A Course on Elation Quadrangles

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Author: Koen Thas

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EMS Series of Lectures in Mathematics Edited by Andrew Ranicki (University of Edinburgh, U.K.) EMS Series of Lectures in Mathematics is a book series aimed at students, professional mathematicians and scientists. It publishes polished notes arising from seminars or lecture series in all fields of pure and applied mathematics, including the reissue of classic texts of continuing interest. The individual volumes are intended to give a rapid and accessible introduction into their particular subject, guiding the audience to topics of current research and the more advanced and specialized literature. Previously published in this series: Katrin Wehrheim, Uhlenbeck Compactness Torsten Ekedahl, One Semester of Elliptic Curves Sergey V. Matveev, Lectures on Algebraic Topology Joseph C. Várilly, An Introduction to Noncommutative Geometry Reto Müller, Differential Harnack Inequalities and the Ricci Flow Eustasio del Barrio, Paul Deheuvels and Sara van de Geer, Lectures on Empirical Processes Iskander A. Taimanov, Lectures on Differential Geometry Martin J. Mohlenkamp and María Cristina Pereyra, Wavelets, Their Friends, and What They Can Do for You Stanley E. Payne and Joseph A. Thas, Finite Generalized Quadrangles Masoud Khalkhali, Basic Noncommutative Geometry Helge Holden, Kenneth H. Karlsen, Knut-Andreas Lie and Nils Henrik Risebro, Splitting Methods for Partial Differential Equations with Rough Solutions Koichiro Harada, “Moonshine” of Finite Groups Yurii A. Neretin, Lectures on Gaussian Integral Operators and Classical Groups Damien Calaque and Carlo A. Rossi, Lectures on Duflo Isomorphisms in Lie Algebra and Complex Geometry Claudio Carmeli, Lauren Caston and Rita Fioresi, Mathematical Foundations of Supersymmetry Hans Triebel, Faber Systems and Their Use in Sampling, Discrepancy, Numerical Integration

Koen Thas

A Course on Elation Quadrangles

Author: Koen Thas Department of Mathematics Ghent University Krijgslaan 281, S25 9000 Ghent Belgium E-mail: [email protected]

2010 Mathematics Subject Classification: 05-02, 20-02, 51-02; 05B25, 05E18, 20B25, 20D15, 20D20, 51B25, 51E12 Key words: Generalized quadrangle, elation group, Moufang condition, p-group

ISBN 978-3-03719-110-1 The Swiss National Library lists this publication in The Swiss Book, the Swiss national bibliography, and the detailed bibliographic data are available on the Internet at http://www.helveticat.ch. This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained. © 2012 European Mathematical Society Contact address: European Mathematical Society Publishing House Seminar for Applied Mathematics ETH-Zentrum SEW A27 CH-8092 Zürich Switzerland Phone: +41 (0)44 632 34 36 Email: [email protected] Homepage: www.ems-ph.org Typeset using the author’s TEX files: I. Zimmermann, Freiburg Printing and binding: Beltz Bad Langensalza GmbH, Bad Langensalza, Germany ∞ Printed on acid free paper 987654321

To Caroline

“Les Perspecteurs” A sketch (85cm 130cm) of the French artist Abraham Bosse (1602–1676) dating from 1648, demonstrating the projecting method of Girard Desargues.

Preface “To every loving, gentle-hearted friend, to whom the present rhyme is soon to go so that I may their written answer know (…)” Translated from A ciascun’alma presa e gentil core, La Vita Nuova Dante Alighieri, 1295

Local Moufang conditions In two famous papers [16], [17], Fong and Seitz showed that all finite Moufang generalized polygons were classical or dual classical. In fact, they obtained this result in group theoretical terms (classifying finite split BN-pairs), but Tits remarked the simple geometrical translation. And of course, the converse was already well known. In a search for a synthetic “elementary” proof of the Fong–Seitz result for the specific case of generalized quadrangles (which is the central and most difficult part in [16], [17]), Payne and J. A. Thas noticed that when one looks at the group generated by all rootelations and dual root-elations which stabilize a given point of a Moufang quadrangle, the group fixes all lines incident with that point, and acts sharply transitively on its opposite points. Let us call a point with this property an elation point, and a generalized quadrangle with such a point an elation generalized quadrangle. Kantor noticed in the early 1980s that, starting from a group with a suitable family of subgroups satisfying certain properties, one can construct an elation quadrangle from this data in a natural way, such that the group acts as an elation group. This process can be easily reversed, so as to obtain such group theoretical data starting from any elation quadrangle. This observation is the precise analogon of the fact that, in a Moufang projective plane, any line is a translation line, and when one singles out the definition of translation line and translation plane, one can also translate the situation in group theoretical terms to a group with certain subgroups, etc. In that case, one obtains a group of order n2 with a family of n C 1 subgroups of order n, two by two trivially intersecting (in the infinite case, one has to require that the product of any two of these subgroups equals the entire group and that the subgroups cover the group). And conversely, starting from such group theoretical data, one readily reconstructs a translation plane for which the group acts as a translation group. The essential difference in this correspondence between planes and quadrangles is that, in the planar case, the translation group necessarily is abelian, and this is not so for elation groups of generalized quadrangles. In the planar case, this property allows one to define a “kernel”, which is some skew field over which

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the translation group naturally becomes a vector space. In quadrangular theory, one has to assume that the elation group is abelian to obtain a similar notion of kernel, and then again, the abelian elation group can be seen as a vector space. In fact, one also assumes that the quadrangle is finite, since there are some nontrivial obstructions when passing to the infinite case. Planes

/ Translation planes

Quadrangles

/ Elation quadrangles

In this monograph, we will focus on general finite elation quadrangles, so without the commutativity assumption on the group. In the commutative case a rich theory is available, and we refer the reader to [59] and the references therein for the (many) details. Another basic difference with the nonabelian case is that an abelian elation group is unique (both for planes and quadrangles). That is, there can only be at most one (“complete”, that is, transitive on the appropriate point set) abelian elation group for a given line in a projective plane or point in a generalized quadrangle, and it necessarily is elementary abelian (in the finite case). As we will see in the present notes, this fact is not true for general elation quadrangles. We will encounter examples which admit different (t-maximal) elation groups with respect to the same elation point, and they even can be nonisomorphic. (As a by-product, we will construct the first infinite class of translation nets with similar properties.) Also, in the planar case and the abelian quadrangular case, any i-root and dual i-root involving the translation line or the elation point is Moufang, and the unique t-maximal elation group is generated by the Moufang elations. In general, such properties do not hold for elation quadrangles. We will obtain the first examples of finite elation quadrangles for which not every (dual) i-root involving the elation point is Moufang. So we first have to handle these standard structural questions as a set up for the theory. After Kantor’s observations, many infinite classes of finite generalized quadrangles were constructed as elation quadrangles, through the identification of “Kantor families” in appropriate groups. Moreover, up to a combination of point-line duality and Payne integration, every known finite generalized quadrangle is an elation quadrangle. This observation lies at the origin of the need for a structural theory for elation quadrangles, which appears to be lacking in the literature. In fact, most of the foundations can be found in Chapters 8 and 9 of [46], and that’s about it. In the present book, I hope to fill up this gap.

Preface

ix

Further outline Let me briefly outline the contents of this book besides what was already mentioned. First of all, let me mention that the three basic references on generalized quadrangles are the monographs “Finite Generalized Quadrangles” [44], [46]; “Symmetry in Finite Generalized Quadrangles” [68] and “Translation Generalized Quadrangles” [59] (on elation quadrangles with an abelian elation group). These works will only have a small overlap with the present notes. I describe, in detail, the beautiful result of Frohardt which solved Kantor’s conjecture in the case when the number of points of the (elation) quadrangle is at most the number of lines. The latter conjecture is the prime power conjecture for elation quadrangles, and states that the parameters of a finite elation generalized quadrangle are powers of the same prime. Along the same lines of Frohardt’s proof, I present a nice proof of X. Chen (which was never published) of a conjecture of Payne on the parameters of skew translation quadrangles (which are elation quadrangles such that any dual i-root involving the elation point is Moufang with respect to the same dual root group). The positivity of this conjecture was independently proven by Dirk Hachenberger (in a more general setting), and his proof is also in these notes. I will also formulate several new questions, often motivated by obtained results. Once the theory on the standard structural questions is worked out, we concentrate on more specific problems, such as a fundamental question posed by Norbert Knarr on the aforementioned local Moufang conditions (motivated by the idea whether there are other, more natural, definitions for the concept of elation quadrangle). Another aim is to emphasize the role of special p-groups and Moufang conditions as central aspects of elation quadrangle theory. In many occasions slightly different proofs are given than those provided in the literature. Also, about seventy exercises of (usually) an elementary character are formulated in the text. Exercises which are somewhat less elementary have been indicated with a superscript “# ”; exercises which come with a superscript “c ” ought to be even more challenging. Mental note. Throughout this work, almost always the generalized quadrangles (and related objects) we consider are finite, even when this is not explicitly mentioned. When this is not the case, the reader will be able to deduce this.

Finally The notes presented here are partially based on several lectures I gave on elation quadrangles. In particular, I think of the lecture I presented at the conference “Finite Geometries” in La Roche (2004, Belgium), and several talks at the “Buildings conferences” in Würzburg, Darmstadt and Münster, Germany. Also, I lectured on this subject

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at the University of Colorado at Denver, USA. These talks were often an inspiration for further research, as were the conversations with members of the audience, such as Bill Kantor, Norbert Knarr, Stanley E. Payne and Markus Stroppel. A first version of the manuscript was finished during a Research in Pairs stay at the Mathematisches Forschungsinstitut Oberwolfach, together with Stefaan De Winter and Ernie Shult, in April 2007. Revised versions were written during the summer of 2010 and the autumn of 2011. In 2010, the counter example of the conjecture stated in [69] was found.

Finally (really) I wish to thank one of the anonymous referees for providing an extremely detailed list of suggestions, remarks and typos which really helped me to write up a better, final, version of the manuscript. I am also extremely grateful to Manfred Karbe of the EMS Publishing House for his exceptional good (and pleasant) help in the process of publishing this work. Finally, during most of the writing, I was a postdoctoral fellow of the Fund for Scientific Research (FWO) – Flanders. Oberwolfach, April 2007, Ghent, December 2011

Koen Thas

Contents

Preface 1

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Generalized quadrangles 1.1 Elementary combinatorial preliminaries 1.2 Some group theory . . . . . . . . . . . 1.3 Finite projective geometry . . . . . . . 1.4 Finite classical examples and their duals

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Elation quadrangles 3.1 Automorphisms of classical quadrangles . . . . 3.2 Elation generalized quadrangles . . . . . . . . 3.3 Maximality and completeness . . . . . . . . . 3.4 Kantor families . . . . . . . . . . . . . . . . . 3.5 The classical GQs as EGQs – second approach

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Some features of special p-groups 4.1 The general Heisenberg group . . 4.2 Exact sequences and complexes . 4.3 Group cohomology . . . . . . . . 4.4 Special and extra-special p-groups 4.5 Another approach . . . . . . . . . 4.6 Lie algebras . . . . . . . . . . . . 4.7 Lie algebras from p-groups . . . .

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Parameters of elation quadrangles and structure of elation groups 5.1 Parameters of elation quadrangles . . . . . . . . . . . . . . . . 5.2 Skew translation quadrangles . . . . . . . . . . . . . . . . . . . 5.3 F -Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Parameters of STGQs . . . . . . . . . . . . . . . . . . . . . . .

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2 The Moufang condition 2.1 Moufang quadrangles . . . . . . . . 2.2 Generators and relations . . . . . . 2.3 Coxeter groups . . . . . . . . . . . 2.4 BN-pairs of rank 2 and quadrangles 3

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Contents

Standard elations and flock quadrangles 6.1 Flock quadrangles . . . . . . . . . . . . . 6.2 Fundamental theorem of q-clan geometry 6.3 A special elation . . . . . . . . . . . . . . 6.4 The nitty gritty . . . . . . . . . . . . . . 6.5 A special elation, once again . . . . . . . 6.6 Standard elations in flock GQs . . . . . . 6.7 The general case . . . . . . . . . . . . .

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Foundations of EGQs 7.1 An application of Burnside’s lemma . . . . 7.2 Implications . . . . . . . . . . . . . . . . . 7.3 Intermezzo – SPGQs . . . . . . . . . . . . 7.4 The classical and dual classical examples . 7.5 Elation groups for flock GQs and their duals 7.6 Dual TGQs which are also EGQs . . . . . . 7.7 GQs of order .k 1; k C 1/ and their duals

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Elation quadrangles with nonisomorphic elation groups 8.1 A nonisomorphism criterion . . . . . . . . . . . . . 8.2 An example: H.3; q 2 /, q even . . . . . . . . . . . . 8.3 Group and GQ automorphisms . . . . . . . . . . . . 8.4 Appendix: GQs not having property ./ . . . . . . .

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9 Application: Existence of translation nets 9.1 Translation nets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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10 Elations of dual translation quadrangles 10.1 Main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Payne’s question in a more general setting . . . . . . . . . . . . . . . 10.3 Recent results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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11 Local Moufang conditions 11.1 Formulation . . . . . . . . . . . . . . . . . . . . 11.2 Proof of the first main theorem . . . . . . . . . . 11.3 Solution of Knarr’s question . . . . . . . . . . . 11.4 Appendix: GQs with a center of transitivity (and s

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Index

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1 Generalized quadrangles

We start this chapter by introducing some combinatorial and group theoretical notions. We then proceed to define the prototypes of finite generalized quadrangles.

1.1 Elementary combinatorial preliminaries We concisely review some basic notions taken from the theory of generalized quadrangles, for the sake of convenience. 1.1.1 Rank 2 geometries. A rank 2 geometry or point-line geometry is a triple D .P ; B; I/, for which P and B are disjoint (nonempty) sets of objects called points and lines respectively, and for which I is a symmetric point-line relation called an “incidence relation”; so I .P B/ [ .B P / and .x; L/ 2 I if and only if .L; x/ 2 I. If .x; L/ 2 I, we also write x I L or L I x. If .x; L/ … I, we write x I L or L I x.

1.1.2 Generalized quadrangles. A generalized quadrangle (GQ) of order .s; t / is a point-line incidence geometry D .P ; B; I/ satisfying the following axioms: (i) each point is incident with t C 1 lines (t 1) and two distinct points are incident with at most one line; (ii) each line is incident with s C 1 points (s 1); (iii) if p is a point and L is a line not incident with p, then there is a unique point-line pair .q; M / such that p I M I q I L. In this definition, s and t are allowed to be infinite cardinals. Exercise. Let D .P ; B; I/ be a GQ of order .s; t / with s; t 2 N. Show that jP j D .s C 1/.st C 1/ and jBj D .t C 1/.st C 1/. This exercise shows that if s and t are finite, then jP j and jBj also are. In that case, we call the quadrangle finite. If s; t > 1, is thick; if one of s, t equals 1, is thin. A thin GQ of order .s; 1/ is also called a grid, while a thin GQ of order .1; t / is a dual grid. A GQ of order .1; 1/ is both a grid and a dual grid – it is an ordinary quadrangle. If s D t, then is also said to be of order s. There is a parameter-free way to introduce generalized quadrangles, as follows. A rank 2 geometry D .P ; B; I/ is a thick generalized quadrangle if the following axioms are satisfied:

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Figure 1.1. A grid of order .3; 1/.

(a) there are no ordinary digons and triangles contained in ; (b) every two elements of P [ B are contained in an ordinary quadrangle; (c) there exists an ordinary pentagon. In (a), (b), (c), ordinary digons, triangles, quadrangles and pentagons are meant to be induced subgeometries. Exercise. Show that if satisfies (a)–(c), there exist constants s and t such that each line is incident with s C 1 points, and each point is incident with t C 1 lines. Exercise. Suppose that (a) and (b) are satisfied, but not (c). Show that has a thin structure, in the sense that either each point is incident with two lines, or each line is incident with two points. Suppose that .p; L/ … I. Then by projL p we denote the unique point on L collinear with p. Dually, projp L is the unique line incident with p concurrent with L. 1.1.3 Duality. There is a point-line duality for GQs of order .s; t / for which in any definition or theorem the words “point” and “line” are interchanged and also the parameters. (If D .P ; B; I/ is a GQ of order .s; t /, D D .B; P ; I/ is a GQ of order .t; s/.) A duality from the GQ to its dual D is a map that bijectively sends points of to lines of D , lines of to points of D , while preserving incidence. (This notion will only be needed in a later chapter for formal reasons.) Exercise. Show that there is a natural one-to-one correspondence between automorphisms of (defined further in this section) and dualities from to D .

1.1 Elementary combinatorial preliminaries

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−→ D

Figure 1.2. Duality. The left-hand side is a grid of order .3; 1/; the right-hand side its dual – a GQ of order .1; 3/.

1.1.4 Inequalities of Higman. Let D .P ; B; I/ be a finite thick GQ of order .s; t /. Then we have t s 2 and, dually, s t 2 (“Inequality of Higman”); see [44], 1.2.3. Also, by [44], 1.2.2, we have that st .s C 1/.t C 1/ 0 mod s C t: 1.1.5 Collinearity and concurrency. Let p and q be (not necessarily distinct) points of the GQ ; we write p q and call these points collinear, provided that there is some line L such that p I L I q. Dually, for L; M 2 B, we write L M when L and M are concurrent. For p 2 P , put p ? D fq 2 P j q pg: Note that p 2 p ? . u

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w 2 fu; vg? Figure 1.3. The “perp” of fu; vg.

If two points are not collinear, we also say they are opposite. Same for lines. A flag is an incident point-line pair.

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1.1.6 Ovoids. An ovoid of a GQ is a set of points O such that each line contains exactly one of its points. Exercise. Let be a GQ of order .s; t /. Show that the number of points of an ovoid is st C 1. T 1.1.7 Regularity. For a set of distinct points S (or lines), we denote w2S w ? also by S ? . In particular, let S D fp; qg be a set of two points; then jfp; qg? j D s C 1 or t C 1, according as p q or p ¦ q, respectively. A set such as fp; qg? is called a trace; it is “trivial” when p q ¤ p. For a set S such as above, we introduce S ?? as ?

S ?? D .S ? / : For p ¤ q distinct points, we have that jfp; qg?? j D s C 1 or jfp; qg?? j t C 1 according as p q or p ¦ q, respectively. If p q, p ¤ q, or if p ¦ q and jfp; qg?? j D t C 1, we say that the pair fp; qg is regular. The point p is regular provided fp; qg is regular for every q 2 P n fpg. Regularity for lines is defined dually. Exercise. Prove that either s D 1 or t s if has a regular pair of noncollinear points (see [44], 1.3.6). A net of order k and degree r is a point-line incidence geometry N D .P ; B; I/ satisfying the following axioms: (i) each point is incident with r lines (r 2) and two distinct points are incident with at most one line; (ii) each line is incident with k points (k 2); (iii) if p is a point and L is a line not incident with p, then there is a unique line M incident with p and not concurrent with L. We say that .k; r/ “are” the parameters of the net. Sometimes we also speak of “.k; r/net”. Exercise. Show that jP j D k 2 and jBj D kr. Exercise. Show that a net N of degree r and order k is an affine plane of order n if and only if r D k C 1. Theorem 1.1 ([44], 1.3.1). Let p be a regular point of a finite GQ D .P ; B; I/ of order .s; t/, s ¤ 1 ¤ t . Then the incidence structure with • point set p ? n fpg, • with line set the set of spans fq; rg?? , where q and r are noncollinear points of p ? n fpg, and with the natural incidence, is the dual of a net of order s and degree t C 1.


5

If in particular s D t, there arises a dual affine plane of order s. (Also, in the case s D t , the incidence structure p with point set p ? , with line set the set of spans fq; rg?? , where q and r are different points in p ? , and with the natural incidence, is a projective plane of order s.) Proof. We leave the proof as a straightforward exercise to the reader.

Exercise. Come up with an “infinite version” of Theorem 1.1. 1.1.8 Antiregularity. The pair of points fx; yg, x ¦ y, is antiregular if jfx; yg? \ z ? j 2 for all z 2 P n fx; yg. The point x is antiregular if fx; yg is antiregular for each y 2 P n x ? . 1.1.9 Triads. A triad of points of a GQ is a set of three pairwise noncollinear points. Let fx; y; zg be a triad of points in a thick finite GQ of order .s; s 2 /. Then jfx; y; zg? j D s C 1; see [44], 1.2.4. Obviously, jfx; y; zg?? j s C 1; if equality holds, the triad fx; y; zg is called 3-regular. Furthermore, a point is 3-regular provided all triads of which it is a member are 3-regular. 1.1.10 Automorphisms. An automorphism or collineation of a GQ D .P ; B; I/ is a permutation of P [ B which preserves P , B and I. The set of automorphisms of a GQ is a group, called the automorphism group of , which is denoted by Aut./. A whorl about a point x is just an automorphism fixing it linewise. A point x is a center of transitivity provided that the group of whorls about x is transitive on the points of P n x ? . An elation with center x is an automorphism of which either is the trivial automorphism, or it fixes x linewise and has no fixed points in P n x ? . Exercise. Show that a (not necessarily thin, nor finite) GQ has the same automorphism group as its point-line dual. 1.1.11 Symmetry. A collineation of , a thick finite GQ of order .s; t /, that fixes all lines meeting a fixed line L is called a symmetry about L. If the group of symmetries about L has the maximum possible order, s, then L is called an axis of symmetry. Dually, one speaks of a center of symmetry. Exercise. Show that not only is a symmetry about L an elation about L, but that it is also an elation about each point incident with L. (One is allowed to use Theorem 1.6 below.) 1.1.12 SubGQs. A subquadrangle, or also subGQ, 0 D .P 0 ; B 0 ; I0 / of a GQ D .P ; B; I/ is a GQ for which P 0 P , B 0 B, and where I0 is the restriction of I to .P 0 B 0 / [ .B 0 P 0 /. A subGQ 0 of order .s; t 0 / of a finite GQ of order .s; t / is called full. Dually we define ideal subGQs.

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The following results will sometimes be used without further reference. Theorems 1.4, 1.5 and 1.6 can be obtained as easy exercises. For the proofs of Theorems 1.2 and 1.3, we refer to [44]. In all of the statements below up to Theorem 1.6, the generalized quadrangles are supposed to be finite. Theorem 1.2 ([44], 2.2.1). Let 0 be a proper subquadrangle of order .s 0 ; t 0 / of the GQ of order .s; t /. Then either s D s 0 or s s 0 t 0 . If s D s 0 , then each external point of 0 is collinear with the st 0 C 1 points of an ovoid of 0 ; if s D s 0 t 0 , then each external point of 0 is collinear with exactly 1 C s 0 points of 0 . Theorem 1.3 ([44], 2.2.2). Let 0 be a proper subquadrangle of the GQ , where has order .s; t/ and 0 has order .s; t 0 / (so t > t 0 ). Then the following hold. (1) t s; if s D t , then t 0 D 1. (2) If s > 1, then t 0 s; if t 0 D s 2, then t D s 2 . (3) If s D 1, then 1 t 0 < t is the only restriction on t 0 . p (4) If s > 1 and t 0 > 1, then s t 0 s and s 3=2 t s 2 . p (5) If t D s 3=2 > 1 and t 0 > 1, then t 0 D s. (6) Let 0 have a proper subquadrangle 00 of order .s; t 00 /, s > 1. Then t 00 D 1, t 0 D s and t D s 2 . Theorem 1.4 ([44], 2.3.1). Let 0 D .P 0 ; B 0 ; I0 / be a substructure of the GQ of order .s; t/ so that the following two conditions are satisfied: (i) if x; y 2 P 0 are distinct points of 0 and L is a line of such that x I L I y, then L 2 B 0; (ii) each element of B 0 is incident with s C 1 elements of P 0 . Then there are four possibilities: (1) 0 is a dual grid, so s D 1; (2) the elements of B 0 are lines which are incident with a distinguished point of P , and P 0 consists of those points of P which are incident with these lines; (3) B 0 D ; and P 0 is a set of pairwise noncollinear points of P ; (4) 0 is a subquadrangle of order .s; t 0 /. The following result is now easy to prove. Theorem 1.5 ([44], 2.4.1). Let be an automorphism of the GQ D .P ; B; I/ of order .s; t/. The substructure D .P ; B ; I / of which consists of the fixed elements of must be given by (at least) one of the following: (i) B D ; and P is a set of pairwise noncollinear points; (i0 ) P D ; and B is a set of pairwise nonconcurrent lines;


7

(ii) P contains a point x so that y x for each y 2 P , and each line of B is incident with x; (ii0 ) B contains a line L so that M L for each M 2 B , and each point of P is incident with L; (iii) is a grid; (iii0 ) is a dual grid; (iv) is a subGQ of of order .s 0 ; t 0 /, s 0 ; t 0 2. Finally, we recall a result on fixed elements structures of whorls. Theorem 1.6 ([44], 8.1.1). Let be a nontrivial whorl about p of the GQ D .P ; B; I/ of order .s; t/, s ¤ 1 ¤ t . Then one of the following must hold for the fixed element structure D .P ; B ; I /. (1) y ¤ y for each y 2 P n p ? . (2) There is a point y, y ¦ p, for which y D y. Put V D fp; yg? and U D V ? . Then V [ fp; yg P V [ U , and L 2 B if and only if L joins a point of V with a point of U \ P . (3) is a subGQ of order .s 0 ; t /, where 2 s 0 s=t t , and hence t < s. Exercise. Create an “infinite version” of Theorem 1.6. 1.1.13 Nets and subquadrangles. The following theorem is taken from [64] and implies that a net which arises from a regular point in a thick finite GQ as earlier explained cannot contain proper subnets of the same degree and different from an affine plane. Theorem 1.7 ([64]). Suppose that D .P ; B; I/ is a GQ of order .s; t /, s; t ¤ 1, with a regular point p. Let Np be the net which arises from p, and suppose Np0 is a subnet of the same degree as Np . Then we have the following possibilities: (1) Np0 coincides with Np ; (2) Np0 is an affine plane of order t and s D t 2 ; also, from Np0 there arises a proper subquadrangle of of order t having p as a regular point. If, conversely, has a proper subquadrangle containing the point p and of order .s 0 ; t/ with s 0 ¤ 1, then it is of order t, and hence s D t 2 . Also, there arises a proper subnet of Np which is an affine plane of order t . Proof. First suppose that contains a proper subquadrangle 0 of order .s 0 ; t /, s 0 ; t ¤ 1, containing the point p. Then p is also regular in 0 and since s 0 ¤ 1, it follows that s 0 t. By Theorem 1.3 this implies that s 0 D t and that s D t 2 . By Theorem 1.1, the net Np0 arising from the point p in 0 is an affine plane of order t , and this net is clearly a subnet of the net which arises from the point p in .

8


Conversely, suppose that Np is the net which arises from the regular point p in the GQ , and that it contains a proper subnet Np0 of the same degree. In the following, we identify points of the net with the corresponding spans of points in the GQ, and we use the same notation. Suppose that P1 ; P2 ; : : : ; Pk are the points of Np0 , define a point set P 0 of as S S consisting of the points of Œ Pi [ Œ Pi? , and define B 0 as the set of all lines of through a point of P 0 . Then it is not hard to check that the following properties are satisfied for the geometry 0 D .P 0 ; B 0 ; I0 /, with I0 D I \ Œ.P 0 B 0 / [ .B 0 P 0 /: (1) any point of P 0 is incident with t C 1 lines of B 0 ; (2) if two lines of B 0 intersect in , then they also intersect in 0 . Then by the dual of Theorem 1.4, 0 is a proper subquadrangle of order .s 0 ; t /, s ¤ 1, and analogously as in the beginning of the proof, we have that s 0 D t and s D t 2 . Also, the affine plane of order t which arises from the regular point p in this subquadrangle is the subnet Np0 . 0

Corollary 1.8 ([64]). A net N which is attached to a regular point of a GQ contains no proper subnet of the same degree as N , other than ( possibly) an affine plane. Corollary 1.9 ([64]). Suppose that p is a regular point of the GQ of order .s; t /, s; t ¤ 1, and let Np be the corresponding net. If s ¤ t 2 , then Np contains no proper subnet of degree t C 1. The following corollary tells us that nets which arise from a regular point of a GQ and which do not contain affine planes are very “irregular”. Corollary 1.10 ([64]). Let p be a regular point of a GQ of order .s; t /, s; t ¤ 1, and suppose that Np is the corresponding net. Moreover, suppose that s ¤ t 2 . If u, v and w are distinct lines of Np for which w ¦ u v, then these lines generate (under the taking of GQ spans) the whole net. Proof. Consider the points of p ? n fpg which correspond to the lines u; v; w of Np , and denote them respectively in the same way. Then by Theorem 1.4, u, v and w generate a (not necessarily proper) subGQ 0 of of order .s 0 ; t /, where s 0 > 1. By Theorem 1.7 this implies that 0 D , since s ¤ t 2 . Hence u, v and w generate Np . Lemma 1.11. Suppose that is a GQ of order .s; s 2 /, s ¤ 1, and suppose that 0 and 00 are two proper subquadrangles of of order s. Then one of the following possibilities occurs: (1) 0 \ 00 is a set of s 2 C 1 pairwise noncollinear points (i.e., an ovoid) of 0 and 00 ; (2) 0 \ 00 consists of a point p of 0 (and 00 ), together with all lines of 0 (and 00 ) through this point, and all points of 0 (and 00 ) incident with these lines;

1.2 Some group theory

9

(3) 0 \ 00 is a GQ of order .s; 1/; (4) 0 D 00 . Exercise. Prove Lemma 1.11. Note that every line of a thick GQ of order .s; s 2 / intersects any subGQ of order s. Then use Theorem 1.4 and a simple counting argument. Theorem 1.12 ([64]). Suppose that is a generalized quadrangle of order .s; t /, s; t ¤ 1, and suppose that is a nontrivial whorl about a regular point p. Also, suppose that fixes distinct points q; r and u of p ? n fpg for which q r and q ¦ u. Then we have one of the following possibilities. (1) We have that s D t 2 and contains a proper subquadrangle 0 of order t . Moreover, if is not an elation, then 0 is fixed pointwise by . (2) is a nontrivial symmetry about p. Proof. It is clear that if v and w are noncollinear points of p ? which are fixed by a whorl about p, then every point of the span fv; wg?? is also fixed by the whorl. Now suppose that Np is the net which arises from p, and suppose that Np0 is the (not necessarily proper) subnet of Np of degree t C 1 which is generated by u, q and r. Then every point of Np0 is fixed by by the previous observation. If Np0 is proper, then by Theorem 1.7 it is an affine plane of order t and s D t 2 . Also, there arises a proper subquadrangle 0 of of order t. If is not an elation, then by Theorem 1.6 it follows that there is a proper subquadrangle of order .s 0 ; t /, s 0 ¤ 1, which is fixed pointwise (and then also linewise) by . Since has a regular point, we have that s 0 t. By Theorem 1.3, 0 is necessarily of order t . From Lemma 1.11 now follows that D 0 . If Np0 D Np , then every point of p ? is fixed by . Since is not the identity, it follows from Theorem 1.6 that is an elation and hence a symmetry about p.

1.2 Some group theory We review some basic notions of group theory. 1.2.1 Identity. We denote the identity element of a group often by id or 1; a group G without its identity is denoted by G . 1.2.2 Permutation groups. We usually denote a permutation group by .G; X /, where G acts on X . We denote permutation action exponentially and let elements act on the right, such that each element g of G defines a permutation g W X ! X of X and the permutation defined by gh, g; h 2 G, is given by gh W X ! X;

x 7! .x g /h :

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1.2.3 Commutators. Let G be a group, and let g; h 2 G. The conjugate of g by h is g h D h1 gh. The commutator of g and h is equal to Œg; h D g 1 h1 gh: Note that the commutator map W G G ! G;

.g; h/ 7! Œg; h;

is not symmetrical; as Œg; h1 D Œh; g, we have that Œg; h D Œh; g if and only if Œg; h is an involution. The commutator of two subsets A and B of a group G is the subgroup ŒA; B generated by all elements Œa; b, with a 2 A and b 2 B. The commutator subgroup of G is ŒG; G, or sometimes G 0 . Two subgroups A and B centralize each other if ŒA; B D fidg. The subgroup A normalizes B if B a D B for all a 2 A, which is equivalent with ŒA; B B. Inductively, we define the n-th central derivative LnC1 .G/ D ŒG; GŒn of G as ŒG; ŒG; GŒn1 , and the n-th normal derivative ŒG; G.n/ as ŒŒG; G.n1/ ; ŒG; G.n1/ . For n D 0, the 0-th central and normal derivative are by definition equal to G itself. The series L1 .G/; L2 .G/; : : : is called the lower central series of G. If, for some natural number n, ŒG; G.n/ D fidg, and ŒG; G.n1/ ¤ fidg, then we say that G is solvable (soluble) of length n. If ŒG; GŒn D fidg and ŒG; GŒn1 ¤ fidg, then we say that G is nilpotent of class n. A group G is called perfect if G D ŒG; G D G 0 . The center of a group is the set of elements that commute with every other element, i.e., Z.G/ D fz 2 G j Œz; g D id for all g 2 Gg. Clearly, if a group G is nilpotent of class n, then the .n 1/-th central derivative is a nontrivial subgroup of Z.G/. 1.2.4 Central products. A group H is an internal central product of its subgroups M and N if both N and M are normal subgroups of H for which N \ M Z.H / and NM D H . Now let M and N be two groups, N Z.N /, M Z.M /, and W N ! M an isomorphism. Then the quotient Q D .N M /=K, where K is the normal subgroup f.n; m/ j n 2 N ; m 2 M ; .n/m D 1g, is the external central product of N and M provided by the data .N ; M ; /. Exercise. Work out the connection between internal and external central product of groups. (The terms “internal/external” will be dropped if it is clear which type of central product is considered.)

1.2 Some group theory

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1.2.5 p-Groups and Hall groups. For a prime number p, a p-group is a group of order p n , for some natural number n ¤ 0. A Sylow p-subgroup of a finite group G is a p-subgroup of some order p n such that p nC1 does not divide jGj. Let be a set of primes dividing jGj for a finite group G. Then a -subgroup is a subgroup of which the set of prime divisors is . The following result is basic. Theorem 1.13 ([19], Chapter 1). A finite group is nilpotent if and only if it is the direct product of its Sylow subgroups. A Hall -subgroup of a finite group G,Q where .G/, and .G/ is the set of primes dividing jGj, is a subgroup of size p2 p np , where p np denotes the largest power of p that divides jGj. Theorem 1.14 (Hall’s Theorem, [19], Chapter 6). Let G be a finite solvable group and a set of primes. Then (a) G possesses a Hall -subgroup; (b) G acts transitively on its Hall -subgroups by conjugation; (c) any -subgroup of G is contained in some Hall -subgroup. Let p and q be primes. A pq-group is a group of order p a q b for some natural numbers a and b. A classical result of Burnside states the following. Theorem 1.15 ([19], Chapter 4). A pq-group is solvable. Let R be a finite group. The Frattini group ˆ.R/ of R is the intersection of all proper maximal subgroups, or is R if R has no such subgroups. Exercise. Let P be a finite p-group. Show that ŒP; P P p D ˆ.P /, where P p D hw p j w 2 P i. 1.2.6 Frobenius groups. Suppose that .G; X / is a permutation group (where G acts on X) which satisfies the following properties: (1) G acts transitively but not sharply transitively on X ; (2) there is no nontrivial element of G with more than one fixed point in X . Then .G; X/ is a Frobenius group (or G is a Frobenius group in its action on X ). Define N G by N D fg 2 G j f .g/ D 0g [ f1g; where f .g/ is the number of fixed points of g in X . Then N is called the Frobenius kernel of G (or of .G; X /), and we have the following well-known result. Theorem 1.16 (Theorem of Frobenius, [19], Chapter 2). Suppose that jGj is finite. The Frobenius kernel N is a normal regular subgroup of G. Moreover, jGx j divides jN j 1 for any x 2 X , and G D N Ì Gx .

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1.2.7 Simple groups. A group is simple if it does not contain any proper nontrivial normal subgroups. A group G is almost simple if S G Aut.S /, with S a simple group and Aut.S / its automorphism group.

1.3 Finite projective geometry 1.3.1 Projective spaces. Below, Fq denotes the finite field with q elements, q a prime power. Let K be any field, and denote by V .n; K/ the n-dimensional vector space over K, n a nonzero natural number. If K D Fq is a finite field, we also use the notation V .n; q/. Define the .n 1/-dimensional projective space PG.n 1; K/ over K as the geometry of all subspaces of V .n; q/ ordered by set inclusion; more precisely, it is V .n; K/ equipped with the equivalence relation “ ” of proportionality, with the induced subspace structure. If K D Fq is finite, we also use the notation PG.n 1; q/. The projective space PG.1; K/ is the empty set, and has dimension 1. In general, if W is a w-dimensional K-vector subspace of V .n; K/, it induces a .w 1/-dimensional projective subspace of V .n; K/= D PG.n 1; K/. Exercise. Let K D Fq . Show that a d (-dimensional)-subspace of PG.n1; q/ contains .q d C1 1/=.q 1/ points. In particular, PG.n 1; q/ has .q n 1/=.q 1/ points. It also has .q n 1/=.q 1/ hyperplanes (= .n 2/-dimensional subspaces). 1.3.2 Collineation groups. An automorphism or collineation of a finite projective space is an incidence and dimension (“type”) preserving bijection of the set of subspaces to itself. It can be shown that any automorphism of a PG.n; q/, n 2 N and n 2, Fq a finite field, necessarily has the following form: W x T ! A.x /T ; where A 2 GLnC1 .q/, is a field automorphism of Fq , the homogeneous coordinate x D .x0 ; x1 ; : : : ; xn / represents a point of the space (which is determined up to a scalar), and x D .x0 ; x1 ; : : : ; xn / (recall that xi is the image of xi under ). Here, vectors are identified with row matrices without any further notice. The set of automorphisms of a projective space naturally forms a group, and in case of PG.n; q/, n 2, this group is denoted by PLnC1 .q/. The normal subgroup of PLnC1 .q/ which consists of all automorphisms for which the companion field automorphism is the identity, is the projective general linear group, and denoted by PGLnC1 .q/. So PGLnC1 .q/ D GLnC1 .q/=Z.GLnC1 .q//, where Z.GLnC1 .q// is the central subgroup of all scalar matrices of GLnC1 .q/. Similarly one defines PSLnC1 .q/ D SLnC1 .q/=Z.SLnC1 .q//, where Z.SLnC1 .q// is the central subgroup of all scalar matrices of SLnC1 .q/ with unit determinant. An elation of PG.n; q/ is an automorphism of which the fixed points structure precisely is a hyperplane (the “axis” of the elation), or the space itself. A homology

1.4 Finite classical examples and their duals

13

either is the identity, or it is an automorphism that fixes a hyperplane pointwise, and one further point not contained in that hyperplane. Exercise. Show that each nontrivial elation of PG.n; q/, n 2 N and n 2, has a unique center, that is, a point which is fixed linewise (and necessarily contained in the axis).

1.4 Finite classical examples and their duals In this section we will introduce some classes of finite rank 2 geometries which are known as the finite “classical generalized quadrangles”. (Tits was the first to identify them as generalized quadrangles – see Dembowski [15].) Their point-line duals are called the dual classical generalized quadrangles. The classical quadrangles are characterized by the fact that they are fully embedded in finite projective space – see Chapter 4 in [44] for details. Recall that a full embedding of a rank 2 geometry D .P ; B; I/ in a projective space P , is an injection W P ,! P .P /; with P .P / the point set of P , such that (E1) h .P /i D P ; (E2) for any line L 2 B (seen as a point set), L is a line of P . Of course, from the point of view of Group Theory, no distinction can be made between a classical quadrangle and its point-line dual – they have the same automorphism group, cf. the exercise in §1.1.10. But from the viewpoint of Incidence Geometry, there is indeed a difference: the dual Hermitian quadrangles H.4; q 2 /D cannot be fully embedded in a projective space PG.`; q 2 /, where ` 2 N [ f1g. (By “1” we mean any infinite cardinal number.) 1.4.1 Orthogonal quadrangles. Consider a nonsingular quadric Q of Witt index 2, that is, of projective index 1, in PG.3; q/, PG.4; q/, PG.5; q/, respectively. So the only linear subspaces of the projective space in question lying on Q are points and lines. The points and lines of the quadric form a generalized quadrangle which is denoted by Q.3; q/, Q.4; q/, Q.5; q/, respectively, and has order .q; 1/, .q; q/, .q; q 2 /, respectively. As Q.3; q/ is a grid, its structure is trivial. Recall that Q has the following canonical form: (1) X0 X1 C X2 X3 D 0 if d D 3; (2) X02 C X1 X2 C X3 X4 D 0 if d D 4; (3) f .X0 ; X1 / C X2 X3 C X4 X5 D 0 if d D 5, where f is an irreducible binary quadratic form.

14


The automorphism group of Q.4; q/ is isomorphic to PO5 .q/, and PSL5 .q/ \ PO5 .q/ μ PSO5 .q/ μ O5 .q/. The automorphism group of Q.5; q/ is isomorphic to PO6 .q/, and PSL6 .q/ \ PO6 .q/ μ PSO6 .q/ μ O 6 .q/. 1.4.2 Hermitian quadrangles. Next, let H be a nonsingular Hermitian variety in PG.3; q 2 /, respectively PG.4; q 2 /. The points and lines of H form a generalized quadrangle H.3; q 2 /, respectively H.4; q 2 /, which has order .q 2 ; q/, respectively .q 2 ; q 3 /. The variety H has the following canonical form (where d D 3 or 4): X0qC1 C X1qC1 C C XdqC1 D 0: The automorphism group of H.3; q 2 / is isomorphic to PU4 .q/, and PSL4 .q/ \ PU4 .q/ μ PSU4 .q/ μ U4 .q/. The automorphism group of H.4; q 2 / is PU5 .q/, and PSL5 .q/ \ PU5 .q/ μ PSU5 .q/ μ U5 .q/. 1.4.3 Symplectic quadrangles. The points of PG.3; q/ together with the totally isotropic lines with respect to a symplectic polarity, form a GQ W .q/ of order q. A symplectic polarity ‚ of PG.3; q/ has the following canonical form: X0 Y3 C X1 Y2 X2 Y1 X3 Y0 : The automorphism group of W .q/ is PSp4 .q/, while PSL4 .q/ \ PSp4 .q/ μ Sp4 .q/ μ S4 .q/. 1.4.4 Some properties. The following results will be very important in this text. For proofs we refer to [44]. Theorem 1.17 ([44], 3.2.1, 3.2.2 and 3.2.3). (i) Q.4; q/ Š W .q/D ; (ii) Q.4; q/ Š W .q/ if and only if q is even; (iii) Q.5; q/ Š H.3; q 2 /D . We sum up the basic combinatorial properties of the classical GQs in the next theorem. Because of Theorem 1.17, we only state the properties for the orthogonal quadrangles and H.4; q 2 /. Theorem 1.18 (Combinatorial properties, cf. [44], §3.3). (i) Let q be odd. Then all lines of Q.4; q/ are regular, and all points are antiregular. In particular, all spans of noncollinear points have size 2. (ii) Let q be even. Then all lines and points of Q.4; q/ are regular. (iii) Any line of Q.5; q/ is regular and any span of noncollinear points has size 2. All points are 3-regular. (iv) For any pair fx; yg of noncollinear points of H.4; q 2 /, we have jfx; yg?? j D q C 1, while all spans of nonconcurrent lines have size 2.

1.4 Finite classical examples and their duals

15

1.4.5 Small GQs. Proofs and references of all the results mentioned in this section can be found in [44], Chapter 6. Let D .P ; B; I/ be a finite thick GQ of order .s; t /, s t . (a) s D 2. By §1.1.4, s C t divides st .s C 1/.t C 1/ and t s 2 . Hence t 2 f2; 4g. Up to isomorphism there is only one GQ of order 2 and only one GQ of order .2; 4/. It follows that the GQs W .2/ and Q.4; 2/ are self-dual and mutually isomorphic. It is easy to show that the GQ of order 2 is unique. (b) s D 3. Again by §1.1.4 we have t 2 f3; 5; 6; 9g. Any GQ of order .3; 5/ must be isomorphic to the GQ T2 .O/ arising from the unique hyperoval in PG.2; 4/, any GQ of order .3; 9/ must be isomorphic to Q.5; 3/, and a GQ of order 3 is isomorphic to either W .3/ or to its dual Q.4; 3/. Finally, there is no GQ of order .3; 6/. (c) s D 4. Using §1.1.4 it is easy to check that t 2 f4; 6; 8; 11; 12; 16g. Nothing is known about t D 11 or t D 12. In the other cases unique examples are known, but the uniqueness question is settled only in the case t D 4.

Figure 1.4. The generalized quadrangle of order .2; 2/. Its points are the black filled circles; its lines are the straight lines, together with the curves that contain one point lying on the middle of some side of the pentagon, and the two closest points to that point lying in the interior.

2 The Moufang condition

The Moufang condition is one of the central group theoretical conditions in Incidence Geometry, and was introduced by Jacques Tits when classifying spherical buildings of rank at least 3, in his lecture notes “Buildings of Spherical Type and Finite BN-Pairs” [85]. It was noted by him that spherical buildings of rank at least 3 satisfy the so-called “Moufang property”, implying these structures to have a lot of symmetry. When the rank of these buildings is 2, i.e., when one is dealing with generalized n-gons [87], this is not necessarily the case; many examples exist which are not Moufang (think of the quadrangular or planar case). Already in the 1960s, Tits started a program to obtain all Moufang generalized n-gons, and much later, J. Tits and R. M. Weiss [86] eventually finished the classification of (finite and infinite) Moufang generalized n-gons. For the finite case, this result was already obtained by P. Fong and G. M. Seitz in [16], [17], the most difficult case being the case n D 4 by far, and for this latter case, there is also a geometrical proof which is a culmination of work by S. E. Payne and J. A. Thas [44] (Chapter 9), W. M. Kantor [28] and the author [61]. We refer to the author and H. Van Maldeghem [83] for a survey on old and new results on Moufang generalized quadrangles. We also refer to Chapter 11 of [59], and especially [58] on that matter. In the aforementioned work of S. E. Payne and J. A. Thas (and the references therein), the importance of local Moufang conditions became obvious – not only numerous characterizations of known classes of generalized quadrangles came out; also the theory of translation generalized quadrangles essentially arose from it, and the abstraction to elation generalized quadrangles eventually led to many new classes of generalized quadrangles. In this chapter, we review some basic facts concerning the Moufang condition.

2.1 Moufang quadrangles 2.1.1 Roots and Moufang roots. Note that an ordinary induced quadrangle in a GQ is just a (necessarily thin) GQ of order .1; 1/ – we call such a subgeometry also an “apartment”. Let A be an apartment of a GQ . A root of A is a set of 5 different elements e0 ; : : : ; e4 in A such that ei I eiC1 (where the indices are taken in f0; 1; 2; 3g), and e0 , e4 are the extremal elements of . There are two types of roots, depending on whether the extremal elements are lines or points; in the second case we speak of dual roots to make a distinction between the types. Also, a (dual) root without its extremal elements – the interior of – is denoted by P and is called a (dual) i-root.

2.2 Generators and relations

17

A root-elation with i-root .x; L; y/ D P (and root ) is an element ˛ of Aut./ which fixes x and y linewise, and L pointwise. We also write ˛ 2 Aut./Œ P , or is the group of all root-elations with i-root P . By ˛ 2 Aut./Œ.x;L;y/ . So Aut./Œ P Theorem 1.6 and the exercise following that theorem, ˛ (if not trivial) cannot fix points of P n L, nor lines of B n L? . Let U ¤ L and U I u, where u is one of x; y. The root and i-root P are called Moufang if Aut./Œ P acts transitively (and so sharply transitively) on the points of U n fug. Exercise. Show that the definition of Moufang (i-)root is independent of the choice of U , or u. (Do not restrict only to the finite case.) Let be thick and finite, of order .s; t/. Show that the root is Moufang if and only if jAut./Œ P j D s. 2.1.2 Moufang quadrangles. A GQ is half Moufang if all its roots, or all its dual roots, are Moufang. It is Moufang if all roots and dual roots are Moufang. We already mentioned that all Moufang quadrangles are classified. We will make this more precise for the finite case later in this chapter. For the general case, we refer to [86].

2.2 Generators and relations Let S ¤ ; be any set (call its elements “letters”), and define the free group F .S / with alphabet S as follows. Put, without loss of generality, S D fsi j i 2 I g, where I is some nonempty index set, and define another set of letters S 1 D fsi1 j i 2 I g. (Note that these are nothing more than symbols!) A word over the alphabet S is a finite sequence of letters taken from S [S 1 ; by definition, the “empty word” (a word with no letters) is also a word. Define the map red as the map which associates with each word w over S the word red.w/ which is w in which all occurrences of a sequence of the type si si1 or si1 si are canceled. For instance, if S D fs1 ; s2 g, then red.s1 s2 s21 / D s1 . A reduced word is a word w for which red.w/ D w (it is a fixed element of red). The free group F .S / over S, or with alphabet S , is the set of all reduced words over S , together with the following operation B; if w, w 0 are words over S, then ww 0 denotes the word over S which is just the concatenation of w and w 0 , and w B w 0 is defined to be the unique reduced word of fredm .ww 0 / j m 2 Ng. If jS j D n < 1, F .S / is called a free group of rank n. Exercise. Let S, S 0 be two sets of the same finite nonzero cardinality. Show that F .S/ Š F .S 0 /. Exercise. Show that a free group of rank 1 is isomorphic to Z; C. Show that a free group of rank greater than 1 is not abelian. Let G be any group generated by the set S G; so G D hSi. For the sake of convenience, write S D fsi j i 2 I g. Then there is a natural homomorphism

18


W F .S/ ! G defined by W F .S/ ! G;

si 7! si ;

i 2 I:

As is onto, G Š F .S/= ker./, where ker./ is the kernel of . So any group G is the quotient of some free group. Motivated by this observation, we can represent groups also as follows. Let S be a set of letters, and R a set of words over S . Then the group which is presented by the generators S and relations R, denoted by hS j Ri, is the quotient of F .S/ by the normal subgroup N.R/ generated by the relations of R. So N.R/ is the smallest normal subgroup of F .S / containing the words of R (it is the intersection of all normal subgroups of F .S/ containing the words in R). Suppose that R D .wi /i2IR ; then sometimes hS j Ri is also denoted by hS j .wi D 1/IR i. Exercise. Let n 2 N, n ¤ 0. Then the cyclic group of order n can be presented as hs j s n i D hs j s n D 1i. The group hS j Ri is the freest group generated by S which is subject to the relations defined by R.

2.3 Coxeter groups To understand the notion of “BN-pair” – which we will need below – we introduce the concepts of “Coxeter groups” and “Coxeter diagrams”. 2.3.1 Coxeter groups and systems. A Coxeter group is a group with a presentation of type hs1 ; s2 ; : : : ; sn j .si sj /mij D 1i; where mi i D 1 for all i , mij 2 for i ¤ j , mij 2 N [ f1g, and i , j are natural nonzero numbers bounded above by the natural number n. If mij D 1, no relation of the form .si sj /mij is imposed. All generators in this presentation are involutions. The natural number n is the rank of the Coxeter group. Exercise. Show that mij D 2 implies that Œsi ; sj D 1. Show that mij D mj i for all i, j . Recall that a dihedral group of rank n, denoted by Dn , is the symmetry group of a regular n-gon in the real plane. (It is also the automorphism group of a generalized n-gon [87] of order .1; 1/.) Exercise. Show that the finite Coxeter groups of rank 2 are precisely the dihedral groups. Note that hs1 ; s2 j s12 D 1 D s22 ; .s1 s2 /m12 D 1i is isomorphic to Dm12 , with m12 < 1. A Coxeter system is a pair .W; S /, where W is a Coxeter group and S the set of generators defined by the presentation.

19

2.4 BN-pairs of rank 2 and quadrangles

2.3.2 Coxeter diagrams. Let .W; S / be a Coxeter system. Define a (weighted) graph, called a “Coxeter diagram”, as follows. Its vertices are the elements of S. If mij D 3, we draw a single edge between si and sj ; if mij D 4, a double edge, and if mij 5, we draw a single edge with label mij . If mij D 2, nothing is drawn. If the Coxeter diagram is connected, we call .W; S / irreducible. If W is finite, we call .W; S / spherical. The irreducible spherical Coxeter diagrams (systems) were classified by H. S. M. Coxeter [13]; the complete list is the following. An :

...

.n 1/

Cn :

...

.n 2/

Dn :

...

.n 4/

En :

...

.n D 6; 7; 8/

F4 :

5

H3 :

5

H4 :

I2 .m/:

m

.m 5/

The subscript n denotes the number of nodes in the diagram. The case Cn is sometimes denoted by Bn , the case I2 .6/ often as G2 .

2.4 BN-pairs of rank 2 and quadrangles In this section we consider a quotient group G=N as consisting of cosets of N in G. With this convention, it makes sense to multiply elements of the quotient group with

20


elements or subgroups of G; it is just the ordinary multiplication of subsets in a group. 2.4.1 Tits quadrangles and systems. Details on the next theorem can be found in [59], Chapter 11. For reasons of convenience, we will call a generalized quadrangle admitting an automorphism group that acts transitively on its ordered ordinary 4-gons, a Tits quadrangle. Two flags F D fx; Lg and F 0 D fy; M g are opposite if x ¦ y and L ¦ M. Proposition 2.1. Let D .P ; B; I/ be a thick Tits generalized quadrangle and let G be a collineation group of such that G acts transitively on the set of ordered pairs of opposite flags of . Let F D fx; Lg be any flag in , with x 2 P and L 2 B, and let † be an apartment containing F . Define B ´ Gx;L , N ´ G† and H D B \ N . Then N \ .Gx n GL / and N \ .GL n Gx / are nonempty. Choose arbitrarily sx and sL , respectively, in these sets. Then .G; B; N / and sx , sL satisfy the following properties. (BN1) G D hB; N i. (BN2) H E N and W ´ N=H D hsx H; sL H i is isomorphic to D4 . (BN3) BsBwB BwB [ BswB, for all w 2 N and s 2 fsx ; sL g. (BN4) sBs ¤ B, for s 2 fsx ; sL g. Also, B does not contain any nontrivial normal subgroup of G and H coincides with T 1 w Bw: w2N

The group W is called the Weyl group of .G; B; N /, and the elements sx and sL representatives of the standard generators of W . We now would like to sketch a converse of the previous proposition. (All details can be found in [59], Chapter 11.) To that aim, we define the notion of a group with a BN-pair, also called a Tits system. Let G be a group, and let B and N be two subgroups of G. Set H D B \ N . Then .G; B; N / is called a Tits system of type B2 , or .B; N / is called a BN-pair of type B2 in G, if there exist elements sx and sL of G, with fsx2 ; sL2 g H , such that (BN1) up to (BN4) of Proposition 2.1 hold. The group W in (BN2) will be called the Weyl group and the cosets sx H and sL H the standard generators of W . If is a Tits quadrangle, and .G; B; N / is as in Proposition 2.1, then we call .G; B; N / a natural Tits system associated with . If .G; B; N / is a Tits system of type B2 with Weyl group W and if sx and sL are corresponding representatives of the standard generators of W , then we define the following incidence structure G;B;N D .P ; B; I/. Define Px D hB; B sx i and PL D hB; B sL i, and call these groups maximal parabolic subgroups. • Points. The elements of P are the right cosets of Px . • Lines. The elements of B are the right cosets of PL .


21

• Incidence. For g; h 2 G, the point Px g is incident with the line PL h if Px g \ PL h ¤ ; (and in this case we may choose g D h). The group G acts (on the right) as a collineation group on G;B;N , and it acts transitively on the flags, since every flag can be written as fPx g; PL gg (and is the image under g of the “standard” flag fPx ; PL g). If we denote the point Px by x, then the point Px g can be written as x g . Similarly, we write PL as L, and every line can be written as Lg , for some g 2 G. 2.4.2 Bruhat decomposition. The Bruhat decomposition states that S GD BwB: w2W g

Consider any flag F . This flag corresponds to the coset Bg. By the foregoing paragraphs we can write g D b 0 nb, with b; b 0 2 B and n 2 N . Hence Bg D Bnb and so b 1 fixes F and maps F g into †. It now follows rather easily that the geometry G;B;N does not contain any digon fx1 ; x2 ; L1 ; L2 g, with x1 ; x2 2 P and L1 ; L2 2 B with xi I Lj , i; j 2 f1; 2g. Indeed, by flag transitivity, we may assume x1 D x and L1 D L. By the previous paragraph, we may also assume that the flag fx2 ; L2 g is contained in †, a contradiction. Suppose that G;B;N contains a triangle x1 I L3 I x2 I L1 I x3 I L2 I x1 , with the xi points and the Li lines, i D 1; 2; 3. We may again assume x1 D x and L2 D L, and by the previous paragraph, we may assume that the flag fx2 ; L1 g belongs to †. Since † is a quadrangle, we must have x2 D x sL sx sL . But then L3 is incident with both x and x sL sx sL , contradicting the fact that these points are not even collinear. Exercise. Show that x and x sL sx sL are not collinear. Now let x 0 be any point not incident with some line L0 . We may assume x 0 D x and L0 2 †. It follows that x 0 is incident with at least two lines (the lines through x in †), that L0 is incident with at least two points, and that there is a flag fy; M g with x 0 I M I y I L0 . The flag fy; M g is unique by the previous paragraph. Hence we have shown that G;B;N is a GQ. The group G acts as a collineation group, by multiplication at the right, on G;B;N . Since every element of the kernel K of that action must in particular fix B, we see that K is a normal subgroup of G contained in B. Conversely, if K 0 E G, and K 0 B, then, since the stabilizer of the flag Bg, g 2 G, clearly coincides with B g , the group K 0 stabilizes every flag of G;B;N , and hence belongs to the kernel K. So K is the biggest normal subgroup of G contained in B, and as such is equal to T g KD B : g2G

The group G=K acts faithfully on G;B;N and the stabilizer of the flag F is B=K. If K D f1g, we say that the Tits system is effective.

22


Now we determine the stabilizer of †. Since N stabilizes † and acts transitively on the flags of †, it suffices to determine the elementwise stabilizer S of †, and then NS is the global stabilizer of †. So suppose that some b 2 B stabilizes †. Since † is uniquely determined by the opposite flags F and F sx sL sx sL , this is equivalent with b 2 B fixing F sx sL sx sL . Hence S D B \ B sx sL sx sL . Note that then b automatically belongs to B w , for every w 2 W . Hence we can also write T w B : SD w2W

We have proved the following theorem. Theorem 2.2. Let .G; B; N / be a Tits system with Weyl group W . Then the geometry G;B;N defined above is a Tits quadrangle. Setting T w T g B and S D B ; KD g2G

w2W

G=K acts naturally and faithfully by right translation on G;B;N . Also, B is the stabilizer of a unique flag F and NS is the stabilizer of a unique apartment containing F , and the triple .G=K; B=K; NS=K/ is a natural Tits system associated with G;B;N . The Tits system .G; B; N / is called saturated precisely when N D NS , with S as above. Replacing N by NS, every Tits system is “equivalent” to a saturated one. 2.4.3 BN-Pairs. More generally, a group G is said to have a BN-pair .B; N /, where B; N are subgroups of G, if the following properties are satisfied: (BN1) hB; N i D G; (BN2) H D B \N E N and N=H D W is a Coxeter group with distinct generators s1 ; s2 ; : : : ; sn ; (BN3) Bsi BwB BwB [ Bsi wB whenever w 2 W and i 2 f1; 2; : : : ; ng; (BN4) si Bsi ¤ B for all i 2 f1; 2; : : : ; ng. The subgroup B, respectively W , is a Borel subgroup, respectively the Weyl group, of G. The natural number n is called the rank of the BN-pair (which corresponds to the rank of the associated “building”). We call the BN-pair spherical if the associated Coxeter group is finite. If the rank is 2 and the BN-pair is spherical, the Weyl group N=.B \ N / is a dihedral group of size 2m for some positive integer m according to the list of spherical connected Coxeter diagrams. The type of the BN-pair is the name of the corresponding Coxeter group. 2.4.4 Classification of finite BN-pairs of rank 2. Using the classification of finite simple groups, the (finite) groups with a BN-pair of rank 2 can be classified. We only state the result for type B2 BN-pairs.


23

Theorem 2.3 (Buekenhout and Van Maldeghem [10]). Let G be a finite group with an effective, saturated BN-pair of type B2 . Then G is an almost simple group related to one of the following classical Chevalley groups of type B2 : (1) S4 .q/ Š O5 .q/; (2) U4 .q/ Š O 6 .q/; (3) U5 .q/. For recent results on finite BN-pairs, we refer the reader to [77], [80], [83], [84]. Exercise# . Show that any of the classical quadrangles and their duals has a BN-pair. 2.4.5 Split BN-pairs and Moufang quadrangles. Let G be a group with a BN-pair .B; N / of type B2 . Put H D B \ N , as before. The BN-pair .B; N / is called split if property ( / below holds: . / There exists a normal nilpotent subgroup U of B such that B D U.B \ N /. In a celebrated work, P. Fong and G. M. Seitz determined all finite split BN-pairs of rank 2 (the B2 -case being, by far, the most complicated type to handle). We only state the result for BN-pairs of type B2 . Theorem 2.4 (Fong and Seitz [16], [17]). Let G be a finite group with an effective, saturated split BN-pair of rank 2 of type B2 . Then G is an almost simple group related to one of the following classical Chevalley groups: (1) O5 .q/; (2) O 6 .q/; (3) U5 .q/. Equivalently, a thick finite generalized quadrangle is isomorphic, up to duality, to one of the classical examples if and only if it verifies the Moufang Condition. Much more recently the conditions of the previous theorem were relaxed still to obtain the same conclusion. The proof is independent of [16], [17]. Theorem 2.5 (K. Thas and Van Maldeghem [84]). A thick finite generalized quadrangle is isomorphic, up to duality, to one of the classical examples if and only if for each point there exists an automorphism group fixing it linewise and acting transitively on the set of its opposite points. There is an interesting corollary: Corollary 2.6 (Configurations of centers of transitivity). If a finite generalized quadrangle is not (dual) classical, it either has precisely one center of transitivity, a line of centers of transitivity, or no such point at all. Later, we will meet generalized quadrangles which are not (dual) classical, having a line of centers of transitivity.

3 Elation quadrangles

We observed in the previous chapter that the classical examples admit interesting automorphism groups. From a local point of view, this motivates us to introduce elation generalized quadrangles.

3.1 Automorphisms of classical quadrangles We indicate some large automorphism groups of the classical GQs. We assume without further reference the well-known fact that the automorphism groups of the classical GQs act transitively on the point set. (See the last section of this chapter for further comments on this property.) By using the explicit forms of the classical GQs, the reader can easily check this fact through straightforward calculation. In this section, we will represent points by row vectors (or 1 -matrices, whenever appropriate). Orthogonal quadrangles. Let f .X2 ; X3 / be an irreducible binary quadratic form over Fq . When q is odd, take f .X2 ; X3 / D X22 C ˛X32 with ˛ a nonsquare of Fq ; when q is even, take f .X2 ; X3 / D X22 C X2 X3 C mX32 with m 2 Fq having trace 1. Let Q.5; q/ be the GQ defined by the equation X0 X5 X1 X4 f .X2 ; X3 / D 0: The tangent hyperplane at x D .0; 0; 0; 0; 0; 1/ is given by X0 D 0. Consider the following element .a; b; c; d / of PGL6 .q/ (with a; b; c; d 2 Fq ): .a; b; c; d / W .x0 x1 x2 x3 x4 x5 /

0

1 B0 B B0 ! .x0 x1 x2 x3 x4 x5 / B B0 B @0 0

a b 1 0 0 1 0 0 0 0 0 0

c d 0 0 0 0 1 0 0 1 0 0

1 ad C f .b; c/ C d C C 2b C: C 2c˛ C A a 1

Observe that .a; b; c; d / stabilizes Q.5; q/ and fixes all lines of Q.5; q/ incident with x. In fact, the abelian automorphism group defined by ˆ D f.a; b; c; d / j a; b; c; d 2 Fq g

25

3.1 Automorphisms of classical quadrangles

acts sharply transitively on the points of Q.5; q/ which are not collinear with x. The hyperplane … with equation X3 D 0 is nontangent, and meets Q.5; q/ in a Q.4; q/-subGQ. Clearly, if an element of ˆ maps a point of … \ Q.5; q/ which is not collinear with x onto another such point, the element fixes …, so also … \ Q.5; q/. So for each point x of Q.4; q/, Q.4; q/ admits an automorphism group fixing x linewise and acting sharply transitively on the points not collinear with x. Symplectic quadrangles. Let ‚ be the symplectic polarity of PG.3; q/ associated with the form X0 Y3 C X1 Y2 X2 Y1 X3 Y0 : Let W .q/ be the GQ defined by ‚. Consider the following element .a; b; c/ of PGL4 .q/ (where a; b; c 2 Fq ): 0 1 1 a b c B0 1 0 bC C .a; b; c/ W .x0 x1 x2 x3 / ! .x0 x1 x2 x3 / B @0 0 1 aA : 0 0 0 1 This induces an automorphism of W .q/ (the matrix commutes with ‚) and it fixes all lines incident with .0; 0; 0; 1/ in the plane X0 D 0. The set f.a; b; c/ j a; b; c 2 Fq g forms an automorphism group of W .q/ that fixes .0; 0; 0; 1/ linewise (in W .q/), and acts sharply transitively on the points of W .q/ not collinear with .0; 0; 0; 1/. Hermitian quadrangles. Let x ! xN be the involutory automorphism of Fq 2 . Define tr.x/ D x C x. N Let U be the unitary polarity of PG.4; q 2 / associated to the following Hermitian form X0 Y4 C X1 Y3 C X2 Y2 C X3 Y1 C X4 Y0 ; and let H.4; q 2 / be the Hermitian quadrangle corresponding to U . following element of PGL5 .q 2 /: 0 1 a b B0 1 0 B 0 .a; b; c; d / W .x0 x1 x2 x3 x4 / ! .x0 x1 x2 x3 x4 / B B0 0 1 @0 0 0 0 0 0

Now consider the 1 c d 0 cN C C 0 bN C C; 1 aN A 0 1

with b bN C tr.d C ac/ N D 0. Then 0 .a; b; c; d / preserves U while fixing all lines incident with .0; 0; 0; 0; 1/ in the hyperplane X0 D 0. The set f 0 .a; b; c; d / j a; b; c; d 2 Fq g

26


constitutes an automorphism group of H.4; q 2 / that fixes .0; 0; 0; 0; 1/ linewise (in H.4; q 2 /), and acts sharply transitively on the points of H.4; q 2 / not collinear with .0; 0; 0; 0; 1/. (Note that the points of H.4; q 2 / which are not collinear with .0; 0; 0; 0; 1/ are of the form .1; a; b; c; d /, with b bN C tr.d C ac/ N D 0.) Consider the point .0; 0; 1; 0; 0/, which is not a point of H.4; q 2 /. Then X2 D 0 is the (nontangent) polar hyperplane of .0; 0; 1; 0; 0/ and .X2 D 0/ \ H.4; q 2 / is a H.3; q 2 /-quadrangle. So for each point x of H.3; q 2 /, H.3; q 2 / admits an automorphism group fixing x linewise and acting sharply transitively on the points not collinear with x. The dual Hermitian quadrangles. Let q be a prime power. For each point of the GQ H.4; q 2 /D there again is an automorphism group which fixes x linewise and acts sharply transitively on the points which are not collinear with x. We leave the explicit calculations to the reader.

3.2 Elation generalized quadrangles We have observed that all finite classical GQs and their point-line duals have, for each point, an automorphism group that fixes it linewise and has a sharply transitive action on the points which are noncollinear with that point. 3.2.1 Elations and quadrangles. Let D .P ; B; I/ be a GQ. If there is an automorphism group H of which fixes some point x 2 P linewise and acts sharply transitively on P n x ? , we call x an elation point, and H “the” associated elation group. If a GQ has an elation point, it is called an elation generalized quadrangle or, shortly, “EGQ”. We frequently will use the notation . x ; H / to indicate that x is an elation point with associated elation group H . Sometimes we also write x if we do not want to specify the elation group. Note that each element of the elation group H is an elation (and moreover, each element of the subgroup hi also is as such). Note that we also allow the thin definition of EGQ (although usually we will exclude this trivial case). 3.2.2 Translation quadrangles. If the elation group of an EGQ is abelian, we speak of a translation generalized quadrangle (TGQ). The elation group is then usually called a “translation group”, the elation point a “translation point”.

3.3 Maximality and completeness

27

3.3 Maximality and completeness We will encounter many situations in which a certain group of elations E, say with center x, in some quadrangle D .P ; B; I/, is proven to act transitively on the points opposite x, in which case the action is sharply transitive. So . x ; E/ is an EGQ. To express this fact, we will say that E is t-complete or t-maximal (the “t” stands for “transitive”). If in a situation it is clear that by “elation group” E of some GQ with elation point x, we mean “t-maximal elation group” (that is, . x ; E/ is an EGQ), then sometimes we will omit the terms “t-maximal”/ “t-complete”.

3.4 Kantor families Suppose that . x ; H / D .P ; B; I/ is a finite (not necessarily thick) EGQ of order .s; t/, and let z be a point of P n x ? . Let L0 ; L1 ; : : : ; L t be the lines incident with x, and define ri and Mi by Li I ri I Mi I z, 0 i t . Define, for i D 0; 1; : : : ; t, Hi D HMi and Hi D Hri , and set J D fHi j 0 i t g. Then we have the following properties: • jH j D jP n x ? j D s 2 t ; • J is a set of t C 1 subgroups of H , each of order s; • for each i D 0; 1; : : : ; t, Hi is a subgroup of H of order st containing Hi as a subgroup. Moreover, the following two conditions are satisfied: (K1) Hi Hj \ Hk D f1g for distinct i; j and k; (K2) Hi \ Hj D f1g for distinct i and j . x Li rig

Mig zg Figure 3.1. From an EGQ to its Kantor family.

28


Conversely, let H be a group of order s 2 t (where s; t 2 N and the value 1 is allowed) and J (respectively J ) be a set of t C 1 subgroups Hi (respectively Hi ) of H of order s (respectively of order st ), and suppose that (K1) and (K1) are satisfied. Exercise. Show that the Hi are uniquely defined by the Hi . Exercise. Show that, for any j 2 f0; 1; : : : ; tg, S H D Hj [ i¤j Hj Hi n Hj is a partition of H . We call Hi the tangent space at Hi , and .J; J / is said to be a Kantor family or 4-gonal family of type .s; t / in H . Sometimes we will also say that J is a (Kantor, 4-gonal) family of type .s; t / in H . Notation. If .J; J / is a Kantor family in H and A 2 J, then A denotes the tangent space at A. Let .J; J / be a Kantor family of type .s; t / in the group H of order s 2 t . Define an incidence structure .H; J/ as follows. • Points of .H; J/ are of three kinds: (i) elements of H ; (ii) left cosets gHi , g 2 H , i 2 f0; 1; : : : ; tg; (iii) a symbol .1/. • Lines are of two kinds: (a) left cosets gHi , g 2 H , i 2 f0; 1; : : : ; tg; (b) symbols ŒHi , i 2 f0; 1; : : : ; tg. • Incidence. A point g of type (i) is incident with each line gHi , 0 i t . A point gHi of type (ii) is incident with ŒHi and with each line hHi contained in gHi . The point .1/ is incident with each line ŒHi of type (b). There are no further incidences. Exercise. Show that the incidence structure .H; J/ is a GQ of order .s; t /. We thus have: Theorem 3.1 ([26]). If we start with an EGQ . x ; H / to obtain J as above, then we have that x Š .H; J/. So a group of order s 2 t admitting a 4-gonal family is an elation group for a suitable elation generalized quadrangle. Exercise. Prove that x Š .H; J/. (Hint: use Figure 3.2.) Show that left multiplication by elements of H guarantees .H; J/ to be an EGQ with elation point .1/ and elation group H . Let be the isomorphism as defined by the figure. Show that for y 2 P and g 2 H , .y g / D g. .y//.

29

3.5 The classical GQs as EGQs – second approach .1/

x

ŒHi

Li rig

−→

gHi

Mig

zg

gHi

g

Figure 3.2. The isomorphism x Š .H; J/.

Exercise. Let . x ; H / be a thick infinite EGQ. Find the correct definition for Kantor family in H so as to recuperate Theorem 3.1.

3.5 The classical GQs as EGQs – second approach Using coordinates, we showed that the finite classical GQs and their duals are EGQs with respect to any point. On the other hand, knowing that these quadrangles are Moufang, there is an easier, synthetic way to show this, which also works for the infinite case. 3.5.1 Moufang quadrangles as EGQs. So let be a thick, not necessarily finite, Moufang quadrangle, and let x be any point of . Let z be a point which is not collinear with x, and let U and V be distinct lines incident with x. Let u D projU z and v D projV z. Let A be the group of root-elations with i-root .x; U; u/, B the group of root-elations with i-root .x; V; v/, and C the group of dual root-elations with dual i-root .U; x; V /. Define K ´ hA; B; C i: Let ˛ 2 A, ˇ 2 B and 2 C ; then Œ˛; 2 A \ C D f1g, Œˇ; 2 B \ C D f1g and ŒA; B C . So AC and BC are (normal) subgroups of K, and K D ABC . Note that K=AC acts sharply transitively on U n fxg and similarly, K=BC acts sharply transitively on V n fxg. So K is a group of elations with center x; if ` 2 K would fix some point y ¦ x, it would have to fix projU y and projV y, so that U and V are fixed pointwise by `. As ` fixes x (and y) linewise, it must be the identity. The fact

30


that K acts transitively on the points not collinear with x is left to the reader as an easy exercise. Exercise. Show that . x ; K/ is an EGQ. 3.5.2 The Knarr condition. Although we have shown that U and V indeed define a t-maximal elation group K forcing x to be an EGQ, there is not much indication to think that the same group would be obtained for different U and V on x. (Still, if W .x/ is the group of whorls about x, then K E W .x/.) Exercise. Show that K E W .x/. If is a Moufang quadrangle, the group M.x/ generated by all root-elations and dual root-elations with (dual) i-root containing x is known to be an elation group, so that in that case, we have a “canonical” way to associate an EGQ to each point of the GQ. But in general, it is not clear as to whether the group M.x/ could be larger. This is the main theme of Chapter 11.

4 Some features of special p-groups

In this chapter we describe several interesting aspects of certain p-groups which model the most important class of nonabelian elation groups known up to present. The starting point of the chapter is the introduction of the general Heisenberg group defined over a finite field. Without proofs, we describe the classification of extra-special p-groups, and make some additional remarks on exact sequences, cohomology and Lie algebras. It is the “odd one out” chapter of this book.

4.1 The general Heisenberg group The general Heisenberg group Hn .q/ D Hn of dimension 2n C 1 over Fq , with n a natural number, is the group of square .n C 2/ .n C 2/-matrices with entries in Fq , of the following form (and with the usual matrix multiplication): 0 1 1 ˛ c @0 In ˇ T A ; 0 0 1 where ˛; ˇ 2 Fqn , c 2 Fq and with In being the n n-unit matrix. Let ˛; ˛ 0 ; ˇ; ˇ 0 2 Fqn and c; c 0 2 Fq ; then 1 0 1 0 1 0 1 ˛0 c 0 1 ˛ c 1 ˛ C ˛0 c C c 0 C ˛ ˇ0 A: @0 In ˇ T A @0 In ˇ 0 T A D @0 In ˇ C ˇ0 0 0 1 0 0 1 0 0 1 Here x y, with x D .x1 ; x2 ; : : : ; xn / and y D .y1 ; y2 ; : : : ; yn / elements of Fqn , denotes x1 y1 C x2 y2 C C xn yn D xy T . Observation 4.1. Hn is isomorphic to the group f.˛; c; ˇ/ j ˛; ˇ 2 Fqn ; c 2 Fq g, where the group operation B is given by .˛; c; ˇ/ B .˛ 0 ; c 0 ; ˇ 0 / D .˛ C ˛ 0 ; c C c 0 C ˛ˇ 0 ; ˇ C ˇ 0 /: T

Throughout these notes, we keep using the latter representation for the general Heisenberg group. Below, 0N denotes the null vector of Fqn . Theorem 4.2. The following properties hold for Hn (defined over Fq ).

32


(i) Hn has exponent p if q D p h with p an odd prime; it has exponent 4 if q is even. (ii) The center of Hn is given by N c; 0/ N j c 2 Fq g: Z.Hn / D f.0; (iii) For each .˛; c; ˇ/ 2 Hn , we have .˛; c; ˇ/1 D .˛; c C ˛ˇ T ; ˇ/. (iv) ŒHn ; Hn D Z.Hn / and Hn is nilpotent of class 2. Proof. (i) Take .˛; c; ˇ/ 2 Hn with ˛; ˇ 2 Fqn and c 2 Fq . Then .˛; c; ˇ/2 D .2˛; 2cC˛ˇ T ; 2ˇ/, while .˛; c; ˇ/n with n > 2 an integer equals .n˛; ncCn˛ˇ T ; nˇ/. This proves (i). (ii) and (iii) These are obvious. (iv) Let .˛; c; ˇ/ 2 Hn be arbitrary. Then .˛; c; ˇ/1 D .˛; c C ˛ˇ T ; ˇ/. So if x; y 2 Hn , their commutator Œx; y is contained in the center. The claim now easily follows.

4.2 Exact sequences and complexes Let I Z be a set of subsequent integers, i.e., an interval (usually we will only consider finite subsets, N or Z). Put ` D sup.I /, { D inf.I / and I ı D I n f`g, I D I n f{g. We will denote the trivial group by 0. Let .Ai /i2I be a sequence of not necessarily nontrivial groups, and for each i 2 I ı , suppose that @i W Ai ! AiC1 is a group homomorphism. These data are presented by the following diagram, which is sometimes also denoted by .Ai ; @i /i2I : @m1

@mC1

@m

! Am ! AmC1 ! : We say that the latter sequence is exact if for every i 2 I ı , im.@i / D ker.@iC1 /. If the latter property is satisfied for some i 2 I ı , we say the sequence is exact at i C 1. In the following easy observation jI j is supposed to be “sufficiently large”. Observation 4.3. Consider a sequence .Ai ; @i /i2I as above. • Let i C 1 2 I ı and Ai D 0, and suppose that the sequence is exact at i C 1. Then @iC1 is injective. • Dually, let i 1 2 I and Ai D 0, and suppose that the sequence is exact at i 1. Then @i2 is surjective. 4.2.1 Short exact sequences. Let I D f0; 1; 2; 3; 4g, and let A0 and A4 be trivial. Suppose that the following sequence is exact: @0

@1

@2

@3

0 ! A1 ! A2 ! A3 ! 0:

33

4.2 Exact sequences and complexes

Such a sequence of groups is called a short exact sequence. Then by Observation 4.3 we have A2 = im.@1 / Š ker.@3 /. As im.@1 / Š A1 and ker.@3 / D A3 , we also write briefly A2 =A1 Š A3 . So a short exact sequence is nothing else than a representation of a group extension (in this case of A3 by A1 ). Often we will not specify @0 and @3 in a short exact sequence. Example. Let N be a normal subgroup of a group H . Then

0!N ! H ! H=N ! 0; where is the natural embedding and projection, defines a short exact sequence of groups in an obvious way. Finally, consider the short exact sequence of groups 0 ! A ! B ! C ! 0: Then we say that the sequence splits if the following commuting diagram exists: 0

/A

/B

1

0

/C

2

/A

/AÌC

/C

/0 3

/ 0,

where is the natural injection and the natural projection, and i is an isomorphism for i D 1; 2; 3. 4.2.2 Chain and cochain complexes. Let I D N with the reverse ordering, and consider a sequence of abelian groups @3

@2

@1

! A2 ! A1 ! A0 : Then A D .Ai ; @i /i2I is a chain complex if for each i 2 N we have that @i B @iC1 D 0; in other words, im.@iC1 / ker.@i /. The homology groups of the chain complex are defined by Hi .A/ D ker.@i /= im.@iC1 /;

i 2 N :

(Note that these groups are well defined.) Define the subgroups Z n .A/ D ker.@n / and B n .A/ D im.@nC1 /. For n D 0 let Z 0 .A/ D 0. The elements of Z n .A/ are the n-cycles; the elements of B n .A/ the n-boundaries.

34


A morphism f D .fi /i2I between chain complexes A D .Ai ; @i /i2I and B D .Bi ; @0i /i2I consists of group homomorphisms fi W Ai ! Bi for all positive integers i such that for all j 2 N the following diagram commutes: Aj @j

Aj 1

fj

fj 1

/ Bj

@j0

/ Bj 1 .

Clearly morphisms between chain complexes naturally induce morphisms between the corresponding homology groups (in a covariant way). Again let I D N with the natural ordering, and consider a sequence of abelian groups @0

@1

@2

A0 ! A1 ! A2 ! : Then A D .Ai ; @i /i2I is a cochain complex if for each i 2 N we have that @iC1 B @i D 0: The cohomology groups of the cochain complex are defined by H i .A/ D ker.@iC1 /= im.@i /;

i 2 N:

The elements of Z n .A/ are now called the n-cocycles; the elements of B n .A/ the n-coboundaries. Morphisms between cochain complexes are defined similarly as above. 4.2.3 Homology and cohomology. At this point there is no intrinsic difference between homology and cohomology – in the setting above, cohomology is just the “homology of cochains” (and so the only difference here is the ordering of the indices). Still, we want to make a contravariant functor out of cohomology, in the following way. Let Ab be the category of abelian groups (with as morphisms group homomorphisms), let G be a fixed object in Ab, and let A D .Ai ; @i /i2I be a chain complex of abelian groups. Let I D N. For each n 2 N, define AnG D Hom.An ; G/, and ın W Hom.An ; G/ ! Hom.AnC1 ; G/;

˛ 7! ın .˛/ D ˛ B @nC1 :

Then AG D .AnG ; ın /n2N defines a cochain complex of abelian groups. Now to a morphism f D .fm /m2N W A ! B of chain complexes corresponds a morphism n ! AG of cochain complexes, where fGn W BG ! AnG , r 7! fG D .fGm /m2N W BG n fG .r/ D r B fn . So considering the map .A ; f / ! .H n .AG /; H n .fG //;

35

4.3 Group cohomology

where H n .fG / are the induced maps in cohomology, we have that for each n 2 N, (n-th) cohomology (“with coefficients in G”) can be seen as a contravariant functor H n from the category of chain complexes of abelian groups (with the morphisms as defined earlier) to Ab. Let ˇ

˛

0!A! B !C !0 be a short exact sequence of cochains A D .Ai ; @i /i2I , B D .Bi ; @0i /i2I , C D .Ci ; @00i /i2I ; by this we mean that for each i 2 I , we have a short exact sequence ˛i

ˇi

0 ! Ai ! Bi ! Ci ! 0 such that the following diagram commutes for all i 2 N: 0

/ Ai @i

0

/ AiC1

˛i

˛i C1

ˇi

/ Bi

@0i

/ BiC1

ˇi C1

/ Ci

/0

@00 i

/ CiC1

/ 0.

Then by the so-called “Zig-Zag Lemma” (see [33], §24) there exists a long exact sequence in cohomology 0 ! H 0 .A/ ! H 0 .B/ ! H 0 .C/ ! H 1 .A/ ! : It can be shown that the cohomology functor is essentially uniquely determined by this property. (Similar properties hold of course for homology.)

4.3 Group cohomology We are ready to introduce some further basics about group cohomology. Proofs will be omitted but can be found in any regular textbook on the subject (such as [1]). 4.3.1 Group modules. Let G be a group. A (left) G-module M is an abelian group (written additively) on which G acts as endomorphisms. In other words, a G-module is an abelian group M together with a map G M ! M;

.g; m/ 7! gm;

such that for all g; h 2 G and m; n 2 M the following properties hold: g.m C n/ D gm C gn;

.gh/m D g.hm/;

1m D m:

36


Example. Let G be a group. The module M D ZŒG with the action P P h. g ng g/ D g ng hg is called the regular G-module. Example. Let G be a group and M be any abelian group. Then M is a trivial G-module if the action is prescribed by G M ! M;

.g; m/ 7! gm D m;

for all m 2 M; g 2 G:

Example. Let M be a G-module, and define the module of invariants M G as M G D fm 2 M j gm D m for all g 2 Gg: M G is of course a submodule of M . 4.3.2 Group cohomology. Let A be a G-module and let C n .G; A/ denote the set of functions of n variables f W G G

G ! A into A. If n D 0, C 0 .G; A/ D Hom.1; A/ Š A. The elements of C n .G; A/ are n-cochains. Clearly, C n .G; A/ is an abelian group with the usual addition and trivial element. Now define homomorphisms @n W C n .G; A/ ! C nC1 .G; A/ as follows. @n .f /.x1 ; : : : ; xnC1 / D x1 f .x2 ; : : : ; xnC1 / n P C .1/i f .x1 ; : : : ; xi1 ; xi xiC1 ; : : : ; xnC1 / iD1

C .1/nC1 f .x1 ; : : : ; xn /: One can prove easily that @nC1 B @n D 0 for all n 2 N. So the sequence @0

@1

@2

A ! C 1 .G; A/ ! C 2 .G; A/ !

is a cochain complex. The n-th cohomology group of G with coefficients in A is then given by the cohomology of the chain complex .C i .G; A/; @i /i . If A D .Fp ; C/ (in this text always as a trivial module), we call it p-modular cohomology. 4.3.3 Low dimensional cohomology. For n D 0 we have H 0 .G; A/ D Z 0 .G; A/ D fa 2 A j xa D a for all x 2 Gg D AG ; the module of invariants. If A is a G-module, then Z 1 .G; A/ D ff W G ! A j f .xy/ D xf .y/ C f .x/g

4.4 Special and extra-special p-groups

37

and B 1 .G; A/ D ff W G ! A j f .x/ D xa a for some a 2 Ag: The 1-cocycles are also called crossed homomorphisms of G into A. More on the next two results can be found in [1]. Theorem 4.4. Let A be a G-module. Then there exists a bijection between H 1 .G; A/ and the set of conjugacy classes of subgroups H G Ë A complementary to A, in which the conjugacy class of G maps to zero. All the complements of A in G Ë A are conjugate if and only if H 1 .G; A/ D 0. Two group extensions 0 ! A ! E ! G ! 0 and 0 ! A ! E 0 ! G ! 0 are said to be equivalent if there is an isomorphism W E ! E 0 such that the diagram below commutes. > E AA AA }} } AA } AA }} } } /A / 0. 0 G AA }> } AA } AA }} A }}} E0 We also mention the following result on the second cohomology group. Theorem 4.5. Let G be a group and A an abelian group, and let M denote the set of group extensions 0!A!E!G!0 with a given G-module structure on A. Then there is a 1-1 correspondence between the set of equivalence classes of extensions of G by A contained in M with the elements of H 2 .G; A/. The class of split extensions in M corresponds to the class Œ0 2 H 2 .G; A/. If A is a trivial G-module, there is a 1-1 correspondence between the set of equivalence classes of central extensions of G by A contained in M with the elements of H 2 .G; A/. Exercise. Let G be a finite group and A be a G-module. Show that every element of H n .G; A/, n 2 N, has a finite order which divides jGj. If A is a finite G-module and .jGj; jAj/ D 1, then H n .G; A/ D 0 for all n 1. So any extension of G by A is split.

4.4 Special and extra-special p-groups A p-group P is special if either ŒP; P D Z.P / D ˆ.P /, and Z.P / is elementary abelian or P itself is. Note that P =ŒP; P is elementary abelian in that case. If jZ.P /j D jŒP; P j D jˆ.P /j D p, P is called extra-special.

38


Example. The general Heisenberg group Hn (n > 0) over Fp is extra-special. Extra-special p-groups and more general p-groups can be classified efficiently. First, let P be a p-group, which satisfies the group extension 0 ! Cp ! P ! E ! 0; where E is an elementary abelian p-group of order p k , k 2 N . (Here, Cp is the cyclic group of order p.) Then P defines a class of groups which is more general than extra-special p-groups (see the following exercise). Exercise. Assume that P is not abelian. Show that Cp D ŒP; P D ˆ.P / Z.P /. Note that the exponent of P is either p or p 2 . Suppose that P has no proper direct factors. Let P be represented by the extension class Œ˛ in the first p-modular cohomology group H 1 .E; Cp /, cf. Theorem 4.4. As Cp is a trivial E-module (since we are dealing with a central extension, cf. the exercise), we have that B 1 .E; Cp / is the trivial group. Also, Z 1 .E; Cp / D ff W E ! Cp j f .xy/ D xf .y/ C f .x/ D f .y/ C f .x/g; so H 1 .E; Cp / Š Hom.E; Cp /. The latter carries the natural structure of a k-dimensional vector space V over Fp . There exists a basis of V such that Œ˛ is in one of the following forms (cf. P. A. Minh [32]): for p D 2 and k D 2n:

(a) X1 Y1 C X2 Y2 C C Xn Yn or (b) X12 C Y12 C X1 Y1 C X2 Y2 C C Xn Yn I

for p D 2 and k D 2n C 1: (c) X02 C X1 Y1 C X2 Y2 C C Xn Yn I for p > 2 and k D 2n: (d) X1 Y1 C X2 Y2 C C Xn Yn or (e) ˇ.X1 / C X1 Y1 C X2 Y2 C C Xn Yn I for p > 2 and k D 2n C 1: (f) ˇ.X0 / C X1 Y1 C X2 Y2 C C Xn Yn : (Here, ˇ is the Bockstein homomorphism [1].) Identify E D P =Cp with V . Define a bilinear map W V V ! ŒP; P ;

.uCp ; vCp / 7! Œu; v:

Then clearly defines a symplectic form over Fp . If is not degenerate, then Z.P / D ŒP; P D ˆ.P /, and P is extra-special. In that case, .V; / defines a symplectic polar space in PG.2n 1; p/. When n D 2, we obtain a symplectic quadrangle W .p/.

4.5 Another approach We now present a different approach for extra-special groups which depends on the knowledge of the nonabelian groups of order p 3 .

4.6 Lie algebras

39

4.5.1 The extra-special groups. There are four nonabelian p-groups of order p 3 – see [19]. First of all, we have M D M.p/: M.p/ D hx; y; z j x p D y p D z p D 1; Œx; z D Œy; z D 1; Œx; y D zi: Next define 2

M3 .p/ D hx; y j x p D y p D 1; x y D x pC1 i: Finally, we have the dihedral group D4 of order 8 and the generalized quaternion group Q8 of order 8. In the next theorem, M denotes M.p/, N denotes M3 .p/, D denotes D4 and Q denotes Q8 . Theorem 4.6 ([19]). An extra-special p-group P is the central product of r 1 nonabelian subgroups of order p 3 . Moreover, we have (1) If p is odd, P is isomorphic to N k M rk , while if p D 2, P is isomorphic to D k Qrk for some k. In either case, jP j D p 2rC1 . (2) If p is odd and k 1, N k M rk is isomorphic to NM r1 , the groups M r and NM r1 are not isomorphic and M r is of exponent p. (3) If p D 2, then D k Qrk is isomorphic to DQr1 if k is odd and to Qr if k is even, and the groups Qr and DQr1 are not isomorphic. All considered products are central products. 4.5.2 M.p/ as a Heisenberg group. Readers should note that M.p/ is isomorphic to H1 .p/.

4.6 Lie algebras A Lie algebra L D .L; Œ; / over a (commutative) field F is an F -vector space L, together with a map Œ; W L L ! L satisfying the following properties: (1) Œ; is bilinear over F ; (2) Œx; x D 0 for all x 2 L (i.e., Œ; is alternating); (3) Œx; Œy; z C Œy; Œz; x C Œz; Œx; y D 0 for all x; y; x 2 L, the “Jacobi identity”. We call Œ; the “Lie bracket” or “Lie multiplication” of the Lie algebra. Note that Œx; y D Œy; x for all x; y 2 L.

40


Example. Let F be a field. Let L D Mn .F / be the vector space of all n n-matrices with entries in F , n 2 N . Define Œ; W Mn .F / ! Mn .F / by Œx; y D x B y y B x; B denoting matrix multiplication. Then .L; Œ; / is a Lie algebra, also denoted by gln .F /. The following (vector) subspaces of Mn .F / constitute Lie algebras with the induced Lie bracket: • sln .F /, consisting of the matrices with trace 0; • bn .F /, consisting of the n n-upper triangular matrices in Mn .F /; • nn .F /, consisting of the n n-upper triangular matrices in Mn .F / with zero diagonal (also called strictly upper triangular matrices). Example. Suppose that F D R, and let L D R3 . Define a Lie bracket by Œ; W .x; y/ ! x y; the vectorial cross-product. Example. Let L be an F -vector space, and define Œ; by Œx; y D 0 for all x; y 2 L. Then .L; Œ; / is a Lie algebra which is, by definition, called “abelian”. The center of a Lie algebra .L; Œ; / is Z.L/ D fx 2 L j Œx; y D 0 for all y 2 Lg: Note that Z.L/ D L if and only if L is abelian. For A; B L, ŒA; B denotes the F -vector subspace hŒa; b j a 2 A; b 2 Bi L. A Lie subalgebra of a Lie algebra .L; Œ; / is a linear subspace M L for which ŒM; M M . A linear operator A of a vector space V is nilpotent if there exists a nonzero natural number m for which Am D 0. Example. Any operator given by a strictly upper triangular matrix is nilpotent. Exercise. Let F be an algebraically closed field. Show that a linear operator given by some matrix B is nilpotent if and only if B has only the eigenvalue 0. The lower central series of a Lie algebra L is the series with terms L1 D ŒL; L D L0

and

Lk D ŒL; Lk1

for k 2:

Then L is nilpotent if Lm D f0g for some positive integer m ¤ 0.

4.7 Lie algebras from p-groups

41

Exercise. Show that a Lie algebra L is nilpotent if and only L=Z.L/ is. Let L and M be Lie algebras over the same ground field. A linear map W L ! M is a Lie algebra homomorphism if .Œx; y/ D Œ.x/; .y/ for all x; y 2 L: If is bijective, it is an isomorphism. Let L be a Lie algebra over F . Then by End.L/ we denote the F -vector space of all endomorphisms W L ! L. A derivation of L is an F -linear map @ W L ! L which satisfies Leibniz’s Law @.Œx; y/ D Œ@.x/; y C Œx; @.y/ for all x; y 2 L. The set Der.L/ is now defined to be the set of all derivations of L; it naturally forms an F -vector subspace of End.L/. Define, for each x 2 L, a linear operator adx (“adjoint action of x on L”) by adx W L ! L;

y 7! Œx; y:

Define ad by ad W L ! Der.L/;

x 7! ad.x/ D adx :

Exercise. Show that ad is a Lie algebra homomorphism. Theorem 4.7 (Engel’s Theorem). Let V be a vector space over the field F . Let L be a Lie subalgebra of gln .F / for which every element is a nilpotent linear transformation of V . Then V has a basis such that every element of L is represented by a strictly upper triangular matrix. A Lie algebra L is nilpotent if and only if for all x 2 L the operator adx is nilpotent. An ideal of a Lie algebra .L; Œ; / is a linear subspace M L for which ŒL; M M . Theorem 4.8 (First Isomorphism Theorem). Let L and M be Lie algebras and let W L ! M be a Lie algebra homomorphism. Then ker./ is an ideal of L and L= ker./ Š im./:

4.7 Lie algebras from p-groups A Lie ring is a set R together with two binary operations, denoted by “C” and “Œ; ” (the latter again called “Lie bracket” or “Lie multiplication”) such that we have the following conditions:

42


(1) Œx C y; z D Œx; z C Œy; z and Œx; y C z D Œx; y C Œx; z for x; y; z 2 R; (2) Œx; x D 0 for all x 2 R; (3) Œx; Œy; z C Œy; Œz; x C Œz; Œx; y D 0 for all x; y; z 2 R. Example. Any associative ring can be made into a Lie ring by defining the Lie product by the rule Œx; y D xy yx: A Lie ring R becomes a Lie algebra over a field F if F operates on R in such a way that R is a vector space over F with respect to addition and aŒx; y D Œax; y D Œx; ay for x; y 2 R and a 2 F . Given any group, there is a natural way of associating a Lie ring to it, which is primarily of value in the theory of p-groups and nilpotent groups. We only introduce it for p-groups. So, consider a p-group P , and let P D L1 .P / > L2 .P / > > LnC1 .P / D f1g be the lower central series of P . In the following lemma, by x y mod H we mean that xy 1 2 H : Lemma 4.9. Let x; x 0 2 Li .P /, y; y 0 2 Lj .P / and z 2 Lk .P /. Then the following properties hold. (i) Œx; y 2 LiCj .P /. (ii) If x x 0 mod LiC1 .P / and y y 0 mod Lj C1 .P /, then Œx; y Œx 0 ; y 0 mod LiCj C1 .P /. (iii) Œxx 0 ; y Œx; yŒx 0 ; y mod LiCj C1 .P / and Œx; yy 0 Œx; yŒx; y 0 mod LiCj C1 .P /. (iv) Œx; y; zŒy; z; xŒz; x; y 0 mod LiCj CkC1 .P /. (v) For any nonnegative integer a, we have Œx; ya Œx a ; y Œx; y a mod LiCj C1 .P /: Exercise. Prove Lemma 4.9. Now define Li D Li .P /=LiC1 .P /;

1 i n:

Each Li is an abelian group which we write additively here. Define L to be the direct sum n L Li LD iD1

4.7 Lie algebras from p-groups

43

of the groups Li . So L is an abelian group itself. Now we define a Lie bracket on L. We first define Œx; N y N for xN in Li and yN in Lj as the image of Œx; y in LiCj , where x, y are representatives of x, N yN in Li .P /, Lj .P /, respectively. By Lemma 4.9 (i), (ii), Œx; N y N is well defined in LiCj . Note that one takes Lk to be trivial if k > n. One now extends the definition of the Lie product to all of L by linearity. Exercise. Show that .L; Œ; / is a Lie ring. The ring L D L.P / is called the associated Lie ring of P . If the exponent of each Li is p, then L.P / will become a Lie algebra over Fp (in this case L.P / is a vector space over Fp ). Exercise. Determine the Lie algebra associated to the general Heisenberg group. Show it is nilpotent.

5 Parameters of elation quadrangles and structure of elation groups

In this chapter we consider the most important known results on the order of a finite elation quadrangle. Among them, we explain Frohardt’s approach to EGQs of order .s; t/ with s t , Chen’s unpublished result on STGQs and various results on so-called “F -factors” by Hachenberger.

5.1 Parameters of elation quadrangles In this section (and thereafter), we use the following notation: if p is a prime divisor of the natural number m, then mp is the largest power of p dividing m, and mp0 is defined as m=mp . Also, .m/ is the set of primes dividing m. Similarly, define m and m 0 for any set of primes (writing mp for mfpg for the sake of convenience). If G is a finite group, .G/ is defined as .jGj/. If R is a finite group, we denote its set of Sylow p-subgroups by Yp .R/ (this set could be the empty set), and for a group R, as before R is the set of nontrivial elements of R. In this entire section, . x ; G/ is a thick finite EGQ of order .s; t /. Starting from a point z ¦ x, construct the 4-gonal family .J; J / as usual, and put J D fH0 ; H1 ; : : : ; H t g, while J D fH0 ; H1 ; : : : ; H t g. Lemma 5.1. (i)T If A and B are distinct elements of J, and g 2 G, then A \B g D f1g. (ii) Let S A2J A , and suppose that S E G. For any subgroup K G, define Kx D KS=S, and if g 2 G, put gN D gS. If A and B are distinct elements of J, and g 2 G, then AS \ BxgN D f1g. In particular, with f1g D S, we obtain (i). Proof. (i) As jGj D s 2 t , jA j D st, jBj D s and A \ B D f1g, we have that G D A B. Let a 2 A and suppose that a 2 B g . Write g 1 D hb with h 2 A and b 2 B. Put c D ah , so that c 2 A and c b 2 B. The latter expression implies that c 2 B, so c D 1. It follows that a D 1. (ii) Suppose that AS \ BxgN ¤ f1g for some A; B ¤ A 2 J and g 2 G. Then there are a 2 A and b 2 B for which AS D b g S. It follows that b g 2 A . By (i) this is only possible when b D 1. The following lemma has a surprisingly easy proof. Lemma 5.2. Let p be a prime, and assume that sp > 1. Then tp0 < sp :

5.1 Parameters of elation quadrangles

45

Proof. Take A 2 J, and let P 2 Yp .G/ contain a Sylow p-subgroup Ap of A . For each B 2 J n fAg, let Bp 2 Yp .B/. Then Bp is G-conjugate to a subgroup QB of P . By Lemma 5.1, we have that the groups Ap and QB for B 2 J n fAg are mutually intersecting in f1g. So P jQB j jP j: jAp j C B2JnfAg

Since jAp j D sp tp , jQB j D sp 1 for all B 2 J n fAg, and since jP j D sp2 tp , this implies that sp tp C t .sp 1/ sp2 tp :

Hence tp tp0 .sp 1/ sp tp .sp 1/. As tp > 0 and sp > 1, we have tp0 < sp (note that tp0 and sp are relatively prime, so that tp0 ¤ sp ). Lemma 5.3. One of the following occurs: (i) s is a prime power; (ii) t < s, .s; t / ¤ 1, s has exactly two prime divisors and every element of J is solvable. Proof. Let k D j.s/j. By the previous lemma, taking products over all p 2 .s/ yields Q Q Q tk D .tp tp0 / < .sp tp / D s tp st: p2.s/

p2.s/

p2.s/

So t k1 < s, and Higman’s inequality leads us to k 2. Suppose now that k D 2, and that .s/ D fp; qg. Then any element of J is a pq-group, so solvable by Burnside’s p a q b -theorem. Finally, since t 2 < t.s/ s t.s/ t 2 , it follows that .s; t / ¤ 1. The following intermediate result is interesting. Lemma 5.4. Assume that G has a normal Hall -subgroup H . Then either s D 1 or .t/ . In particular, if G is nilpotent, then G is a p-group. Proof. For each A 2 J, define AH D A \ H and AH D A \ H . Set JH D fAH j A 2 Jg and JH D fAH j A 2 J g:

Then either s D 1, or we have the following conditions for each distinct AH , BH , CH 2 JH : j D s t ; • jAH j D s and jAH • AH AH ;

• jH j D s2 t ; • AH BH \ CH D f1g and AH \ BH D f1g.

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An easy exercise (cf. also the exercise in §3.4) yields that t D jJj 1 D jJH j 1 t ; so that t D t (that is, .H; JH / is a Kantor family of type .s ; t / in H ) and .t / . Suppose now that p divides s and that G has a normal Sylow p-subgroup. Then .t/ D fpg. The lemma follows since .t / D .s/. We are ready to obtain Frohardt’s proposition on Kantor’s conjecture. The proof differs a bit from the original one. Theorem 5.5. If either j.s/j D 1 or G is solvable, then .G/ .s/. In particular, if .s/ D fpg, then G is a p-group. Proof. For every A 2 J choose a Hall (or Sylow) .s/-subgroup SA of G such that A SA and SA contains a Hall -subgroup of A . Since G has at most jGj 0 D t 0 distinct Hall -subgroups and jJj D t t 0 C 1, the pigeonhole principle shows that there is a Hall -subgroup S of G with S D SA for at least t C 1 members of J. Fix such an S, and let JS D fA 2 J j SA D Sg: If we set AC D A \ S for all A 2 JS and JS D fAC j A 2 JS g; then .JS ; JS / is a Kantor family of type .s; t 0 /, where t 0 t . So t 0 D t . Let 0 be the corresponding subGQ of order .s; t /. Of course, 0 x is an EGQ with elation group S. Consider a C 2 J which is not contained in JS ; then we use Lemma 5.2 to obtain jSC j jGj D s 2 t D s 2 t t 0 < s 2 t s jSj jC j: Whence jS \ C j 2 for any such C . Since 0 is a subGQ, this implies readily that C 2 JS – in other words, 0 D and t D t . Putting Lemma 5.3 and Theorem 5.5 together, we obtain Theorem 5.6. If is a thick EGQ of order .s; t / with s t , then st is a prime power.

5.2 Skew translation quadrangles Let us call a thick finite EGQ . .1/ ; G/ of order .s; t /, for which G contains a group of symmetries about .1/ of maximal order (which is then t ) a skew translation generalized quadrangle (STGQ). For s D t, the latter condition is equivalent to saying that .1/ is a regular point by Theorem 1.12; see [64]. And in general, the point .1/ is of course always regular.

5.3 F -Factors

47

5.3 F -Factors Let be a thick EGQ, and let .F ; F / be the associated Kantor family. Let F D F [ F . A nontrivial subgroup X of G is an F -factor of G if .U \ X /.V \ X / D X for all U; V 2 F satisfying U V D G: Let X be an F -factor. Define FX D fU \X jU 2 F g and FX D fU \X jU 2 F g. We say that X is “of type .; /” if jX j D 2 , jA \ X j D and jA \ X j D for all A 2 F (and in [21] it is shown that such integers ; always exist). Theorem 5.7 (Hachenberger [21]). Let X be an F -factor of type .; / in G. Then necessarily one of the following cases occurs: T (a) D 1, jX j D t and X is a subgroup of A2F A ; (b) > 1, D t and .FX ; FX / is a Kantor family in X of type .; /. If we are in case (b) of Theorem 5.7, we call X a thick F -factor. An F -factor X in G is normal if X is a normal subgroup of G. In [21] D. Hachenberger obtained the following partial classification of normal F factors in Kantor families. Since case (a) of Theorem 5.7 is not of particular interest, one may suppose essentially without loss of generality that D t . Theorem 5.8 (Hachenberger [21]). Let G be a group of order s 2 t admitting a Kantor family .F ; F / of type .s; t /, with s; t > 1, and having a normal F -factor X of type .; / with D t. Then one of the following cases occurs: (a) G is a group of prime power order; (b) > 1, jGj has exactly two prime divisors, and X is a Sylow subgroup of G for one of these primes. Proof. By Theorem 5.6 we have that and t are powers of the same prime p, since .FX ; FX / is a Kantor family of type .; t / by Theorem 5.7, and t by Theorem 1.3. Letpr be any prime dividing jG=X j, and note that for any Sylow r-subgroup P in G=X , a . jP j; t C 1/-net in P is induced by fAX=X j A 2 F g. So if jG=X j would pnot be a prime power, we could choose a Sylow subgroup Q in G=X such that jQj < jG=X j. But this is easily seen to violate Higman’s inequalities. Hence jG=X j is a prime power. So either G is a p-group, or X is a Sylow p-subgroup of G. Theorem 5.8 led D. Hachenberger to prove a well-known conjecture of S. E. Payne, which amounts precisely to showing that G is a p-group if X is of type .1; t /: Corollary 5.9 (Hachenberger [21]). The parameters of any thick finite STGQ are powers of one and the same prime. Proof. If x is a thick finite STGQ of order .s; t / with elation group G, and S G is the group of symmetries with center x, then S D X is a normal F -factor in G of type .1; t/. Now apply Theorem 5.8 (a).

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In the next section, we will give another proof of this result. In [21] D. Hachenberger conjectured that case (b) of Theorem 5.8 cannot occur. In [74], we “completed” his classification by proving that this conjecture is indeed true. While I was writing up the present manuscript, I was not able to reconstruct the combinatorial lemma (on subquadrangles) stated in [74] (erroneously) without proof. Its main theorem still remains true however in the category of generalized quadrangles described by Theorem 5.8 (b) which satisfy an even much weaker form of the aforementioned lemma – see the exercises below. Suppose that we are in the situation of Theorem 5.8 (b). The GQ defined by .F ; F / is denoted by ; it is an EGQ of order .s; t / with elation point .1/ and elation group G. Since X defines a Kantor family .FX ; FX / of type .; t / in X with > 1, has a thick subGQ 0 of order .; t / which is an EGQ with the same elation point as , and with elation group X G. We also have that t by Theorem 1.3. By letting G act on the points of not collinear with .1/, each point y of the latter set is contained in at least one subGQ y of order .; t /. Suppose that z0 and z00 are different subGQs containing .1/ and sharing some point z not collinear with .1/; then by Theorem 1.3, D t , s D t 2 , and 0 \ 00 is a subGQ of order .1; t /. By Theorem 5.6, t is the power of some prime p, and so s also is. Hence in this case G is a p-group. So we may assume that each point y not collinear with .1/ is contained in precisely one subGQ y of order .; t /. Now define an incidence structure … as follows. • Points are the .s= /2 subGQs of order .; t / of 0 G (call subGQs in this orbit “G-subGQs”). • Lines are the point sets 00 \ M , where 00 2 0 G and M is a line incident with .1/. • Incidence is inverse containment. Exercise. Prove that if … is an affine plane, we are not in case (b) of Theorem 5.8. Exercisec . Let be a thick finite GQ of order .s; t /, and let 0 and 00 be thick subGQs, both of order .s 0 ; t /, and both containing the same point r. If 0 \ 00 does not contain points not collinear with r, is it true that 0 \ 00 consists of s 00 C 1 points (including r) of some line incident with r, and all lines incident with these points, with s 00 ¤ 0? Exercise. Show that if the property of the previous exercise holds true for the GQ in Theorem 5.8 (b), then … is an affine plane, so that the exercise above leads to a contradiction. Exercise# . Show that an F -factor X of type .; / in G, > 1, is normal if .1/ is not a regular point. (Hint: suppose, for instance, that G acts on the group coset representation by the right, and note that all right cosets Xh of X yield subquadrangles of order .; / with elation group X h . Now observe that the “same” conclusion is true when replacing X by any conjugate X g , and that if .1/ is not regular, the set .g/ ´ fX g h j h 2 Gg must be independent of g.)

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5.4 Parameters of STGQs

5.4 Parameters of STGQs We will describe the elegant proof of X. Chen, who obtained the result of Corollary 5.9 independently. In the next lemma, we use the notation of Lemma 5.1 (ii). T Lemma 5.10. Suppose that S D A2F A is a normal subgroup of G, and that jSj D r. If sp > 1, then kp0 r sp where k D t =r. x D s 2 k (G x being G=S). If P is a Sylow Proof. We know that jGj D s 2 t and jGj 2 x p-group of G then its size is sp kp . If B is an element of F , clearly B \ S D f1g, so Bx D BS=S Š B=.B \ S/ Š B: x then its size is sp . Take A 2 F , and now let P So if R is a Sylow p-subgroup of B, x that contains a Sylow p-subgroup PA of AS . For each be a Sylow p-subgroup of G x B 2 F n fAg let PB be a subgroup of P that is conjugate to a Sylow p-subgroup of B. By Lemma 5.1 (ii) the members of fPA g [ fPB j B 2 F n fAgg have pairwise trivial intersection. Then P jPA j C jPB j jP j; B2F nfAg

so that jPA j D sp kp ; jPB j D sp , and jP j D sp2 tp lead us to sp kp C t .sp 1/ sp2 kp : Whence rkp kp0 .sp 1/ sp kp .sp 1/. If sp > 1, then rkp0 sp .

Theorem 5.11. LetT G be a group of order s 2 t admitting a Kantor family p .F ; F / of type .s; t/. If S D A2F A is a normal subgroup of G and jSj s, then G is a p-group.

Proof. Suppose that G is not a p-group; then we already know that s has at least two prime divisors. Let p and q be such primes. Then from Lemma 5.10 we have rkp0 sp and rkq 0 sq . One of the equalities holds only if r D sp and kp0 D 1, or r D sq and kq 0 D 1. So one of the inequalities is strict. It follows that r 2 kp0 kq 0 < sp sq s: Note that r 2 kp0 kq 0 D t rkq 0 =kp , so that t r < s. But then r Higman’s inequality.

p

s would contradict

Corollary 5.12 (X. Chen). If . x ; G/ is an STGQ, then G is a p-group.

6 Standard elations and flock quadrangles

In this chapter we introduce the concept of “standard elation” and study the set of standard elations for certain classes of EGQs. Most of the results presented here are taken from [47].

6.1 Flock quadrangles A flock of the quadratic cone K in PG.3; q/ is a partition of K without its vertex into q irreducible conics. From such a flock, one can construct an interesting class of EGQs, as we will see in the next subsection. 6.1.1 Flock GQs. Let F D fC1 ; C2 ; : : : ; Cq g be a flock of the quadratic cone K in PG.3; q/. The planes i , i D 0; 1; : : : ; q, generated by Ci , are the flock planes of F . We shall use the following notation: the set of all elements m of Fq , q even, for which the polynomial X 2 C X C m is reducible over Fq , respectively irreducible, is denoted by C0 , respectively C1 . Theorem 6.1. With each flock F of the quadratic cone K of PG.3; q/ there corresponds a set of q ordered triples .ai ; bi ; ci /; ai ; bi ; ci 2 Fq , such that for q odd .ci cj /2 4.ai aj /.bi bj / is a nonsquare whenever i ¤ j , and for q even ci ¤ cj and .ai C aj /.bi C bj /.ci C cj /2 2 C1 whenever i ¤ j . Conversely, with each such set of q ordered triples there corresponds a flock F . Proof. Let X0 X1 D X22 be the equation of the cone K. Now we consider q planes i , with i D 1; 2; : : : ; q, which do not contain the vertex x.0; 0; 0; 1/ of K. Such a plane i has equation ai X0 Cbi X1 Cci X2 CX3 D 0. The plane ij projecting i \j , with i ¤ j , from the vertex x has equation .ai aj /X0 C .bi bj /X1 C .ci cj /X2 D 0. Let q be odd. Then the plane ij has no line in common with K if and only if .ci cj /2 4.ai aj /.bi bj / is a nonsquare. Hence the conics i \ K form a flock of K if and only if .ci cj /2 4.ai aj /.bi bj / is a nonsquare, whenever i ¤ j . Let q be even. Then the plane ij has no line in common with K if and only if the homogeneous quadratic polynomial X 2 .ai C aj /2 C X Y .ci C cj /2 C Y 2 .bi C bj /2 is

6.1 Flock quadrangles

51

irreducible, that is, if and only if ci ¤ cj and .ai C aj /2 .bi C bj /2 .ci C cj /4 2 C1 , if and only if ci ¤ cj and .ai C aj /.bi C bj /.ci C cj /2 2 C1 . Hence the conics i \ K form a flock if and only if ci ¤ cj and .ai C aj /.bi C bj /.ci C cj /2 2 C1 , whenever i ¤ j. Remark 6.2. Clearly coordinates can be chosen in such a way that 1 has equation X3 D 0, that is, .a1 ; b1 ; c1 / D .0; 0; 0/. In this section we will consider a particular class of GQs .G; J/ with G Š H2 .q/. If u D .u1 ; u2 /; v D .v1 ; v2 / 2 Fq2 , then put u v D u1 v1 C u2 v2 . Represent H2 .q/ as G D f.˛; c; ˇ/ j ˛; ˇ 2 Fq2 ; c 2 Fq g; with the group operation defined by .˛; c; ˇ/ .˛ 0 ; c 0 ; ˇ 0 / D .˛ C ˛ 0 ; c C c 0 C ˇ ˛ 0 ; ˇ C ˇ 0 /: N 0; ˇ/ 2 G j ˇ 2 Fq2 g (here 0N D .0; 0/). For each t 2 Fq , let Put A.1/ D f.0; xt yt At D 0 zt be an upper triangular .2 2/-matrix over Fq , with the convention that A0 be the zero matrix. For each t 2 Fq , put K t D A t C ATt , and let A.t / D f.˛; ˛A t ˛ T ; ˛K t / j ˛ 2 Fq2 g: It follows that A.t / is a commutative subgroup of G having order q 2 , for each t 2 N c; 0/ N 2 G j c 2 Fq g, and put A .t / D A.t / C , t 2 Fq [ f1g. Fq [ f1g. Let C D f.0; Then A .t/ is a commutative subgroup of G having order q 3 , t 2 Fq [ f1g. Moreover A .t/ A.t/. Further, let J D fA.t / j t 2 Fq [ f1gg

and

J D fA .t / j t 2 Fq [ f1gg:

With the foregoing notations we have the following two important theorems. Theorem 6.3 (Kantor [27]). If q is odd, then G, J, J define a Kantor family, so that .G; J/ is an EGQ of order .q 2 ; q/, if and only if det.K t Ku / D .y t yu /2 4.x t xu /.z t zu / is a nonsquare of Fq whenever t; u 2 Fq ; t ¤ u. Theorem 6.4 (Payne [37]). If q is even, then G, J, J define a Kantor family, so that .G; J/ is an EGQ of order .q 2 ; q/, if and only if y t ¤ yu and .x t C xu /.z t C zu /.y t C yu /2 2 C1 whenever t; u 2 Fq , t ¤ u.

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A set of q upper triangular .2 2/-matrices over Fq satisfying the conditions of Theorems 6.3 or 6.4 is called a q-clan. (In fact, as we will see in the next section, the assumption that the matrices be upper triangular is not essential.) Comparing Theorems 6.3 and 6.4 with Theorem 6.1 yields the following result. Theorem 6.5 (J. A. Thas [52]). .G; J/ is a GQ of order .q 2 ; q/ if and only if the planes x t X0 C z t X1 C y t X2 C X3 D 0, t 2 Fq , define a flock F of the quadratic cone with equation X0 X1 D X22 . Each flock F of the quadratic cone of PG.3; q/ gives us a GQ, and each GQ .G; J/ of the type described above gives us a flock of the quadratic cone. The GQ .G; J/ is also denoted by .F / and is called a flock GQ. 6.1.2 Linear flocks. A flock F is linear if all the flock planes contain a common line. The interest in linear flocks is reflected in the next characterization of classical flock GQs. Theorem 6.6 (J. A. Thas [52]). A flock F of the quadratic cone in PG.3; q/ is linear if and only if .F / Š H.3; q 2 /. We have introduced flock GQs as a particular class of EGQs. As the following result shows, in general the elation point is unique. Theorem 6.7 (Payne and J. A. Thas [45]). If a flock F is not linear, .F / has one and only one elation point. Often, we will speak of the “special point” of a flock GQ .F /; if F is not linear, this refers to the unique elation point. If F is linear, all points are special.

6.2 Fundamental theorem of q-clan geometry In this section, we describe the so-called “Fundamental theorem of q-clan geometry”. Starting with a natural definition for equivalent q-clans (defined below), it interprets the equivalence of q-clans C1 and C2 as an isomorphism between G .C1 / and G .C2 /, where G .Ci / is either the flock F .Ci / associated to Ci , or the flock quadrangle .F .Ci //, i D 1; 2. 6.2.1 The fundamental theorem. Suppose q is a prime power, and assume further that A D .xz wy / and B D .rt us / are .2 2/-matrices over Fq . We write A B to mean that x D r, w D u and y C z D s C t . It follows that A B if and only if ˛A˛ T D ˛B˛ T for all ˛ 2 Fq2 . If C D fA t j t 2 Fq g is a q-clan, the matrices A t 2 C are used to construct quadratic forms ˛A t ˛ T , with ˛ 2 Fq2 , so an A t 2 C may be replaced by an A0t whenever A t A0t , without effectively “changing” C. (The

6.2 Fundamental theorem of q-clan geometry

53

conditions of Theorem 6.3 and Theorem 6.4 keep being satisfied, and the A.t /s and A .t/s rest unchanged.) If q is odd, one could adjust each A t 2 C to be symmetric, say, for instance, y t =2 xt At D ; t 2 Fq : y t =2 zt For any q we can adjust each A t 2 C to be upper triangular, so, for instance, xt yt At D ; t 2 Fq : 0 zt The q-clan is normalized if A0 is the zero matrix. For any prime power q, let ˚ C D A t x0t yz tt j t 2 Fq and

0 ˚ x C 0 D A0t 0t

y t0 z t0

j t 2 Fq

be two not necessarily distinct q-clans. We say that C and C 0 are equivalent and write C C 0 provided there exist 0 ¤ 2 Fq , B 2 GL2 .q/, 2 Aut.Fq /, M a .22/-matrix over Fq and a permutation W t ! tN on Fq such that the following holds: A0tN BAt B T C M

for all t 2 Fq :

If for a q-clan C each A t is replaced by At D A t A0 , then there arises a normalized q-clan equivalent to C . Theorem 6.8 (Fundamental theorem of q-clan geometry [41]). Assume that C D fA t j t 2 Fq g and C 0 D fA0t j t 2 Fq g are normalized q-clans. The following are equivalent: (i) C C 0 . (ii) The flocks F .C / and F .C 0 / are projectively equivalent. (iii) The GQs .F .C // and .F .C 0 // are isomorphic by an isomorphism mapping N 0; 0/ N to .0; N 0; 0/. N .1/ to .1/, ŒA.1/ to ŒA0 .1/, and .0; (Here 0N is again the null vector of Fq2 , not to be confused with .0/ below.) These three equivalent conditions hold if and only if the following exist: (i) a permutation W t ! tN of the elements of Fq ; (ii) 2 Aut.Fq /; (iii) 2 Fq , ¤ 0; (iv) D 2 GL2 .q/ for which AtN D T A t D A0N is skew-symmetric (with zero diagonal) for all t 2 Fq .

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Given , D, and the permutation W x ! xN satisfying condition (iv), an isomorphism D . ; D; ; / of the GQ .F .C // onto .F .C 0 // arises, as follows: D . ; D; ; / W .˛; c; ˇ/ ! .1 ˛ D T ; 1 c C 2 ˛ .D T A0N D 1 /.˛ /T ; ˇ D C 1 ˛ D T K0N /; writing D T for the matrix .D 1 /T D .D T /1 . We have that sends ŒA.1/ to N 0; 0/; N also, defines an automorphism of G. ŒA0 .1/ and fixes .0; For D . ; D; ; /, write D D ac db . Define a projective semilinear collineation T of PG.3; q/ as follows (defined on the planes of PG.3; q/, and with a general plane having equation xX0 C zX1 C yX2 C X3 D 0): 1 2 3 2 3 0 2 ab b 2 x0N x x a 6y 7 B2ac .ad C bc/ 2bd y N C 6y 7 0C 6 7: 7 B T W 6 4 z 5 ! @ c 2 cd d 2 z0N A 4 z 5 0 0 0 1 1 1 Then T fixes the cone K with equation X0 X1 D X22 , and leaves invariant F D F .C/ N 0; 0/. N precisely when defines a collineation of .F / fixing .1/, ŒA.1/ and .0; The following statement is a direct corollary of the fundamental theorem. Corollary 6.9. Let F and F 0 be projectively equivalent flocks of the quadratic cone in PG.3; q/. Then .F / Š .F 0 /. Proof. Let C be the q-clan associated to F and C 0 the q-clan associated to F 0 . Then C C 0 , and .F / D .F .C// Š .F .C 0 // D .F 0 /. 6.2.2 The kernel of a flock GQ. The map T W ! T N 0; 0/ N is a homomorphism from the subgroup of Aut..F // leaving .1/; ŒA.1/ and .0; invariant onto the subgroup of PL4 .q/ leaving the flock F invariant. The kernel N.T / of T is N.T / D fa j a W .˛; c; ˇ/ ! .a˛; a2 c; aˇ/; 0 ¤ a 2 Fq g: N 0; 0/. N Note that N.T / is a group of whorls about .1/ and .0; The following observation is taken from S. E. Payne [41] – Kantor–Knuth flocks, and the associated are defined in the next section. Lemma 6.10 (S. E. Payne [41]). Suppose that .F / is a thick nonclassical flock GQ of order .q 2 ; q/, and let H be the full group of automorphisms of .F / fixing .1/ and N 0; 0/ N linewise. .0; (i) If q D 2h , then H D N.T /. (ii) If q is odd, and H D f. ; D; ; / 2 H j D 1g, then H D N.T /, except if F is a Kantor–Knuth semifield flock. In that case, H D H if 2 ¤ 1, and then ŒH W N.T / D 2. If 2 D 1 ¤ , then ŒH W H D ŒH W N.T / D 2.

6.3 A special elation

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6.3 A special elation Now define the following q-clan C : C D fA t D

t

0 0 mt

j t 2 Fq g;

where q is odd, m a given nonsquare and 2 Aut.Fq /. Then the GQ arising from C is the Kantor–Knuth semifield flock GQ. Note that if D 1, then Š H.3; q 2 /. 1 0 Let Q D 0 1 , so Q D QT D Q1 . It is noted in [40] that the map

W .˛; c; ˇ/ ! .˛Q; c; ˇQ/ N 0; 0/ N and .1/ is an automorphism of G which induces a collineation of that fixes .0; 0 0 0 linewise. Let g D .˛ ; c ; ˇ / be a fixed element of G to be determined later. Put D B .g/ D B .˛ 0 ; c 0 ; ˇ 0 /; where .g/ stands for right multiplication by g, so that for h 2 G, h D h B.g/ D h g: For the appropriate choice of g we will establish the following: (1) is an elation about .1/ that is not in G; (2) p , where q D p h for some h and the odd prime p, is an involution whose fixed element structure is a subquadrangle of order q, so that in particular p is not an elation about .1/; (3) 2 2 G. The fact that some power of (i.e., p ) is not an elation about .1/ suggests that possibly we have been using an unsatisfactory definition of elation. Let us say that a standard elation about .1/ is an elation about .1/ that acts semiregularly on the points not collinear with .1/, i.e., a standard elation about .1/ is a collineation of that generates a cyclic group of elations about .1/. In Section 6.6, we will then obtain that if is a flock GQ, the set of standard elations about the point .1/ is a group. Whence the usual elation group G is the entire set of standard elations about .1/.

6.4 The nitty gritty Let be the Kantor–Knuth semifield flock GQ given in the previous section, and let Q be as defined there. Lemma 6.11. If j is a positive integer, let j2 denote the element of f0; 1g to which j is congruent modulo 2. Similarly, let j2 denote 1 if j is odd and 0 if j is even.

56


Also, I is the 2 2 identity matrix.

j C1 0 ; I C Q C Q C

C Q D 0 .j C 1/2 j 0 Q C Q2 C C Qj D ; 0 j2 ! .j 1/j 0 j 1 j 2 j 3 1 2 Q C 2Q C 3Q C C .j 1/Q D : 0 b j2 c

2

j

(6.1) (6.2) (6.3)

(In (6.1) and (6.2) we take j 1. In (6.3) we must take j 2.) Proof. All three equations are established by routine induction arguments (note that 2 ). b j2 c D j j 2 By definition we have D B .˛ 0 ; c 0 ; ˇ 0 / W .˛; c; ˇ/ ! .˛Q C ˛ 0 ; c C c 0 C ˇQ ˛ 0 ; ˇQ C ˇ 0 /: Then by an easy calculation we have 2 W .˛; c; ˇ/ ! .˛Q2 C˛ 0 .QCI/; cC2c 0 Cˇ.QCQ2 / ˛ 0 Cˇ 0 Q ˛ 0 ; ˇQ2 Cˇ 0 .QCI//: It now follows by a routine induction that i W .˛; c; ˇ/ ! ˛Qi C ˛ 0 .Qi1 C Qi2 C C Q C I/; c C i c 0 C ˇ.Q C Q2 C Q3 C C Qi / ˛ 0 C ˇ 0 .Qi1 C 2Qi2 C 3Qi3 C C .i 1/Q1 / ˛ 0 ;

ˇQi C ˇ 0 .Qi1 C Qi2 C C Q C I/ 0 D ˛ 10 .1/ C ˛ 0 0i i02 ; i .i 1/i i 0 0 0 0 0 2 c C i c C ˇ 0 i2 ˛ C ˇ

˛0; 0 b 2i c 0 ˇ 10 .1/ C ˇ 0 0i i02 : i At this stage it is convenient to have written out the image of i separately for odd and even i : 0 ; c C 2kc 0 C ˇ 2k 0 ˛ 0 C ˇ 0 .2k1/k 0 2k W .˛; c; ˇ/ ! ˛ C ˛ 0 2k

˛0; 0 0 0 0 0 k 0 ˇ C ˇ 0 2k I 0 0 0 C ˛ 0 2kC1 0 ; c C .2k C 1/c 0 C ˇ 2kC1 0 ˛ 0 2kC1 W .˛; c; ˇ/ ! ˛ 10 1 0 1 0 1 1 0 0 0 2kC1 0 0 C ˇ

˛ C ˇ 0 k.2kC1/ ; ˇ : 0 1 0 1 0 k

6.5 A special elation, once again

57

We now determine whether or not i fixes some point .˛; c; ˇ/. First, note that 2k fixes .˛; c; ˇ/ if and only if 0 (i) ˛ 0 2k 0 0 D .0; 0/; 0 ˛ 0 C ˇ 0 .2k1/k 0

˛ 0 D 0I and (ii) 2kc 0 C ˇ 2k 0 0 0 k 0 (iii) ˇ 0 2k 0 0 D .0; 0/. It follows readily that if ˛ 0 D .a1 ; a2 / with a1 ¤ 0, then we have proved the following lemma: Lemma 6.12. Assume that a1 ¤ 0. Then 2k fixes some .˛; c; ˇ/ if and only if k 0 mod p, in which case 2k D 1. In particular 2 is an elation, even a standard elation. Note that it is also easy to check that 2 2 G. Now we consider whether or not 2kC1 fixes some .˛; c; ˇ/. Since we are assuming easily that if 2kC1 fixes some .˛; c; ˇ/ that ˛ 0 D .a1 ; a2 / with a1 ¤ 0, it follows 0 2kC1 0 0 0 it must be the case that ˛ 0 2 D ˛ , which forces 2k C 1 0 mod p. So 0 1 consider the fixed points of p . It is routine to check that p fixes the point .˛; c; ˇ/ if and only if .˛; c; ˇ/ D ..a; ka2 /; c; .b; kb2 //, a; c; b 2 Fq , where ˛ 0 D .a1 ; a2 /, ˇ 0 D .b1 ; b2 / and p D 2k C 1. It now follows that p is an involution with a subquadrangle of order q as its fixed element structure.

6.5 A special elation, once again In view of Lemma 6.10 and of the rest of this chapter, we point out the existence of the following remarkable collineation if q is odd (for all flocks!). Suppose q is odd, and put D 1 . So W .˛; c; ˇ/ ! .˛; c; ˇ/: N 0; 0/ N linewise, and which Then is an involution of .F / which fixes .1/ and .0; ?? N c; 0/ N j c 2 Fq g [ f.1/g. N 0; 0/g N D f.0; fixes each point of f.1/; .0; Remark 6.13. In the Kantor–Knuth case (or more generally, in the case of the translation duals of the semifield flock GQs in odd characteristic – cf. the next chapter for definitions), the existence of this collineation was pointed out in [66], [67] without coordinates. Now suppose that is an automorphism of the flock GQ .F / of order .q 2 ; q/, q N 0; 0/ N linewise, and each point of f.1/; .0; N 0; 0/g N ?? D odd, and that fixes .1/ and .0; N N f.0; c; 0/ j c 2 F g [ f.1/g. We can write as . ; D; ; /. As each point of the N c; 0/ N is fixed by , the explicit form of . ; D; ; / above leads to the fact form .0;

58


that D 1 (and D 1). Hence by Lemma 6.10, is an element of N.T / if F is not Kantor–Knuth. It follows immediately that D W .˛; c; ˇ/ ! .˛; c; ˇ/; and thus is an involution.

6.6 Standard elations in flock GQs The reader is referred to [54] for a (short, combinatorial) proof of the next theorem, which will come in handy. Theorem 6.14 (J. A. Thas [54]). Let 0 be a thick subGQ of order s of a GQ of order .s; s 2 / which is fixed pointwise by an involutory automorphism of . Furthermore, let s be even. Then 0 Š W .s/. We proceed with a general lemma. Lemma 6.15. Suppose D .P ; B; I/ is a thick GQ of order .s; t /, with s and t powers of the same prime p. Suppose that .1/ is a regular point which is a center of transitivity, and let H be the full group of whorls about .1/. Let G be a Sylow p-subgroup of H . Then we either have (i) jGj D s 2 t , or (ii) p D 2, jGj D 2s 2 t , and contains a ( proper) subGQ of order t isomorphic to W .t/; consequently, s D t 2 . Proof. As H acts transitively on P n f.1/g? , s 2 t is a divisor of jH j. Suppose that s 2 t is not the largest power of p dividing jH j. Consider a point x ¦ .1/. Then Hx contains a nontrivial element of order p, say . As s is a power of p, fixes a point of L n f.1/; projL xg, where L is an arbitrary line incident with .1/. By Theorem 1.12, the fixed element structure of is a subGQ 0 of order t , and s D t 2 . Also, necessarily is an involution, hence p D 2 in that case. By Theorem 6.14, it follows that 0 Š W .t/. It also follows easily that jGj D 2s 2 t . Exercise. Show that indeed is an involution. (Hint: use the fact that spans of nonconcurrent lines are trivial in finite GQs of order .t 2 ; t /.) Exercise. Show that jGj D 2s 2 t in (ii). (Hint: suppose that jGj=s 2 t D 2h , h 1, h 2 N. Then jGx j D 2h and Gx must fix some subGQ 00 of order t pointwise. Now use the proof of the previous exercise.) From now on, we suppose that .F / is a flock GQ of order .q 2 ; q/, where q D p h for the prime p, and F is not a linear flock. By G we denote the elation group as

6.6 Standard elations in flock GQs

59

defined in Section 6.1, and by S, its center (as defined in the same section), which is a maximal group of symmetries of size q about the regular point .1/. Also, H denotes the full group of whorls about .1/. Lemma 6.16. Let fG0 D G; G1 ; : : : ; Gr g be the set of all t-complete elation groups with elation point .1/. Then each Gi is a Sylow p-subgroup, thus all t-maximal elation groups with elation point .1/ are conjugate. Proof. For q odd, the lemma follows from Lemma 6.15. Let q be even. As F is not linear, by C. M. O’Keefe and T. Penttila [34], .F / contains no classical subGQs of order q containing .1/. So the lemma follows again from Lemma 6.15. Corollary 6.17. Assume that .F / is a flock GQ of order .q 2 ; q/, where q D p h for the prime p, and F is not a linear flock. Then each whorl about .1/ of order p k for k ¤ 0 is a standard elation, and conversely. Proof. Immediate.

Corollary 6.18. S is the center of each of the elation groups Gi , i D 0; 1; : : : ; r. Proof. Immediately from the fact that the Gi are conjugate, that S is the center of G0 D G, and that S E H . Now we introduce property .M /.1/ with respect to Gj , j 2 f0; 1; : : : ; rg: Property .M /.1/ . Let M be an arbitrary line not incident with .1/. Put m D projM .1/. If 2 Gj fixes M , then fixes each line incident with m . Lemma 6.19. For each j 2 f0; 1; : : : ; rg, property .M /.1/ holds for Gj . Proof. Suppose M and m are as above, and let 2 Gj fix M . Suppose that N I m is a line for which N ¤ N . Let 2 S be so that M D N . Then M D N , while M D N ¤ N , contradicting the fact that S is the center of Gj . Before proceeding with the following lemma, recall that the dual net Hq3 is constructed as follows: • the points of Hq3 are the points of PG.3; q/ not on a given line U of PG.3; q/; • the lines of Hq3 are the lines of PG.3; q/ which have no point in common with U ; • the incidence in Hq3 is the natural one. We will need the following result taken from J. A. Thas and H. Van Maldeghem [60]. Theorem 6.20 ([60], Theorem 7.2). Let be a thick flock GQ of order .s 2 ; s/ with special point x, where s is a prime power. Then NxD is isomorphic to Hs3 if and only if is isomorphic to a Kantor–Knuth flock GQ.

60


Lemma 6.21. Let y 2 .1/? n f.1/g. Suppose that Hy is the subgroup of H which fixes y linewise. Then Hy has a unique (Sylow p-)subgroup H.y/ of size q 2 . Proof. By Lemma 6.19, it is clear that jHy jp q 2 , where jHy jp is the largest power of p dividing jHy j. First suppose equality does not hold. Then there is some nontrivial element of Hy fixing some point x y, x ¦ .1/, whence fixing the dual grid defined by .1/ and x elementwise. When q is even, this already contradicts Lemma 6.10. Now suppose that q is a power of the odd prime p and that F is not Kantor–Knuth. Then by the end of the previous section, it follows immediately that jHy j D 2q 2 . Suppose that H.y/ and H.y/0 are two distinct Sylow p-subgroups of Hy . Then clearly jHy j pq 2 by Sylow’s Theorem, contradiction. Finally, suppose that F is a Kantor–Knuth semifield flock. As the point .1/ is a D D and by Theorem 6.20, N.1/ is isomorphic regular point, there arises a dual net N.1/ 1 2 3 to Hq . Suppose there are distinct groups H .y/ and H .y/ which have size q 2 , both D fixing .1/ and y linewise. Interpreted in N.1/ Š Hq3 , the H i .y/ faithfully induce collineation groups of PG.3; q/ fixing U and a point y 0 I U (corresponding to y). Also, the q C 1 planes through U (corresponding to the lines through .1/) are fixed by both groups. Thus they are subgroups of PGL4 .q/U (cf. the exercise below). Consider the action of hH 1 .y/; H 2 .y/i μ F on the set S D ffy; y 00 g?? j y 00 2 .1/? ; y 00 ¦ yg. Note that F acts transitively on S. If some element ˛ of F fixes at least 2 distinct elements of S , then, since ˛ fixes y linewise, we have ˛ D 1 (otherwise ˛ fixes a subGQ of order t pointwise, implying that ˛ induces a Baer involution in PGL4 .U /, contradiction). So .F; S / is a Frobenius group which is faithful (a hypothetical kernel of the action of F on S would be a group of symmetries with center .1/ fixing y linewise, contradiction). Let E be the Frobenius kernel of .F; S / – it has size q 2 and acts sharply transitively on S; E is generated by the fixed-elements-free members of F , so that H 1 .y/ [ H 2 .y/ E. As F is generated by H 1 .y/ [ H 2 .y/, we have obtained our desired contradiction (F D E). It follows that H 1 .y/ D H 2 .y/.

Exercise. Let q be any prime power. Show that PGL2 .q/, in its natural action on the projective line PG.1; q/ D X , has the property that once an element fixes three distinct points of X, it is the identity. Exercise. Let ˛ ¤ ; be an m-subspace of a finite projective space PG.n; q/, m < n, and suppose that some automorphism ˇ of PG.n; q/ fixes all r-subspaces containing ˛, for some r with m < r < n. Show that ˇ is an element of PGLnC1 .q/. (Hint: first show that if an automorphism of a finite Desarguesian projective space fixes some line pointwise, it must be a linear automorphism. Then apply this to the dual statement of the exercise.) Remark 6.22. (i) It is clear that H.y/ acts sharply transitively on the q 2 points different from y of an arbitrary line N I y, N ¤ .1/y.

61

6.6 Standard elations in flock GQs

(ii) Suppose that .F / is as in the preceding lemma and that q is even. Suppose that D H.y/1 ¤ H.y/2 . By J. A. Thas and H. Van Maldeghem [60], N.1/ Š Hq3 . Taking over the proof of the second part of Lemma 6.21, we find again a contradiction. We are finally ready to obtain the following result. Theorem 6.23. Let .F / be a nonclassical flock GQ of order .q 2 ; q/, where q D p h for the prime p. Then the set of standard elations about the point .1/ is a group. Whence the usual elation group G D G0 is the entire set of standard elations about .1/. Proof. Suppose that F is not linear (in particular, q > 2). It is clear that, with the notation of Lemma 6.16, it suffices to prove that G0 D G1 D D Gr . Fix a j 2 f0; 1; : : : ; rg. For each point y .1/ ¤ y, by Lemma 6.19, Gj has a subgroup Gj .y/ of size q 2 , each element of which fixes y linewise (the stabilizer .Gj /M of any line M I y in Gj , M ¤ .1/y, has size q 2 ). But by Lemma 6.21, this group Gj .y/ is unique in H , and thus G0 .y/ D G1 .y/ D D Gr .y/ for each such y. Now define for each i D 0; 1; : : : ; r the group Gi0 D hGi .y/ j y .1/ ¤ yi: It should be clear to the reader that if Gi0 coincides with Gi for each i , then G0 D G1 D

D Gr . Suppose that x and x 0 are arbitrary but different points not collinear with .1/. Then by a result of A. Brouwer [6], see also the exercise after the theorem, there are points x D x0 ; x1 ; : : : ; xn D x 0 , all not collinear with .1/, so that xj xj C1 ¤ xj for 0 j < n. .1/

y0

yn1

x2

x D x0

xn2

x 0 D xn

x1

xn1 Figure 6.1. Transitivity of Gi0 .

Fix i 2 f0; 1; : : : ; rg. For each j D 0; 1; : : : ; n 1, define yj as the point of .1/? incident with xj xj C1 . Then clearly there is a j 2 Gi .yj / mapping xj to xj C1 .

62


Whence

x 0 1 :::n1 D x 0 ;

and Gi0 acts transitively on the points not collinear with .1/. It follows that Gi D Gi0 for each i D 0; 1; : : : ; r, and hence that G0 D G1 D D Gr . If F is linear and q is odd, the reader will notice that we have the same conclusion as in Theorem 6.23. When q is even, we refer the reader to Chapter 8. Exercise (On Brouwer’s result [6]). Let D .P ; B; I/ be a thick, not necessarily finite, GQ, and let x 2 P . Define a rank 2 geometry .x/ D .P .x/; B.x/; I.x// as follows: P .x/ is P n x ? , B.x/ is B without the lines on x, and I.x/ is induced by I. Determine when precisely .x/ is connected. This is the part of Brouwer’s paper [6] that is used in the proof of the previous theorem.

6.7 The general case We now present some generalizations. The proofs are related but slightly different – they all use the Theorem of Frobenius. We can say the following about the case s D t : Theorem 6.24. Suppose that D .P ; B; I/ is a thick GQ of order s. Suppose that .1/ is a regular point which is a center of transitivity. Then the set of all standard elations about .1/ is a group. Proof. Let H be the full group of whorls about .1/. Suppose that T is the set of spans of noncollinear points in .1/? . Then from Theorem 1.12 follows that .H=S; T / is a Frobenius group, where S E H is the group of symmetries about .1/ (as .1/ is a regular point which is a center of transitivity, by Theorem 1.12 the group S of symmetries about .1/ is of maximal size s). Let F=S be the Frobenius kernel of H=S; then F H is a group of size s 3 which acts freely, and hence sharply transitively, on the points not collinear with .1/. Whence is an EGQ, and by Theorem 5.6, s is a prime power (say of the prime p). Let G be a Sylow p-subgroup of H . As .jH=F j; p/ D 1 by the Theorem of Frobenius, it follows that G D F is the only Sylow p-subgroup in H . The theorem follows. Application to STGQs Theorem 6.25. Let . .1/ ; G/ be an STGQ of order s > 1. Then the set of all standard elations about .1/ is a group. Exercise (A stronger result). By applying the main result of [21], one can obtain the following more general result, the proof of which is an exercise:

6.7 The general case

63

Theorem 6.26. Let . .1/ ; G/ be an STGQ of order .s; t /, s; t > 1. Then we have one of the following possibilities: (i) the set of all standard elations about .1/ is a group; (ii) s D t 2 , s is a power of 2, and there is a subGQ of order t isomorphic to W .t / which contains the point .1/.

7 Foundations of EGQs

When [44] first appeared, it was an open question as to whether the set of elations about a point of a generalized quadrangle must be a group, and the basic attitude was to prove as generally as possible that indeed this is the case. It was proved in [44] that for finite TGQs, the translation group is the entire set of elations about the translation point. (A related result is the following: if x is a point of a GQ of order s, and if x is incident with a regular line, then the set of elations about x is a group ([36] or see [44], 8.2.6 (ii)).) As the set of examples of known GQs was enlarged, eventually it was noticed that an EGQ with base point x may have too many elations about x for these to form a group – this is what we considered in the previous chapter. In this chapter we are interested in the following question: Question. Given a thick GQ x with elation point (respectively center of transitivity) x, when does the set of all elations about x form a group?

7.1 An application of Burnside’s lemma In this section we develop a criterion to decide, given an elation point x of some EGQ, precisely when the set of all elations about x forms a group. 7.1.1 Burnside’s lemma. Let .G; X / be a finite faithful permutation group. Suppose that for j 2 N, ı.j / denotes the number of elements of G that fix precisely j points of X. If m is the number of G-orbits in X , then mjGj D

jXj P

i ı.i /:

iD1

7.1.2 Criterion. Setting. In this section, D .P ; B; I/ is a GQ of order .s; t /, s; t > 1, and x is an elation point for the elation group G. We let W be the group of all whorls about x, and suppose that, for o an arbitrary point not collinear with x, jfx; og?? j D r C 1, r 2 N. Note that as x is an elation point, o indeed is arbitrary. For the moment, suppose that has a proper thick subGQ 0 of order .s 0 ; t / (so s … f1; sg) that contains x, and suppose that 00 is a subGQ of order .s 00 ; t /, s > s 00 > 1, also containing x but with s 0 ¤ s 00 . Without loss of generality, we can assume that both 0 and 00 contain the point o ¦ x. Then ´ 0 \ 00 is a subGQ of 0 and 00 0

7.1 An application of Burnside’s lemma

65

which is proper in at least one of 0 , 00p . By Theorem 1.3, it follows directly that is of order .1; t/, and that s 0 D s 00 D t D s, contradiction. Hence, if has proper thick subGQs containing x and all lines of through x, there is only one s 0 for which s 0 C 1 is the number of points incident with any line of these subGQs. We will therefore use the notation s 0 for that purpose. Suppose that ˛ 2 W is nontrivial. Then one notes that, by Theorem 1.6, ˛ either fixes at most r points in P n x ? , or ˛ fixes s 02 t points in P n x ? . Now let H W be any subgroup of W that acts transitively on P n x ? . We apply Burnside’s lemma on the permutation group .H; P n x ? / to obtain jH j D s 2 t C

r P

i ı.i / C s 0 2 t ı.s 0 2 t /:

iD1

Let E be the number of elations in H (not including 1) about x; then, as 1CE C

r P

ı.i / C ı.s 0 2 t / D s 2 t C

iD1

r P

i ı.i / C s 0 2 t ı.s 0 2 t /;

iD1

it follows that the number of elations (now including 1!) is given by 1 C E D s2t C

r P

.i 1/ı.i / C .s 0 2 t 1/ı.s 0 2 t /;

iD2

which is clearly at least s 2 t .1 Moreover, we obtain the following criterion: Observation 7.1 (Criterion). Let H be a group of whorls about x acting transitively on the set X D P n x ? . The set of elations in H does not form a group if and only if (at least) one of the following conditions is satisfied: (1) there is a j 2 f2; 3; : : : ; rg for which ı.j / > 0; (2) there is a proper thick ideal subGQ of containing x which is fixed pointwise by a nonidentity element of H (that is, ı.s 0 2 t / > 0). Proof. The “if part” is obvious. Now suppose that neither (1) nor (2) holds. Then .H; X/ is a Frobenius group, and the set of elations is precisely the Frobenius kernel of H , which is a group. Remark 7.2. The criterion of Observation 7.1 underlines the importance of spans and subGQs in the context of automorphism questions. This fact will be encountered at several other places in these notes. 1 One notes that straightforward generalizations could be made for groups of whorls about x that do not act transitively on P n x ? .

66


7.2 Implications The easy but fundamental Observation 7.1 yields several (sometimes surprising) corollaries which deserve separate mentioning. Theorem 7.3. Let be a GQ of order .s; t /, s; t > 1, and let x be a center of transitivity for the group H . If H contains a nontrivial element that fixes pointwise some proper subGQ of order .s 0 ; t / containing x, then the set of elations in H does not form a group. The next theorem easily follows from Observation 7.1. Theorem 7.4. Let be a GQ of order .s; t /, s; t > 1, and let x be a center of transitivity for the group H . Then the set of elations of H about x is a group if and only if H is a Frobenius group in its action on the points noncollinear with x. If this is the case, the Frobenius kernel consists precisely of all elations about x. In particular, the theorem applies if t > s 2 =2, or if jfx; og?? j D 2 and s t , where o ¦ x. (If s < t, Theorem 1.2 implies that cannot have thick proper subGQs of order .s 0 ; t/. Now use [44], 1.4.1.) The following theorem will be useful for the rest of this chapter. It generalizes results of M. R. Brown [7] and J. A. Thas [54]. Theorem 7.5. Let be a thick GQ of order .s; s 2 / having a thick subGQ of order .s; t 0 /, fixed pointwise by a nontrivial automorphism of . Then t 0 D s and is an involution. If s is even, 0 Š Q.4; s/ Š W .s/. If s is odd, each point of 0 is antiregular. Proof. First note that fixes no point outside 0 ; this follows from Theorem 1.3 and the fact that t 0 > 1. Let x be a point of n 0 , and let Ox be the set of points of 0 that are collinear with x, so that Ox is an ovoid of 0 (by Theorem 1.2). (Such an ovoid is called a subtended ovoid – it is subtended by x; if it is subtended by two distinct points, we call it doubly subtended.) No three distinct points u; v; w outside 0 can subtend the same ovoid of 0 ; otherwise jfu; v; wg? j jOu j D st 0 C 1 > s C 1, contradiction (cf. Chapter 1, §1.1). It follows easily now that necessarily is an involution. Let s be even. Let fu; v; wg be an arbitrary triad of points in 0 , so that jfu; v; wg? j D s C 1 in by Chapter 1, §1.1.9. Since is an involution, the number of centers of fu; v; wg outside 0 is even. So fu; v; wg has a center in 0 , and each triad of 0 is centric. Now count in two ways the number of pairs .fu; v; wg; z/, where fu; v; wg is a triad which is a subset of a fixed subtended ovoid Ox , x 2 n 0 , and z is a center of fu; v; wg in 0 . Then .st 0 C 1/st 0 .st 0 1/:1 .st 0 C 1/s.t 0 C 1/t 0 .t 0 1/; yielding t 0 D s. The result now follows from Theorem 6.14.

7.3 Intermezzo – SPGQs

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Let s be odd, and let fu; v; wg be an arbitrary triad of points in 0 . Then the number of centers of fu; v; wg in 0 is even. As t 0 s by Theorem 1.3, 1.3.6 (iii) of [44] implies that t 0 D s, and that each point of 0 is antiregular. In fact, one also notes the following. Let be a GQ of order .t 2 ; t /, t even, with center of transitivity x, so that for at least one point o ¦ x, fx; og?? D fx; og (so that this property holds for all points not collinear with x). Let have a proper thick subGQ 0 of order .s 0 ; t / that contains x. Suppose that 0 were fixed pointwise by some nonidentity whorl about x. Then by Theorem 7.5, s 0 D t , and 0 Š Q.4; t / Š W .t /. But in that case, each point of 0 is regular, contradicting the assumption that in there is a point o ¦ x, such that fx; og?? D fx; og. Whence the full group of whorls about x is a Frobenius group, and we can use Observation 7.1 such as in the proof of Theorem 7.4 to conclude the following: Theorem 7.6. Let be a GQ of order .t 2 ; t / with t even. Let x be a center of transitivity for which fx; og?? D fx; og for some point o ¦ x. Then the full group of whorls about x is a Frobenius group on the points not collinear with x, and the Frobenius kernel is the set of all elations about x.

7.3 Intermezzo – SPGQs A span-symmetric generalized quadrangle (SPGQ) is a thick GQ containing nonconcurrent axes of symmetry U and V . The set fU; V g?? is called a “base-span” of the SPGQ. Exercise. Let be an SPGQ with U and V as above. Show that each line of fU; V g?? is an axis of symmetry. Show that the group H.U; V / generated by the symmetries with axis U or V acts doubly transitively on the lines of fU; V g?? . The following results can be found in W. M. Kantor [30] and K. Thas [63]. The reader is also referred to [68] for an elaborate account of the theory of SPGQs. (i) An SPGQ of order .s; t / with t ¤ s 2 is isomorphic to Q.4; s/. (ii) Let U; V be any two nonconcurrent axes of symmetry of , and let H.U; V / be as above. Then H.U; V / Š SL2 .s/. If N is the kernel of the action of H on X ´ fU; V g?? , then H.U; V /=N Š PSL2 .s/ (and the action is the natural one on the projective line PG.1; s/). In particular, if s is odd, jN j D 2. In [66], [68] it was proven that (ii) holds for general SPGQs (without restricting the order).

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7.4 The classical and dual classical examples We start with the classical GQs of order q: Theorem 7.7 (Q.4; q/ and W .q/). The following are all consequences of Observation 7.1. (1) Let x be an arbitrary point of Q.4; q/, q an odd prime power. Then the set of elations about x is a group. (2) Let x be an arbitrary point of W .q/, q an odd prime power. Then the set of elations about x is not a group. (3) Let x be an arbitrary point of W .q/ Š Q.4; q/, q an even prime power. Then the set of elations about x is a group. Proof. (1) This part follows from Observation 7.1 and the fact that the span of any two noncollinear points has size 2. (Another way to do this is to observe that each point of Q.4; q/ is a translation point, and then to apply [44], 8.6.4.) (2) Let o be an arbitrary point not collinear with x. Then W .q/ is a dual SPGQ with base-span P ´ fx; og?? . By §7.3 the group generated by the symmetries about the points of P is isomorphic to SL2 .q/, and contains a unique involution that fixes P pointwise. Now the criterion of Observation 7.1 applies to obtain (2). (3) It is well known (and easy to observe, as all lines of W .q/ are regular if q is even) that a collineation as in (2) cannot exist. The subgroup of PL4 .q/ which stabilizes W .q/ and fixes x and o linewise acts naturally as a subgroup of PSL2 .q/ Š SL2 .q/ on P n fx; og (the subgroup of PSL2 .q/ fixing 2 points of PG.1; q/ in its natural action), hence as soon as distinct points of P n fxg are fixed by an element of this group, it is the identity on P . Observation 7.1 applies. Recall from Theorem 1.17 that Q.5; q/ and H.3; q 2 / are point-line duals of each other. Theorem 7.8 (Q.5; q/ and H.3; q 2 /). (1) Let x be an arbitrary point of Q.5; q/. Then the set of elations about x is a group. (2) Let x be an arbitrary point of H.3; q 2 /. Then the set of elations about x is not a group. Proof. (1) This follows from the fact that each point of Q.5; q/ is a translation point. (It also follows from the criterion by noting that there are no proper ideal subGQs, and that all spans of noncollinear points have size 2. As case (2) of the criterion cannot occur, we are in case (1), by way of contradiction. But this case can only occur if Q.5; q/ has point spans of size at least 3, giving a contradiction. This proof also works for TGQs of order .s; t / with s < t, since the trivial span condition is also satisfied by [44], 8.6.1.) (2) This follows from the fact that for each W .q/-subGQ there is an involution fixing it pointwise.

7.5 Elation groups for flock GQs and their duals

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Let be a GQ, and let x; y be noncollinear points. Let L I x be arbitrary; if the group of whorls about x and y acts transitively on the points of L which are different from x and not collinear with y, then we say that is fx; yg-transitive. Exercise. Show that the definition is indeed independent of the choice of L. Recall that H.4; q 2 / is classical, i.e., it is embedded in PG.4; q 2 /, but H.4; q 2 /D is not classical. In the next theorem, we will use the fact that H.4; q 2 / is fx; yg-transitive for any two noncollinear points x and y [51]. Theorem 7.9 (H.4; q 2 / and H.4; q 2 /D ). (1) Let x be an arbitrary point of H.4; q 2 /. Then the set of elations about x is not a group. (2) Let x be an arbitrary point of H.4; q 2 /D . Then the set of elations about x is not a group. Proof. (1) Consider an arbitrary point y ¦ x. As H.4; q 2 / is fx; yg-transitive, the group of whorls about x and y has a size divisible by q 2 1. As jfx; yg?? j D q C 1, it follows that H.4; q 2 / admits whorls about x and y that fix at least another point of fx; yg?? (besides x and y). Observation 7.1 applies. (2) This follows from the fact that each Q.5; q/-subGQ of H.4; q 2 /D is fixed pointwise by q nontrivial automorphisms of H.4; q 2 /D . Remark 7.10. The GQ H.4; q 2 / behaves differently than the other classical GQs, relative to the theory of elation groups. For the other classical examples, one could say that the set of elations about a fixed point forms a group “up to duality” (i.e., the result holds for the classical GQ or for its point-line dual). This is not true for H.4; q 2 /. Finally, note that in contrast to, for example, §6.7, we do consider all elations (with a given center) in this section, and not just standard elations. It can perfectly happen (as in the case of flock GQs) that an EGQ admits a unique t-complete elation group (of standard elations), while the set of all elations is not a group.

7.5 Elation groups for flock GQs and their duals 7.5.1 Flock GQs. For flock GQs, we have the following interesting corollary which has a rather remarkable and easy proof. Theorem 7.11. (1) Let .F / be a flock generalized quadrangle of order .q 2 ; q/, q odd, with special (elation) point .1/. Then the set of elations about .1/ does not form a group. (2) Let .F / be a nonclassical flock generalized quadrangle of order .q 2 ; q/, q even. Then the set of elations about .1/ does form a group.

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Proof. (1) Consider the following whorl about .1/ (and recall Chapter 6, §6.5): W .˛; c; ˇ/ ! .˛; c; ˇ/: N 0; 0/g N ?? D f.0; N c; 0/ N j c 2 Fq g, and so ObserThen fixes each point of f.1/; .0; vation 7.1 applies. (2) If .F / has a subGQ of order q containing .1/ which is fixed pointwise by some nontrivial automorphism of .F /, then that subGQ is isomorphic to W .q/ (cf. Theorem 7.5), and .F / is well known to be isomorphic with H.3; q 2 / (see, for instance, [34]). So we assume that such subGQs do not exist. The group of all whorls N 0; 0/ N is given by all collineations of type about .1/ and .0; a W .˛; c; ˇ/ ! .a˛; a2 c; aˇ/; N 0; 0/g N ?? n f.1/; .0; N 0; 0/g, N then as such a 2 Fq . If a a would fix a point of f.1/; .0; N N a point has the form .0; c; 0/, c ¤ 0, it follows that a D 1 and a D 1. Now the criterion of Observation 7.1 applies. In particular, the theorem applies for .F / Š H.3; q 2 / when q is odd (as was first noted by the author and H. Van Maldeghem in [83] for any characteristic, but with a different (though related) proof). Remark 7.12. The reader will note the similarity between the involution in the proof of Theorem 7.11 and the involution in (2) of the proof of Theorem 7.7 – in fact, when F is a semifield flock if q is odd, these involutions are essentially the same. 7.5.2 Dual flock GQs. Suppose that .F / is a flock GQ of order .q 2 ; q/, and suppose that .F /D is an EGQ for some elation point x. By Theorem 7.4, the set of all elations of .F /D about x is a group.

7.6 Dual TGQs which are also EGQs Let be a thick GQ which has a center of transitivity x. Suppose that D also has a center of transitivity x 0 . If ı is a duality between and D , then either x 0 ı I x, or each point of is a center of transitivity, and then is classical or dual classical by Theorem 2.5. In this section, we will henceforth always assume that x 0 ı I x. 7.6.1 More on TGQs. We need some more definitions at this point.

7.6 Dual TGQs which are also EGQs

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Representation in projective space. Suppose that H D PG.2n C m 1; q/ is the finite projective .2nCm1/-space over Fq , and let H be embedded in a PG.2nCm; q/, say H 0 . Now define a set O D O.n; m; q/ of subspaces as follows: O is a set of q m C 1 .n 1/-dimensional subspaces of H , denoted by PG.n 1; q/.i/ , so that (i) every three generate a PG.3n 1; q/; (ii) for every i D 0; 1; : : : ; q m , there is a subspace PG.n C m 1; q/.i/ of H of dimension n C m 1, which contains PG.n 1; q/.i/ and which is disjoint from any PG.n 1; q/.j / if j ¤ i . By (i), we have n m. If O satisfies these conditions for n D m, then O is called a pseudo-oval or a generalized oval or an Œn 1-oval of PG.3n 1; q/. A Œ0-oval of PG.2; q/ is an oval of PG.2; q/. For n ¤ m, O.n; m; q/ is called a pseudo-ovoid or a generalized ovoid or an Œn 1-ovoid or an egg of PG.2n C m 1; q/. A Œ0-ovoid of PG.3; q/ is an ovoid of PG.3; q/. Remark 7.13. From [44], 8.7.2, we know that O.n; m; q/s only can exist for n D m or n.a C 1/ D ma for some odd positive integer a, and a D 1 when q is even. From any O D O.n; m; q/ there arises a GQ T .n; m; q/ D T .O/ which is a TGQ of order .q n ; q m / for some special point .1/. This goes as follows. • The points are of three types: (1) a symbol .1/; (2) the subspaces PG.nCm; q/ of H 0 which intersect H in a PG.nCm1; q/.i/ ; (3) the points of H 0 n H . • The lines are of two types: (a) the elements of O.n; m; q/. (b) the subspaces PG.n; q/ of PG.2n C m; q/ which intersect H in an element of the egg. • Incidence is defined as follows: the point .1/ is incident with all the lines of type (a) and with no other lines; a point of type (2) is incident with the unique line of type (a) contained in it and with all the lines of type (b) which it contains (as subspaces); finally, a point of type (3) is incident with the lines of type (b) that contain it. Conversely, any TGQ is isomorphic to a T .n; m; q/ associated to an O.n; m; q/ in PG.2n C m 1; q/. We refer to the monograph [59] for all details and further references. Tits quadrangles. As noted, for the values .n; m/ D .1; 1/ we obtain an oval O D O.1; 1; q/ in PG.2; q/. The corresponding TGQ is also denoted by T2 .O/. When

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.n; m/ D .1; 2/, we obtain an ovoid in PG.3; q/, and the corresponding TGQ is also denoted by T3 .O/. Theorem 7.14 ([59], 1.5.2 (i)). Let O be an oval in PG.2; q/. A T2 .O/ is isomorphic to Q.4; q/ if and only if O is a nonsingular conic. Theorem 7.15 ([59], 1.5.2 (ii)). Let O be an ovoid in PG.3; q/. A T3 .O/ is isomorphic to Q.5; q/ if and only if O is a nonsingular elliptic quadric. The following is a recent characterization of classical TGQs. Theorem 7.16 (Brown and Lavrauw, [9]). A TGQ x of order .q; q 2 / with q even is isomorphic to Q.5; q/ if and only if x is contained in some subGQ isomorphic to Q.4; q/. Translation dual. Let n; m 2 N , and consider an O D O.n; m; q/ in PG.2n C m 1; q/. If n ¤ m, or if q is odd when n D m, the spaces PG.i/ .n C m 1; q/ define an egg, respectively generalized oval, in the dual space of PG.2n C m 1; q/ [59], §3.9 and §3.10. It is usually denoted by O D O .n; m; q/, and the TGQ T .O / is the translation dual of T .O/. Remark 7.17. Note that if F is a Kantor–Knuth semifield flock of the quadratic cone in projective 3-space over Fq , .F /D is a TGQ (see, e.g., [38], [59]). Also, by [38] – see also [59] – this TGQ is isomorphic to its translation dual. Automorphisms. The following theorem was independently obtained in L. Bader, G. Lunardon and I. Pinneri [2] and J. A. Thas and the author [57]. For a discussion on the notion of “kernel”, which is used in the statement of the next theorem, see §3.4 of [59], and the exercise after the theorem. Theorem 7.18 ([2], [57]). Suppose that D T .O/ is a TGQ of order .q n ; q m / with translation point .1/, and let Fq be a subfield of the kernel Fq 0 of T .O/, where O is a generalized ovoid in PG.2n C m 1; q/ PG.2n C m; q/. Then every automorphism of which fixes .1/ is induced by an automorphism of PG.2n C m; q/ which fixes O, and conversely. Exercisec . Let . x ; H / be a finite TGQ, and let .F ; F / be an associated Kantor family. Define K as the set of all endomorphisms of H for which A A for all A 2 F . Show that, endowed with the natural addition and multiplication (composition) of endomorphisms, K is a finite field. This is, by definition, the kernel quoted above. Now show that each element of fH g [ F [ F can be interpreted as a vector space (or projective space) over K in a natural way, and reconstruct the notion of O.n; m; q/s. Can you adapt this theory to the infinite case?


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Good TGQs. Consider an egg O D O.n; 2n; q/ in PG.4n 1; q/. It is called good at its element if for any distinct elements 0 and 00 in O n fg, the .3n 1/-space h; 0 ; 00 i μ contains precisely q n C 1 elements of O. (It is easy to see that this is the maximal number of subspaces can contain.) TGQs arising from good eggs are also called “good”. To understand the geometric implications of this property, consider the next exercise. Exercise. We work in PG.4n 1; q/, and consider an egg O D O.n; 2n; q/ in this space. Let O be good at , and let be generated by distinct elements ; 0 00 , so that contains q n C 1 elements of O. Show that these elements define a generalized oval O 0 in . As a consequence, observe that for any point z 2 PG.4n; q/ n PG.4n 1; q/, hz; i is a PG.3n; q/ in which a TGQ T .O 0 / arises. It is a full subGQ of T .O/. Now show that the number of subGQs arising in this way (by letting vary) is q 3n C q 2n . Exercise. More generally, let be a thick GQ of order .s; s 2 /, and let L be any line. Show that the number of full subGQs containing L is at most s 3 C s 2 . Show that equality holds, if and only if for any distinct lines U; V incident with a given point z I L, U ¤ L ¤ V , and any point v ¦ z, there is a full subGQ containing L; U; V; v. We will need three more results in the discussion below. The first one concerns SPGQs. Theorem 7.19 ([67], [68]). Let T .O/ be a good TGQ in odd characteristic, where O is good at . Then the line L corresponding to is a line of translation points. So, if U and V are nonconcurrent lines in L? , T .O/ is an SPGQ with base-span fU; V g?? . Conversely, if a thick GQ of order .s; t / has a line L of translation points, then we have the following possibilities: • s D t is a prime power and Š Q.4; s/; • t D s 2 is a power of 2 and Š Q.5; s/; • t D s 2 is an odd prime power and x is a good TGQ for any point x I L, L being the “good line”. (If x is not classical, there is a unique good line.) Theorem 7.20 ([71]). Let O D O.n; 2n; q/ be an egg in PG.4n1; q/ with q odd, and suppose that either O or O is good. If T .O/ contains a thick full subGQ which is fixed pointwise by some nontrivial automorphism of T .O/, then T .O/ is isomorphic to the point-line dual of .F / with F a Kantor–Knuth flock. (It follows that T .O/ Š T .O /.) The next theorem connects good TGQs to flocks. Theorem 7.21 ([53], [55]). A TGQ of order .s; s 2 / with s odd is good if and only if it is the translation dual of the point-line dual of a flock GQ .F /. If is not classical, its good element corresponds to the special elation point of .F /.

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Further combinatorics. Let . x ; H / be a TGQ. An easy exercise shows that all lines incident with x are regular. (Points with this property are usually called coregular.) We will need the following two results in order to proceed. (We refer to the cited works for proofs.) Theorem 7.22 ([44], 1.5.2). A coregular point in a GQ of even order s is regular. Theorem 7.23 ([62]). Let . x ; H / be a thick EGQ of even order s. Then x is a TGQ if and only if some line incident with x is regular. 7.6.2 The case s D t. Let x D T .O/ be a thick TGQ of order s, and suppose that D is also an EGQ with elation point `, where ` corresponds to some line through x. Suppose that s is even. Then x is a regular point by Theorem 7.22. Whence the elation point ` in D is incident with some regular line, and by Theorem 7.23, ` is a translation point (in fact, one only has to assume that ` is a center of transitivity to have the same conclusion). Whence the set of all elations about ` is a group. Exercise. Show that the assumption that ` is a center of transitivity is indeed sufficient. There are no nonclassical TGQs of odd order s known. 7.6.3 The case s < t. Let x D T .O/ be a thick TGQ of order .s; t /, and suppose that D is also an EGQ with elation point `, where ` corresponds to some line through x. Suppose first that D admits some nontrivial collineation that fixes a thick subGQ 0 of order .t 0 ; s/ pointwise. If 0 does not contain the point x, it readily follows that each point of is a translation point, and so Š p Q.5; s/ since is Moufang. So we may assume that 0 contains x. Then t 0 D s D t by Theorem 1.3 and the fact that has regular lines. If s is even, 0 Š Q.4; s/ by Theorem 7.5, and then Theorem 7.16 implies that Š Q.5; s/. Suppose that s is odd. Then each known class of TGQs of order .s; s 2 / either is a good TGQ or the translation dual of a good TGQ. By Theorem 7.20, the existence of a doubly subtended subGQ as above then implies that Š .F /D , where F is a Kantor–Knuth semifield flock of the quadratic cone in PG.3; s/. Whence D is a flock GQ, and we can apply Section 7.5. Now suppose that D admits some whorl about ` which fixes some point o ¦ `, and a point o0 2 f`; og?? n f`; og (observe that ` is regular in D ). We again have a separate look at the known classes in the even and odd case. 7.6.3.1 Even case Each known TGQ of order .s; t / with s < t and s even arises as a T3 .O/ from an ovoid of PG.3; s/. In that case t D s 2 . If ` corresponds to the line X I x, then there is a group of whorls about X that acts sharply transitively on the lines incident with x and different from X . It follows from literature (see e.g. [4]) that O either is an elliptic quadric or a Suzuki–Tits ovoid, which can be defined as follows.


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Let q D 2e be an odd power of 2, and let 2 Aut.Fq / be such that 2 D 2. Now define a map f W Fq2 ! Fq ; .a; b/ 7! aC2 C ab C b : The Suzuki–Tits ovoid of PG.3; q/ is then given by O D f.0; 1; 0; 0/g [ f.1; f .a; b/; a; b/ j a; b 2 Fq g: It is well known that this ovoid admits the natural 2-transitive action of the Suzuki group Sz.q/. Embed PG.3; q/ as a hyperplane in PG.4; q/ by .x; y; z; w/ ! .0; x; y; z; w/. Construct the TGQ D T3 .O/, and call it the Suzuki–Tits quadrangle. Then Aut./.1/ (where .1/ is the special point of T3 .O/) admits the natural 2-transitive action of Sz.q/ on the q 2 C1 lines incident with x. Now define collineations .a; b; c; d; e/ W .u; x; y; z; w/ ! .u; x; y; z; w/Œa; b; c; d; e, with a; b; c; d; e 2 Fq , of PG.4; q/ as follows: 1 0 1 0 c d e B0 1 f .a; b/ a b C C B 1 0 0C Œa; b; c; d; e D B C: B0 0 @0 0 aC1 C b 1 a A 0 0 a 0 1 Then G D f.a; b; c; d; e/ j a; b; c; d; e 2 Fq g is a group of order q 5 with binary operation Œa; b; c; d; eŒa0 ; b 0 ; c 0 ; d 0 ; e 0 D Œa C a0 ; b C b 0 C aa0 ; c C c 0 C d.a0

C1

C b 0 / C ea0 ; d C d 0 ; e C e 0 C d a0 :

The group G leaves O invariant; it fixes the line L of corresponding to .0; 0; 1; 0; 0/ 2 O pointwise, and acts sharply transitively on the lines not concurrent with L. So D indeed is an EGQ with elation group G. Let x be a T3 .O/ of order .s; s 2 /, where O is a Suzuki–Tits ovoid of PG.3; s/, and suppose X I x is so that there is some line M ¦ X , and an element N 2 fX; M g?? n fX; M g, for which there is a nontrivial automorphism of x that fixes X pointwise, and also M and N . Without loss of generality, we may assume that the order of is a prime. Take an arbitrary point o x ¤ o, with o not contained in the .s C 1/ .s C 1/grid defined by M and N . Then o and o ¤ o are collinear with at least three distinct points of (note that o ¤ o as we may assume that no subGQ of order s is fixed pointwise by some nontrivial automorphism). As x is a T3 .O/, there are s C 1 distinct subGQs 0 ; 1 ; : : : ; s of order s which contain . If one interprets as an element of PL5 .s/ (or even PGL5 .s/!) – which may be done by Theorem 7.18 – then it easily follows that fixes pointwise. Thus acts semiregularly on the lines on x which are not contained in , and hence induces (on hOi D PG.3; s/) a nontrivial element in a two point stabilizer of PGL3 .s/O Š Sz.s/. Hence the order of divides s 1.

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So as s is even, there is at least one i 2 f0; 1; : : : ; sg for which i D i . Without loss of generality, we assume that o 2 i (and so o 2 i ). As i is a TGQ of even order s, the point x is regular in i by Theorem 7.22. This contradicts the fact that o and o are centers of some triad fx; u; vg of points on . For, as then v 2 fx; ug?? , we must have projX u D projX v, which is impossible since u; v 2 . 7.6.3.2 Odd case Each known TGQ of order .s; t / with s ¤ t and s odd either is a good TGQ or the translation dual of a good TGQ. In the latter case, D is a flock GQ, and we refer to an earlier section for the answer to the problem in that case. Let x be a good TGQ of order .s; s 2 /, s odd. Suppose X is the line incident with x that corresponds to ` in D . As we may take not to be classical, X is the (unique) good line incident with x (cf. Theorem 7.19). Now consider an arbitrary line M ¦ X . Then by Theorem 7.19 and §7.3 there is an involution in Aut./ that fixes each line of fX; M g?? pointwise. Whence Observation 7.1 applies to conclude that the set of elations of D about x does not form a group. 7.6.4 Conclusion. We have obtained the following theorem. Theorem 7.24. Let x be a nonclassical thick TGQ of order .s; t /, and suppose that ` 2 D is an elation point. (1) If s D t is even, the set of all elations about ` is a group. (2) If t D s 2 is even and is a T3 .O/ for some ovoid O of PG.3; s/, then O is a Suzuki–Tits ovoid and the set of elations about ` is a group. (3) If t D s 2 is odd and is a good TGQ, then the set of elations about x is not a group. Theorem 7.24 together with the section on flock GQs covers all the known examples of dual TGQs.

7.7 GQs of order .k 1; k C 1/ and their duals For GQs of order .k 1; k C 1/ and their duals we have the following theorem: Theorem 7.25. Let be a GQ of order .k 1; k C 1/ or .k C 1; k 1/, k > 3, and consider an arbitrary point x. Then x cannot be an elation point. Proof. First let be of order .k 1; k C 1/, k > 3, and suppose that x is an elation point. (Note that we may assume k > 3 by §1.4.5.) Then k 1 and k C 1 are powers of the same prime by Theorem 5.6, contradiction. Next, let be of order .k C 1; k 1/, k > 3. Let p be a prime dividing k C 1, and assume again that x is an elation point. Suppose that p e divides k C 1 but p eC1 does

7.7 GQs of order .k 1; k C 1/ and their duals

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not. Let p r be the largest power of p that divides k 1. Then k1 < pe pr by Lemma 5.2. If k C 1 is a power of p, Theorem 5.5 implies that k 1 also is, contradiction. If k is even, p e .k C 1/=3 and r D 0, and we have a contradiction. Let k be odd. We may assume that k C 1 has at least two distinct prime divisors, one of which is 2. Assuming that p is an odd prime divisor of k C 1, we obtain the same contradiction as before (from k 1 < p e < kC1 ). 2

8 Elation quadrangles with nonisomorphic elation groups

The following fundamental question was posed by S. E. Payne in [42]: Question. Let x be an EGQ. Can x be an elation point for nonisomorphic elation groups? In this chapter, we will consider a class of GQs which do admit nonisomorphic elation groups, thus answering Payne’s question affirmatively. The only known examples of this class are H.3; q 2 /-GQs with q even. The details are taken from [69] but were independently obtained by Rob Rostermundt in [48].

8.1 A nonisomorphism criterion First consider the following lemma. Lemma 8.1. Let 0 and 00 be distinct W .q/-subGQs in a GQ of order .q; q 2 /. Suppose y 2 0 \ 00 is such that the lines of 0 through y are those of 00 through y. Furthermore, suppose ¤ 1 is an involution that fixes 0 pointwise. Then 00 is stabilized by . Proof. Let z ¦ y be a point of 00 ; then fy; z; z g is a triad of , so that jfy; z; z g? j D q C 1 and jfy; z; z g?? j q C 1, cf. §1.1. As y is regular in W .q/, it follows that fy; z; z g?? 00 , whence z 2 00 . We deduce that 00 D 00 . 8.1.1 Standing hypothesis 1. For now, x D is an EGQ of order .q 2 ; q/, q even, with elation group H . Also, 0 is a thick subGQ of order .s 0 ; q/ which is fixed elementwise by a nontrivial collineation of , and which contains x. Note that by Theorem 7.5, we have that 0 Š W .q/, and that is an involution. As 0 Š W .q/, and as each point of W .q/ is regular, one observes that for each point z ¦ x, the pair fx; zg is regular, so that x is a regular point of . This implies that two distinct subGQs of order q containing x can only intersect in a very restricted manner: • either they share a unique line through x, some points incident with this line (including x), and the lines of through these points, or • they intersect in the points and lines of a dual grid of order .1; q/.

8.1 A nonisomorphism criterion

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Suppose that W is the group of all whorls with center x, and let S2 be a Sylow 2subgroup of W which contains H . Then S2 clearly has size 2q 5 (by, e.g., Lemma 6.15). Put H 0 D H , so that S2 D H [ H 0 . Let 0 and 00 be two distinct nontrivial involutions in S2 that respectively fix the subGQs 0 and 00 (of order q) pointwise. Suppose that they intersect in a dual grid as above. Then there is a point z ¦ x for which fx; zg?? 0 \ 00 . Since both 0 and 00 fix z, we immediately have a contradiction since 0 ¤ 00 and jŒS2 z j D 2. So all subGQs of order q that are fixed pointwise by a nontrivial involution in S2 mutually do not share points not collinear with x. This implies that if 1 and 2 ¤ 1 are two such subGQs, there is some line M I x so that M ? \ 1 D M ? \ 2 : Also, it follows easily that the number of such subGQs is q 2 , and that the associated involutions are mutually conjugate in S2 . Note also that all whorls of S2 which are not elations about x are contained in H 0 . The group S2 is noncyclic; if it were cyclic, then H would be abelian, implying in its turn that there are more lines through a point than points incident with a line. This contradicts the fact that .s; t / D .q 2 ; q/. As S2 is noncyclic, the following result of P. Deligne [14] implies that S2 has at least three subgroups of size q 5 (one of which is H ). Theorem 8.2 ([14]). Let G be a group of order p s h, p a prime and .p; h/ D 1. Denote by dk the number of subgroups of G of order p sk , 0 k s. Then (i) dk 1 mod p, (ii) if a Sylow p-subgroup S is elementary abelian or cyclic, then dk is congruent mod p kC1 to the number of subgroups of S of order p sk , and (iii) if S is not cyclic then d1 1 C p mod p 2 . Suppose that H 00 ¤ H is a subgroup of S2 of order q 5 . If H 00 does not contain any of the q 2 involutions of above, then H 00 is an elation group (“first case”). If H 00 contains at least one such involution (“second case”), it contains all of them since they are mutually conjugate, and since H 00 is a normal subgroup of S2 (as a group of index 2). In that case, put H1 D H 00 \ H , and H2 D H 0 \ H 00 . So jH1 j D jH2 j D q 5 =2. Then it is straightforward to see that H1 [ ŒH n H1 μ H is an elation group of size q 5 . As the first and second case are equivalent, we keep using the notation of the second case. We put H4 D H n H1 and H3 D H4 . Suppose that L I x, and let y I L I x ¤ y. By H.x; L; y/, we denote the subgroup of H of root-elations with i-root .x; L; y/. 8.1.2 Standing hypothesis 2. For all L I x and y I L I x ¤ y, we have that jH.x; L; y/j D q 2 . Also, H 2 Z.H /, where H 2 D fh2 j h 2 H g and Z.H / is the center of H .

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Since jH.x; L; y/j D q 2 for all L and y as above, and since these groups generate H , it is straightforward to show that Z.H / is the group of symmetries about x. Exercise# . Show that Z.H / is the group of symmetries with center x. (Hint: the hard bit is showing that Z.H / can only contain symmetries about x. Note (and check) first that since the order of the GQ is .q 2 ; q/, all spans of nonconcurrent lines have size 2. Deduce that NH .H.x; L; y// Hy for all y x ¤ y. Now note that T Z.H / yx¤y NH .H.x; L; y//.) In fact, one observes now easily that Z.H / D Z.H /. Let H.x; L; y/ be a root group; then H.x; L; y/2 Z.H /, so that H.x; L; y/2 D f1g. So all such root groups are elementary abelian. Now consider 2 H , where 2 H4 is a nontrivial rootelation in H.z; M; x/ with z 2 which does not fix (it is an easy exercise that such a exists for suitable z). Then . /2 D Œ; 1 D Œ; clearly cannot be the identity, while it fixes z linewise. So . /2 … Z.H /, so that H © H . We have obtained the following theorem. Theorem 8.3. Let D . x ; H / be an EGQ of order .q 2 ; q/, where q is even, which contains a thick subGQ 0 of order .s; q/, fixed pointwise by a nontrivial automorphism of , and containing x. Furthermore, assume the following properties: (i) H 2 Z.H /; (ii) for all L I p and x I L I p ¤ x, we have that jH.x; L; p/j D q 2 . Then there is an automorphism group H 0 of such that H 0 © H and . x ; H 0 / is an EGQ. Proof. By Theorem 7.5, we have that 0 Š W .q/ (so that in particular s D q), and that is an involution. The rest follows from the part of this section occurring before this theorem. Corollary 8.4. Let D . x ; H / be an EGQ of order .q 2 ; q/, where q is even, which contains a thick subGQ 0 of order .s; q/, fixed pointwise by a nontrivial automorphism of , and containing x. Let z ¦ x and suppose z zi x for i D 0; 1; : : : ; q. If all groups H.x; xzi ; zi / H are elementary abelian and have size q 2 , then there is an automorphism group H 0 of such that H 0 © H and . x ; H 0 / is an EGQ. Proof. The root-groups are elementary abelian if and only if H 2 Z.H / (easy exercise). Exercise. Prove the “easy exercise”. In the next section we will show that H.3; q 2 / with q even satisfies the assumptions of Theorem 8.3.

8.2 An example: H.3; q 2 /, q even

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8.2 An example: H.3; q 2 /, q even Consider Š H.3; q 2 /, q even, and suppose that x is a point of . We will show that all the assumptions of Theorem 8.3 hold. Suppose that L I x, and let y I L I x ¤ y; then the group of all root-elations H.x; L; y/ has size q 2 , and is isomorphic to the additive group of Fq 2 . By putting H equal to the group generated by all such root-elations (so that . x ; H / is an EGQ), the assumptions of Theorem 8.3 are indeed satisfied (H 2 D Z.H / for this H ). Remark 8.5. The previous result was independently obtained by R. Rostermundt [48] in an entirely different fashion. He represents H.3; q 2 / (q even) as a group coset geometry in the special group H2 D f.˛; c; ˇ/ j ˛; ˇ 2 Fq 2 ; c 2 Fq g, in other words, as a flock quadrangle. He then constructs q 2 1 distinct elation groups Ki D 1; 2; : : : ; q 2 1 of size q 5 , and shows that all Ki are mutually isomorphic. The Ki ’s have nilpotency class 3, while H2 has nilpotency class 2, so that K © Ki for all i. The proofs are long and technical. For details and several other results, see R. Rostermundt [48]. It is easy to see that the q 2 elation groups obtained form the complete set of such groups.

8.3 Group and GQ automorphisms Suppose that . x ; G/ is a thick finite EGQ. Then there is associated a 4-gonal family .J; J / to x (and some point z opposite x), and, conversely, each 4-gonal family yields an EGQ. It is clear that any automorphism of G that fixes J as a set – and then also J – induces in a natural way an automorphism of x fixing x (and z). Exercise. Let . x ; G/ be a thick finite EGQ, and let .J; J / be as above. Show that any automorphism of G which fixes J as a set, also fixes J . It is therefore a basic question whether the following converse holds: Question. Is any automorphism of x fixing x and z induced by a group automorphism of G (fixing J)? In this section, we answer this question by constructing a class of counter examples. (For translation generalized quadrangles, an answer to a stronger version of this question is known, cf. Theorem 7.18. Whence there is a satisfactory treatment for TGQs.) We note that if is an element of .Aut./x /z , then is induced by an automorphism of G if and only if fixes G under conjugation in Aut./, that is, if and only if G D G. In that case, we must have that J D J. Now suppose that D . x ; H / satisfies the hypotheses of §8.1, and suppose that S2 is the Sylow 2-subgroup of W which is generated by H and H . Suppose that ˛ ¤ 1 is an involution in W that fixes a subGQ of order q pointwise containing z,

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and that is not contained in S2 (the latter is an extra hypothesis!). Then S2˛ ¤ S2 , and as H W D H , it follows that .H /˛ ¤ H . So there are indeed elements in Wz .Aut./x /z which are not induced by automorphisms of H . Again, H.3; q 2 / with q even yields an example. Exercise. Show that H W D H (that is, H E W ).

8.4 Appendix: GQs not having property ./ Let D . x ; H / D .G; J/ be a thick EGQ of order .s; t /. We introduce property ./ as follows. Property ./. For each Hi 2 J we have Hi E H . (In [69], we called this property “property (F)”.) Before the papers [48], [69], each known EGQ . x ; H / satisfied this property. Theorem 8.6 ([69]). The examples . x ; H / considered in §8.1, and in particular .H.3; q 2 /x ; H / with q even, do not have property ./. Proof. First note that .H.3; q 2 /x ; H / has ./. Property ./ is satisfied if and only if for each z x ¤ z, Hz fixes zx pointwise. Clearly an element of the form 2 H , with 2 Hy n Hy , y x ¤ y, does not have this property. In [69], motivated by the construction of the class of examples . x ; H /, the following conjecture was made. Conjecture. If property ./ does not hold for . x ; G/, then (which has order .s; t /) has nonisomorphic (t-complete) elation groups, and has a subGQ of order .s=t; t / fixed pointwise by some nontrivial collineation ( possibly under some mild extra assumption). Remark 8.7. As [81] shows, property ./ truly is a very important instance for understanding the structure of EGQs. It is also closely related to Kantor’s conjecture. 8.4.1 A counter example. While revising this manuscript, it occurred to me that the Suzuki–Tits quadrangles yield counter examples to this conjecture. Let q D 2e be an odd power of 2, and let 2 Aut.Fq / be such that 2 D 2. Define, as before, the map f W Fq2 ! Fq ;

.a; b/ 7! aC2 C ab C b :

The Suzuki–Tits ovoid of PG.3; q/ is given by O D f.0; 1; 0; 0/g [ f.1; f .a; b/; a; b/ j a; b 2 Fq g:

8.4 Appendix: GQs not having property ./

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We suppose that e > 1, so that O is not an elliptic quadric. Let T3 .O/ be the corresponding TGQ of order .q 2 ; q/, and let L I .1/ be arbitrary. Define the group G as before (see §§7.6.3.1), so that D is an EGQ with elation group G and elation point L. Now define for t 2 Fq : A.t / D fŒa; b; tf .a; b/; t a; t b j a; b 2 Fq g; A .t / D fŒa; b; c; t a; t b j a; b; c 2 Fq g; and A.1/ D fŒ0; 0; 0; d; e j d; e 2 Fq g; A .1/ D fŒ0; 0; c; d; e j c; d; e 2 Fq g: Then F D fA.t / j t 2 Fq [ f.1/gg and F D fA .t / j t 2 Fq [ f.1/gg define the Kantor family of G corresponding to the point h.1; 0; 0; 0; 0/; .0; 1; 0; 0; 0/i D x of D . The following observation is straightforward. Observation 8.8. (i) A .1/ and A.1/ are elementary abelian and A .1/ E G, so that A .1/ fixes ŒA.1/ pointwise. (ii) For t 2 Fq , A .t / and A.t / are nonabelian of exponent 4; moreover, for t ¤ t 0 , A .t/ Š A .t 0 / and A.t / Š A.t 0 /. Also, no A .t / is normal in G. (iii) G is the entire set of elations about x. Proof. (i) and (ii) follow by a simple calculation. (iii) was already proved before.

By (ii), property ./ is not satisfied for .F ; F /. Let have a subGQ 0 of order q fixed pointwise by an involutory automorphism of . By Theorem 7.5, 0 Š Q.4; q/. By Theorems 1.3 and 1.5, fixes no points outside 0 , and if z is such a point, z is not collinear with z since otherwise there would be triangles. So if 0 would not contain .1/, then .1/ and .1/ would be noncollinear translation points. Hence each point of would be a translation point, leading to the fact that is classical (cf. Corollary 2.6). So 0 contains .1/. It follows that 0 is a T2 .O 0 / for some oval O 0 lying on O (by, for instance, the exercises in §7.6). By Theorem 7.14, O 0 is a conic, and whence by the following result of M. R. Brown [8], O is an elliptic quadric, contradiction. Theorem 8.9 ([8]). An ovoid in PG.3; q/ is an elliptic quadric if and only if it contains a conic. Whence cannot contain subGQs of order q fixed pointwise by some nontrivial automorphism, while .F ; F / does not satisfy ./.

9 Application: Existence of translation nets

In this chapter, we exhibit a class of .22k ; 2k C 1/-translation nets with nonabelian translation group, for any natural k, following [76]. It is the first infinite class of translation nets known to admit nonisomorphic translation groups for each of its elements. The result will follow from observations of the previous chapter.

9.1 Translation nets A translation net N is a net for which there is a group G of automorphisms of N each element of which fixes each parallel class of N , and so that G acts sharply transitively on the points of N . (If N has order k and degree r, k; r 2, and if k D r 1, then N is a translation plane.) Quite in contrast to the theory of translation planes, translation groups of translation nets need not be abelian. In [20], the first infinite series of large translation nets is constructed with nonabelian translation group. (We refer the reader to [20] for a discussion on the term “large”.) Also, in [20] “nonabelian translation nets” of order q 2 and degree q C 1 are constructed for q any odd prime power, for q D 2n with n not a power of 2, and for q 8 an odd power of 2. No other .q 2 ; q C 1/-nets with q a power of 2 are known. In this chapter, which is based on [76], we aim at constructing translation nets N of order q 2 and degree q C 1 with nonabelian translation groups G for q any power of 2, thus completing the series of examples with parameters .q 2 ; q C 1/, q a prime power. These nets also provide the first infinite class of translation nets which admit nonisomorphic translation groups. Only a few of such nets seem to be known – see [49]. We will address the results of the previous chapter for our purpose.

9.2 Construction Suppose that . x ; G/ is an STGQ of order .s; t /. Then clearly Nx is a translation net with parameters .s; t C1/, the translation group being naturally induced by G=S, where S is the group of symmetries about x. Now let H be a nonsingular Hermitian variety in the projective space PG.3; q 2 /; the points and lines of H form the generalized quadrangle H.3; q 2 /, which has order .q 2 ; q/, all of whose points are regular elation points. Fix some point x. If q is odd, we have seen at the end of Chapter 6 that there is a unique t-maximal elation group H

9.2 Construction

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with respect to this point. Moreover, G=S (using the notation of above) is elementary abelian. When q is even the situation is entirely different: there are precisely two classes of nonisomorphic t-maximal elation groups with respect to x, say C1 and C2 , with the following properties. • jC1 j D 1 and jC2 j D q 2 1. • For each R 2 C1 [ C2 , S D Z.R/. • Elements of C1 are of nilpotency class 2, while elements of C2 have class 3. (All relevant information can be found in the previous chapter.) So we can conclude that if R 2 C1 , R=S is an (even elementary) abelian group for Nx , and if R 2 C2 , R=S is of class 2. This completes the proofs of the results announced in the introduction.

10 Elations of dual translation quadrangles

In 2001, W. M. Kantor [29] asked, at the conference “Finite Simple Groups (Manhattan, Kansas)”, whether dual good translation generalized quadrangles in odd characteristic always have an elation point, motivated by a result on which the author lectured at that conference. In 1989, S. E. Payne [39] showed that the Roman GQs, which are dual good TGQs of order .32r ; 3r / for some natural r, have this property. Moreover, very recently Havas et al. [23] showed that the elation group of a nonclassical Roman GQ is nonisomorphic to the standard elation group of the corresponding flock GQ, thus answering a question of S. E. Payne at the conference “Finite Geometries, Groups and Computation (Pingree Park, 2004)” in honor of W. M. Kantor. In this chapter, we first describe an affirmative answer to William Kantor’s question, based on [72]. We then sketch some very recent results (written at the time the present monograph was finished), without proofs, which provide a satisfying answer to a generalized version of the question of Payne.

10.1 Main result We may suppose that the TGQs under consideration are nonclassical, since otherwise the answer is known; for Š Q.5; q/, q odd, D is an elation generalized quadrangle for each point. Also, in this case each point of D is an elation point as well with a unique elation group. Note that it is required that q be odd; when q is even, we already described that there are nonisomorphic t-maximal elation groups. Theorem 10.1 ([72]). Let D be the point-line dual of a nonclassical good TGQ of order .q; q 2 / with q odd. Then there is a unique elation point in D , and it admits a unique elation group. Proof. Let D x be a TGQ of order .q; q 2 /, q odd, with translation point x, for which the translation dual . x / is the point-line dual of a flock GQ .F / of order .q 2 ; q/. Equivalently, is the translation dual of a TGQ which is the point-line dual of a flock GQ. Also, by Theorem 7.21, the class of TGQs with this property is precisely the class of good TGQs. Let L I x be the line of at which the TGQ is good. By Theorem 7.19, the automorphism group of fixes L and acts 2-transitively on the points of L. So each point z incident with L is a translation point, and we denote the corresponding translation group by Gz . Now note that, if . y ; H / is a general TGQ (with translation point y and translation group H ), then the following property is satisfied:

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10.1 Main result

• For each point y 0 y ¤ y 0 , Hy 0 fixes yy 0 pointwise. This property follows easily from the fact that H is abelian. (Note that this is precisely property ./ of Chapter 8.) All groups Gz have this property. Clearly, if z 0 ILIz 00 ¤ z 0 , with z … fz 0 ; z 00 g, then .Gz /z 0 D .Gz /z 00 . Hence we may denote the latter group by GzL for each z. Now define the group K by K D hGzL j z I Li: Note that the group of symmetries about L, which we denote by SL , is a subgroup of K, and that K fixes L pointwise. Let B denote the line set of . Now let U and V be different lines in B n L? . By [6] there exist a natural number n and lines U0 ; U1 ; : : : ; Un , all contained in B n L? , so that U0 D U , Un D V , and Ui UiC1 for i D 0; 1; : : : ; n 1. As in the proof of Theorem 6.23, it follows that K acts transitively on B n L? . The regular line L of defines a net NL as in Theorem 1.1. The points of NL are the sets fY; Zg?? , with Y ¦ Z and Y; Z 2 L? . We want to consider the action of K on the point set P 0 of NL . Clearly, the kernel of that action is SL . By Theorem 1.12 and Theorem 7.5 we have two possibilities: (a) The permutation group .K=SL ; P 0 / is a Frobenius group. (b) There is an involution in the automorphism group of that fixes a subGQ of order q pointwise which contains L. Note that Theorem 1.12 in fact allows a third possibility at first sight, but that it implies (b). Namely, the possibility that there is an automorphism of inducing a symmetry (with axis L) in some subGQ 0 of order q containing L, but which is not a symmetry itself for . Let M 2 L? \ 0 , and suppose u D v with u I L, u I M . Let s be the symmetry in SL mapping v to u; then s fixes 0 elementwise, and s is as in (b). If we are in case (b), Theorem 7.20 leads to the fact that is a Kantor–Knuth TGQ and Š , so that D is an EGQ, since the flock GQ . /D is as such. Then there is a unique elation group for the elation point by Theorem 6.23. Suppose now we work in case (a). Let F be the Frobenius kernel of .K=SL ; P 0 /. So F consists of the identity together with the elements of K=SL fixing no point of P 0 . Let K # be the subgroup of K for which K # =SL D F . Then it is clear that K # is a group of elations about L containing all elations about L (as soon as an elation fixes a point of NL , it is a symmetry about L, by Theorem 1.12). Moreover, since the number of points of NL is q 4 , the size of K # is q 5 (as jSL j D q). Whence, D is an EGQ with elation point L, and K # clearly is the only group of elations about L of size q 5 . Finally, suppose that D has distinct elation points. Then has distinct lines each point of which is a translation point, so each point is a translation point. Corollary 2.6 finishes the proof of Theorem 10.1.

In particular, Theorem 10.1 applies when D is a Roman GQ (as introduced by S. E. Payne in [39], and also described in §3.4.2 of [68]).

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10.2 Payne’s question in a more general setting Payne’s question can now be seen as a special case of the following (still using the notation of §10.1). Question. When is the group K # isomorphic to the standard elation group H2 .q/ of the associated flock quadrangle? Havas et al. [23], [22] handled the question when D is the Roman GQ (Payne’s question), and for the Kantor–Knuth GQs the groups are isomorphic (as indicated above). In the next section, we describe a recent answer to the general question.

10.3 Recent results We have seen that the Roman construction of Payne appears to be a special case of a more general one: each flock quadrangle D .F / for which the dual D is a translation generalized quadrangle gives rise to another generalized quadrangle (which is the dual of the translation dual . D / ) which is in general not isomorphic to D , and which, by §10.1, also arises from a Kantor family. Denote the class of such flock quadrangles by C . (By reasons to be explained later, we only consider odd characteristic.) 2C H2

D

/ D

/ . D /

D

/ .. D / /D

‹

/G

10.1

In a recent paper [78], the author resolved the question of Payne for the complete class C , by showing that the flock quadrangles are characterized by their groups (in any characteristic), a question which was open for quite some time. As an application of the main result, the special case of prime q yields an alternative proof of the main result of [3]. One of the main results of [78] is: Theorem 10.2 ([78]). Let be an EGQ of order .q 2 ; q/, q any prime power, with elation group H2 .q/. Then is a flock quadrangle. In terms of Kantor families, this result reads as follows: Corollary 10.3. A Kantor family of type .q 2 ; q/ in H2 .q/ arises from a q-clan. Passing from the latter theorem to the solution of Payne’s question goes as follows. Theorem 10.4. Let D .F / be a flock GQ of order .q 2 ; q/ for which the dual D is a TGQ, q odd. Let G be defined as in the diagram above. Then G Š H2 .q/ if and only if F is a Kantor–Knuth flock if and only if D Š . D / .

10.3 Recent results

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Proof. If .F / is a Kantor–Knuth GQ, D Š . D / , and the result follows from the fact that flock GQs in odd characteristic admit unique t-maximal elation groups (cf. the end of Chapter 6). If .F / is not a Kantor–Knuth GQ, it is known that the dual of . D / is not a flock quadrangle [59], Chapter 4. Theorem 10.2 now implies that its elation group cannot be H2 .q/. If .F / is the so-called Ganley flock quadrangle, the dual of . D / is the Roman quadrangle [39] (which is not isomorphic to a Kantor–Knuth quadrangle), yielding thus the result of Havas et al. [22], [23]. If D .F / is a flock GQ of order .q 2 ; q/ for which the dual D is a TGQ, and q is even, Š H.3; q 2 / (cf. N. L. Johnson [25], [11] or [59]), Theorem 5.1.11. In that case, the analogous question is reduced to the main results of [48], [69], which can be found in Chapter 8. Other implications can be found in [78]. In order to describe the true main result of [78], we need to make an observation. Let H2 .q/ be a 5-dimensional Heisenberg group, and let Z be its center. Let V be the elementary abelian p-group H2 .q/=Z. The map W V V ! Fq ;

.aZ; bZ/ 7! Œa; b;

naturally defines a nonsingular bilinear alternating form over Fq Z. So V can be seen as a 4-dimensional space over Fq , and in the corresponding projective 3-space over Fq , defines a symplectic polar space W .q/ of rank 2 (projective index 1). Theorem 10.5 below is a converse of this observation. We will start from a Kantor family .F ; F / in some special p-group H , and define a bi-additive alternating map W V V ! Fq ;

.aZ.H /; bZ.H // 7! Œa; b;

where we see H=Z.H / as a vector space V over Fp . We assume that “defines” a nonsingular bilinear alternating form over Fq Z.H /; by this we simply mean that the commuting structure of is a (nonsingular) symplectic polar space over Fq . Then is an Fq -bilinear form whatever the Fq -structure is put on Z.H / (it should be remarked that this can be done in many ways, although the choice is irrelevant here). The assumption implies that the dimension m of V over Fq is even, and defines a symplectic polar space W .2n1; q/ of rank n1 in the projective space PG.2n1; q/ associated to V , with m D 2n. We are ready to state the main theorem of [78]. Note that it also characterizes 5-dimensional Heisenberg groups, and that it is independent of the characteristic. Theorem 10.5 ([78]). Suppose that H is a special p-group of order q 5 for which Z.H / D ˆ.H / D ŒH; H is elementary abelian of order q. Suppose that H admits a Kantor family of type .q 2 ; q/ and that defines a nonsingular bilinear alternating form over Fq . Then H Š H2 .q/, and the corresponding generalized quadrangle of order .q 2 ; q/ is a flock quadrangle. In the aforementioned paper [3], the following theorem, which complements the result of Bloemen et al. [5] (classifying EGQs of order .p; t /), was obtained.

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Theorem 10.6 (Bamberg et al. [3]). An EGQ . x ; H / of order .s; p/ with p a prime, is either isomorphic to W .p/, or to a flock quadrangle, in which case s D p 2 . In [78], Theorem 10.5 is applied to obtain an alternative and very short proof of Theorem 10.6. (The proof of that result in [3] also contains a flaw – see [78] for details – and this is another justification for the alternative proof.)

11 Local Moufang conditions

About fifteen years ago, Norbert Knarr studied generalized quadrangles which satisfy Moufang conditions locally at one point. He then posed the fundamental question as to whether the group generated by the root-elations and dual root-elations with (dual) i-root containing that point is always a sharply transitive group on the points opposite this point, that is, whether this group is an elation group and the point an elation point [31]. One of his motivations to do so was to find a good definition for an elation generalized quadrangle, as an alternative to the existing one, and as a natural generalization of the concept of a translation plane [24] (and so also as an alternative to the theory of translation generalized quadrangles developed in Chapter 8 of [44] in order to classify finite Moufang quadrangles). The reader will recall the discussion at the end of Chapter 2. The solution of this question, which appeared in [73], is the topic of this chapter. When I was writing up the final version of the present work, I noticed that a part of the proof that appeared in [73] could be slightly simplified – in fact, a part can be deleted at small cost. It is this simplified version (but with more details than in the presentation of [73]) that we sketch here.

11.1 Formulation Let be a Moufang quadrangle – all its roots and dual roots are Moufang. In that case, the group H generated by all root-elations and dual root-elations corresponding to i-roots and dual i-roots containing a fixed point x, is a group that fixes x linewise and acts sharply transitively on the points not collinear with x. Whence is an elation generalized quadrangle with elation point x and elation group H . Also, if the number of points of the quadrangle is finite, H can be easily proved to be nilpotent, and even a p-group, without using any big classification result: Proposition 11.1. Let be a finite Moufang quadrangle and define H as above. Then H is a p-group. Proof. Let .s; t/ be the order of . If s t , H is a p-group by Theorem 5.6. So let t < s. As D is also an EGQ for any of its points, again Theorem 5.6 implies that st is a prime power. We can now formulate the question of Knarr as follows:

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Knarr’s question. Let be a thick generalized quadrangle, and let .1/ be a point of . Suppose that any i-root and dual i-root containing .1/ is Moufang. Is . .1/ ; W / an elation generalized quadrangle with elation point .1/, where W is the group generated by the root-elations and dual root-elations associated to the i-roots and dual i-roots on .1/? In this chapter, we solve the question affirmatively for finite generalized quadrangles. We will first show Theorem 11.2 ([73]). Let be a thick finite generalized quadrangle, and let .1/ be a point of . Suppose that any i-root containing .1/ is a Moufang i-root. Then . .1/ ; W / is an elation generalized quadrangle with elation point .1/, where W is the group generated by the root-elations associated to the i-roots on .1/. From Theorem 11.2 and its proof, we will deduce the solution to Knarr’s question. Moreover, we solve another long-standing open question by showing that the group W necessarily is nilpotent, both in the situation of Theorem 11.2 and in that of Knarr’s question. As already mentioned, this was only known up to now when both Moufang conditions are satisfied for all points and lines, that is, when the quadrangle is a Moufang quadrangle. In fact, we will even prove that these groups always have to be p-groups for some prime p. However, the nilpotency will be obtained prior to the fact that W is a p-group.

11.2 Proof of the first main theorem Setting. D .P ; B; I/ is a thick GQ of order .s; t / satisfying the hypotheses of Theorem 11.2. We suppose that s > 2, since s D 2 implies that the GQ is isomorphic to Q.d; 2/, d 2 f4; 5g, and then the theorem is well known to hold. Suppose that r .1/ ¤ r; then the group of s root-elations with i-root .r; r.1/; .1// is denoted by Rr throughout. The group generated by the Rr is W . We proceed in a number of steps to obtain Theorem 11.2. First of all we note it is easy to prove that W is a center of transitivity, cf. the proof of Theorem 6.23 using Brouwer’s result on connectedness (a stronger statement will be obtained later on). The following result will be used in the first step of the proof. It is taken (in a slightly different form but with essentially the same proof) from K. Thas and H. Van Maldeghem [84]. Proposition 11.3 ([84], Proposition 7.3). Let be a thick finite GQ with center of transitivity v, and order .s; t /. Denote by H.v/ the group of whorls with center v. If v is a regular point, and for each x v ¤ x there is a group H.v; x/ E H.v/x of elations with centers v and x of order s, then either there is a subGQ of order t containing v and s D t 2 , or v is a center of symmetry.

11.2 Proof of the first main theorem

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Proof. First we assume that there is a subquadrangle 0 of order .s 0 ; t /, with 1 < s 0 < s. Then by Theorem 1.3, s D t 2 and s 0 D t (as t s 0 , since 0 has regular points). Now suppose that does not admit any subquadrangle of order .s 0 ; t /, with 1 < s 0 < s. The group H.v/ of whorls about v acts transitively on the set of points of not collinear with v, as v is a center of transitivity. If an element of H.v/ fixes at least two distinct nontrivial traces in v ? , then by Theorem 1.12, either it fixes a subquadrangle of order t – impossible by our assumption – or it fixes v ? pointwise. Hence the action of H.v/ on the set of nontrivial traces in v ? has a Frobenius kernel and so we obtain a group F , which is a normal subgroup of H.v/, containing the normal subgroup N E H.v/ which fixes v ? pointwise and, modulo N , acting sharply transitively on the set of nontrivial traces in v ? (clearly, H.v/ acts transitively on those traces). So F is defined such that F=N precisely is the Frobenius kernel of H.v/=N in its action on the nontrivial traces in v ? . It follows that F acts freely on the set of points opposite v. But, by the definition of Frobenius kernel, H.v; x/N=N F=N , so that H.v; x/ F , for all points x collinear with v. Hence F D H.v/ acts sharply transitively on the set of points opposite v. Since the elements of F fixing at least one nontrivial trace in v ? are symmetries about v, we deduce that F contains t symmetries about v (in fact, jN j D t ). So v is a center of symmetry. We now distinguish several cases. 11.2.1 If .1/ is a regular point, the theorem is proved. Suppose that .1/ is regular. By Proposition 11.3 (where H..1/; x/ is Rx if x .1/ ¤ x), either .1/ is a center of symmetry, or there is a subGQ 0 of order t which contains .1/, and then s D t 2 . Suppose we are in the latter case. Suppose that L is a line of 0 not containing .1/, and let L I z .1/. Then the t elements of Rz which map a point of L \ 0 onto a point of L \ 0 stabilize 0 (since the intersection of 0 with the image of 0 under such an element contains all its lines on .1/ and all its points of .1/z). Whence 0 satisfies the hypotheses of Theorem 11.2, and so .1/ is also a center of transitivity for 0 . By Theorem 6.24, 0 is an EGQ and by Theorem 5.6, it now follows that s is the power of a prime, say p. Combining Theorem 7.4 and Theorem 7.5, we then have two possibilities. (a) W has a subgroup which acts sharply transitively on P n.1/? , and this subgroup contains all elation subgroups of the full group of automorphisms that fix .1/ linewise, so also all root and dual root groups with (dual) i-root containing .1/. In particular, in this case Theorem 11.2 and an affirmative answer to Knarr’s question are verified. Note that since s D t 2 , W is a p-group. (b) p D 2, and we may suppose 0 to be isomorphic to W .t /. Moreover, there is an involutory automorphism fixing 0 pointwise, and .1/ is a center of symmetry. Exercise. Show that .1/ is a center of symmetry in (b). Now suppose that .1/ is a center of symmetry, and denote the group of symmetries with center .1/ by S. Take two arbitrary points x; y 2 .1/? for which x ¦ y. Then

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clearly ŒRx ; S D ŒRy ; S D f1g: As W is generated by the root groups for which the i-root contains .1/, and as S commutes with these root groups, S is contained in the center Z.W / of W . Also, ŒRx ; Ry fixes .1/x and .1/y pointwise, so that by Theorem 1.12, ŒRx ; Ry S. It now easily follows that Rx SRy D hRx ; S; Ry i is a group of order s 2 t . Suppose 2 hRx ; S; Ry i fixes some point l of P n.1/? . Write D gx gc gy , with gx 2 Rx ; gc 2 S; gy 2 Ry . As fixes each point of f.1/; lg? , it follows that gx D 1 D gy . But a symmetry gc can only fix l if it is the identity. So, hRx ; S; Ry i is an elation group which acts sharply transitively on P n .1/? . Take a 2 Rx and b 2 Ry . Then Œa; b 2 S. So hRx ; S; Ry i has the property that as soon as an element fixes a point different from .1/ on a line through .1/, the line is pointwise fixed by the element (hRx ; S; Ry i induces a sharply transitive abelian group on such a line minus .1/). It now follows easily that all (dual) root-elations with (dual) i-root containing .1/ are contained in hRx ; S; Ry i, so that again Theorem 11.2 and Knarr’s question are affirmatively verified for this case. Moreover, Corollary 5.12 now tells us that W D hRx ; S; Ry i is a p-group for some prime p. 11.2.2 Fixed points structure of elements of W . Let x be not collinear with .1/, and put H D Wx . Let M I x be a fixed line, and suppose r; r 0 ¤ r are on M so that there is some element ˛ in H mapping r onto r 0 . Let z be the point of .1/? incident 1 0 1 0 with M . Let ; 0 be root-elations of Rz . Then r ˛ D r ˛ if and only if 0 ˛ .˛ /1 D fixes r. It is clear that fixes r and z linewise, so that fixes a subGQ of order .s 0 ; t/. But also fixes z.1/ pointwise, leading to D 1. So ˛ D ˛

0

(or 0 1 ˛ 0 1 D ˛). As ˛ fixes x, this is only possible if ˛ fixes some subGQ of order .s 00 ; t/ pointwise, where s 00 > 1 (cf. Theorem 1.6). In the rest of the proof, we will distinguish two cases for x: (1) There is no ˛ 2 Wx fixing a thick subGQ of order .s 0 ; t / pointwise. (2) There is an element of Wx that fixes some thick subGQ of order .s0 ; t / pointwise. If any two thick ideal subGQs on x and .1/ would be distinct, they would have to intersect in a subGQ of order .1; t / (by Theorem 1.3), leading to the fact that .1/ is regular. But by the previous section we can exclude this case. So s0 D s0 0 D s 0 for any such ; 0 2 Wx , and all elements of Wx which fix a subGQ elementwise fix the same subGQ elementwise.


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11.2.3 Case 1: Suppose that there is no ideal subGQ containing .1/ fixed ele0 mentwise by elements of W . Let ˛ 2 Wx as before. Then ˛ ¤ ˛ for ¤ 0 . So Œa; ˛ D Œb; ˛, with a; b root-elations in Rz , if and only if a D b. Note that such a commutator Œa; ˛ is itself a root-elation. It follows now that hŒa; ˛ j a 2 Rz i D Rz : As the root-elations generate W , it holds that W equals its derived group. Lemma 11.4. W equals its derived group. Recall an elementary property for commutators: if x, y, z are elements of a group G, then Œxy; z D Œx; zy Œy; z: Let E be the set of all root-elations in W of which the i-root contains .1/. Then this is a normal set in W (that is, for any w 2 W , E w D E). Call an E-commutator of W a commutator of the form Œa; b, where a; b 2 E. So, if g; h; i are elements of E, then Œgh; i D Œg; i h Œh; i D Œg h ; i h Œh; i ; hence Œgh; i is a product of E-commutators. Since E generates W , the following lemma now follows inductively: Lemma 11.5. For any g; h 2 W , the commutator Œg; h is a product of E-commutators. Whence any element of the derived group W 0 is also a product of E-commutators. Exercise. Prove Lemma 11.5. Let a be an element of E, We write A R a or a R A, with A I .1/, if A is the unique line through .1/ of “the” i-root of a. Before proceeding, we first observe a property. Lemma 11.6. Let L be a line incident with .1/. Then W is generated by the rootelations (with i-root on .1/) of which the i-root does not contain L. Proof. Let L0 L be a line not incident with .1/. Let W .L/ be the group generated by the root-elations of which the i-root does not contain L. We must show that W .L/ contains all root groups of type Rz , where .1/ ¤ z I L. Let x; y be arbitrary distinct points on L0 not collinear with .1/. Suppose that X I x is different from L0 and that Y I y also different from L0 . Let o ¤ x be a point on X not collinear with .1/. Since s > 2, we can choose o such that the unique line O through o that meets Y has O \ Y ¦ .1/? . Also, it is easy to see that we can choose X and Y in such a way that O ¦ L. Put X \ .1/? D x 0 ; Y \ .1/? D y 0 and O \ .1/? D o0 . 0 Take ˛ 2 Rx 0 such that x ˛ D o, ˛ 0 2 Ro0 such that o˛ D Y \ O and ˛ 00 2 Ry 0 00 such that .Y \ O/˛ D y. Then ˛˛ 0 ˛ 00 2 W .L/ and it maps x onto y. So W .L/L0 acts transitively on the points of L0 not in .1/? . Call the latter point set L0 . As

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no nontrivial element fixes a thick subGQ pointwise, W .L/L0 in its action on L0 is a Frobenius group, so that the Frobenius kernel consists of all elements of W .L/L0 that do not fix a point of L0 . On the other hand, WL0 is also a Frobenius group on L0 , so that the Frobenius kernel of this action coincides with that of the action of W .L/L0 on L0 . It follows easily that the elements of Rz induce this Frobenius kernel, so that W .L/ contains Rz . The lemma follows. Lemma 11.7. Suppose that some nontrivial automorphism ˛ 0 fixes the lines F; F 0 ¤ F incident with .1/ pointwise, and let ˇ ¤ 1 be a root-elation for which ˇ R F . Then Œ˛ 0 ; ˇ D 1. Proof. Clearly, Œ˛ 0 ; ˇ fixes F and F 0 pointwise, and .1/ linewise. Since ˇ is a root-elation, there is a point .1/0 I F , different from .1/, fixed linewise by ˇ. So Œ˛ 0 ; ˇ also fixes .1/0 linewise. By Theorem 1.6 we conclude that Œ˛ 0 ; ˇ is the identity element. Fix A I .1/. Now suppose that a, b, c are (nonidentity) root-elations for which a R A R b R A R c: By Lemma 11.6, we can write aD

Q

gi ;

i

where i ranges over a finite set f1; 2; : : : ; rg and the gi 2 E have i-roots not containing A. So Q Q Q Œa; Œb; c D Œ i gi ; Œb; c D i Œgi ; Œb; c j >i gj ; Q Q 1 where in the latter expression we in fact mean . g / j k gk when writing j i Q 0 “ j >i gj ”. Let Ai R gi , and note that for k ¤ k in f1; 2; : : : ; rg, Ak can equal Ak 0 . Q Then Œgi ; Œb; c j >i gj D ì fixes Ai and A pointwise. Now let d be a root-elation with again d R A; we have Œd; Œa; Œb; c D ŒŒa; Œb; c; d 1 D Œ

Q i

ì ; d 1 D .

Q

Q

i Œì ; d

j >i `j

/1 :

Consider Œì ; d for i 2 f1; 2; : : : ; rg. As d is a root-elation with i-root containing A, Œì ; d also is as such. On the other hand, Œì ; d fixes A and Ai ¤ A pointwise. It Q `j j >i follows that Œì ; d is the identity, and so also Œì ; d , and hence Œd; Œa; Œb; c D 1: Lemma 11.8. For a; b; c; d 2 E nontrivial root-elations with i-root containing A, we have Œd; Œa; Œb; c D 1:


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Now we calculate the lower central series .Li .W //i of W (despite Lemma 11.4, in search for a contradiction!). One notes that L2 .W / is generated by E-commutators, while L3 .W / is generated by commutators of the form Œg; Œh; i (call such commutators E 2 -commutators), with g; h; i 2 E, L4 .W / is generated by commutators of the form Œg; Œh; Œi; j (E 3 commutators) with g; h; i; j 2 E, etc. Let a; b 2 E be arbitrary and not trivial, and suppose A R a and B R b. Then we have two possibilities for Œa; b: (1) either A D B and Œa; b fixes A pointwise, or (2) A ¤ B and A and B are pointwise fixed by Œa; b. Take c 2 E nontrivial, C R c, and consider Œc; Œa; b. First suppose that A D B. Then either (1.a) C ¤ A and Œc; Œa; b fixes A and C pointwise, or (1.b) C D A and A is fixed pointwise by Œc; Œa; b. Now suppose A ¤ B; then either (2.a) C 2 fA; Bg and Œc; Œa; b is the identity by Lemma 11.7, or (2.b) C … fA; Bg and Œc; Œa; b fixes A, B, C pointwise. Finally, let d 2 E be nontrivial, D R d . Then we obtain the following cases: Case (1.a) if D 2 fA; C g, Œd; Œc; Œa; b D 1 by Lemma 11.7; if D … fA; C g, A, C , D are fixed pointwise by this commutator; Case (1.b) if D D C D B D A, Œd; Œc; Œa; b D 1 by Lemma 11.8; if D ¤ A, D, A are fixed by Œd; Œc; Œa; b; Case (2.b) if D 2 fA; B; C g, Œd; Œc; Œa; b D 1 by Lemma 11.7; if D … fA; B; C g, A, B, C , D are fixed pointwise by Œd; Œc; Œa; b. So either Œd; Œc; Œa; b D 1 or it fixes (at least) A; D ¤ A pointwise. It now easily follows inductively, using Lemma 11.7, that E i1 -commutators, i 4, which generate Li .W /, fix at least i 2 distinct lines incident with .1/ pointwise. Put i D t C 4. Then L tC4 .W / is generated by the set of E tC3 -generators, which are of the form Œa; `; with a 2 E and ` an E tC2 -commutator. So ` fixes each point collinear with .1/. By Lemma 11.7, Œa; ` D 1, and L tC4 .W / D f1g. Theorem 11.9. W is nilpotent of class at most t C 3. But this latter theorem contradicts Lemma 11.4, so that in the setting of case 1, the stabilizer in W of a point opposite .1/ is trivial, thus answering Knarr’s question affirmatively. Note that the nilpotency still holds, however. 11.2.4 Case 2: Suppose that some ¤ 1 fixes a thick ideal subGQ 0 elementwise. This part is presented slightly differently than in [73]. In fact, we apply the proof of Lemma 11.6 to shorten the reasoning. (And it becomes as such an implicit way to define the “subGQ-plane” as in [73], which is still more explicit than in [73], since much is left to the reader. An exercise at the end of §§11.2.4 alludes to the subGQ-plane.)

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Consider Wx ; as we may assume there is only one thick subGQ of order .s 0 ; t / containing x (in fact, there is only one thick subGQ containing x with t C 1 lines through a point), Wx stabilizes 0 . We first state a lemma which is closely related to Lemma 11.6. Lemma 11.10. Let L be any line incident with .1/, and define W .L/ as before (see Lemma 11.6). Then either (a) W .L/ is an elation group acting sharply transitively on P n .1/? , or (b) W .L/ D W , or (c) some element ˛ of W .L/ fixes an ideal subGQ on .1/ pointwise. Proof. Let ˛ 2 W .L/ fix some point z 0 of P n .1/? , and let z ´ projL z 0 . Then if 2 Rz , as before, Œ˛; D 1 if and only if ˛ fixes some ideal proper subGQ pointwise (containing .1/). So if this is not the case, jŒ˛; Rz j D s, so that Œ˛; Rz D Rz (since obviously Œ˛; Rz Rz ). As W .L/ E W , we have that Rz W .L/, so that W .L/ D W . Take ˛ 2 Wx as in case (c) of the lemma, x ¦ .1/, note that such an element also exists in case (b), and consider again the set fŒ; ˛ j 2 Rz g; where L I z .1/, x z. Then we observed earlier that Œ; ˛ D 1 if and only if 2 .Rz / 0 , that is, if and only if is one of the s 0 elements in the stabilizer of 0 in Rz . So jfŒ; ˛ j 2 Rz gj D s=s 0 : Note that all of these elements comprise a root-elation, and that hŒ; ˛ j 2 Rz i D T stabilizes 0 (as ˛ 2 W .L/!), so T can contain at most s 0 root-elations. Exercise# . Prove that each element of fŒ; ˛ j 2 Rz g indeed stabilizes 0 . (Hint: as ˛ 2 W .L/, each element of fŒ; ˛ j 2 Rz g is a product of (conjugates of) commutators of the form Œ; 0 , where 2 Rz and 0 2 Rz 0 , .1/ z 0 I L. Note now that each such commutator fixes each element of 0 W , since it is a dual rootelation.) A slightly stronger property will be obtained in an exercise below (concerning case (a) of Lemma 11.10), from which this exercise also essentially follows.

Whence

s s0: s0 As s=s 0 s 0 , s 0 t and s 0 t s (the latter two inequalities coming from Theorem 1.3), we have 2 s D s0t D s0 : It is straightforward to see that h.Ry / 0 j y .1/ ¤ yi induces an elation group of 0 (by case 1 of our treatment), so that s 0 D t is a prime power by Theorem 5.5, say of


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the prime p. Let Sp be a Sylow p-subgroup of W , and note that either p is odd and Sp is a normal (elation) subgroup of W , or p D 2, 0 Š W .t /, and ˛ is an involution. Exercise. Prove that if Sp is a Sylow p-subgroup of W , either p is odd and Sp is a normal (elation) subgroup of W , or p D 2, 0 Š W .t /, and ˛ is an involution. (Hint: we have encountered similar situations before when .1/ is a regular point, but the proof is the same.) In the former case, all root groups Ry , y .1/ ¤ y, are inside Sp , so that W indeed is an elation group. In the latter case, it follows that .1/ is also regular, and we already handled this. When we are in the first case of Lemma 11.10, a positive answer of Knarr’s question is not hard to deduce along the same lines as in case (b) and (c) of Lemma 11.10. We leave this as an interesting exercise (below) for the reader. Exercise# . Obtain a positive answer to Knarr’s question when we are in the first case of Lemma 11.10. (Hint: since there are thick ideal subEGQs 0 of order .s 0 ; t /, Theorem 5.6 yields that s 0 t is a prime power. Also, as we may assume .1/ to be not regular, an exercise in §5.3 says that W .L/ 0 D X is a normal F -factor in W .L/. Now construct the .s=s 0 ; t C 1/-net as in the proof of Theorem 5.8, and show that W .L/=X is abelian. Assuming not to be in case (b) nor (c) of Lemma 11.10, it follows that this conclusion is independent of the choice of L I .1/. So W 0 E W and W =W 0 is abelian, in other words, ŒW; W W 0 . Now repeat the same reasoning as in case (b) and (c).) This ends the proof of Theorem 11.2. Remark 11.11. (i) Note that the nilpotency of W immediately follows from the proof. By Lemma 5.4 we have that W is a p-group for some prime p. (ii) Note that the method explained in the previous exercise can be easily adapted to case (b) and (c)! Exercise (SubGQ-plane). Suppose that we are in case 2 (hypotheses of §§11.2.4). Define an incidence structure … as follows. • Points are the .s=s 0 /2 subGQs of order .s 0 ; t / of 0 W . • Lines are the point sets 00 \ M , where 00 2 0 W and M a line incident with .1/. • Incidence is inverse containment. Show that … is an affine plane of order s=s 0 D t . Show that W induces a group of translations of … which forces … to be a translation plane with translation group W =(kernel of the action).

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11.3 Solution of Knarr’s question In order to give a complete answer to Knarr’s question, we still have to show that if .1/ is not a regular point, Theorem 11.2 and its proof imply that W , as defined in Theorem 11.2, contains all dual root groups for which the dual i-root contains .1/, if satisfies the conditions of Knarr’s question. Suppose that A; B ¤ A are any two lines incident with .1/, and let R0 be the group of dual root-elations with dual i-root .A; .1/; B/. First note that W is a normal subgroup of the stabilizer of .1/ in the automorphism group of (as E is a normal set in this group). Since W is a p-group, W # D hW; R0 i D WR0 also is a p-group. If W ¤ W # , the stabilizer of any point z ¦ .1/ in W # is a p-group, and Wz# fixes z and .1/ linewise, while fixing at least three points on any line incident with .1/ (since s is a power of p). So Theorem 1.6 implies that Wz# fixes a subGQ 0 of order .s 0 ; t/ of pointwise, with 1 < s 0 < s. Since we may suppose .1/ not to be regular, the last part of the proof of Theorem 11.2 (case 2) implies that t 2 D s, t is even, and 0 Š W .t/. But then again 0 , and so also , contains a regular point, contradicting our assumption. Remark 11.12. In a very recent paper [79], I have shown that under the conditions of Theorem 11.2, and using the theorem itself, one can prove that all dual i-roots on .1/ automatically are Moufang, with corresponding dual root groups contained in the elation group. As a direct corollary, one obtains a short “local proof” of the finite Half Moufang Theorem of [56] (which states that if all roots of a finite GQ are Moufang, then all dual roots are Moufang as well). We refer the reader to [79] for a detailed discussion.

11.4 Appendix: GQs with a center of transitivity (and s t) In this appendix we mention some results taken from [75], that generalize Frohardt’s result to GQs with a center of transitivity. Setting. In this entire section, x is a (thick) GQ of order .s; t /, and G is an automorphism group that acts transitively on the points opposite x, while it fixes each line incident with x. Denote the lines incident with x by L0 ; L1 ; : : : ; L t . Starting from a point z ¦ x, let Mi be the line on z which meets Li , for i running through f0; 1; : : : ; tg. Also, denote Li \ Mi by xi . Put Hi D GMi and J D fHi j 0 i t g; while

Hi

D Gxi and J D fHi j 0 i t g:

Let jGj D s 2 tk, k 1. If A D Hj 2 J, we put A D Hj .

11.4 Appendix: GQs with a center of transitivity (and s t )

101

11.4.1 Some technical lemmas Lemma 11.13. Let p be a prime and assume that sp > 1. If A and B are distinct elements of J, Ap a Sylow p-subgroup of A , Bp a Sylow p-subgroup of B, and g 2 G, then either (i) kp D 1 and Ap \ Bpg D f1g, or (ii) kp ¤ 1 and st is a power of p. Proof. First suppose that kp ¤ 1; then p divides the size of Gy for each y ¦ x. So since p divides s, by Theorem 1.6 we know that an element ˛ of Gy of order p fixes a thick subGQ 0 of order .s 0 ; t / elementwise. Suppose that fx; yg is contained in two distinct subGQs of order .s 0 ; t /; then a standard argument together with Theorem 1.3 leads to the fact that s D t 2 D s 0 t (these subGQs intersect in a dual grid with GQ parameters .1; t /). Now Theorem 7.5 tells us that t is a power of 2, so also st . (In fact, it also tells us that kp D k2 D 2 and that contains a subGQ isomorphic to W .t / which is fixed pointwise by an involution in G.) Now suppose that fx; yg is contained in only one such subGQ, and note that y is essentially arbitrary. Then by the first part of §11.2.4, s 0 t D s and s is a power of p, so also st . Suppose that kp D 1. Clearly, Ap \ Bpg is a p-group that fixes some point opposite x, so (i) follows from (ii). The proof of the next lemma is the one we already met for the sharply transitive case. We repeat it for the sake of convenience. Lemma 11.14. Let p be a prime, and assume that sp > 1. Then tp0 < sp : Proof. If kp ¤ 1, st is a power of p by Lemma 11.13, so the lemma is trivial. We proceed by supposing that kp D 1. Take A 2 J, and let P 2 Yp .G/ contain a Sylow p-subgroup Ap of A . For each B 2 J n fAg, let Bp 2 Yp .B/. Then Bp is G-conjugate to a subgroup QBp of P . By Lemma 5.1, we have that the sets Ap and QB are mutually disjoint, while QBp \ QCp D f1g if B ¤ C in J. So p jAp j C

P B2JnfAg

jQB j jP j: p

j D sp 1 for all B 2 J n fAg, and since jP j D sp2 tp , this Since jAp j D sp tp , jQB p implies that sp tp C t .sp 1/ sp2 tp :

Hence tp tp0 .sp 1/ tp sp .sp 1/. As tp > 0 and sp > 1, we have tp0 < sp .

Note that all the results of this section prior to the following theorem do not require any assumption on the order of the GQ.

102


Theorem 11.15. If s t, st is a prime power. Proof. We may suppose that kp D 1. Put r D j.s/j. Then tr D

Q

.tp tp0 /

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