MICHAEL ASCHBACHER Department of Mathematics California Institute of Technology
Sporadic groups
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MICHAEL ASCHBACHER Department of Mathematics California Institute of Technology
Sporadic groups
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CAMBRIDGE UNIVERSITY PRESS
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Published by the Press Syndicate of the University of Cambridge The Pitt BuiIding, Trumpington Street, Cambridge CB2 IRP 40 West 20th Street, New York, NY 10011-4211, USA 10 Stamford Road, Oakleigh, Victoria 3166, Australia
Contents
@ Cambridge University Press 1994
First published 1994
Preface
page i x
Printed in the United States of America
PART I Library of Congress Cataloging-in-PublicationData Aschbacher, Michael. Sporadic groups / Michael Aschbacher. p. cm. - (Cambridge tracts in mathematics ; 104) Includes bibliographical references and indexes. ISBN 0-521-420440 1. Sporadic groups (Mathematics) I. Title. 11. Series. QA177.A83 1994 512l.2 - dc20 92-13653 CIP
1. Preliminary Results 1. Abstract representations 2. Permutation representations 3. 4.
Graphs Geometries and complexes 5. The general linear group and its projective geometry 6. Fiber products of groups
2.
2-Structure i n F i n i t e G r o u p s 7. Involutions 8. Extraspecial groups
3.
Algebras, Codes, and Forms 9. Forms and algebras 10. Codes 11. Derived forms
4.
Syrnplectic 2-Loops 12. Symplectic 2-loops 13. Moufang symplectic Zloops 14. Constructing a 2-local from a loop
46 47 54 57
5.
T h e Discovery, Existence, a n d Uniqueness of the Sporadics 15. History and discovery 16. Existence of the sporadics 17. Uniqueness of the sporadics
65 65 70
A catalog record for this book is available from the British Library. ISBN 0-521-42049-0 hardback
6.
The M a t h i e u Groups, T h e i r Steiner Systems, a n d t h e Golay C o d e 18. Steiner systems for the Mathieu groups 19. The Golay and Todd modules
74
Contents
Contents
vi 7.
The Geometry and Structure of MZ4 20. The geometry of M24 21. The local structure of M24
8.
The Conway Groups and the Leech Lattice 22. The Leech lattice and - 0 23. The Leech Iattice mod 2
9.
Subgroups of - 0 24. The groups Co3, Mc, and H S 25. The groups Col, Coz, SUZ,and J2 26. Some local subgroups of Col
10. The Griess Algebra and the Monster 27. The subgroups C and N of the Monster 28. The Griess algebra 29. The action of N on B 30. N preserves the Griess algebra 31. The automorphism group of the Griess algebra 11.
Subgroups of Groups of Monster Type 32. Subgroups of groups of Monster type
PART I11 12.
Coverings of Graphs and Simplicia1 Complexes 33. The fundamental groupoid 34. Triangulation 35. Coverings of graphs and simplicial complexes
13. The Geometry of Amalgams 36. Amalgams 37. Uniqueness systems 38. The uniqueness system of a string geometry 14. The Uniqueness of Groups of Type M24, He, and L5(2) 39. Some 2-local subgroups in L5(2),Mz4, and He 40. Groups of type L5(2),M24, and He 41. Groups of type L5(2) and M24 42. Groups of type He 43. The root 4-group graph for He 44. The uniqueness of groups of type He
15. The Group U4(3) 45. U4(3)
16. Groups of Conway, Suzuki, and Hall-Janko Type 46. Groups of type Col, Suz, J2, and J3 47. Groups of type J2 48. Groups of type Suz 49. Groups of type Col 17.
Subgroups of Prime Order in Five Sporadic Groups 50. Subgroups of Suz of prime order 51. Subgroups of Col of prime order 52. Subgroups of prime order in He
Symbols Bibliography Index
Preface
The classification of the finite simple groups says that each finite simple group is isomorphic to exactly one of the following:
A group of prime order An alternating group A, of degree n A group of Lie type One of twenty-six sporadic groups
I I
,
I
I
I
I
As a first step in the classification, each of the simple groups must be shown to exist and to be unique subject to suitable hypotheses, and the most basic properties of the group must be established. The existence of the alternating group An comes for free, while the representation of An on its n-set makes possible a first uniqueness proof and easy proofs of most properties of the group. The situation with the groups of Lie type is more difficult, but while groups of Lie rank 1 and 2 cause some problems, Lie theory provides proofs of the existence, uniqueness, and basic structure of the groups of Lie type in terms of their Lie algebras and buildings. However, the situation with the sporadic groups is less satisfactory. Much of the existing treatment of the sporadic groups remains unpublished and the mathematics which does appear in print lacks uniformity, is spread over many papers, and often depends upon machine calculation. Sporadic Groups represents the first step in a program to provide a uniform, self-contained treatment of the foundational material on the sporadic groups. More precisely our eventual aim is to provide complete proofs of the existence and uniqueness of the twenty-six sporadic groups subject to appropriate hypotheses, and to derive the most basic structure of the sporadics, such as the group order and the normalizers of subgroups of prime order. While much of this program is necessarily technical and specialized, other parts are accessible to mathematicians with only a basic knowledge of finite group theory. Moreover some of the sporadic groups are the automorphism groups of combinatorial objects of independent interest, so it is desirable to make this part of the program available to as large an audience as possible. For example, the Mathieu groups are the automorphism groups of Steiner systems and Golay codes while the largest Conway group is the automorphism group of the Leech lattice.
x
Preface
Preface Sporadic Groups begins the treatment of the foundations of the sporadic groups by concentrating on the most accessible chapters of the subject. It is our hope that large parts of the book can be read by the nonspecialist and provide a good picture of the structure of the sporadics and the methods for studying these groups. At the same time the book provides the basis for a complete treatment of the sporadics. The book is divided into three parts: Part I, introductory material (Chapters 1-5); Part 11, existence theorems (Chapters 6-11); and Part 111, uniqueness theorems (Chapters 12-17). The goal of the existence treatment is to construct the largest sporadic group (the Monster) as the group of automorphisms of the Griess algebra. Twenty of the twenty-six sporadic groups are sections of the Monster. We establish the existence of these groups via these embeddings. To construct the Griess algebra one must first construct the Leech lattice and the Conway groups, and to construct the Leech lattice one must first construct the Mathieu groups, their Steiner systems, and the binary Golay code. There are many constructions of the Mathieu groups. Our treatment proceeds by constructing the Steiner systems for the Mathieu groups as a tower of extensions of the projective plane of order 4. This method has the advantage of supplying the extremely detailed information about the Mathieu groups, their Steiner systems, and the Golay code module and Todd module necessary both for the construction of the Leech lattice and the Griess algebra, and for the proof of the uniqueness of various sporadics. The construction given here of the Leech lattice and the subgroups stabilizing various sublattices is the standard one due to Conway in [Col] The construction of the Griess algebra combines aspects of and [CO~]. and Tits [T2], plus the treatments due to Griess [Gr2], Conway [CO~], a few extra wrinkles. The basis of the construction is Parker's loop and Conway's construction via the Parker loop of the normalizer N of a certain Csubgroup of the Monster. Chapter 4 contains a discussion of a general class of loops which includes the Parker loop. This discussion contains much material not needed to construct the Parker loop or the Griess algebra, but the extra discussion provides a context which hopefully makes the Parker loop and Conway's construction of N more natural. The majority of the sporadic groups contain a large extraspecial 2subgroup. Such subgroups provide one of the unifying features of our treatment. The basic theory of large extraspecial subgroups is developed
xi
in Chapter 2. The theory is used to recognize and establish the simplicity of the sporadics contained in the Monster that are not symmetry groups of any nice structure. The eventual object of the uniqueness treatment is to prove each sporadic is unique subject to suitable hypotheses. Here is a typical hypothesis; let w be a positive integer and L a group. (See Chapter 2 for terminology and notation.) Hypothesis Z(w, L): G is a finite gmup containing an involution z such that F*(CG(z))= Q is an extraspecial &subgroup of order 22W+1, CG(z)/Q L, and z is not weakly closed in Q with respect to G .
I
1 \
,
I
For example, Hypothesis 7f(12,Col) characterizes the Monster. Hypotheses of this sort are the appropriate ones for characterizing the sporadic~for purposes of the classification. Sporadic Groups lays the foundation for a proof of the uniqueness of each of the sporadics and supplies actual uniqueness proofs for five of the sporadic groups: M24, He, J2, Suz, and Col. Our approach to the uniqueness problem follows Aschbacher and Segev in [ASl]. Namely given a group theoretic hypothesis 31 we associate t o each group G satisfying 7f a coset graph A defined by some family F of subgroups of G. We prove the amalgam A of 3 is determined up t o isomorphism by 31 independently of G, and form the free amalgamated product G of A and its coset graph A. Now there exists a covering d :A A of graphs. To complete the proof we show A is simply connetted so d is an isomorphism and hence G = is determined up t o isomorphism by H. After developing the most basic part of the conceptual base for our treatment of the sporadic groups in Part I, Chapter 5 closes the first part of the book with an overview of the sporadic groups including the hypotheses by which we expect each group to be characterized, the approach for constructing each of the twenty sporadics involved in the Monster, and a number of historical remarks. While Sporadic Groups concentrates on some of the most accessible and least technical aspects of the study of the sporadic groups, a complete treatment of even this material sometimes requires some difficult specialized arguments. The reader wishing to minimize contact with such arguments can do so as follows. As a general rule the book becomes progressively more difficult in the later chapters. Thus most of the material in Part I should cause little difficulty. A possible exception is Chapter 4, containing the discussion of loops. However, much of this material is not
-
xii
Preface
needed in the rest of the book, and none is needed outside of Chapter 10, where the Griess algebra is constructed. As Chapter 10 is the most technical part of Part 11, some readers may wish to skip both Chapter 4 and Chapter 10. Part I1 contains constructions df the Mathieu groups, the Conway group Col and its sporadic sections, and the Monster and its sporadic sections. Two chapters are devoted to the Mathieu groups and two to the Conway groups. In each case the second of the two chapters is the most technical. Thus the reader may wish to read Chapters 6 and 8, while skipping or skimming Chapters 7 and 9. As suggested in the previous paragraph, dilettantes should skip the construction in Chapter 10 of the Griess algebra and the Monster. The existence proofs for the sporadic sections of the Monster not contained in Col appear in the very short Chapter 11. The Steiner systems and Golay codes associated to the Mathieu groups and the Leech lattice associated to the Conway groups are beautiful and natural objects. Most of the discussion of these objects appears in Chapters 6 and 8. There is some evidence that the Griess algebra is also natural, in that it is the 0-graded submodule of a conformal field theory preserved by the Monster (cf. [FLM]). However, the construction of the Griess algebra in Chapter 10 is not particularly natural or edifying. The first two chapters of Part I11 provide the conceptual base for proving the uniqueness of the sporadic groups. These chapters are fairly elementary. Sections 39 through 41 establishing the uniqueness of M24 and L5(2)probably provide the easiest example of how to apply this machinery to establish uniqueness. On the other hand the proofs of the uniqueness of He,J2, Suz, and Col, while more difficult, are also more representative of the complexity involved in proving the uniqueness of the sporadic groups. The book closes with tables describing the basic structure of the five sporadic groups considered in detail in Sporadic Groups: M 2 4 7 He, J2, Suz, and Gol. These tabIes enumerate the subgroups of prime order of each group G and the normalizers of these subgroups. Much of this information comes out during the proof of the uniqueness of G, but some of the loose ends are tied up in Chapter 17.
PART I
Chapter 1 Preliminary Results
We take a s our starting point the text Finite Group Theory [FGT], although we need only a fraction of the material in that text. Requently quoted results from [FGT]will be recorded in this chapter and in other of the introductory chapters. Chapters 1 and 2 record some of the most basic terminology and notation we will be using plus some elementary results. The reader should consult [FGT] for other basic group theoretic terminology and notation, although we will try to recall such notation when it is first used, or at least give a specific reference to [FGT] at that point. There is a "List of Symbols" at the end of [FGT] which can be used to help hunt down notation. We begin in Section 1 with a brief discussion of abstract representations of groups. Then in Section 2 we specialize to permutation representations. In Section 3 we consider graphs and in Section 4 geometries (in the sense of J. Tits) and geometric complexes. In the last few sections of the chapter we record a few basic facts about the general linear group and fiber products of groups.
1. Abstract representations Let C be a category. For X an object in C, we write Aut(X) for the group of automorphisms of X under the operation of composition in C (cf. Section 2 in [FGT]).A ~presentatzonof a group G in the category C is a group homomorphism ?r; G -+ Aut(X). For example, a permutation representation is a representation in the category of sets and a linear
2
Chapter 1 Preliminary Results representation is a representation in the category of vector spaces and linear maps. If a :A -+ B is an isomorphism of objects in C then a induces a map
2. Permutation representations
3
Let n : G -+ S y m ( X ) be a permutation representation of a group G on X. Usually we suppress n and write xg for the image x(gn) of a point x E X under the permutation gn, g E G. For S C G, we write Fix(S) = Fixx(S) for the set of fixed points of S on X . For Y X,
c
and a* restricts to an isomorphism a* : Aut(A) -+ Aut(B). Thus in particular if A r B then Aut(A) E Aut(B). A representation n : G -t Aut(A) is faithful if n is injective. Two representations n : G -, Aut(A) and u : G + Aut(B) in C are equivalent if there exists an isomorphism a : A --, B such that a = na* is the composition of n with a*. Equivalently for all g E G, (gn)a= a(ga). Similarly if Ti : Gi -+ Aut(Ai), i = 1,2, are representations of groups Gi on objects Ai in C, then nl is said to be quasiequivalent to n2 if there exists a group isomorphism f l : GI -, G2 and an isomorphism cr : Al -+ A2 such that 1r2 = p-lnlat. Observe that we have a permutation representation of Aut(G) on the equivalence classes of representations of G via cr : n H a n with the orbits the quasiequivalence classes. Write Aut(G), for the stabilizer of the equivalence class of n under this representation. The following result is Exercise 1.7 in [FGT]:
Lemma 1.1: Let n,u : G -+ Aut(A) be faithful representations. Then
is quasiequivalent to a if and only if Gn is conjugate to Ga in Aut (A). (2) Aut,,t(,) (Gn)2 Aut(G),. (1)
R
If H < G then write AutG(H) = Na(H)/CG(H)for the group of automorphiims of H induced by G. Also
CG(H)= ( c E G :ch=hcfor all h~ H ) is the centralizer in G of H and NG(H) is the normalizer in G of H, that is, the largest subgroup of G in which H is normal.
2. Permutation representations In this section X is a set. We refer the reader to Section 5 of [FGT] for our notational conventions involving permutation groups, although we record a few of the most frequently used conventions here. In particular we write S y m ( X ) for the symmetric group on X and if X is finite we write Alt(X) for the alternating group on X . Further S,, A, denote the symmetric and alternating groups of degree n; that is, Sn = S y m ( X ) and A, = Alt(X) for X of order n.
G y = { g E G : yg = y for all y E Y) is the pointwise stabilizer of Y in G,
is the global stabilizer of Y in G, and G~ = G ( Y ) / G y is the image of G ( Y ) in Sym(Y) under the restriction map. In particular Gp denotes the stabilizer of a point y E X. Recall the orbit of x E X under G is XG = {xg : g E G ) and G is transitive on X if G has just one orbit on X. If G is transitive on X then our representation n is equivalent to the representation of G by right multiplication on the coset space G/G, via the map Gzg t+ xg (cf. 5.9 in [FGT]). A subgroup K of G is a regular normal subgroup of G if K G and K is regular on X; that is, K is transitive on X and Kx = 1 for x E X. Recall a transitive permutation group G is primitive on X if G preserves no nontrivial partition on X . Further G is primitive on X if and only if Gx is maximal in G (cf. 5.19 in [FGT]).
Lemma 2.1: Let G be transitive on X , x
E X,
and K < G. Then
(1) K is transitive on X if and only if G = GZK.
If 1 # K q G and G is primitive on X then K is transitive on X . (3) If K is a regular normal subgroup of G then the representations of Gx on X and on K by conjugation are equivalent.
(2)
Proof: These are all well known; see, for example, 5.20, 15.15, and 15.11 in [FGT].
Recall that G is t-transitive on X if G is transitive on ordered t-tuples of distinct points of X . In Chapter 6 we will find that the Mathieu group Mm+t is t-transitive on m+t points for m = 19 and t = 3,4,5 and m = 7 and t = 4,5.
Lemma 2.2: Let G be t-transitive on a finite set X with t 2 2, x E X, and 1 # K G. Then
(1) G is primitive on X . (2) K is transitive on X and G = G,K.
3. Graphs
Chapter 1 Pmliminary Results
(3) If K is regular on X then 1KI = 1x1 = pe is a power of some prime p, and if t > 2 then p = 2. (4) If t = 3 < 1x1 and IG : KI = 2 then K is 2-transitive on X . Proof: Again these are well-known facts. See, for example, 15.14 and 15.13 in [FGT]for (1) and (3), respectively. Part ( 2 ) follows from ( 1 ) and 1.1. Part (4) is left as Exercise 1.1.
3. Graphs A graph A = (A,*) consists of a set A of vertices (or objects or points) together with a symmetric relation * called adjacency (or incidence or something else). The ordered pairs in the relation are called the edges of the graph. We write u * v to indicate two vertices are related via * and say u is adjacent to v. Denote by A ( u ) the set of vertices adjacent to u and distinct from u and define uL = A ( u ) LJ{u). A path of length n from u to v is a sequence of vertices u = uo, u l , . ., u,= v such that ui+l E uf for each i. Denote by d(u,v ) the minimal length of a path from u to v. If no such path exists set d(u,v) = oo. d(u,v ) is the distance from u to v. The relation on A defined by u v if and only if d(u, v ) < OCI is an equivalence relation on A. The equivalence classes of this relation are called the connected components of the graph. The graph is connected if it has just one connected component. Equivalently there is a path between any pair of vertices. A morphism of graphs is a function a : A -+ At from the vertex set of A to the vertex set of At which preserves adjacency; that is, uLa C (ua)' for each u E A. A group G of automorphisms of A is edge transitive on A if G is transitive on A and on the edges of A. Representations of groups on graphs play a big role in this book. For example, we prove the uniqueness of some of the sporadics G by considering a representation of G on a suitable graph. The following construction supplies us with such graphs. Let G be a transitive permutation group on a finite set A. Recall the orbitals of G on A are the orbits of G on the set product A2 = A x A. The permutation rank of G is the number of orbitals of G; recall this is also the number of orbits of Gz on A for x E A. Given an orbital R of G , the paired orbital Rp of R is
.
-
N
Stp = { ( Y ,x ) : (2,y)
E R).
5
Evidently RP is an orbital of G with (Rp)P = R. The orbital R is said to be self-paired if RP = 0. For example, the diagonal orbital { ( x , x ) : x E A ) is a self-paired orbital.
Lemma 3.1: (1) A nondiagonal orbital ( x ,y)G of G is self-paired i f and only i f ( x ,y) is a cycle i n some g E G. (2) If G is finite then G possesses a nondiagonal self-paired orbital i f and only i f G Gs of even order. (3) If G is of even order and pennutation mnk 3 then all orbitals of G are self-paired. Proof:See 16.1 in [FGT].
Lemma 3.2: (1) Let R be a self-paired orbital o f G . Then R is a symmetric relation on A, so A = (A,51) is a graph and G is an edge transitive group of automorphisms of A. (2) Conversely i f H is an edge transitive group of automorphisms of a graph A = ( A , *) then the set * of edges of A is a self-paired orbital of G on A, and A is the graph determined by this orbital. Many of the sporadics have representations as rank 3 permutation groups. Indeed some were discovered via such representations; see Chap ter 5 for a discussion of the sporadics discovered this way. See also Exercise 16.5, which considers the rank 3 representation of J2, and Lemmas 24.6, 24.7, and 24.11, which establish the existence of rank 3 representations of Mc, U4(3),and H S . In the remainder of this section assume G is of even order and permutation rank 3 on a set X . Hence G has two nondiagonal orbitals A and I' and by 3.1, each is self-paired. Further for x E X , G z has two orbits A ( x ) and r ( x ) on X - { x ) , where A ( x ) = { y E X : (2,y) E A } and r ( x ) = { z E X : ( x , z ) E I?}. By 3.2, X = ( X , A ) is a graph and G is an edge transitive group of automorphisms of X. Notice A ( x ) = X ( x ) in our old notation. The following notation is standard for rank 3 groups and their graphs: k = lA(x)l, 1 = Ir(x)l, X = l A ( x ) n A(y)l for y E A ( x ) , and p = I A ( x ) n A ( z )( for z E r ( x ) . The integers k , 1, A, p are the parameters of the rank 3 group G. Also let n = 1x1 be the degree of the representation.
Lemma 3.3: Let G be a mnk 3 permutation group of even order on a finite set of order n with parameters k , 1, A, p. Then
6
Chapter 1 Preliminary Results If p # 0 or k then G is primitive and the graph B of G is connected. (4) Assume G is primitive. Then either (a) k = 1 and p = X + 1 = k/2, or (b) d = (A - p)2 4(k - p) is a square and setting D = 2k + (A - p)(k l), d1f2 divides D and 2d1I2 divides D i f and only i f n is odd. (3)
+
+
Proof: See Section 16 of [FGT].
4. Geometries and complexes In this book we adopt a notion of geometry due to J. Tits in [TI]. Let I be a finite set. For J I, let J' = I - J be the complement of J in I. A geometry over I is a triple ( l ? , ~*), where I' is a set of objects, 7 : I' + I is a surjective type function, and * is a symmetric incidence relation on I' such that objects u and v of the same type are incident if and only if u = v. We call ~ ( uthe ) type of the object u. Notice (I?,*) is a graph. We usually write I'for the geometry (I?, 7, *) and ri for the set of objects of I? of type i. The rank of the geometry l? is the cardinality of I. A flag of I? is a subset T of I' such that each pair of objects in T is incident. Notice our one axiom insures that if T is a flag then the type function T : T -t I is injective. Define the type of T to be T(T) and the rank of T to be the cardinality of T. The chambers of I? are the flags of type I. A morphism a : I' -+ I" of geometries is a function a : I' -+ I" of the associated object sets which preserves type and incidence; that is, ) ~ ' ( u a )and ua *' var. A group G of if u, v E I? with u * v then ~ ( u = automorphisms of I' is edge transitive if G is transitive on flags of type J for each subset J of I of order at most 2. Similarly G is flag transitive on I? if G is transitive on flags of type J for all J E I. Representations of groups on geometries also play an important role in Sporadic Groups. For example, the Steiner systems in Chapter 6 are rank 2 geometries whose automorphism groups are the Mathieu groups. Here are some other examples: Examples (1) Let V be an n-dimensional vector space over a field F. We associate a geometry PG(V) to V called the projective geometry of V. The objects of PG(V1 are the proper nonzero subspaces of V, with incidence defined by inclusion. The type of U is T(U) = dim(U). Thus
4.
Geometries and complexes
7
PG(V) is of rank n - 1. The projective general linear group on V is a flag transitive group of automorphism of PG(V). (2) A projective plane is a rank 2 geometry I' whose two types of objects are called points and lines and such that: (PP1) Each pair of distinct points is incident with a unique line. (PP2) Each pair of distinct lines is incident with a unique point. (PP3) There exist four points no three of which are on a common line.
Remarks. (1) Rank 2 projective geometries are projective planes. (2) If I'is a finite projective plane then there exists an integer q such that each point is incident with exactly q + 1 l i i , each lime is incident with exactly q + 1 points, and I' has q2 + q + 1 points and lines. Examples (3) If f is a sesquilinear or quadratic form on V then the totally singular subspaces of V are the subspaces U such that f is trivial on U.The set of such subspaces forms a subgeometry of the projective geometry. See, for example, page 99 in [FGT]. (4) Let G be a group and 3 = (Gi : i E I ) a family of subgroups of G. Define I'(G,3) to be the geometry whose set of objects of type i is the coset space GIGi and with objects Gix and Gjy incident if Gix n Gjy # 0.Observe: Lemma 4.1: (1) G is represented as an edge transitive group of automorphisms of r(G, 3)via right multiplication and r(G, 3 ) possesses a chamber. (2) Conversely if H is an edge transitive group of automorphisms of a geometry I' and I' possesses a chamber C, then I' r I'(H,3), where 3 = ( H , : c E C).
The construction of 4.1 allows us to represent each group G on various geometries. The construction is used in Chapter 13 as part of our machine for establishing the uniqueness of groups. Further the construction associates to each sporadic group G various geometries which can be used to study the subgroup structure of G. The latter point of view is not explored to any extent in Sporadic Groups;see instead [A21 or [RS] where such geometries are discussed. We do use the 2-local geometry of M24 to study that group in Chapter 7. Define the direct sum of geometries Pi on Ii,i = 1,2, to be the geometry rl @r2over the disjoint union I of Il and I2whose object set is the disjoint union of rl and I'2, whose type function is TI U 72, and whose incidence is inherited from rl and I'2 with each object in incident with each object in I'2.
8
4.
Chapter 1 Preliminary Results
Example (5) A generalized digon is a rank 2 geometry which is the direct sum of rank 1 geometries. That is, each element of type 1 is incident with each element of type 2.
Lemma 4.2: Let G be a group and 3 = (GI, G2) a pair of subgroups of G . Then I'(G,F) is a generalized digon if and only if G = G1G2. Proof: As G is edge transitive on I', I? is a generalized digon if and only if G2 is transitive on I'l if and only if G = GIG2 by 2.1.1.
Given a flag T, let r ( T ) consist of all v E I' - T such that v * t for all t E T. We regard I'(T) as a geometry over I - T(T). The geometry r ( T ) is called the residue of T.
Example (6) Let I' = PG(V) be the projective geometry of an ndimensional vector space. Then for U E I', the residue r ( U ) of the object U is isomorphic t o PG(U) $ PG(V/U). The category of geometries is not large enough; we must also consider either the category of chamber systems or the category of geometric complexes. A chamber system over I is a set X together with a collection of equivalence relations y, i E I. For J E I and x E X , let N J be the equivalence relation generated by the relations -j, j E J, and [ x ] the ~ equivalence class of J containing x. Define X t o be nondegenerate if for ~ [x]j = n i E j t [ x ] iA. morphism each x E X , and j E I , {XI = n i [ x ] iand of chamber systems over I is a map preserving each equivalence relation. The notion of "chamber system" was introduced by J. Tits in [TI]. Recall that a simplicial complex K consists of a set X of vertices together with a distinguished set of nonempty subsets of X called the simplices of K such that each nonempty subset of simplex is a simplex. The morphiims of simplicial complexes are the simplicial maps; that is, a simplicial map f : K --, K' is a map f : X --,X' of vertices such that f (s) is a simplex of K' for each simplex s of K.
-
Example (7) If A is a graph then the clique complex K(A) is the simplicial complex whose vertices are the vertices of A and whose simplices are the finite cliques of A. Recall a clique of A is a set Y of vertices such that y E xL for each x, y E Y . Conversely if K is a simplicial complex then the gmph of K is the graph A = A(K) whose vertices are the vertices of K and with x * y if { x , y) is a simplex of K. Observe K is a subcomplex of K(A(K)). Given a simplicial complex K and a simplex s of K , define the star of s to be the subcomplex s t K ( s ) consisting of the simplices t of K such that
,
Geometries and complexes
9
s U t is a simplex of K. Define the link LinkK(s) t o be the subcomplex of s t K ( s ) consisting of the simplices t of s t K ( s ) such that t n s = 0. A geometric complex over I is a geometry I? over I together with a collection C of distinguished chambers of I' such that each flag of rank 1 or 2 is contained in a member of C. The simplices of the complex are the subflags of members of C. A morphism cu :C -,C' of complexes over I is a morphism of geometries with Ca G C'. Notice a geometric complex is just a simplicial complex together with a type function on vertices that is injective on simplices.
Example (8) The flag complex of a geometry I? is the simplicial complex on I? in which all chambers are distinguished. Notice the flag complex is a geometric complex if and only if each flag of rank a t most 2 is contained in a chamber. Further as a simplicial complex, the flag complex is just the clique complex of I'regarded as a graph. Many theorems about geometries are best established in the larger categories of geometric complexes or chamber systems. Theorem 4.11 is an example of such a result. We find in a moment in Lemma 4.3 below that the category of nondegenerate chamber systems is isomorphic t o the category of geometric complexes. I find the latter category more intuitive and so work with complexes rather than chamber systems. But others prefer chamber systems and there is a growing literature on the subject. Given a chamber system X define rx to be the geometry whose objects of type i are the equivalence classes of the relation - i t with A* B if and only if A n B # 0.For x E X let C, be the set of equivalence classes containing x; thus C, is a chamber in I?x. Define CX to be the set of chambers C,, x E X , of I'x. If a : X --, X' is a morphism of chamber systems define ac : CX 4 CXt t o be the morphism of complexes such that ac : A H A' for A a -it equivalence class of X and A' the equivalence class containing Aa. Conversely given a geometric complex C over I let -i be the equivrtlence relation on C defined by A -i B if A and B have the same subflag of type it. Then we have a chamber system XCwith chamber set C and k r t h e r if a : C -+ C' is a morphism of comequivalence relations plexes let ax : XC--, Xct be the morphism of chamber systems defined by the induced map on chambers.
Lemma 4.3: The categoy of nondegenemte chamber systems over I is isomorphic to the category of geometric complexes over I via the maps X w C x andC-XC.
10
Chapter 1 Preliminary Results
4.
Example (9) Let G be a group and 3 = (Gi : i E I ) a family of subgroups of I. For J 5 I and x E G define SJ,Z = {Gjx :j E J ) . Thus Sj,, is a flag of the geometry I'(G, 3)of type J . Observe that the Gj. Define stabilizer of the flag SJ = SJllis the subgroup G j = C(G,3) to be the geometric complex over I with geometry I'(G,3) and distinguished chambers Sz,,, x E G. Then C(G,3) is a geometric complex with simplices Sj,,, J E I, x f G, and G acts as an edge transitive group of automorphisms of C(G, 3 ) via right multiplication, and transitively on C(G, 3 ) . Indeed:
njE
Lemma 4.4: Assume C is a geometric complex over I and G is an edge transitive group of automorphisms with C = CG for some C E C. Let Gi = G,,, where xi E C is of type i, and let 3= (Gi : i E I). Then the Gig is an isomorphism of C with C(G, 3 ) . map x,g Further we have a chamber system X(G,3) whose chamber set is GIGz and with Gzx ~i Gzy if and only if xy-' E Gil. Observe that the map GIs I+ S1,, defines an isomorphim of the chamber systems X(Gl 3 ) and XC(G,F). The construction of 4.4 allows us to represent a group G on many complexes. We make use of this construction in Chapter 13 as part of our uniqueness machine. Let C = (I?, C) be a geometric complex over I. Given a simplex S of type J, regard the link Linkc(S) of S to be a geometric complex over J'; thus the objects of Linkc(S) of type i E J' are those u E ri such that S U {v) is a simplex and with v * u if S U {u, v) is a simplex, and the chamber set C(S) of Linkc(S) consists of the simplices C - S with S E C E C. For example, C = Linkc(@) is the link of the empty simplex. Notice that if all flags are simplices then the geometry of Linkc($) is the residue r(S) of S in the geometry r. We say C is residually connected if the link of each simplex of corank at least two (including 0 if 111 1 2) is connected. A geometry 'I is residually connected if each flag is contained in a chamber and the flag complex of I? is residually connected.
Lemma 4.5: Let 3= (Gi : i E I ) be a family of subgroups of G. Then (1) I?(G,3 ) is connected if and only if G = (3). (2) Linkc ( Sj ) C(Gj,3 j ) for each J E I, where
11
Geometries and complexes
(3) C(G,3) is residually connected if and only if GJ = ( 3 j ) for all
J E I. Proof: Notice (1) and (2) imply (3) so it remains to prove (1) and (2). As 3is a chamber, the connected component A of G, in I' is the same for each i, and H = ( 3 ) acts on A. Conversely as Gi is transitive on rj(Gi) for each j, A A' = Uj GjH, so A = A' and H is transitive on ri n A for each i . Thus as G is transitive on I?,, I' is connected if and only if H is transitive on I'i for each i, and as Gi < H this holds if and only if G = H. Thus (1) is established. In (2) the desired isomorphim is Gkx w SK,, for x E G j , K = J U {k).
Lemma 4.6: Assume C is a residually connected geometric complex over I, J C_ I with IJI 2 2, and x, y E I?. Then there exists a path x = vo, vm = y in I? with 7(vi) E J for all 0 < i < m.
...
Proof: Choose x, y to be a counterexample with d = d(x, y) minimal. As the residue l? of the simplex 0 is connected, d is finite, and clearly d > 1. Let x = vo vd = y be a path. By minimality of d there is a path vl = ug . .um = y with 7(ui) E J for 0 < i < m. Thus if r(vl) E J then xu0 % is the desired path, so assume 7(vl) 4 3. We also induct on the rank of C; if the rank is 2 the lemma is trivial, so our induction is anchored. Now Linkc(vl) is a residually connected complex and x,u1 E Linkc(ul), SO by induction on the rank of C, there is a path x = wo-..wk = u1 with 7(wi) E J for 0 < i < k. NOW ~ the job. x = w ~ " ' w ~ u ~="y 'does Given geometric complexes C over J and over 3 define C €D to be the geometric complex over the disjoint union I of 3 and whose geometry is I? @ and with chamber set {C U : C E C, C E The basic diagram for a geometric complex C over I is the graph on I obtained by joining distinct i, j in I if for some simplex T of type {i,j}' (including 0 if III = 2), Linkc(T) is not a generalized digon. The basic diagram of a geometry is the basic diagram of its flag complex. Diagrams containing more information can also be associated to each geometry or geometric complex. The study of such diagrams was hitiated b y ' ~ i t s[TI] and Buekenout [Bu]. A graph on I is a string if we can order I = (1,. , n ) so that the edges of I are {i,i 11, 1 i < n. Such an ordering will be termed a string ordering. A string geometry is a geometry whose basic diagram is a string. Most of the geometries considered in Sporadic Groups are string geometries; for example:
. ..
c
+
2, Rn contains a subgroup of Q of order at least 4, so for each v E Q, 1# CRn(v). However, r E CR,(v) projects only on L and LU, so v acts on {L, LU). Thus as Y = (LQ), Y = LLU.
1, x E X , C(x) the set of blocks of X not containing x, and Y = D(X,x) the residual design of X at x. Then C(x) is an extension subset of Y.
A Steiner system S(v, k, t) or t-design is a rank 2 geometry (X,U) (cf. Section 4) whose objects are a set X of points and a collection U of k-subsets of X called blocks such that each t-subset of X is contained in a unique block. Of course incidence in this geometry is inclusion. Example (1) Recall the definition of a projective plane from Exarnple 2 in Section 4. Each projective plane of order q is a Steiner system s(q2 + q + l , q + l , 2 ) . Let (X, U) be a Steiner system S(v, k, t). Given a point x E X define the residual design D(X, x) of X at x to be the geometry (X(x), U(x)), where X(x) =X
- {x),
U(x) ={B- {x) : x E B €23).
79
Lemma 18.3: Let C be an extension subset of X. Then (1) There exists an extension Z of X such that C is the extension subset C(z) induced by Z and the point z in Z - X. (2) The restriction map a I-+ alx defines an isomolphism of Aut(Z)Z with NAU~(X) Proof: The blocks of Z are the members of C together with the blocks {z)uB, B E U. Remark 18.4. If Z is an extension of X at some point z we identify Aut(Z), with NAUt(x)(C(z)) via the isomorphism of 18.3.2. For example, this convention is used in the statement of the following hypothesis: Extension Hypothesis: Y is a Steiner system S(v, k,t) with t y E Y, and A A U ~ ( Y )Further ~.
1 then the residual design D(X,x) of X at x is a Steiner system S(v - 1, k - 1,t - 1).
(Exl)
Next define an extension of X to be a Steiner system (2, A) such that (X, 23) = D(Z, z) is the residual design of Z at some point z E Z. Notice if Z is an extension of X then by 18.1, Z has parameters (v+l, k+1, t+l). Our object is to construct a tower
(Ex2) (ExJ)
of extensions of Steiner systems beginning with the projective plane of order 4, such that the Mathieu group Mv is an automorphism group of the Steiner system S(v, k, t). The remainder of this section is devoted to this construction and to the generation of properties of these Steiner systems that we will need to analyze the Mathieu groups. -Define a subset I of X to be independent if no (t 1)-subset of I is contained in a block of X .
+
Example (2) If X is the projective plane of a vector space V, then the independent subsets are just the sets O of points such that each triple of points in 0 is linearly independent in V in the usual sense.
(Ex4)
> 1,
Aut(Y) is transitive on the extension subsets of Y invariant under some Aut(Y)-conjugate of A, and there exists such an extension subset. I f A S A ' < A U ~ ( Y then ) ~ A=A'. If Y' is an extension of D(Y, y) with A 5 Aut(Y1) then there exists an isomorphism ?r :Y -t Y' acting on D(Y, y). NAUt(Y)(C)is t-transitive on Y for each A-invariant extension subset C of Y.
Lemma 18.5: Let Y be a Steiner system S(v, k, t) and A Then
< Aut(Y).
(1) If A,Y satisfies (Exl) then, up to isomorphism, there exists a unique extension X of Y with A 5 NAut(x)(Y). Moreover NAut(x)(Y) = NAut(Y)(C), where C is the extension subset of Y induced by X. (2) Assume t > 1 and A,Y satisfies the Extension Hypothesis with respect to some y E Y. Then Aut(X) is (t 1)-transitive on X and transitive on the blocks of X.
+
+
Proof: For i = 1,2, let Xi be (t 1)-designs, xi E Xi, & = D(Xi, xi), and Ci the extension subset of Yi induced by Xi, and assume ai :Y -+ &
80
Chapter 6 The Mathieu Groups
18. Steiner systems for the Mathieu groups
c
30.S0 C1(B)is empty and Co(B)is of order 30. Thus (1) and (3)hold. Let A E Co(B).As Q is regular on X - B, Q # NQ(A).On the other hand as (Co(B)I-= 2 mod 4, we may choose A so that 1 AM1 is not divisible by 4. Hence a Sylow 2-subgroup of NM(A)is of index 2 in a Sylow 2-group of M and NQ(A)is regular on A. Indeed A and A+ B + X are the two orbits of NQ(A)on X - B, and g E Q - NQ(A)interchanges . these orbits, so A f B X = Ag is also an octad, establishing (7). Finally as NQ(A)is regular on A, NM(A)= NQ(A)NM=(A). Also NM=(A) L N M (NQ(A)), ~ so
-
(62)
+
+ +
+
1.4
&J,K
PK
196
Chapter 13 T h e Geometry of Amalgams
A completion P : A -,G for A is a family P = ( P j : Pj -+ Gj t i :roup homomorphims such that G = ( P j P j : J c I ) and for all J C K c I
36. Amalgams
! I
the obvious diagram commutes:
Proof: Suppose /3 : A --+ G is a faithful completion. Then injective, so Lj is injective.
197
P j = L j$ is
Lemma 36.6: Isomorphic amalgams have isomorphic universal completions.
4 :A + A is an isomorphism of amalgams and L : A --P G(A) and : A --+ G(A) are universal completions, then $b : A -+ G(A) is a completion, so there exists $J : G(A) -+ G(A) with Lj$ = +i;j for each J. Similarly we have 4 : G(A) -+ G(A) with = 4-I/, j. Then = $-I.
Proof: If
The completion
: A --+
G is said to be faithfil if each PJ is an injection.
Example 36.2 Let 3 = (Gi : i f I ) be a family of subgroups of a group G with G = (3). Form the amalgam A(3) of Example 36.1. Then the identity maps idJ : Pj -+ Pj form a faithful completion id = (idJ :J c I ) with id : A(3) -+ G.
I
4
The free product F(A) of the groups Pj, J C I, in an amalgam A is the free group on the disjoint union of the sets Pj modulo defining relations for the groups Pj. We have the following universal property:
Given a completion P : A 4 G of A let F(P) = (Gi : i E I ) , where Gi = P , I / ~ ~I?(@ , , = r(G, 3(P)), and C(P) = C(G,3(P)) be the geometry and geometric complex of P, as defined in Examples 4 and 9 of Section 4. Further for i E I , define the collinearity graph A(P, i ) of C(P) at i to be the graph on the set GIGi of objects of r ( P ) of type i with x adjacent to y if there exist chambers C, of C(P) for u = x, y with u E Cu and C, n C, a flag of type it.
Lemma 36.3: If (cpJ : Pj --+ G) is a family of group homomorphisms then there exists a unique group homomorphism cp : F(A) -+ G with gcp = gcpJ for all g f P j and all J c I.
Lemma 36.7: Let /3 : A -+ G be a faithful wmpletion of an amalgam A and L : A -+ G(A) the universal completion of A. Let cp : G(A) --+ G be the surjection of 36.4. Then
Define the free amalgamated product G(A) of the amalgam A to be the free product F(A) of the groups Pj, J c I , modulo the relations g - l ( g ~ j , K )= 1 for 'J c K c I and g E Pj. Write i j for the image of g f F(A) in G(A) and let L = ( ~: Pj j -+ G(A)) be defined by g~J = i j for g E Pj. Then
Lemma 36.4: L : A G(A) is a universal completion for A. That is, if p : A --+ G is a completion of A then there exists a unique group = PJ for all J C I . homomorphism $ : G(A) + G such that ~ j $ --+
Proof: By construction
L :A
4
G(A) is a completion of A. Suppose
p : A -t G is a completion. By the universal property of the free product recorded in 36.3, there exists a group homomorphism cp : F(A) + G defined by gcp = gPJ for g E Pj. As 0 is a completion of A, ga jKPK = gPJ for all J C K C I , so g - l ( g ~ j K ) E ker(cp). for each g E Pj. Thus cp induces a group homomorphism $J : G(A) --+ G defined by fi$ = g v = g P j for g E Pj. This is the map of 36.4.
Lemma 36.5: If A possesses a faithfil completion then the universal completion L : A -+ G(A) is faithful.
(1) cp : G(A)i
--+ Gi is an isomorphism with cp(G(A)j ) = G j for each i E J E I . (2) cp induces a morphism cp : C(L)+ C(P) of geometric complexes which is a covering of simplicia1 complexes, via cp : G(A)ig I--+ Gi~(g). (3) Assume for some &ed i E I that: (*) Gi, = GrtGr and Gi n Gi = (Gij n qj : j E it)for some t E Gil - GI. Then the covering cp of (2) restricts to a fibering cp : A(L,i ) --+ A(P, i ) of collinearity graphs. (4) Assume (*) and (**) The closure of the set of cycles of A(P,i) conjugate under G to a cycle of the wllinearity graph at i of Linkqp) ( G j ) ,as j ranges over i', is the set of all cycles of A(P,i). Then cp : G(A) + G is an isomorphism.
Proof: Let H = G(A). By 36.5, L is faithful. Thus cp : Hi Gi is the composition cp = L;'/~~Iof isomorphisms, so (1) holds. As cp(H,) = Gi, the map cp : Hig I--+ Giq(g) of (2) is well defined and as cp : H -+ G is surjective, cp : C(L)--, C(P) is surjective on vertices. -+
198
37. Uniqueness systems
Chapter 13 The Geometry of Amalgams
From the definition of C ( L ) in Example 9 of Section 4, the chambers of C(L) are of the form SI,, = {His : i E I ) , x E H , and P ( S ~ ,=~Sz,rp(x) ) is a chamber of C ( P ) , so cp : C ( L ) -+ C ( P ) is a morphism of geometric com~ ) Link(Gi) C(Gil3 ( P ) i ) 1 plexes. By 4.5, Link(Hi) C(HilJ ( L )and while by ( I ) , cp : C ( H i 1 3 ( ~ ) * ) C(Gi13(P)i)is an isomorphism, SO cp : Link(Hi) -+ Link(Gi) is an isomorphism and hence cp : C ( L ) C ( P ) is a covering of simplicia1 complexes. That is, (2) is established. ) A = A(P,i).By (2), cp restricts to a surjective Let A = A ( L , ~and morphim of graphs cp : A -+ A. Further as G is transitive on chambers of C ( P ) , each chamber through Gi is conjugate under Gi to 3 ( P ) . Also Gi, is the stabilizer of the wall W = 3 ( P ) - {Gi) of type i' and GI is the stabilizer of F(P). Assume (*). Then Gr = GztGz,so Git is 2-transitive on the chambers through W and GI is transitive on Link(W)- {G;). Hence Gi is transitive on A(Gi).Then by (1)and (2),HI is transitive on Link(U)- {Hi), where U = 3 ( ~ - {Hi), ) so Hi is transitive on A(Hi). Next Git E A(Gi) and by (*), Gi n G: = (Gij n : j E i f )is the stabilizer in Gi of Git. Let s E Hi, with cp(s) = t. By ( I ) , cp : Hi -+ Gi is an isomorphism with v(Hij n Hfj) = Gij n GIj SO (p(Hijn Hfj) : j E it)= Gi nGi. Of course q(Hi nH,S) Gi nGf,SO p(Hi n Hf ) = Gi nGi. Thus by ( I ) , cp :Hi/(Hi n H f ) -+ Gi/(Gi nGI) is a bijection and hence cp : A(Hi) -+ A(Gi) is a bijection and (3) is established. anjisomorphism ) of Finally by (2),cp : LinkC(')(Hj)-+ L ~ n k ~ ( ~ ) is( G simplicia1 complexes and hence induces an isomorphism of the collinearity graphs at i of these links. Hence by 35.16.3, under the hypotheses of (4),cp : A -+ A is an isomorphism. That is, the map cp : Hi + Gi is an isomorphism and the map cp : H/Hi -+ G/Gi is a bijection. Hence cp : H -+ G is an isomorphism and (4) is established. -+
-+
qj
7.
and iQ= i Q 6 , where : Q = C g ( i ) and Q; = [Q,i]. Pick Y E AH;by 46.5, C G ( Y ) / Y 2 U4(3), so there is an element t E C H ( Y )with t2 = z and t* E c2. Now t centralizes Y [ Q , Y ]G SL2(3) with Q t = C Q ( Y )n Q?, so by the previous paragraph each involution i E t Q is conjugate under Y Q to t u for some fixed u E [Q, Y ]of order 4. In particular there are 24 involutions in t Q , so as H* is transitive on c2 of order 270, H is transitive on r of order 24 270. Thus C H ( i ) = IH1/(24 270) = 2'.
-
Lemma 46.8: H is determined up to isomorphism independently of G. Proof: Let Ho '.
9
the universal covering group of H and
HI ). Then , by 46.4 and 8.17, P % Q and Let Z = Z ( H )and P = [ o ~ ( H H = H / U for some complement U to Z ( P ) in Z and H / P 2 L , where L = Lo/O(Lo)ia(02(Z(Lo)))and Lo is the universal covering group of H*. Let (n)= Z ( P ) . Suppose first w = 2. Then L Z SL2(5) (cf. 33.15 in [FGT]).Thus Z 2 Ed. Also H* has one class of involutions and m([Q, t ] )= 2 for each such involution t, so I? is transitive on involutions in I? - Q by Exercise 2.8. Thus if z Z is an involution in H / Z - P Z / Z then x2 = a is the unique element of Z such that t2 = a for some t E H - P Z with t2 E Z. Noticeas Z E E4, Z = ( a , ~and ) IUI = 2 . We claim U = (a).For if not U = ( a n ) , so for each t E H Q with t2 E ( z ) , t2 = z. That is, Q contains all involutions in H. This contradicts 46.3.3 which says H contains an S3 subgroup. Suppose next that w = 3. The argument is similar. This time by Exercise 16.6, Z Y Eq and each involution in H* of type c2 l i i s to an element of H I P of order 4. Now by 46.7 and its proof, H has two orbits t f , i = 1,2, on the set f' of t E 81 with t2 E Z and t P Z of type q in H I P 2 r H* with t 2 = t l x for some element x of order 4 in C p ( t 2 ) , Itrl = 8, and It;[ = 24. Then as involutions of H* of type c2 lift to = a x 2 = an. In this elements of order 4 in H I P , cr = # 1; then case we claim U = ( a n ) . If not U = ( a ) and t = tlU E I' (where I' is defined in 46.7), so 46.7 supplies a contradiction as ltQl = Itr( = 8. Finally take w = 4. We lift X E A H to a Sylow 3-group in its preimage in H, and hence regard X also as a subgroup of H. Now X centralizes
-
ti
ti
254 Chapter 16 Groups of Conway, Suzulci, and Hall-Janko Type an involution i E H with i* of type c2, and by the previous paragraph i lifts to t E H of order 4 with t2 E U. By Exercise 2.11, H* has two classes of involutions of type a4 with representatives jl and j2, where [Q,!,] = Ai are the two classes of maximal totally singular subspaces of Q. Further if a is a transvection in O(Q)then cu induces an outer automorphism on H* with A? = A2 and jr = j2. By Exercise 16.7, m ( Z ) = 3 and ji lifts to an element ti of order 4 in H I P with Z = (al,ag,n),where t; = ai. By 8.17.4, a l i i s to an automorphism of H, which we also denote by a. Then tq E t3-,Z, so a interchanges a1 and 0 2 . Hence up to conjugation under a , U = (al,a2), ( a l ,nu2),or (nal,na2). Next Q induces the full group of transvections on Ai with center ( z ) , so cH(Ai) = QCG(Ai)and CG(Ai)/Ai E cH*(Ai) g Efj4. Further the are of type c2 or jl, so as t2 E U , @(CG(A,)) =1 involutions in cH*(Ai) if and only if ui E U. Now if G = Col then (cf. the proof of 46.12) @(CG(Ai))= 1 for exactly one of i = 1,2, say i = 1. Thus in that case a1 E U but a2 $ 4. Therefore U = (ol,n q ) . Further t2 E U is fixed by a , so t2 = n a l w . Now in the general case we know t 2 E U, but of the three possibilities for U, only the second contains t2 = nsla2, so indeed H is determined up to isomorphism and the proof is complete. Notice in the process of proving 46.8 we have also proved several other facts. First Lemma 46.9: Let L 3 0 = Aut(A) be the automorphism group of the Leech lattice, K < L be quasisimple with K / Z ( K ) E Suz, and Z ( L ) 5 J 5 K with J/Z(L) a mot J2-subgroup of L/Z(L). Then the root 4involutions of K / Z ( K ) % Suz, L/Z(L) %' Col, and J/Z(L) r J2 lift to elements of K , L, and J of order 4, and J is quasisimple.
See 49.1.1 for the definition of a root J2-subgroup of Col. Now Z ( K ) %' and Z ( L ) G Z2, so from the proof of 46.8, CK(z)/O3(Z(K))= I;TK is HO,K/O(HO,K), where HO,Kis the universal covering group of C K ( z ) / Z ( K )Similarly . fiL = & , L / ( o l ) ~ ( ~ Oin, lthe ) language of the - -prooE of 46.8. Our analysis in Section 48 will show that each root 4 involution i of K / Z ( K ) contained in H is of type c2 in H* and hence a lift ;in riICor HL is of order 4 from the proof of 46.8. Also Z2 generates Z ( L ) ,so as we can pick ;E J , J is quasisimple. Zg
Lemma 46.10: If w = 4 and jl and jz are representatives for the two
1
255
46. Groups of type Col, Suz, J2, and J3 classes of involutions of H* of type a4 then there exist involutions i E H with i* = jl, but no involution j E H with j* = j2.
I
For if ti E H with tiPZ = ji then we showed that t: E U but t i
4 U.
L e m m a 46.11: If w = 2 then H is transitive on the involutions in H - Q. Namely we saw that H is transitive on involutions in H - Q, whiie since an involution t E H -Q is of type c2 on Q, t z E tQ by Exercise 2.11, completing the proof.
Lemma 46.12: Assume w = 4 and let T E Sy12(H).Then (1) J ( T ) = A r E 2 i i . (2) NG(A) is the split extension of A by M24 with A the Golay code module for NG(A)/A. NG(A) has two orbits on A#: the octad involutions with repre(3) sentative z and the dodecad involutions. (4) The isomorphism type of the amalgam H
6
N H ( A )+ NG(A)
is determined independently of the group G of type Col. Proof: By 46.8, there is an isomorphism a : H -, H of H with the centralizer H of a 2-central involution f in G = Col. By 22.5, G contains a subgroup M which is the split extension of A Z E211 by M24 with A the 11-dimensionalGolay code module for M24. By 23.10, CA(A)is a point and N = N ~ ( A ;= )~ ~ ( 2 and then by 23.1 and 23.2, that point is By Exercise 7.4, A = J ( M ) . Let A = Aa-l. The isomorphism a : A -+ A induces an isomorphism a* : GL(A)A -t GL(A)Aof the semidirect product of A by GL(A)with the semidirect product of A by GL(A).Now M = N ~ ( A ) satisfies conclusions ( 2 ) and (3) of the lemma. In particular we may regard M as a subgroup of GL(A)A;let M = ~ a - li: GL(A)A. Then N H ( A ) = ~ ~ ( A ) a - lM is isomorphic to N ~ ( Avia ) a , so as A = J ( M ) , also A = J ( T ) and hence (1) holds. Similarly NH(A)is the split extension of A by the split extension Ho of El6 by L4(2). Further NH(A)is the stabilizer of the octad involution z in the Golay code module for a complement Mo to A in M containing Ho. Let h E 02(Ho); then by 8.10, CHo(h)= CM,(h) and hence by 8.8 is isomorphic to the centralizer of a transvection in L5(2). Let N = NG(A) and N* = NIA. We claim C N . ( ~ *= ) CEi,(h)*Z CHo(h).First CG(A) = CH(A) = A, SO N* is faithful on A and
1 and then as CHo(h)is irreducible on B/(z), B = [A,h]. So CN(h)* S CHo(h) is the centralizer of a transvection in L5(2). Further O~(HO)#= hHo C 02(CHo(h))= Qo, so by 44.4, either N = NH(A) or N* G L5(2), M24, or He. The first case is impossible as A = J(T), so A is weakly closed in T with respect to G, and hence by 7.7, N controls fusion in A. Thus z is fused to zg in N. As Ho 2 Hz is a 2-local subgroup of N*, 40.5 says N* is not He. Suppose N* 2 L5(2). Then IN : NH(A)I = 31 = IQ n A#I, so as Q n A# zG n A = zN, Q n A# = zG n N. This is a contradiction as NG(R) n NG(A) does not act on Q n A from 46.3. So N* E M24. As Ho is a complement to A in NH(A) and NH(A) contains the Sylow 2-subgroup T of M, N splits over A by Gaschutz's Theorem (cf. 10.4 in [FGT]). Next A = (zN) with HG = CN. (2) and m(Q n A) = 5 with Hz the stabilizer in GL(Q nA) of z. Thus by Exercise 7.2, A is the Golay code module as an N*-module. Thus (2) holds. By 19.8, (3) holds. We have shown there is an isomorphism : N -, IV with ( H n N)(' = H n = ( H n N)a. To prove (4) it remains to show we can pick a and so that a = C on H nN = I. For this we observe I/A = Aut(I/A). For example, I/Oz(I) = GL(O2(I)/A) and H~(GL(o~(I)/A),02(I)/A) = 0 by the Alperin-Gorenstein argument (cf. Exercise 6.4 in [FGT]) applied to the class of subgroups of order 3 that are fixed point free on 02(I)/A. Then the claim is easy. Thus by Exercise 14.3.1 we may assume a = C on IIA. Finally A is generated by the orbits of I on zN of length 30 and 240 (cf. 19.2) and as a = on I/A, a = on these orbits, so a = 6 on I.
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[Tm] F. Timmesfeld, Finite simple groups in which the generalized Fitting group of the centralizer of some involution is extra-special, Ann. Math. 107 (1978), 297-369. [Toll J. Todd, On representations of the Mathieu groups as collineation groups, J. London Math. Soc. 34 (1959), 406-16. [To21 J. Todd, A representation of the Mathieu group M2.4 89 a collineation group, Annali di Math. Pure ed App. 71 (1966), 199-238. [Wall H. Ward, On Ree's series of simple groups, h n s . AMS 121 (1966), 62-89. [Wa2] H. Ward, Combinatorial polarization, Discrete Math. 26 (1979), 18597. [Wa3] H. Ward, Multilinear forms and divisors of codeword weights, Quart. J. Math. 34 (1983), 115-28. [Wl] E. Witt, Die 5-fach transitiven Gruppen von Mathieu, Abl. Math. Hamburg 12 (1938), 256-64. [W2] E. Witt, ~ b e Steinersche r Systeme, Abl. Math. Hamburg 12 (1938), 265-74.
309
Index
adjacent, 4 algebra, 37 algebra, commutative, 37 algebra, symmetric, 37 amalgam, 195 collinearity graph, 197 completion, 196 faithful completion, 196 free amalgamated product, 196 geometric complex, 197 morphism, 195 rank, 195 residually connected, 199 Baby Monster, 68 basic relation, 178 bilinear form, radical, 43 binary Golay code 41, 71 extended, 90 Brauer, R., 66 Brauer-Fowler Theorem, 26, 66 Buekenout, F., 17 Burnside, W., 66 central product, 23 centralizer, 2 chamber, 6 chamber system, 8 nondegenerate, 8 Chevalley, C., 66 clique, 8, 185 clique complex, 8, 185 code, 40 word, 40 (m,n), 40 distance, 40 doubly even, 40 even, 40 minimum weight, 40 perfect, 40 weight, 40 collinearity graph, 197 complement, 25 completion, 196
faithful, 196 components of a group, 26 connected component, 4 Conway group, 67, 71, 72, 74, 76, 116, 128, 130,290, 293 Conway, J., 67, 68, 108, 121, 139, 140, 170, 171, 174,290 Curtis, R., 105 cycle, 176 derived form, 41 degree, 42 hyperbolic subspace, 43 singular subspace, 43 subhyperbolic subspace, 43 diagonal orbital, 5 distance, 4 dodecad, 88 dodecad involution, 255 elementary abelian pgroup, 22 extension, 25 split, 25 extraspecial group, 22 width, 22 faithful, 2 fiber product, 15 fibering, 185 field automorphism, 13 Fischer, B., 67, 68 Fischer groups, 67,69 Fitting subgroup, 26 flag, 6 flag complex, 9 form, 36 %form, 44 alternating, 36 symmetric, 36 Frattini subgroup, 21 free amalgamated product, 196 free product, 196 Fkenkel, I., 72 Frobenius, 66
-
Index
Index F, type, 172, 173 fundamental groupoid, 180, 191 Galois, 65 general linear group, 13 generalized diagon, 8 generalized Fitting subgroup, 26 geometric complex, 9 basic diagram, 11 coset, 10 direct sum, 11 geometry, 6 coset. 7 direct sum, 7 edge transitive, 6 flag transitive, 6 rank, 6 string, 11 truncation, 206 global stabilizer, 3 Golay, M., 94 Golay code, 35, 41, 71 Golay code module, 91 graph, 4 closed subset. 176 closure of cycles, 178 coset, 5 covering, 75, 175, 185, 191, 192 cycle, 176 deletion, 178 edge transitive, 4 fundamental group, 180 insertion, 178 invariant relation, 176 morphism, 4 n-generated, 182 simply connected, 175, 185 triangulable, 182 Griess algebra, 35, 70, 72, 142, 151, 1 fia
standard basis, 162 Griess, R., 68, 70, 169, 170, 174 groupoid, 179 Hall, M., 67, 290
Hall, P., 32, 66 Hall-Janko group, 69, 74, 135 Harada group, 68, 69, 74 Harada, K., 68, 174 Held, D., 68, 212, 238 Held group, 26, 68, 69, 74, 174, 212, 293 root 4-subgroup, 225 Higrnan, D., 17,67, 140 Higrnan, C.,67,68, 140, 238 Higman-Sims group, 67, 69, 74, 119, 131, 140 inner automorphism, 23 invariant relation, 175, 176 basic, 178 kernel, 176 involution, 18 %central, 19 isometry, 36 Janko groups, 66, 67,69, 70, 73, 74, 76, 290, 293 Janko, Z., 32,66, 70 large extraspecial %subgroup, 23, 71 Leech, J., 108, 121 Leech lattice, 67, 71, 108 coordinate frame, 116 mod 2, 116 shape, 111 Leon, J., 68 Lepowski, J., 72 linear representation, 1 link, 9, 185 local bijection, 185 local subgroup, 66 local system, 187 loop, 46, 47 associator, 49 central isomorphism, 50 coboundary, 49 cocvcle. . 48 commutator, 49
diassociative, 47 diassociative cocycle, 53 even automorphism, 58 Moufang, 47 parameters, 49 power map, 49 symplectic, 48 Lyons group, 68, 69, 73, 74 Lyons, R., 68 MacKay, J., 67, 68, 238 Mathieu, E., 77, 94 Mathieu group, 26, 65,69, 74, 77, 82, 84, 85, 94, 212, 238, 293 2-local geometry, 96, 99 McLauglin group, 67,69, 71, 74, 118, 128, 130, 140 McLauglin, J., 67, 140 Meierfrankenfeld, U., 70 Meurman, A., 72 Miller, G., 94 modular function, 72 module, core, 42 monomial, 36 Monster, 35, 63, 68, 69, 70, 72, 74, 142, 169, 174 Monster type, 172, 173 n-linear form, 36 n-simplex, 185 n-skeleton, 185 Norton, S.,68, 70, 170, 174 octad, 85 involution, 255 collinear, 99 coplanar, 99 O'Nan group, 68 O'Nan, M., 68 orbit, 3 orbital, 4 paired, 4 Paige, N., 94
Parker loop, 63, 72, 142, 144, 170 path, 4 basic degree, 181 end, 176 origin, 176 T-gon, 183 reduced, 182 trivial, 182 Patterson, N., 140, 290 permutation rank, 4 permutation representation, 1 Phan, K., 249 pointwise stabilizer, 3 pregroupoid, 179 inversion, 179 morphism, 179 primitive, 3 projective geometry, 6 projective plane, 7, 71 quadratic form, singular vector, 43 quasiequivalent, 2 quasisimple, 26 Quillen complex, 76 rank 3 group, 5 parameters, 5 regular representation, 3 regular normal subgroup, 3 representation, 1 residually connected, 10 residue, 8 Ronan, M., 105 root Ar-subgroup, 269 root As-subgroup, 269 root As-subgroup, 279 root A,-subgroup, 288 root 4-involution, 260, 269 root 4-subgroup, 260, 269 root Jz-subgroup, 280 root U3(3)-subgroup, 288 Rudvalis, A., 67 Rudvalis group, 67,69,73,74
313
Index
-
Segev, Y., 70, 75, 210 self-paired orbital, 4 semilinear transformation, 13 sextet, 91 simplex, 8, 185 dimension, 185 simplicial complex, 8, 184 covering, 185 F-homotopy, 187 fundamental group, 186,191 graph of, 8, 185 link, 185 local system, 187, 188 n-skeleton, 185 star, 185 vertex, 184 simplicial map, 185 Sims, 67,68, 140 Smith, P., 69 Smith, S., 105, 170 special linear group, 13 split extension, 25 star, 8, 185 Steiner system, 70, 71, 78 block, 78 extension, 78 Extension Hypothesis, 79 independent subset, 78 point, 79 residual design, 78 Stellrnacher, B., 290 string diagram, 11 string geometry, 11, 204 collinearity graph, 205 line, 205 plane, 205 point, 205 string ordering, 11 strongly embedded subgroup, 20 Suzuki group, 67, 69, 71, 74, 76, 135, 290, 293 h&&i, M., 67, 290 symmetric form, 36
t-transitive, 3 Thompson group, 68,69, 74 Thompson, J., 32, 66, 68, 70, 140, 174, 290 Thompson Order Formula, 19 3-transposition, 67 Timmesfeld, F., 32 Tits, J., 17, 70, 72, 170, 171, 191 Todd, J., 94 Todd module, 92 totally singular subspace, 7 transitive, 3 transvection, 14 triangulable, 182 trio, 97 type Col, 250, 258 type He, 219 type Jz, 250,258 type J3, 258 type L5(2), 219 type Mz4, 219 type Suz, 250,258 type U4(3), 244 uniqueness system, 198 amalgam, 199 base, 200 equivalence, 199 geometry, 199 morphism, 200 similarity, 199 unitary group, 128, 241 vertex, 4 Wales, 67 Wales, D., 290 Ward, 45, 66 Witt, 71, 94 Wong, S., 290 wreath product, 26
)