}-transposition groups

(2) If a, b, c ∈ D with

a

c

b

e

375

e

e

then c = a b and a, b, c D8 . (3) If a, b, c ∈ D with

a

e eb JJ

J

e

J c

then a, b, c is isomorphic to W (C3 ) or W (A3 ) ( (4) If a, b, c ∈ D with

a

4 !).

e eb

J J e

c

then a, b, c is isomorphic to

4

or

3.

Proof. Let X = a, b, c. Then by condition (a) of the theorem X is an image of a Weyl group W of rank ≤ 3 and a, b, c are images of reflections. Now there is no rank 3 Weyl group with 3 reflections satisfying (1), so (1) is impossible. In case (2) W is irreducible. Hence it is easy to see that rank W = 3 and then that W W (C2 ). Now (2) follows immediately. In (3) W is also irreducible and thus W W (C3 ), which implies (3). Finally in (4) W W (A3 ) or W (A2 ), since W (C3 ) 4 ×ZZ2 is not generated by 3 reflections satisfying (4). This proves (2.1). Lemma 2.2. Let E ⊆ D be a connectivity component of F (D) with |E| > 1. Then E is a class of 3-transpositions of R = E and E g = E for all g ∈ G. Proof. Suppose false and under all pairs e, f ∈ E with o(ef ) = 4 choose e, f so that e and f are connected by a path of minimal length in F (D) (and whence in F (E)!). Let ei ∈ E, i = o, . . . , n such that e0 = e, en = f and (e0 , e1 , . . . , en ) is such a minimal path from e to f in F (D). Then by (2.1) (1) n > 2. By the minimality of d(e, f ) (d denotes the distance in F (D)) we have o(ei ej ) = 4 for all i, j ≤ n with (i, j ) = (o, n). Moreover by the minimality of the path from e to f we have o(ei ej ) = 3 for all i, j ≤ n with j = i ± 1. Hence o(ei ej ) = 2 for all i, j ≤ n with j = i ± 1 and (i, j ) = (o, n); i.e. we have

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Franz G. Timmesfeld

e2 e1 e

e

e

· e

·

·

·

·

een−2 \\ en−1 e \\ f e

.

Let a be the highest reflection in Y = e1 , . . . , en−1 n . Then o(ea) = 3, since H = e, Y n+1 and thus CH (e) = e × e2 , . . . , en−1 . By the same argument also o(f a) = 3, a contradiction to (2.1) (1). This shows that E is a class of 3-transpositions of R. It remains to show E g ⊆ E for all g ∈ G. Since G = D it suffices to show E d ⊆ E for all d ∈ D. If now d ∈ D centralizes an element e of E then clearly E ∩ E d ⊇ {e} and thus E = E d . So we may assume o(ed) = 4 for all e ∈ E. But then, if e, f ∈ E with o(ef ) = 3, we obtain e e ef JJ

J

e

. J d

Hence by (2.1)(3) o(def ) = 2, a contradiction to ef ∈ E. Corollary 2.3. Let E and R be as in (2.2). Then R is a factor group of W (A ), ≥ 2, W (D ), ≥ 4 or W (E ), 6 ≤ ≤ 8 and E is the image of the reflections. Proof. This is a direct consequence of (2.2) and theorem (6.7) of [5]. Lemma 2.4. Let E and R be as in (2.2) and suppose R W (A ) +1 , ≥ 2. Let d ∈ D − E. Then either d ∈ C(R) or R 4 W (A3 ), R, d is a factor group of W (C3 ) 4 ×ZZ2 and E ∪ d R is the image of the reflections of W (C3 ). Proof. Pick d ∈ D − E. Then, since d normalizes E, d induces even in case = 5 an inner automorphism on R. Hence z = td ∈ C(R) for some t ∈ R. Now o(tt r ) = o(dzd r z) = o(dd r ) ≤ 4 for each r ∈ R. If now d is an isolated vertex of F (D), then o(tt r ) = o(dd r ) ∈ {2, 4} for each r ∈ R. Hence by Baer’stheorem, see [A, (39.6)] t ∈ O2 (R). Hence either t= 1 and d = z ∈ C(R) or R 4 . In the second case obviously R, d = R, z ×Z Z or 4 4 and the conclusion of (2.4) holds. So we may assume that d is not isolated in F (D) and thus we have by (2.2) that o(tt r ) ≤ 3 for each r ∈ R. Hence in case = 5 we obtain t ∈ E (or t = 1 which is to show). But then o(t r d) = o(t r tz) = 6 for some r ∈ R, a contradiction to t r , d ∈ D. Thus we may assume that = 5, t is a conjugate of (12)(34)(56) and E is the class of transpositions of R. But then, if we let e = (12), f = (23) and t r = (14)(25)(36), we obtain

A classification of Weyl groups as finite {3, 4}-transposition groups e

377

e ef JJ

J

e

J dr

and e, f, d r 3 2 , a contradiction to (2.1)(3). Thus in any case t = 1 and d = z ∈ C(R), which proves (2.4). Lemma 2.5. Let E and R be as in (2.2) and suppose R is isomorphic to W (D ) or W (D )/Z(W (D )). Pick d ∈ D − E. Then one of the following hold: (1) d ∈ C(R). (2) d, R W (C ) or W (C )/Z(W (C )) and E is (the image of) the class of short reflections and d R the (image of) the class of long reflections of W (C ). Proof. Suppose first > 4. Then by (2.4) applied to G/O2 (R) we have [R, d] ≤ O2 (R). Hence either (1) holds or there exists an e ∈ E with e = ed ∈ eO2 (R) ∩ E = eO2 (R) . Thus there exists e1 , . . . , ee ∈ E with e−1 e1

e

e−2

e2

e H HH

e …

e

R = e1 , . . . , e and ed = e−1 . e

e

Now by (2.1)(2) we obtain o(dei ) = 2 for i ≤ − 3. Moreover by (2.1)(4) either o(e−2 d) = 2 or 4. In the first case we get e1

e2

e

e …

e−2

e−1

e

e

d

e e

−1 , we and (2) holds. So we may assume o(e−2 d) = 4. But then, setting f = e−2 obtain by (2.1) (3)

e1

e

e2

e …

e−3

e

f

e

e−1

e

d

e

and again (2) holds. Next assume = 4. Then there exist exactly three normal 2-subgroups Ni , i = 1, 2, 3 of R with R/Ni 4 . Thus one of these normal 2-subgroups, say N1 , must be d-invariant. Set G0 = R, d and D0 = D ∩ G0 . Then D0 is a set of {3, 4}-transpositions of G0 satisfying conditions (a) and (b) of the theorem. Hence applying (2.4) to G0 /N1 we obtain [R, d] ≤ N1 . Now arguing in exactly the same manner as in case > 4 we see that either [R, d] = 1 or R, d W (C4 ) or W (C4 )/Z(W (C4 )) and (2) holds.

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Lemma 2.6. Let E and R be as in (2.2) and suppose R is isomorphic to a factor group of W (E ), 6 ≤ ≤ 8. Then D − E ⊆ C(R). Proof. Pick d ∈ D − E. Then d induces an inner automorphism on R. Since E is the only class of 3-transpositions of R satisfying condition (a) of the theorem we obtain either d ∈ C(R) or de ∈ C(R) for some e ∈ E (W (E6 ) U4 (2) contains another class of 3-transpositions. But this class contains 3 involutions satisfying e T e T e which generate a group of order 33 · 2!). But in the second case we obtain as in (2.4) o(deg ) = 6 for some g ∈ R, a contradiction. Lemma 2.7. Let E and R be as in (2.2) and suppose that R W (D ) (resp. W (D )/Z(W (D )). Let d ∈ D − E such that Y = R, d W (C ) or W (C )/Z(W (C ). Then for each x ∈ D − Y one of the following holds: (1) x ∈ C(Y ). (2) = 4, o(xd) = 3 and H = Y, x W (F4 ) or W (F4 )/Z(W (F4 )). Moreover H ∩ D consists of the long and short reflections of W (F4 ). Proof. Suppose first x ∈ C(R). Then o(xd) = 2 or 4 and (2.1)(2) shows o(xd) = 2. Hence (1) holds. So we may by (2.4) assume R, x W (C ) or W (C )/Z(W (C )). Hence if = 4 then d and x centralize O2 (R) and R/O2 (R). But then (2.1)(1) and (2) imply either x ∈ Y or x ∈ C(Y ). Finally if = 4 the same argument shows that d, x permutes the three normal 2-subgroups N1 , N2 and N3 of R with R/Ni 4 as a 3 or x ∈ C(Y ) resp. x ∈ Y . Hence in the first case d, x induces the full group of diagram automorphisms on R W (D4 ) and it is easy to see that Y, x W (F4 ) or W (F4 )/Z(W (F4 )). Since 3 | o(dx), this shows that (2) holds.

3. Proof of the theorem We prove the theorem by induction on |D|. First assume D ∩ Z(G) = ∅. Pick d ∈ D ∩Z(G) and set D1 = D −{d}, G1 = D1 . Then D1 is a set of {3, 4}-transpositions of G1 satisfying the hypothesis of the theorem. So, by induction assumption, there exists a finite Weyl group W1 with set of reflections R1 and a surjective homomorphism ϕ1 : W1 → G1 with ϕ1 (R1 ) = D1 . Hence W1 × t, t ZZ2 , is a Weyl group of type W1 × A1 and it is easy to see that there exists a surjective homomorphism

A classification of Weyl groups as finite {3, 4}-transposition groups

379

ϕ : W1 × t → G with ϕ|W1 = ϕ1 and ϕ(t) = d. (To define ϕ one has to distinguish between the cases d ∈ G1 and d ∈ G1 !) So we may assume Z(G)∩D = ∅. Next assume that there are two isolated vertices a, b of F (D) with o(ab) = 4. Let A = D ∩ a, b and D1 = D − A. If c ∈ D1 , then by (2.1)(1) and (2) c commutes with a and b. Hence G1 = D1 centralizes a, b and moreover, again by induction assumption, we may assume that there exists a finite Weyl group W1 with set of reflection R1 and a surjective homomorphism ϕ1 ·W1 → G1 with ϕ1 (R1 ) = D1 . Thus it is easy to show that G is an image of a Weyl group of type W1 × C2 resp. of type W1 × C2 × A1 with D the image of the reflections. Hence we may assume that L is abelian, where L is the set of isolated vertices of F (D). Let, as in §2, Ei , i = 1, . . . , n be the connectivity components of F (D) of cardinality > 1. We distinguish between the cases: (1) L = ∅. (2) L = ∅. ˙ . . . ∪E ˙ n . If now [Ei , Ej ] = 1 for i = j , then D is a set of In case (1) D = E1 ∪ 3-transpositions of G and it is easy to see that the theorem is a consequence of (2.3). (One just has to look how the Z(Gi ), Gi = Ei are amalgamated!) So assume [E1 , E2 ] = 1. Then by (2.4) - (2.6) H = E1 , e is an image of W (C ), G1 = E1 an image of W (D ) and E1 ∪ eH is the image of the reflections of W (C ) for each e ∈ E2 . Now E2 = eH . Hence by (2.7) we obtain for f ∈ E2 − eH that Y = H, f is an image of W (F4 ) and o(ef ) = 3. Now |Y ∩ E2 | = 12 = |E1 | since G1 W (D4 ). Reversing the roles of E1 and E2 this implies E2 ⊆ Y and thus Y = E1 , E2 and E1 ∪ E2 is the image of the reflections of W (F4 ). We now claim [Ei , Ej ] = 1 for i ≤ 2 and j > 2. Namely, if a ∈ E1 , b ∈ E2 with o(ab) = 4, then by (2.1)(1) and (2) each c ∈ Ej , j > 2 commutes with a and b. Since each Ei is a conjugacy class in Ei and a normal set in G, this proves our claim. So if we set D1 = E1 ∪ E2 , D2 = E3 ∪ · · · ∪ En and Xi = Di for i = 1, 2, then [X1 , X2 ] = 1, X1 is an image of W (F4 ), D1 is the image of the reflections of W (F4 ) and D2 satisfies the hypotheses (a) and (b) of the theorem in X2 . Hence applying the induction assumption to X2 it follows easily that the theorem holds. Finally assume that L = ∅. Since L ∩ Z(G) = ∅ and since L is elementary abelian, there exists for each xi ∈ L a connectivity component Ei of F (D) with [Ei , xi ] = 1. Let Yi = Ei , xi and Li = xiYi . Then by (2.5) Yi is an image of W (Ci ) and Ei ∪ Li is the set of reflections of Yi . By (2.7) D0 ⊆ C(Yi ) for D0 = D − (Ei ∪ Li ), since Li consists of isolated vertices of F (D). Hence applying the induction assumption to D0 and D0 , this shows again that the theorem holds. 2

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References [1] M. Aschbacher, Finite group theory, Cambridge University Press, Cambridge 1986. [2] B. Fischer, Finite groups generated by 3-transpositions, I, Invent. Math. 13 (1971), 232–246. [3] F. G. Timmesfeld, A characterization of Chevalley and Steinberg groups over F2 , Geom. Dedicata 1 (1973), 269–321. [4] F. G. Timmesfeld, Abstract Root Subgroups and Simple Groups of Lie-Type, Monogr. Math. 95, Birkhäuser Verlag, Basel 2001. [5] F. G. Timmesfeld, Presentations for certain Chevalley-groups, Geom. Dedicata 73 (1998), 85–117. F. G. Timmesfeld, Mathematisches Institut, Arndtstraße 2, 35392 Giessen, Germany E-mail: [email protected]

Powerful subgroups of 2-groups Lawrence E. Wilson

Abstract. We improve the bound on the index of a powerful subgroup of a 2-group of rank r. We also prove that certain dimension subgroups are powerful under a natural hypothesis. 2000 Mathematics Subject Classification: 20

1. Introduction When Lubotzky and Mann introduced powerful p-groups in [LM], one immediate goal was to prove results about p-groups in which every subgroup can be generated by at most r elements, called a p-group of rank r. Their primary technique was to prove that such groups have a characteristic powerful subgroup of index bounded by a function of p and r. If we let λ(r) be an integer such that 2λ(r)−1 < r ≤ 2λ(r) , then Theorem 1.14 of [LM] states that a p-group of rank r with p odd has a characteristic powerful subgroup of index at most prλ(r) . In the case of p = 2 the result is weaker. Theorem 4.1.14 of the same paper states that there is a characteristic powerful subgroup of index at most 2r(λ(r)+1) . The primary result of this paper is the following improvement on this bound for 2-groups. Theorem A. Let G be a finite 2-group such that every characteristic subgroup of G can be generated by at most r elements. If λ(r) is defined by 2λ(r)−1 < r ≤ 2λ(r) then λ(r) G has a characteristic powerful subgroup of index at most 2r(λ(r)+1)−(2 −r) . Lubotzky and Mann use their bound to prove (Proposition 4.2.6) that the order of a 2-group of rank r is at most 2r(λ(r)+1)+re where the exponent of the group is 2e . (We note that they use m where we use λ(r) and that there is a typo in the proposition; the m2 should be m.) The same proposition also contains the fact that the derived length of the group is at most λ(e + 1) + λ(r) + 1. Using their proof with our bound does not improve this bound for the derived length; to do so the bound would need to be reduced to at most 2rλ(r) . However, substituting our bound leads to the following improvement in the bound on the order of a 2-group of rank r. Corollary B. Let G and λ(r) be as in T heoremA and let 2e be the exponent of G. λ(r) Then |G| ≤ 2r(λ(r)+1)−(2 −r)+re .

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The proofs of some important results on p-groups involve bounding the rank and deducing the existence of a characteristic powerful subgroup of bounded index. For example, the derived length of a finite p-group is bounded in terms of the order and number of fixed points of a p-automorphism which acts on the group. An early step in the proof is to deduce a bound on the rank of this group and from this that a bounded term of the derived series is contained in a powerful subgroup. The proof then proceeds by considering the special case of powerful p-groups. Again, our improvement is not sufficient to improve this bound on derived length.

2. Notation and tools We will be primarily concerned with the dimension subgroups of a given p-group. The dimension subgroup series is the fastest descending series of G beginning at D1 (G) = G such that [Di (G), Dj (G)] ≤ Di+j (G) and Di (G)p ≤ Dpi (G). Lazard j found a closed form for the dimension subgroups, Dk (G) = ipj ≥k γi (G)p . Let us pause now to review the notation. j For a p-group G and a positive integer j the subgroup Gp is the subgroup generj ated by the elements g p for g ∈ G. Commonly, this subgroup will contain elements j not of the form g p . Another important series in the group G is the lower central series. This is the series defined by γ1 (G) = G and γi+1 (G) = [γi (G) , G]. For subgroups H and K of G we recursively define [H, K; 0] = H and [H, K; + 1] = [[H, K; ], K]. Therefore, γ+1 (G) = [G, G; ]. If p is odd then the p-group G is powerful if γ2 (G) ≤ Gp . A 2-group is powerful if γ2 (G) ≤ G4 . As in Exercise 2.4.i of [DdMS], this condition is equivalent to the condition that γ2 (G) ≤ (G2 )2 . A normal subgroup N of G is said to be powerfully embedded in G if [N, G] ≤ N p for odd primes or if [N, G] ≤ N 4 if p = 2. Again, for p = 2 it suffices to prove that [N, G] ≤ (N 2 )2 . The following lemma was known to Zassenhaus and Jennings. It is a consequence of Lazard’s closed formula for the dimension subgroups and also a consequence of Proposition 11.12 of [DdMS]. We adopt the notation that n∗ = n/p is the least integer such that pn∗ ≥ n. Lemma 2.1. If G is a finite p-group then Dn (G) = γn (G) Dn∗ (G)p . Many results on commutators of dimension subgroups are known. Each is essentially a consequence of the following result of P. Hall, [Ha]. Lemma 2.2. For any group G, elements x and y of G, prime p, and positive integer k the following relations hold: k k k k k−i (1) (xy)p ≡ x p y p modulo γ2 (x, y )p · ki=1 γpi (x, y )p , k k k k−i (2) [x p , y] ≡ [x, y]p modulo γ2 (x, [x, y] )p · ki=1 γpi (x, [x, y] )p .

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We need the following result of [W]. Lemma 2.3. Let G be a finite p-group. Then for positive integers a ≥ b, [Da (G), Db (G)] ≤ γa+b (G) γa ∗ +b (G)p Da+p2 b (G)Dpa+pb (G) Lemmas 2.3 and 2.1 are used to prove the following basic result. In the proof, we first adopt the standard practice of writing Di for Di (G) and i for γi (G). Lemma 2.4. [Di (G), Di (G)] ≤ γ2i (G) γi+i ∗ (G)2 (Di (G)2 )2 for all 2-groups G and all positive integers i. 2 Proof. Lemma 2.3 implies that [Di , Di ] ≤ 2i i+i ∗ D4i . Lemma 2.1 implies that 2 2 D4i ≤ 4i D2i and that D2i ≤ 2i Di . Hence D4i ≤ 2i (Di2 )2 . Substituting this for 2 2 2 D4i in the conclusion from Lemma 2.3, we find that [Di , Di ] ≤ 2i i+i ∗ (Di ) as desired.

A very important tool in this study is Theorem 2.3 (1) of [SS]. Theorem 2.5. If G is a finite p-group then [Dk (G), G; ] =

γi+ (G)p

j

ip j ≥k

Lastly, we include a very basic lemma which will be of great help. Lemma 2.6. Let M and N be normal subgroups of a p-group G and suppose N ≤ N p [N, G]M. Then N ≤ M.

3. Powerful and potent dimension subgroups Chapter 11 of [DdMS] includes a proof of Lazard’s result that if a finitely generated h pro-p group G satisfies γm (G) ≤ Gp with m < p h then G is p-adic analytic. The technique used there is to conclude that Dn (G) = Dn+1 (G) for some n (specifically, n = m if p does not divide m and n = m + 1 otherwise). Lazard’s result is proven by finding that certain subgroups of G are powerful. One way to do this is to use the following theorem of David Riley, Theorem 11.5 of [DdMS]. Theorem 3.1. If G is a finitely generated pro-p group and Dn (G) = Dn+1 (G) then Di (G) is powerful for i ≥ n − n+1 p if p is odd or for i ≥ n for p = 2. We will prove that more dimension subgroups are powerful under the stronger hypothesis of Lazard’s result. Actually, we will be working with the slightly weaker hypothesis that γm (G) ≤ D2h (G). A similar result for odd primes can be found in [W]. We assume that h ≥ 2 as m < 2 produces an assumption met only by the trivial group.

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Theorem 3.2. Let G be a finite 2-group such that γm (G) ≤ D2h (G). (1) If m < 2h−1 + 2h−2 then Di (G) is powerful if i ≥ (2m − 1)/3. (2) If 2h−1 + 2h−2 ≤ m < 2h then Di (G) is powerful if i ≥ 2(m − 2h−1 ). Proof. The proof of this result will be divided into two cases as in the statement of the theorem. Consider first the case that m < 2h−1 + 2h−2 and suppose i ≥ (2m − 1)/3. 2 2 2 Lemma 2.4 implies that [Di , Di ] ≤ 2i i+i ∗ (Di ) . Note that as i ≥ (2m − 1)/3 we have that 2i ≥ (4m − 2)/3 and hence 2i ≥ m. Therefore, 2i ≤ [D2h , G; 2i − m] = 2h +2i−m 22h−1 +2i−m

h

j

22h−j +2i−m

j =2 h

≤

2h +2i−m 22h−1 +2i−m

=

2h +2i−m 22h−1 +2i−m D24h−2 +2i−m

j =2

(Theorem 2.5)

D24h−2 +2j −2 (2i−m)

As m < 2h , Lemma 2.6 implies that 2i ≤ 22h−1 +2i−m D24h−2 +2i−m . We claim that 2h−1 + 2i − m ≥ i + i ∗ . As m ≤ 2h−1 + 2h−2 − 1, we have that 4m ≤ 2h+1 + 2h − 4 = 3 · 2h − 4. Hence 6m − 3 · 2h + 3 ≤ 2m − 1 and thus 2m − 2h + 1 ≤ (2m − 1)/3 ≤ i. Thus 3i + 1 ≤ 2h + 4i − 2m and 2i ∗ ≤ i + 1 so 2i + 2i ∗ ≤ 2h + 4i − 2m and therefore i + i ∗ ≤ 2h−1 + 2i − m. We also claim that 2h−2 + 2i − m ≥ i. As m + 1 ≤ 2h−1 + 2h−2 = 3 · 2h−2 , we have that 3m − 3 · 2h−2 ≤ 2m − 1 ≤ 3i. Therefore i ≥ m − 2h−2 and hence 2h−2 + 2i − m ≥ i. 2 4 2 2 2 We have now proven that 2i ≤ i+i ∗ Di and hence [Di , Di ] ≤ i+i ∗ (Di ) . We ∗ now wish to argue that i + i ≥ m. We take cases on the residue of m mod 3. If m = 3 then i ≥ (6−1)/3 and hence i ≥ 2. Therefore, i ∗ ≥ and i +i ∗ ≥ m. If m = 3 + 1 then i ≥ (6 + 2)/3 and hence i ≥ 2 + 1. Therefore i ∗ ≥ + 1 and so i + i ∗ ≥ 3 + 2 > m. If m = 3 + 2 then i ≥ (6 + 3)/3 = 2 + 1 and so i ∗ ≥ + 1. Therefore i + i ∗ ≥ 3 + 2 = m. Therefore, i+i ∗ ≤ [D2h , G; i + i ∗ − m] = 2h +i+i ∗ −m ≤ 2h +i+i ∗ −m

h j =1 h j =1

j

22h−j +i+i ∗ −m

(Theorem 2.5)

D22h−1 +2j −1 (i+i ∗ −m) = 2h +i+i ∗ −m D22h−1 +i+i ∗ −m

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As m < 2h , Lemma 2.6 implies that i+i ∗ ≤ D22h−1 +i+i ∗ −m . We claim that 2h−1 + i + i ∗ − m ≥ i which is equivalent to i ∗ ≥ m − 2h−1 . As 2i ∗ ≥ i it suffices to argue that i ≥ 2m − 2h . However, we have already shown that i ≥ 2m − 2h + 1. Therefore, we have proven that i+i ∗ ≤ Di2 and therefore [Di , Di ] ≤ (Di2 )2 as desired, completing the proof of (1). Assume now that 2h−1 + 2h−2 ≤ m < 2h and let i ≥ 2(m − 2h−1 ). Lemma 2.4 2 2 2 implies that [Di , Di ] ≤ 2i i+i ∗ (Di ) . We have that 2i ≥ 4m − 2h+1 = m + 3m − 2h+1 ≥ m + 9 · 2h−2 − 2h+1 = m + 2h+1 + 2h−2 . Therefore, 2i ≤ [D2h , G; 2i − m] = 2h +2i−m 22h−1 +2i−m ≤ 2h +2i−m 22h−1 +2i−m =

h

j

22h−j +2i−m

j =2 h j =2

(Theorem 2.5)

D24h−2 +2j −2 (2i−m)

2h +2i−m 22h−1 +2i−m D24h−2 +2i−m

As m < 2h , Lemma 2.6 implies that 2i ≤ 22h−1 +2i−m D24h−2 +2i−m . We claim that 2h−1 + 2i − m ≥ i + i ∗ . If i > 2(m − 2h−1 ) then i ≥ 2(m − 2h−1 ) + 1 and hence 2i ≥ 2(m − 2h−1 ) + i + 1. As 2i ∗ ≤ i + 1 we have that 2i ≥ 2(m − 2h−1 ) + 2i ∗ and hence i ≥ m − 2h−1 + i ∗ . Therefore 2h−1 + 2i − m ≥ i + i ∗ . If i = 2(m − 2h−1 ) then i ∗ = m−2h−1 and i +i ∗ = 3(m−2h−1 ) = 4(m−2h−1 )−m+2h−1 = 2h−1 +2i −m. We also claim that 2h−2 + 2i − m ≥ i which is equivalent to i ≥ m − 2h−2 . As m ≥ 2h−1 + 2h−2 we have that m − 2h ≥ −2h−2 . We have assumed that i ≥ 2 4 2(m − 2h−1 ) = m + m − 2h ≥ m − 2h−2 . Therefore, we have that 2i ≤ i+i ∗ Di 2 2 2 and hence [Di , Di ] ≤ i+i ∗ (Di ) . As i ≥ 2(m−2h−1 ) we have that i ∗ ≥ m−2h−1 and hence i +i ∗ ≥ 3m−3·2h−1 = m + 2m − 3 · 2h−1 . As m ≥ 3 · 2h−2 we know that 2m ≥ 3 · 2h−1 and hence i + i ∗ ≥ m. Therefore, i+i ∗ ≤ [D2h , G; i + i ∗ − m] = 2h +i+i ∗ −m ≤ 2h +i+i ∗ −m

h j =1 h j =1

j

22h−j +i+i ∗ −m

(Theorem 2.5)

D22h−1 +2j −1 (i+i ∗ −m) = 2h +i+i ∗ −m D22h−1 +i+i ∗ −m

As m < 2h , Lemma 2.6 implies that i+i ∗ ≤ D22h−1 +i+i ∗ −m . We claim that 2h−1 + i + i ∗ − m ≥ i which is equivalent to i ∗ ≥ m − 2h−1 . As 2i ∗ ≥ i this

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Lawrence E. Wilson

follows from the assumption that i ≥ 2(m − 2h−1 ). Therefore i+i ∗ ≤ Di2 and hence [Di , Di ] ≤ (Di2 )2 as desired. This completes the proof of Theorem 3.2.

4. Proof of the main result We can now prove Theorem A. Recall that we assume that G is a 2-group of rank r and that λ(r) is determined by 2λ(r)−1 < r ≤ 2λ(r) . We wish to prove that G has a λ(r) powerful subgroup of index at most 2r(λ(r)+1)−(2 −r) . Proof of Theorem A. Notice that we make no claim beyond the known results if r = 2λ(r) . We will assume that r < 2λ(r) . Let us write λ = λ(r). As [Di , Di ]Di2 ≤ D2i each Di /D2i is elementary abelian and hence has size at most 2r . Therefore, |G : D2λ | ≤ 2rλ . We will find a characteristic powerful subgroup λ P of G contained in D2λ with |D2λ : P | ≤ 22r−2 and this will complete the proof. We first note that D2λ /D2λ+1 is elementary abelian and hence of size at most 2r . Therefore, [D2λ , G; r] ≤ D2λ+1 . In particular, 2λ +r ≤ D2λ+1 . Theorem 3.2 (2) implies that D2r is powerful. This provides the base case in an inductive proof of the following claim: either λ D2λ contains a powerful subgroup of index at most 22r−2 or D2(r−2i +1) is powerful for 0 ≤ i < λ(r − 2λ−1 + 1). Assume now that this holds for i < λ(r − 2λ−1 + 1) − 1 and we will attempt to prove it for i + 1. λ If D2λ contains a powerful subgroup of index at most 22r−2 then we are done. λ Otherwise, |D2λ : D2(r−2i +1) | > 22r−2 . As |D2λ : D2λ+1 | ≤ 2r we can conclude that λ

λ

|D2(r−2i +1) : D2λ+1 | < 2r−(2r−2 ) = 22 −r . Hence, [D2(r−2i +1) , G; 2λ − r − 1] ≤ D2λ+1 . Also, [D2λ , G; 2(r −2i +1)−2λ ] ≤ D2(r−2i +1) and so [D2λ , G; r −2i+1 +1] ≤ D2λ+1 . In particular, 2λ +r−2i+1 +1 ≤ D2λ+1 . As i + 1 < λ(r − 2λ−1 + 1) we know that i+1 2 < r − 2λ−1 + 1 and hence 2λ−1 < r − 2i+1 + 1 and hence Theorem 3.2 (2) implies that D2(r−2i+1 +1) is powerful. This completes the induction proof. If we let = λ(r − 2λ−1 + 1) then we know that either D2λ contains a powerful λ subgroup of index at most 22r−2 or D2(r−2−1 +1) is powerful. Arguing as above, λ

we find that either D2λ contains a powerful subgroup of index at most 22r−2 or 2λ +r−2 +1 ≤ D2λ+1 . If 2λ + r − 2 + 1 = 2λ + 2λ−1 then Theorem 3.2 (2) implies that D2λ is powerful. If 2λ + r − 2 + 1 < 2λ + 2λ−1 then Theorem 3.2 (1) implies that Di is powerful for i ≥ (2(2λ + r − 2 + 1) − 1)/3. We know that 2 ≥ r − 2λ−1 + 1 and hence 2+1 ≥ 2r − 2λ + 2 > 2r − 2λ + 1. Therefore, 2λ > 2r − 2+1 + 1 and hence 3 · 2λ > 2 · 2λ + 2r − 2+1 + 2 − 1. We conclude that 2λ > (2(2λ + r − 2 + 1) − 1)/3. Therefore, in either case D2λ is powerful.

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Hence, D2λ contains a powerful subgroup of index at most 22r−2 or D2λ is powerful itself. Either way, we can now conclude that there is a powerful subgroup of λ D2λ of index at most 22r−2 and hence a powerful subgroup of G of index at most λ 2r(λ+1)−(2 −r) . It is worth noting that the only place we used the hypothesis in the proof of the theorem was to deduce that |D2λ(r) (G) : D2λ(r)+1 (G)| ≤ 2r . Hence the result could be about the least such that |D2 (G) : D2+1 (G)| < 22 . In this case, we would get a bound on the index of a powerful subgroup of D2 but without the assumption on the rank we could not get a bound on the index of a powerful subgroup in G. We would also like to point out that all of the results of this paper also hold in finitely generated pro-p groups. One needs to take the quotient by successively large terms of the dimension series. One then finds the results up to closure, but all of the groups in this paper are closed by standard results. Acknowledgment. I would like to thank the referee for a suggestion which simplified the proof of Theorem 3.2.

References [DdMS]

J. D. Dixon et. al., Analytic Pro-p groups, 2nd. ed., Cambridge Stud. Adv. Math. 61, Cambridge University Press, Cambridge 1999.

[Ha]

P. Hall, A contribution to the theory of groups of prime-power order, Proc. London Math. Soc. (2) 36 (1933), 29–95.

[LM]

A. Lubotzky and A. Mann, Powerful p-groups. I. Finite Groups, J. Algebra 105 (1987), 484–505.

[SS]

C. Scoppola and A. Shalev, Applications of dimension subgroups to the power structure of p-groups, Israel J. Math. 73 (1991), 45–56.

[W]

L. Wilson, Dimension subgroups and p-th powers in p-groups, to appear in Israel J. Math.

Lawrence E. Wilson, Department of Mathematics, University of Florida, PO Box 118105, 358 Little Hall, Gainesville, FL 32611–8105, U.S.A. E-mail: [email protected]

Regular orbits of induced modules of finite groups Thomas R. Wolf

Abstract. If W is a H -module for a group H and if S is a transitive permutation group of degree n, then the wreath product K of H by S acts naturally on a module V of dimension n times that of W . Here, we give an exact count of the number of regular K-orbits on V . This can be expressed in a formula dependent only on the number of regular orbits of H on W and the permutation action of S on a set of n elements. This situation arises frequently, e.g. if V is an irreducible G-module for a finite group G and V is not quasi-primitive, then G is isomorphic as a linear group to a subgroup of a wreath product K (as above). The number of regular G-orbits is at least |K : G| times the number of regular K-orbits. 2000 Mathematics Subject Classification: 20C40

1. Introduction If W is a H -module for a group H and if S is a transitive permutation group on a set with | | = n, then the wreath product H S acts naturally on a module V , which as a vector space is the direct sum of n copies of W . Here, we give an exact count of the number of regular H S -orbits on V . This can be expressed in a formula dependent only on the number of regular orbits of H on W and the action of S on the partitions of the set . Our counting arguments also give an exact count of the number of orbits of H S on V . Our emphasis here is on the partitions of the set , versus partitions of the integer n that are frequently used in the representation theory of the symmetric group Sn . The difference in emphasis is presumably due to the fact that any two partitions of the set of the same “type” are necessarily conjugate in Sn , but not necessarily conjugate in S. The above situation arises frequently, e.g. if V is an irreducible G-module for a finite group G and V is not quasi-primitive (i.e. VN is not homogeneous for some normal subgroup N of G), then G is isomorphic as a linear group to a subgroup of H S for a group H (isomorphic to J /I for some I < J < G) and a transitive permutation group S (isomorphic to G/K for a normal subgroup K of G). Furthermore, S may be even chosen to be a primitive permutation group. Of course, if H has infinitely many regular orbits on W , then G has infinitely many regular orbits on V . The formula is essentially for when H has finitely many regular orbits. Indeed, the motivation for this is when |W | and hence |H | are finite, where there are applications to many well-known problems, including conjectures of Brauer.

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Recall = {1 , . . . , l } is a partition of length l of the set if is the disjoint union 1 ∪ · · · ∪ l and each i is non-empty. If S acts on , we let NS () = {s ∈ S | s = } and CS () = {s ∈ S | si = i for all i}. We say that is trivially centralized by S if CS () = 1. While the formula for the number of regular orbits of G is exact, it is probably more easily used to give a lower bound. The following is Corollary 2.6 below. Theorem 1.1. Let S = 1 act transitively on a set and let l ( , S ∗ ) denote the set of partitions of length l of that are trivially centralized by S. If the group H has k regular orbits on the H -module W and V is the “natural” module for G = H S, then the number of regular G-orbits on V is 1

P (k, l)|l ( , S ∗ )|. |S| 2≤l≤| |

Note that P (k, l) = k!/(k − l)! (and equals 0 if k < l). In particular (see Corollary 2.8), the number of regular orbits of G when S = Sn is “k choose n”. When S is a solvable primitive permutation group on , a Theorem of Gluck [1] shows that S has a trivially centralized partition of length two, unless | | < 10. Indeed there are often many of these. In all cases, S has a trivially centralized partition of length four (or less). We discuss this more in Section 3.

2. Counting orbits Hypothesis 2.1. Suppose that W is an H -module and that S = 1 is a transitive permutation group on n letters. We let V be the direct sum of n copies of W and let G be the wreath product H S, so that G acts naturally on V . We let m denote the number of orbits of H on W and let k denote the number of regular orbits of H on W . Assuming Hypotheses 2.1, we have that V = W1 ⊕ · · · ⊕ Wn for subspaces Wi transitively permuted by S G/K where K = H1 ⊕ · · · ⊕ Hn is a normal subgroup of G and the Hi (isomorphic to H ) are permuted transitively by S. Also the action of Hi on Wi is isomorphic to that of H on W , while Hi centralizes Wj for i = j . We identify H with H1 , W with W1 , and with {W1 , . . . , Wn }. Observe that V is a faithful G-module if and only if W is a faithful H -module. Likewise V is an irreducible G-module if and only if W is an irreducible H -module. We first note that x and y in W are G-conjugate if and only if they are H -conjugate. This is because x and y are G-conjugate if and only if they are NG (W )-conjugate and NG (W ) = KNS (W ) = KCS (W ) = H CK (W )CS (W ) = H CG (W ). Also the Gconjugacy class C of x is the disjoint union ∪i (C ∩ Wi ) and each (C ∩ Wi ) is an Hi -orbit on Wi (and is as well a K-orbit on V ). Let O1 , . . . , Om be the distinct H -orbits on W . Then every element of W1 ∪· · ·∪Wn is conjugate to the elements of exactly one Oj . For v = v1 +· · ·+vn ∈ W1 +· · ·+Wn ,

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we let j (v) = {Wi | vi ∈ OjG }. In particular, for each v in V , we have that {W1 , . . . , Wn } is the disjoint union 1 (v) ∪ · · · ∪ m (v). The non-empty i (v) form a partition of = {W1 , . . . , Wn }, which we call the partition of v. It is easy to see that u = u1 + · · · + un and v = v1 + · · · + vn in V are conjugate in K = H1 × · · · × Hn if and only if each pair ui and vi are Hi -conjugate; or equivalently if and only if j (u) = j (v) for all j , 1 ≤ j ≤ m. In particular, [j (v)]k = j (v) for all k ∈ K, all v ∈ V , and for all j , 1 ≤ j ≤ m. We claim that u and v in V are G-conjugate if and only if there is some s ∈ S such that (j (u))s = j (v) for all j , 1 ≤ j ≤ m. If such an s exists, then j (us ) = j (v) for all j and so us and v are K-conjugate by the last paragraph, whence u and v are G-conjugate. On the other hand, if u and v are G-conjugate; there exist k ∈ K and s ∈ S such that v = uks . By the last paragraph, (j (u))k = j (u) for all j . Thus j (v) = (j (u))ks = (j (u))s for all j , 1 ≤ j ≤ m, establishing the claim. For u and v to be G-conjugate, we must have the partitions determined by u and v are G-conjugate, but that is not sufficient. If the partitions of u and v are equal, we still have that u and v are G-conjugate if and only if there is some s ∈ S such that (j (u))s = j (v) for all j for all non-empty j (u). For a given partition of {W1 , . . . , Wn }, the set of vectors v in V that have partition is a union of K-conjugacy classes. If the length of is l, then the number of K-conjugacy classes of elements of V with partition is P (m, l) = m(m − 1)(m − 2) . . . (m − l + 1) = m!/(m − l)!. In particular, there are no elements with partition if the length of is larger than m (we use the convention that P (m, l) = 0 when l > m). Assume that is the partition of v ∈ V . If g = ks ∈ G with k ∈ K and s ∈ S, then v g is K-conjugate to v if and only if v s is K-conjugate to v and this occurs if and only if s centralizes . Of course, in this case, v g has partition . But more generally, v g has partition if and only if s ∈ NS () (i.e. g ∈ NG ()). If we let O be the set of those K-conjugacy classes of elements of V whose partition is , then NG () = KNS () acts on O. For each [v] ∈ O, we have that the stabilizer in G of [v] is precisely KCS (). Thus the number of K-conjugacy classes in O that are NG ()-conjugate to [v] is |NS () : CS ()|. Hence the number of NG ()-orbits on O is |O|/|NS () : CS ()| = P (m, ||)/|NS () : CS ()|. This is precisely the number of NG ()-orbits on O ∗ = {v ∈ V | v has partition }. Since two elements of O ∗ are G-conjugate if and only if they are NG ()-conjugate, P (m, ||)|CS ()|/|NS ()| is the number of G-conjugacy classes C of V for which the partitions of the elements of C are the S-conjugates class of . The number of G-conjugacy classes of V is then P (m, ||)|CS ()|/|NS ()|, where the sum is taken over a set of representatives of each S-conjugacy class of partitions of . Of course, the number of S-conjugates of a partition is |S : NS ()|. Summing over all partitions, we then get that the number of G-orbits on V is then

P (m, ||)|CS ()|/|S|. P (m, ||)|CS ()|/|NS ()||S : NS ()| =

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Summarizing we have: Theorem 2.2. Assume Hypothesis 2.1. The number of orbits of G on V is exactly 1

P (m, ||)|CS ()|, |S| ∈( )

where ( ) is the set of partitions of . The theorem is most useful in giving lower bounds for the number of orbits of G and applied to partitions of short length. Note that the partition of an element v in V is determined by the action of H on W and not dependent upon the action of S on . Corollary 2.3. Assume Hypotheses 2.1. Then the number of G-orbits of elements v in V whose partition has length q is 1

P (m, q)|CS ()|, |S| l()=q

where the sum is taken over all partitions of whose length l() is q. Next we count the number of regular orbits. For v = v1 + · · · + vn to be in a regular G-orbit, then v must be in a regular K-orbit. Now v is in a regular K-orbit if and only if each vi is in a regular Hi -orbit, i.e. whenever j (v) is non-empty, then Oj is a regular orbit of H on W . Again the non-empty j (v) form a partition of = {W1 , . . . , Wn }. If k is the number of regular orbits of H on W and is a partition of , we argue as above that the number of G-orbits of K-regular elements of V with partition or an S-conjugate of is P (k, ||)|CS ()|/|NS ()|. Let v ∈ V be in a regular K-orbit and let be the partition of v. We claim that v is in a regular G-orbit if and only if CS () = 1. For g = ks ∈ G with k ∈ K and s ∈ S, we have from above that v g is K-conjugate to v if and only if s ∈ CS (). If 1 = t ∈ CS (), then v t = v x for some x ∈ K and 1 = tx −1 ∈ CG (v), whence v is not in a regular G-orbit. For the converse, assume that v is not in a regular G-orbit and 1 = g = ks centralizes v. Then v g is K-conjugate to v and so s ∈ CS (). Since v is in a regular K-orbit and 1 = g = ks centralizes v, indeed s = 1 and CS () = 1. This proves the claim. Thus v is in a regular G-orbit if and only if v is in a regular K-orbit and the partition of v is in a regular orbit of S. To count regular G-orbits, we apply the same counting argument used above to count all orbits, except we restrict ourselves to regular H -orbits (vs. all H -orbits) and to partitions that are trivially centralized by S (vs. all partitions). Since CS () = 1 for a trivially centralized partitions of , we get: Theorem 2.4. Assume Hypothesis 2.1 and let ( , S ∗ ) denote the set of partitions of that are trivially centralized by S. Then the number of regular G-orbits on V is exactly ∈( ,S ∗ ) P (k, ||)/|S|. Corollary 2.5. Assume Hypothesis 2.1. G has a regular orbit if and only if there is a trivially centralized partition of whose length is at most the number of regular

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orbits of H on W . In particular, for G to have a regular orbit when | | > 1, H must have at least two regular orbits on W and S must act faithfully on . The necessity that H must have two regular orbits on W above is because S = 1 will centralize the unique partition of length one. When S is a solvable and primitive permutation group, S has a trivially centralized partition of length two, except in a few cases where | | < 10. This is Gluck’s Theorem, which we discuss in more detail later. The following is more convenient for counting regular orbits. Corollary 2.6. Assume Hypothesis 2.1 and let l ( , S ∗ ) denote the set of partitions of length l of that are trivially centralized by S. Then the number of regular G-orbits on V is 1

P (k, l)|l ( , S ∗ )|. |S| 2≤l≤| |

We even note that |S| divides each term P (k, l)|l ( , S ∗ )| in the sum above 1 P (k, l)|l ( , S ∗ )| is the number of regular G-orbits of elements of V whose since |S| partition of {W1 , . . . , Wn } has length l. If = 1 ∪ · · · ∪ l is a partition of length l, then NS ()/CS () faithfully permutes the i and so |NS ()/CS ()| divides l!. If is also trivially centralized, then even |NS ()| divides l!. Every conjugate of is k! trivially centralized and so |l ( , S ∗ )| ≥ |S|/ l!. Now P (k, l)/ l! = l!(k−l)! = kl , known commonly as “k choose l”. Note kl = 0 for k < l. Corollary 2.7. If S has at least one trivially centralizedpartition of length l and if H has k regular orbits on W , then G = H S has at least kl regular orbits of elements of V with partition length l. Corollary 2.8. Assume Hypothesis 2.1 and that S is Sn or An . (1) The number of regular orbits of H Sn on V is nk ; and k . (2) The number of regular orbits of H An on V is 2 nk + (n − 1) n−1 Proof. The only partition of {1, . . . , n} that is trivially centralized by Sn is {{1}, . . . , {n}} and it has, of course, length n. By Corollary 2.6 the number of regular 1 k! P (k, n) = (k−n)!n! = nk . The alternating group has exactly orbits of H Sn is n! one other orbit of trivially centralized partitions of {1, . . . , n}, namely the family of partitions of length n − 1, each such partition consisting of a doubleton and n − 2 singletons. By Corollary 2.6, the number of regular orbits of H An is thus n k k 2 P (k, n) + P (k, n − 1) = 2 + (n − 1) . n! 2 n n−1

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3. Trivially centralized partitions of solvable S Here we give a brief accounting of the number of trivially centralized orbits when S is a primitive solvable permutation group. We restrict ourselves to primitive permutation groups since most applications (e.g. V a quasi-primitive G-module) can be reduced to Hypothesis 2.1 with the additional condition that S is a primitive permutation group. Gluck [1] showed that a solvable primitive permutation group on with | | = n necessarily has a regular orbit on the power set of if n > 9. Note that a non-empty ⊂ is in a regular orbit of S if and only if ∪ ( − ) is a partition of (of length two) that is trivially centralized by S. While there are certainly exceptions to Gluck’s lemma on the existence of trivially centralized partitions of length two, an inspection of the proof of Gluck’s lemma shows that the number of trivially centralized partitions is an exponential function of n. Indeed the argument of Proposition 5.2 of [3] shows that S has at least (2n − |S|23/4 )/2 trivially centralized partitions of length two, and |S| ≤ n13/4 /2 via Corollary 3.6 of [3]. But this says nothing for values of n < 83. Indeed, there maybe none when n < 10. Note that n must be a power of a prime as S is a primitive solvable permutation group. Theorem 3.1. Suppose that S is a solvable primitive permutation group of degree n > 1. Then (i) if n > 9, then there are at least 4|S| trivially centralized partitions of length two; (ii) if |S| is odd, then there are at least |S| trivially centralized partitions of length two; (iii) if 5 ≤ n ≤ 9, there are at least |S| trivially centralized partitions of length three; and (iv) there is a trivially centralized partition of length l ≤ 4. There is one of length l ≤ 3 unless n = 4 and S = S4 . Proof. If n > 9, a refinement of Gluck’s lemma by Zhang [4] shows that S has at least 8 regular orbits on the power set P ( ). Thus there exist 8|S| subsets of such that stabS () = 1. For each such , the stabilizer of the complement c of is also 1 and ∪ c forms a partition of length two of that is trivially centralized by S. Thus there are at least 8|S|/2 trivially centralized partitions of length two. Suppose that |S| is odd. Then Gluck’s Lemma even shows that S has a regular orbit on the power set of (even if S is not primitive and even when S is not transitive). Since n > 1, we may choose a proper non-empty subset of in a regular orbit of S on P ( ). Then c is also in a regular orbit of S and ∪ c forms a partition of length two of that is trivially centralized by S. Since |S| is odd, then and c are not S-conjugate and the partition {, c } of has |S| distinct conjugates. Part (ii) follows. For n = 5 or 7, Part (iii) follows from Corollary 3.4 (below). For n = 8 or 9, Part (iii) follows from Proposition 3.5 or 3.6 (below), respectively. Since n must be a prime power,

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the proof of (iii) is complete. That S has a trivially centralized partition of length at most four is also a direct consequence of Gluck’s lemma and a proof is given in of [2]. However this follows from Part (iii). To see this, we may assume via iii that n < 5. Hence the partition of singletons has length at most four and must be trivially centralized by S. We note part (ii) of Theorem 3.1 remains valid if S is imprimitive or even intransitive. The proof of Theorem 3.1 (iii) will be complete upon examination of small permutation groups below. But first, we show how the results could be used “inductively”. Corollary 3.2. Assume Hypothesis 2.1 and that S is a solvable primitive permutation group. Assume that H has k ≥ 2 regular orbits on W . Then (i) If k > 4, then G has at least k regular orbits on V . Indeed G has at least 2k regular orbits on V , except when k = 5 and S = S4 (and G has 5 regular orbits); (ii) If n > 9, then G has at least 2k regular orbits on V ; (iii) If n > 4 and k ≥ 3, then G has at least 2k regular orbits on V Proof. If n > 9, then G has at least 4P (k, 2) = 4k(k − 1) regular orbits by Theorem 3.1, Part (i) and Corollary 2.6. If n > 4 and k ≥ 3, then G has at least P (k, 3) = k!/(k − 3)! regular orbits by Theorem 3.1, Part (iii) and Corollary 2.6. This proves ii and iii. Now S has a trivially centralized partition of length l with either l < 4 or with l = 4 and S = S4 . By Corollary 2.7, G has at least kl regular orbits. For k > 4 and l < 5, note that kl ≥ 2k, except when k = 5 and l = 4. We have exceptional cases when k and n are small. For n < 5, S is either alternating or symmetric and so the number of regular orbits is given by Corollary 2.8. We let S(n, l) be the number of partitions of length l of a set when | | = n. The numbers S(n, l) are called Sterling numbers of the second kind. If α ∈ , the partitions of may be obtained by modifying the partitions of − {α}, and so we have the following recursive relationship: S(n, l) = S(n − 1, l − 1) + l · S(n − 1, l). These can be easily calculated for small n (e.g. by spreadsheet) and these calculations will be used in Corollary 3.4. We adopt the notation S(n, l) = 0 for l > n. Of course S(n, 1) = 1 = S(n, n). For l = 2, we even have that S(n, 2) = 2n−1 − 1. This is because each of the 2n − 2 non-empty proper subsets of is a member of exactly one partition of length two of , namely = ∪ ( − ). Lemma 3.3. Suppose that S is a solvable transitive permutation group on of prime degree p, so that Sα is cyclic and |Sα | divides p − 1. The number of partitions of

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of length l > 1 that have trivial centralizers in S is

S(1 + p−1 S(p, l) − p q , l) q∈π(Gα )

−

distinct

S(1 +

q,r∈π(Gα )

p−1 qr , l) +

distinct

S(1 +

q,r,s∈π(Gα )

p−1 qrs , l) − . . .

where π(Gα ) is the set of prime divisors of |Gα |. The terms in the alternating sum in parentheses decrease. Proof. Since | | is a prime p, S must be primitive and a subgroup of Zp · Zp−1 and has a normal transitive subgroup M of order p. For a partition of of length larger than one, CM () = 1. It follows that CS () must be a subgroup of one the p point stabilizers, each such stabilizer being cyclic of order dividing p − 1. If g ∈ Gα for some α ∈ and g has order s, then g is a product of a one-cycle and p−1 s s-cycles, whence g has 1+ p−1 orbits on . If a subgroup H of S centralizes = 1 ∪· · ·∪l , s then each i must be a union of H -orbits. If H has r orbits on , then the number of partitions of length l centralized by H is S(r, l), understanding S(r, l) = 0 for r < l. Recalling that Gα is cyclic (of order dividing p − 1), a partition of with nontrivial centralizer in Gα must be centralized by a subgroup of prime order. We may use an inclusion-exclusion argument to get an exact count of the number of partitions of length l, 1 < l < p, that have a non-trivial stabilizer in Gα , namely:

q∈π(Gα )

S(1 +

p−1 q , l) −

distinct

S(1 +

q,r∈π(Gα )

p−1 qr , l) +

distinct

S(1 +

q,r,s∈π(Gα )

p−1 qrs , l) − . . .

.

Each pair of the p point stabilizers intersect trivially, whence the formula in the statement of the lemma is exact. Corollary 3.4. Let S be a solvable transitive permutation group on , with | | = p, a prime. (i) If |S| = p, then S has S(p, l) trivially centralized partitions of length l for all l > 1. In particular, there are 2p−1 − 1 such of length two. (ii) If p ≥ 19, then S has at least 2(p−1)/2 |S| trivially centralized partitions of length two. (iii) If p = 11, 13, or 17 (respectively), then S has at least 6|S|, 20|S|, or 225|S| trivially centralized partitions of length two. (iv) If p = 7, then S has at least 6|S| trivially centralized partitions of length three. If a point stabilizer Sα has order 2, 3, or 6; then the number of trivially centralized partitions of length two is |S|, 2|S|, or 0 (respectively), and (v) If p = 5 < |S|, there are exactly 0, 20, 10 and 1 (respectively) trivially centralized partitions of length 2, 3, 4, or 5 (respectively).

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Proof. Since Gα is cyclic of order dividing p − 1, all parts save Part (ii) can be calculated by the formula in Lemma 3.3. The argument of the lemma shows that the number of partitions of length l with trivial centralizer is at least

p − 1 , l) . S(1 + S(p, l) − p q q∈π(Gα )

Now S(k, 2) = 2k−1 − 1. Also the number of prime divisors of |Gα | is less than log2 (p) and 1 + (p − 1)/q ≤ (p + 1)/2 for all q. Then the number of trivially centralized partitions of length two is at least: 2p−1 − 1 − p log2 (p) ∗ (2(p+1)/2) − 1) ≥ 2((p−1)/2) [2(p−1)/2 − 2p log2 (p)]. For p ≥ 19, note that 2(p−1)/2 − 2p log2 (p) > p(p − 1) ≥ |S|. Proposition 3.5. Let S be a solvable primitive permutation group on , | | = 8. Then Sα is cyclic of order 7 or a Frobenius group of order 21. Then (i) The number of trivially centralized partitions of length three is at least (9/2) · |S|; and (ii) The number of trivially centralized partitions of length two is 0 if |Sα | = 21 and is 3|S|/2 if |Sα | = 7. Proof. Since S is a primitive solvable permutation group of degree 8, S has a unique minimal normal subgroup that is elementary abelian of order 8 and S is the semi-direct product of M and Sα , that acts faithfully an irreducibly of M. The actions of Sα on M and are permutation isomorphic. That Sα is cyclic of order 7 or a Frobenius group of order 21 is routine (e.g. Theorem 2.12 of [3]). A subgroup H of S with k orbits centralizes exactly S(k, l) partitions of length l. The only subgroups of A(23 ) with more than three orbits on are the seven subgroups of order two and the 28 subgroups of order three, and all of these 35 subgroups have exactly four orbits. The only subgroups with exactly three orbits are the 28 cyclic groups of order 6, each containing 2 subgroups of prime order. Thus have the number of trivially centralized partitions of length 3 is S(8, 3) − 35 · S(4, 3) + 28 · S(3, 3) ≥ (9/2) · 168 ≥ (9/2) · |S|. We now consider the partitions of length two. If ∪ c is in a trivially centralized partition of , then is in a regular orbit of S on P ( ). If || = t, then is even in a regular orbit of S on Pt ( ), the collection of all subsets of of size t. Unless t = 3, 4, or 5 and |S| = 8 · 7, then |S| > |Pt ( )| = 8t and S has no regular orbit on Pt ( ). Hence there are no trivially centralized partitions of length two if |S| = 8 · 7 · 3. When |S| = 8·7, note the subgroups of S of order 7 have two orbits on , while those of order two have cycle structure 24 with 4 orbits. Thus the number of trivially centralized partition of length two is at least S(8, 2) − 8 · S(2, 2) − 7 · S(4, 2) = 70 > |S| = 56. We note no element of prime order of S fixes a subset of of size three (or five), and thus the 56 partitions of type (3,5) are all trivially centralized by S. Thus there is at least one partition of of type (4,4) that is trivially centralized by S. As NS ()

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has order 1 or 2, there are at least 28 conjugates of , each trivially centralized by S. Indeed, as there are only 35 partitions of type (4,4), exactly 28 partitions of type (4,4) are trivially centralized. Proposition 3.6. Let S be a solvable primitive permutation group on , | | = 9. (i) If Sα is cyclic or quaternion, then S has at least 2 · |S| or 38 · |S| trivially centralized partitions of length two or three (respectively); (ii) If Sα is not cyclic nor quaternion, then S has no trivially centralized partitions of length two; and (iii) S has at least |S| trivially centralized partitions of length three. Proof. Let M be the minimal normal subgroup of S so that S = MSα and M ∩Sα = 1. Then Sα acts irreducibly on M, so that Sα is an irreducible subgroup of GL(2, 3) and |S| divides 9 · 48. In particular, Sα is cyclic, quaternion of order eight, dihedral of order eight, semi-dihedral of order sixteen, or is isomorphic to SL(2, 3) or GL(2, 3). Now Sα has a unique central involution t that inverts M, whence the cycle structure of t on is 24 · 1. In particular, < t > has 5 orbits and belongs to exactly one point stabilizer. Assume that Sα is cyclic or quaternion. Every element of prime order in S = MSα indeed lies in S0 = M < t >. A partition of with non-trivial centralizer in S has a non-trivial centralizer in S0 . Each of the nine involutions of S0 has 5 orbits on , while each of the four subgroups of S0 order 3 has 3 orbits on . A subgroup with r orbits centralizes exactly S(r, l) partitions of length l. The number of trivially centralized partitions of length l > 1 for S (or S0 ) is at least S(9, l) − 4 · S(3, l) − 9 · S(5, l). As |Sα | divides 8 and |S| divides 72, there are more than 38|S| trivially centralized partitions of length three and at least 3|S|/2 of length two. But if = {, c } is a trivially centralized partition of of length two, then NS () = 1 because || = |c | and so |S| divides the number of trivially centralized partitions of length two. This proves i. Part (iii) follows from Part (i) when Sα is cyclic or quaternion. To prove (ii) and (iii), we may assume that Sα is dihedral of order 8 or has order at least 16. If = ∪ c is a trivially centralized partition of and || = t, then is in a regular 9 orbit of S in its action on Pt ( ) = { ⊆ ||| = t}. If |Sα | > 8, then |S| > t = | t | for all t and so S has no trivially centralized partition of of length two. A dihedral group D of order 8 in GL(2, 3) contains all involutions in a semidihedral Sylow-2-subgroup T of GL(2, 3) and consequently MD contains all elements of prime order belonging to MT . Thus MD has no trivially centralized partition of length two since MT has none. Thus S has no trivially centralized partitions of length two. Part (ii) follows. Finally to prove (iii), it is no loss to assume that Sα GL(2, 3). Fix α, β ∈ with α = β. Now the two-point stabilizer Sαβ is isomorphic to S3 and has orbit pattern 6·13 . In particular, Sαβ also fixes a third point γ ∈ {α, β}. Note that S acts transitively on the 36 subsets of with two elements. Thus |NS {α, β}| = |S|/36 = 12 = 2|Sαβ | =

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399

2|CS {α, β}|. Set N = NS {α, β} and note that the orbits of N are {α, β}, {γ }, and − {α, β, γ }. Now N ≤ Sγ and N is the normalizer of a Sylow-3-subgroup of Sγ and is isomorphic to Z2 ⊕ S3 . Subgroups of N of prime order with their orbit pattern on − {α, β, γ } are: • four subgroups of order two with orbit pattern 23 on − {α, β, γ }; • three subgroups of order two with orbit pattern 22 · 12 on − {α, β, γ }; and • one subgroup of order three with orbit pattern 32 on − {α, β, γ }. Thus the number of subsets of − {α, β, γ } with exactly threeelements fixed by some nonidentity element of N is at most 4 · 0 + 3 · 4 + 1 · 2 < 63 . Thus we may choose ⊆ − {α, β, γ } such that || = 3 and NN () = 1. It follows that the partition = {{α, β}, , [ − {α, β} − ]} of is trivially centralized by N. Now {α, β} is the only subset in of size two and so CS () ⊆ NS ({α, β}) = N . Thus is a partition of of length three and type (2,3,4) that is trivially centralized by S, with γ in the subset of size four. Since the subsets of the partition have distinct sizes, NS () = CS () = 1. Thus has |S| conjugates, each trivially centralized by S.

References [1] D. Gluck, Trivial set-stabilizers in finite permutation groups, Canad. J. Math. 35 (1983), 59–67. [2] A. Moretó and T. Wolf, Orbit sizes, character degrees and Sylow subgroups, Adv. Math. 184 (2004), 18–36; Erratum ibid. p. 409. [3] O. Manz and T. Wolf, Representations of Solvable Groups, London Math. Soc. Lecture Note Ser. 185, Cambridge University Press, Cambridge 1993. [4] J. Zhang, Finite groups with few regular orbits on the power set, Algebra Colloq. 4 (1997), 471–480. Thomas R. Wolf, Department of Mathematics, Ohio University, Athens, OH 45701, U.S.A E-mail: [email protected]

Radical subgroups and p-local ranks Jiping Zhang∗

Abstract. In this paper we investigate some problems on radical p-subgroups and p-local ranks, in particular, conditions on the existence of p-blocks with a given maximal Sylow p-intersection as defect group, and the influence of p-local ranks on the group structure will be provided. 2000 Mathematics Subject Classification: 20C20

Let p be a prime integer and let G be a finite group. An important question in modular representation theory is: When is the given p-subgroup D of G a defect group for some p-block of G? By the First Main Theorem of Brauer we see that if D is a defect group for some p-block then D = Op (NG (D)), that is to say, D is radical. Also, as proved by Green[3], defect groups are Sylow p-intersections, i.e. D = P ∩ P x , where P ∈ Sylp (G) and x ∈ G. Note that Sylow p-intersections are not necessarily radical, thus a Sylow p-intersection is not necessarily a defect group for some p-block. However it is easy to verify the following result. Proposition 1. Suppose that G is a finite group with an abelian Sylow p-subgroup P . Then a proper p-subgroup D of P is radical if and only if D is a Sylow p-intersection in G. Let p and q be two different primes such that p > 2 and p | (q − 1). Then the affine semi-linear group A(q p ) has a non-cyclic abelian Sylow p-subgroup with Op (A(q p )) = 1. Note that A(q p ) has no defect-zero p-blocks. Thus even for finite groups with an abelian Sylow p-subgroup a given radical subgroup is not necessarily a defect group, see [12] for more information. For any finite group G set SIp (G) = {P ∩ P x : P ∈ Sylp (G), x ∈ G \ NG (P )} to be the set of all Sylow p-intersections (= P ). A maximal element (under inclusion) in SIp (G) is called a maximal Sylow p-intersection. We see that maximal Sylow p-intersections are radical. It is thus very interesting to study the following Problem I. When is the given maximal Sylow p-intersection of a finite group G a defect group for some p-block of G ? In the case p = 2, G. Robinson [6] first proved some deep result on Problem I. ∗ Supported by Cheung Kong Scholar’s Programme, National 973 project and RFDP.

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Theorem 2 (Robinson, [6]). Let Q be a maximal Sylow 2-intersection of a finite group G. Then either Q is a defect group for some 2-block of G or (i), (ii) and (iii) are all true: (i) Q is a Sylow 2-subgroup of O2 ,2 (NG (Q)), (ii) NG (Q) is 2-constrained , and CG (Q) has a normal 2-complement, (iii) O2 (NG (Q)) = O2 (NG (T )) whenever T is a Sylow 2-subgroup of G containing Q. Let Q be a maximal Sylow p-intersection of a finite group G. Suppose that Q is not a defect group for any p-blocks of G, then every p-block B of NG (Q) is of the highest defect; in other words, B has a Sylow p-subgroup of NG (Q) as its defect group. We call such groups full p-defective groups. Thus full p-defective groups are of great importance in investigating Problem I. Finite full p-defective groups were first introduced by the author, and the following general structure theorem was proved in [10]. Theorem 3 (Zhang, [10]). Let G be an arbitrary finite group, P a Sylow p-subgroup of G and M a p -subgroup of G such that M = Op (CG (P )). Then G is full p-defective if and only if the following two statements hold: (1) Op (G) = ∪x∈G M x ; (2) F ∗ (G/Op (G)) = (Op (G) · Op (G))/Op (G), and for p = 2, we may have besides F ∗ (G/O2 (G)) = (F × O2 (G))/O2 (G), where F is a normal subgroup of G and all components of F are of type M22 or M24 . From this theorem we have the following corollary which solves Problem I for finite groups with an abelian Sylow p-subgroup Corollary 4. Let G be a finite group with an abelian Sylow p-subgroup. Then every maximal Sylow p-intersection of G is a defect group for some p-block. Proof. Let D be a maximal Sylow p-intersection of G. If D is not a defect group for any p-blocks of G, then, as we noted before, the normalizer H := NG (D) of D in G is a full p-defective group. By the structure theorem above we see that F ∗ (H /Op (H )) = (Op (H ) · Op (H ))/Op (H ). Since a Sylow p-subgroup P of G is abelian and contained in H and by the typical property of generalized Fitting subgroups it follows that P = D is normal in H , which is a contradiction as D is a proper subgroup of P . We are done. From a different approach an answer to Problem I can be derived from the following theorem. To this end we need to introduce the concept of strong radical p-subgroups. A p-subgroup D of a finite group G is called strong radical if NG (D)/D contains a strongly p-embedded subgroup. We see that maximal Sylow p-subgroups are strong radical.

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Theorem 5 (Zhang, [11]). Suppose D is a strong radical p-subgroup of a finite group G then D is a defect group for some p-block of G if and only if there exists a p-regular element x in G such that D is a Sylow p-subgroup of the centralizer CG (x) of x in G. For further studies of Problem I more information are needed on the inner structure of finite full p-defective groups. Let G be a finite full p-defective group, and P a Sylow p-subgroup of G with CG (P ) = Z(P ) × M where Z(P ) is the center of P . In Theorem 3, we see from Lemma 9.2 of ([4], p. 118) that if p = 2 then M = Op (G). In general, for finite full p-defective group G with M = Op (G) one can not control the structure of the quotient of G by the normal subgroup Op ,p (G). In fact any finite group can be embedded into such a quotient of some finite full p-defective group G. However, if M = Op (G) then the group structure of K/CK (Op (G)) can be very much restricted where K is the normal closure of the Sylow p-subgroup P of G, i.e., K = P G = P x : x ∈ G. This is indeed the case if G is a finite solvable full p-defective group. Theorem 6 ([8]). Let G be a finite solvable full p-defective group, and P a Sylow ¯ the quotient group G/(G) p-subgroup of G. Suppose that G = P G . Denote by G ¯ = F (G)H ¯ where (G) is the Frattini subgroup of G. We may assume G with ¯ and the Vi ’s are ¯ ∩ H = 1 and F (G) ¯ = W × V1 × · · · × Vs , where W = Op (G) F (G) ¯ ¯ contained in Op (F (G)). Set M := Op (CG¯ (P¯ )). minimal normal subgroups of G ¯ ¯ Then M = Op (G) unless H /CH (Op (F (G))) isomorphic to a subgroup of the direct product G1 × · · · × Gs , where the Gi ’s satisfy the following statements: (1) If H acts trivially on Vi , then Gi = 1. (2) If Vi is not pseudo-primitive as G-module, then p = 3 and there exists a normal subgroup C in Gi such that Gi /C is isomorphic to A(23 ); if Vi = W1 ⊕ · · · ⊕ Wn where Wj are C-invariant spaces, then Gi /C primitively and faithfully permutes {W1 , . . . , Wn } and furthermore n = 8. (3) If Vi is pseudo-primitive as G-module, then p ≥ 3, Gi ≤ (V ) such that O p (Gi ) is a cyclic p -group; Vi , as an irreducible Gi -module, is faithful and p does not divide |Vi |. Corollary 7. Let G be a finite solvable full p-defective group then a Sylow p-subgroup P of G/CG (Op (G)) is abelian unless p = 3 and P is metabelian. We note that such investigation on finite full p-defective groups has an important connection with the Kronecker equivalence of algebraic number fields; see [5] for reference. In order to get a deeper understanding on Alperin’s weight conjecture and the conjectures by Dade, Robinson [7] introduces the concept of the p-local rank of a finite group, which agrees with the rank of the (B, N )-pair for a group with a (B, N )pair of characteristic p. Let G be a finite group and p a prime divisor of |G|. Given a chain of p-subgroups σ : Q0 < Q1 < · · · < Qn

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of G, define the length |σ | = n, the k-th initial sub-chain σk : Q0 < Q1 < · · · < Qk , and the normalizer of the chain Gσ = NG (Q0 ) ∩ NG (Q1 ) ∩ · · · ∩ NG (Qn ). We say that the p-chain σ is radical if Qi = Op (NG (σi )) for each i, i.e., Q0 is a radical p-subgroup of G and Qi is a radical p-subgroup of NG (σi−1 ) for each i = 0. Write R = R(G) for the set of radical p-chains of G and write R(G)/G for a set of orbit representatives under the action of G. Following [7] we define the p-local rank, plr(G), of G to be the length of a longest chain in R(G). We say that a subgroup H of G is a trivial intersection (T.I.) set in G if H g ∩H = 1 for every g ∈ G \ NG (H ). By [7, 7.1] if plr(G) > 0, then plr(G) = 1 if and only if G/Op (G) has T. I. Sylow p-subgroups. Eaton [2] has verified Dade’s projective conjecture (and a related conjecture of G. R. Robinson) for p-blocks of finite groups of p-local rank 1. It generalizes results by H. I. Blau and G. O. Michler on groups with a T. I. Sylow p-subgroup [1]. Eaton’s proof makes use of the classification of finite simple groups. It seems possible to determine to some extend the finite groups with p-local rank 2 and verify the related conjecture for such groups. However this problem is very difficult and little progress has been made. We call a p-subgroup Q of a given finite group G nontrivially radical if Q is radical in G and Op (G) = Q = P where P is a Sylow p-subgroup of G containing Q. Now we are able to prove the following result which characterizes finite groups with p-local rank 2. Theorem 8. Let G be a finite group with p-local rank plr(G) > 1. Then plr(G) = 2 if and only if every nontrivially radical p-subgroup Q of G is a maximal Sylow pintersection in NG (Q). Proof. Suppose first that G is a finite group with plr(G) = 2. Let Q be a nontrivially radical p-subgroup of G and let R be a maximal Sylow p-intersection in NG (Q). If Q is not a maximal Sylow p-intersection in NG (Q) then it is easy to verify that Op (G) < Q < R < L is a radical chain in G, where L is a Sylow p-subgroup of NG (Q). Thus plr(G) > 2, which is a contradiction. So Q is a maximal Sylow pintersection in NG (Q). Now suppose that every nontrivially radical p-subgroup Q of G is a maximal Sylow p-intersection in NG (Q). Let Op (G) < Q < R2 < ··· < Rn be a radical chain of G such that plr(G) = n. Since Q is a maximal Sylow p-intersection in NG (Q), we see that R2 ∈ Sylp (NG (Q)), so R2 = Rn , n = 2, and thus plr(G) = 2. We are done. The following theorem gives some information on the group structure of finite groups with small p-local rank.

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Theorem 9. Let G be a finite group such that every maximal Sylow p-intersection of G is a T. I. set in G. Then the p-local rank is at most 2. Proof. If plr(G) ≤ 1 then we have nothing to prove. So we now suppose that plr(G) > 1. Let Q be a nontrivially radical p-subgroup of G. If Q is a maximal Sylow pintersection of G then it is also a maximal Sylow p-intersection in NG (Q). If Q is not a maximal Sylow p-intersection of G, then Q is properly contained in a maximal Sylow p-intersection R of G. Now R being a T. I. set in G we have that NG (Q) ≤ NG (R), thus Q < R ∩ NG (Q) ≤ Op (NG (Q)), which is a contradiction since Q is radical. It follows from Theorem 8 that plr(G) = 2. We are done. Theorem 10. Let G be a finite group such that every maximal Sylow p-intersection of G is a T. I. set in G. Then the defect group of any p-block of G is either a maximal Sylow p-intersection of G or isomorphic to {1} or a Sylow p-subgroup of G. Proof. Let B be a p-block of G with defect group D. Suppose D is not trivial or a Sylow p-subgroup of G. Since D is radical, we see from the proof of Theorem 9 that D is a maximal Sylow p-intersection of G. We are done. It seems possible to determine the finite groups with every maximal Sylow pintersection a T. I. set. Please note that finite groups containing a Sylow p-subgroup of order at most p 2 satisfy such condition. Now we consider the relationship between the p-local rank and other invariants of finite groups G. The following result is a typical example of such relationship. Proposition 11. Let G = H P with Op (G) = 1 , where H = Op (G) and P is an elementary abelian p-subgroup. Then the p-rank mp (G) = plr(G) . Proof. Let x be a nontrivial element of P such that the centralizer CG (x) has the maximal possible order among the centralizers of nontrivial elements of P . We claim that Op (CG (x)) = x. Let Tx = CH (x) and note that CG (x) = Tx P . Suppose, on the contrary, that x is a proper subgroup of Px := Op (CG (x)). Then Px is not cyclic of order at least p 2 . Note that Px centralizes Tx . Since H = CH (A) : A ≤ Px , |Px : A| ≤ p one can choose y ∈ Px such that y = 1 and CH (y) is not contained in Tx . Since Tx ≤ CH (y) we have CG (x) = Tx P < CH (y)P = CG (y), which is a contradiction to the choice of x. Thus Op (CG (x)) = x, as claimed. Let P = x × P1 . Repeating the above process for the group Tx P1 we obtain a radical chain of p-subgroups 1 < Q1 < Q2 < · · · < Qn where Q1 = x, Qj = x, x2 , . . . , xj for j = 2, 3, . . . , n, and pn = |P |. Thus mp (G) = n = plr(G). We are done. Theorem 12 ([9]). Let G be a finite p-solvable group with Op (G) = 1. Then we have: (1) The p-rank mp (G) ≤ plr(G). (2) The p-length p (G) ≤ plr(G); and if p (G) = plr(G) with p > 2 then p (G) = 0 or 1.

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Acknowledgements. We thank Professors W. Feit and G. Robinson for their help during the preparation of this paper.

References [1]

H. I. Blau and G. O. Michler, Modular representation theory of finite groups with T. I. Sylow p-subgroups, Trans. Amer. Math. Soc. 319 (1990), 417–468.

[2]

Ch. W. Eaton, On finite groups of p-local rank one and conjectures of Dade and Robinson. J. Algebra 238 (2001), 623–642.

[3]

J. A. Green, Some remarks on defect groups. Math. Z. 107 (1968), 133–150.

[4]

O. Manz and T. R. Wolf, Representations of Solvable Groups, London Math. Soc. Lecture Note Ser. 185, Cambridge University Press, Cambridge 1993.

[5]

Ch. E. Praeger, Covering subgroups of groups and Kronecker classes of fields. J. Algebra 118 (1988), 455–463.

[6]

G. R. Robinson, The number of blocks with given defect group, J. Algebra 84 (1983), 493–502.

[7]

G. Robinson, Local structure, vertices and Alperin’s conjecture, J. London Math. Soc. 72 (3) (1996), 312–330.

[8]

B. S. Wang, J. P. Zhang, On finite solvable full p-defective groups, Algebra Colloquium V10 (2003), 405–412.

[9]

B. S. Wang, J. P. Zhang, On the p-local rank of finite groups, Acta Math. Sin. 19 (2003), 29–34.

[10] J. P. Zhang, On finite groups all of whose p-blocks are of the highest defect, J. Algebra 118 (1988), 129–139. [11] J. P. Zhang, Studies on defect groups, J. Algebra 166 (1994), 310–316. [12] J. P. Zhang, p-regular orbits and p-blocks of defect zero. Comm. Algebra 21 (1993), 299–307. Jiping Zhang, Lmam, The School of Mathematical Sciences, Peking University, Beijing, P. R. China E-mail: [email protected]

List of contributors

Michael Aschbacher, Department of Mathematics, California Institute of Technology, Pasadena, CA 91125-0001, U.S.A. E-mail: [email protected] Christopher P. Bendel, Department of Mathematics, Statistics and Computer Science, University of Wisconsin-Stout, Menomonie, WI 54751, U.S.A. E-mail: [email protected] David J. Benson, Department of Mathematics, University of Georgia, Athens, GA 30602, U.S.A. E-mail: [email protected] Stephen Doty, Mathematics and Statistics, Loyola University Chicago, Chicago, IL 60626, U.S.A. E-mail: [email protected] Norberto Gavioli, Dipartimento di Matematica Pura ed Applicata, Università degli Studi di L’Aquila, Via Vetoio, 67010 Coppito (L’Aquila) AQ, Italy E-mail: [email protected] Meinolf Geck, Institut Girard Desargues, Université Claude Bernard-Lyon I, 21 Avenue Claude Bernard, 69622 Villeurbanne Cedex, France E-mail: [email protected] George Glauberman, Department of Mathematics, University of Chicago, 5734 S. University Ave., Chicago, IL 60637, U.S.A. E-mail: [email protected] Ralf Gramlich, TU Darmstadt, FB Mathematik / AG 5, Schloßgartenstraße 7, 64289 Darmstadt, Germany E-mail: [email protected] Robert M. Guralnick, Department of Mathematics, University of Southern California, DRB 155, 1042 W. 36th Place, Los Angeles, CA 90089-1113, U.S.A. E-mail: [email protected] Allen Herman, Department of Mathematics and Statistics, University of Regina, Regina, SK Canada S4S 0A2, Canada E-mail: [email protected] Gerhard Hiss, Lehrstuhl D für Mathematik, RWTH Aachen, Templergraben 64, 52062 Aachen, Germany E-mail: [email protected] Andrei Jaikin-Zapirain, Departamento de Matemáticas, Facultad de Ciencias, Universidad Autónoma de Madrid, Cantoblanco Ciudad Universitaria, 28048 Madrid, Spain E-mail: [email protected]

408

List of contributors

Ben Klopsch, Mathematisches Institut, Heinrich-Heine-Universität, 40225 Düsseldorf, Germany E-mail: [email protected] Inna Korchagina, Department of Mathematics, Rutgers University, New Brunswick, NJ 08903, U.S.A. E-mail: [email protected] Mark Lewis, Department of Mathematics, Kent State University, Kent, OH 44242, U.S.A. E-mail: [email protected] Frank Lübeck, Lehrstuhl D für Mathematik, RWTH Aachen, 52056 Aachen, Germany E-mail: [email protected] Gunter Malle, Fachbereich Mathematik/Informatik, Universität Kassel, HeinrichPlett-Straße 40, D-34132 Kassel, Germany E-mail: [email protected] Barry Monson, Department of Mathematics and Statistics, University of New Brunswick, Fredericton, NB Canada E3B 5A3, Canada E-mail: [email protected] Valerio Monti, Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate, Università degli Studi di Roma ‘La Sapienza’, via Scarpa, 16, 00161 Roma, Italy E-mail: [email protected] Alexander Moretó, Departamento de Matemáticas, Facultad de Ciencias, Universidad del País Vasco, Apartado 644, Bilbao, Spain E-mail: [email protected] Daniel K. Nakano, Department of Mathematics, University of Georgia, Athens, GA 30602, U.S.A. E-mail: [email protected] Gabriel Navarro, University of Valencia, Dr. Moliner 50, 46100 Burjassot, València, Spain E-mail: [email protected] Alan R. Prince, School of Mathematical and Computer Sciences, Heriot-Watt University, Edinburgh EH14 4AS, United Kingdom E-mail: [email protected] Cornelius Pillen, Department of Mathematics and Statistics, ILB 325, University of South Alabama, Mobile, AL 36688, U.S.A. E-mail: [email protected] Urmie Ray, Université de Reims, 34 Rue des Cheneaux, Sceaux 92330, France E-mail: [email protected] Geoffrey R. Robinson, University of Birmingham, School of Mathematics and Statistics, Edgbaston, Birmingham B15 2TT, United Kingdom E-mail: [email protected]

List of contributors

409

Mark Ronan, Department of Mathematics (M/C 249), University of Illinois at Chicago, 851 S. Morgan, Chicago, IL 60607-0745, U.S.A. E-mail: [email protected] Josu Sangroniz, Departamento de Matemáticas, Facultad de Ciencias, Universidad del País Vasco, Apartado 644, Bilbao, Spain E-mail: [email protected] Carlo Scoppola, Dipartimento di Matematica Pura ed Applicata, Università degli Studi di L’Aquila, Via Vetoio, 67010 Coppito (L’Aquila) AQ, Italy E-mail: [email protected] Ronald M. Solomon, Department of Mathematics, Ohio State University, 231 W. 18th St, Columbus, OH 43210-1101, U.S.A. E-mail: [email protected] M. Chiara Tamburini, Dipartimento di Matematica e Fisica, Università Cattolica del Sacro Cuore, Via Musei 41, 25121 Brescia, Italy E-mail: [email protected] Franz G. Timmesfeld, Mathematisches Institut, Arndtstraße 2, 35392 Giessen, Germany E-mail: [email protected] Lawrence E. Wilson, Department of Mathematics, University of Florida, PO Box 118105, 358 Little Hall, Gainesville, FL 32611-8105, U.S.A. E-mail: [email protected] Thomas R. Wolf, Department of Mathematics, Ohio University, Athens, OH 45701, U.S.A. E-mail: [email protected] Alexandre E. Zalesski, School of Mathematics, University of East Anglia, Norwich NR4 7TJ, United Kingdom E-mail: [email protected] Jiping Zhang, Lmam, The School of Mathematical Sciences, Peking University, 100871, Beijing, China E-mail: [email protected]

List of participants Hossein Andikfar, Department of Mathematics (M/C 249), University of Illinois at Chicago, 851 S. Morgan, Chicago, IL 60607-0745, U.S.A. E-mail: [email protected] Michael Aschbacher, Department of Mathematics, California Institute of Technology, Pasadena CA 91125-0001, U.S.A. E-mail: [email protected] Christine Bachoc, Laboratoire d’Algorithmique Arithmétique, Université Bordeaux 1, 33405 Talence, France E-mail: [email protected] Aron Bereczky, Sultan Qaboos University, Department of Mathematics, P.O.B. 36, Al Khodh 123, Sultanate of Oman E-mail: [email protected] Robert Boltje, Department of Mathematics, University of California, Santa Cruz, CA 95064, U.S.A. E-mail: [email protected] Michel Broué, Institut Henri Poincaré, University of Paris VII, 11 rue Pierre et Marie Curie, 75005 Paris, France E-mail: [email protected] Douglas Brozovic, Department of Mathematics, University of North Texas, PO Box 311430, Denton, TX 76203, U.S.A. E-mail: [email protected] Wai Kiu Chan, Department of Mathematics, Wesleyan University, Middletown, CT 06459, U.S.A. E-mail: [email protected] David Chandler, Department of Mathematics, University of Delaware, Newark, DE 19716-2553, U.S.A. E-mail: [email protected] Naoki Chigira, Department of Mathematical Sciences, Muroran Institute of Technology, Hokkaido 050-8585, Japan E-mail: [email protected] Sarah Cunningham, Department of Mathematics, University of California, Santa Cruz, CA 95064, U.S.A. E-mail: [email protected] Ulrich Dempwolff, Universität Kaiserslautern, Fachbereich Mathematik, Postfach 3049, 67653 Kaiserslautern, Germany E-mail: [email protected]

412

List of participants

Eloisa Detomi, Dipartimento di Matematica pura ed aplicata, Università di Padova, Via G. Belzoni, 7, 35131 Padova, Italy E-mail: [email protected] Richard Dipper, Mathematisches Institut B Fakultät Mathematik, Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany E-mail: [email protected] Gary Ebert, Department of Mathematical Sciences, University of Delaware, Newark, DE 19716, U.S.A. E-mail: [email protected] Harald Ellers, Department of Mathematical Sciences, Northern Illinois University, DeKalb, IL 60115-2888, U.S.A. E-mail: [email protected] Jacqueline Espina, Department of Mathematics, University of California, Santa Cruz, CA 95064, U.S.A. E-mail: [email protected] Clara Franchi, Dipartimento di Matematica e Fisica, Università Cattolica del Sacro Cuore, via Trieste 17, 25121 Brescia, Italy E-mail: [email protected] Michael D. Fried, Department of Mathematics, University of California Irvine, Irvine, CA 92697-3875, U.S.A. E-mail: [email protected] Daniel Frohardt, Department of Mathematics, Wayne State University, Detroit, MI 48202, U.S.A. E-mail: [email protected] Meinolf Geck, Institut Girard Desargues, Université Claude Bernard-Lyon I, 21 Avenue Claude Bernard, 69622 Villeurbanne Cedex France E-mail: [email protected] Stephen P. Glasby, Department of Mathematics, Central Washington University, Ellensburg, WA 98926, U.S.A. E-mail: [email protected] George Glauberman, Department of Mathematics, University of Chicago, 5734 S. University Ave., Chicago, IL 60637, U.S.A. E-mail: [email protected] Adam Glesser, Department of Mathematics, University of California, Santa Cruz, CA 95064, U.S.A. E-mail: [email protected] David Gluck, Department of Mathematics, Wayne State University, Detroit, MI 48202, U.S.A. E-mail: [email protected]

List of participants

413

Jon González, Departamento de Matematicas, Facultad de Ciencias, Universidad del País Vasco, Apartado 644, Bilbao, Spain E-mail: [email protected] Roderick Gow, National University of Ireland, Belfield, Dublin 4, Ireland E-mail: [email protected] Ralf Gramlich, TU Darmstadt, FB Mathematik / AG 5, Schlossgartenstrasse 7, 64289 Darmstadt, Germany E-mail: [email protected] Robert M. Guralnick, Department of Mathematics, University of Southern California, DRB 155, 1042 W. 36th Place, Los Angeles, CA 90089-1113, U.S.A. E-mail: [email protected] David Hemmer, Department of Mathematics, University of Georgia, Athens, GA 30602, U.S.A. E-mail: [email protected] Allen Herman, Department of Mathematics and Statistics, University of Regina, Regina, SK, Canada S4S 0A2 E-mail: [email protected] Marcel Herzog, School of Mathematical Sciences, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel-Aviv University, Tel-Aviv, Israel E-mail: [email protected] Gerhard Hiss, Lehrstuhl D für Mathematik, RWTH Aachen, Templergraben 64, 52062 Aachen, Germany E-mail: [email protected] Chat Yin Ho, Department of Mathematics, University of Florida, PO Box 118105, 358 Little Hall, Gainesville, FL 32611-8105, U.S.A. E-mail: [email protected] Corneliu Hoffman, Department of Mathematics and Statistics, Bowling Green State University, Bowling Green, OH 43403, U.S.A. E-mail: [email protected] Alexander Hulpke, Department of Mathematics, Colorado State University, Fort Collins, CO 80523-1874, U.S.A. E-mail: [email protected] Andrei Jaikin-Zapirain, Departamento de Matemáticas, Facultad de Ciencias, Universidad Autónoma de Madrid, Cantoblanco Ciudad Universitaria, 28048 Madrid, Spain E-mail: [email protected] Michael Kallaher, Department of Pure and Applied Mathematics, Washington State University, Pullman, WA 99164-3113, U.S.A. E-mail: [email protected]

414

List of participants

Thomas Keller, Department of Mathematics, Southwest Texas State University, 601 University Drive, San Marcos, TX 78666-4604, U.S.A. E-mail: [email protected] Radha Kessar, Ohio State University, Department of Mathematics, 231 W 18th St., Columbus, OH 43210-1101, U.S.A. E-mail: [email protected] Alexander S. Kleshchev, Department of Mathematics, University of Oregon, Eugene, OR 97403, U.S.A. E-mail: [email protected] Ben Klopsch, Mathematisches Institut, Heinrich-Heine-Universität, 40225 Düsseldorf, Germany E-mail: [email protected] Inna Korchagina, Department of Mathematics, Rutgers University, New Brunswick, NJ 08903, U.S.A. E-mail: [email protected] Paul Lescot, INSSET, Université de Picardie, 48 Rue Raspail, 02100 Saint-Quentin, France E-mail: [email protected] Mark Lewis, Department of Mathematics, Kent State University, Kent, OH 44242, U.S.A. E-mail: [email protected] Klaus Lux, Department of Mathematics, University of Arizona, 617 N Santa Rita, Tucson, AZ 85721, U.S.A. E-mail: [email protected] Frank Lübeck, Lehrstuhl D für Mathematik, RWTH Aachen, 52056 Aachen, Germany E-mail: [email protected] Richard Lyons, Department of Mathematics, Rutgers University, New Brunswick, NJ 08903, U.S.A. E-mail: [email protected] Kay Magaard, Department of Mathematics, Wayne State University, Detroit, MI 48202, U.S.A. E-mail: [email protected] Mario Mainardis, Dipartimento di Matematica ed Informatica, Università di Udine, via delle Scienze 206, 33100 Udine, Italy E-mail: [email protected] Gunter Malle, Fachbereich Mathematik/Informatik, Universität Kassel, HeinrichPlett-Straße 40, 34132 Kassel, Germany E-mail: [email protected] David McNeilly, Department of Mathematical and Statistical Sciences, University of Alberta, 632 Central Academic Building, Edmonton, Alberta T6G 2G1, Canada E-mail: [email protected]

List of participants

415

Alexander Moretó, Departamento de Matematicas, Facultad de Ciencias, Universidad del País Vasco, Apartado 644, Bilbao, Spain E-mail: [email protected] Marta Morigi, Dipartimento di Matematica, Università di Bo