Sporadic Groups

MICHAEL ASCHBACHER Department of Mathematics California Institute of Technology

Sporadic groups

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Published by the Press Syndicate of the University of Cambridge The Pitt BuiIding, Trumpington Street, Cambridge CB2 IRP 40 West 20th Street, New York, NY 10011-4211, USA 10 Stamford Road, Oakleigh, Victoria 3166, Australia

Contents

@ Cambridge University Press 1994

First published 1994

Preface

page i x

Printed in the United States of America

PART I Library of Congress Cataloging-in-PublicationData Aschbacher, Michael. Sporadic groups / Michael Aschbacher. p. cm. - (Cambridge tracts in mathematics ; 104) Includes bibliographical references and indexes. ISBN 0-521-420440 1. Sporadic groups (Mathematics) I. Title. 11. Series. QA177.A83 1994 512l.2 - dc20 92-13653 CIP

1. Preliminary Results 1. Abstract representations 2. Permutation representations 3. 4.

Graphs Geometries and complexes 5. The general linear group and its projective geometry 6. Fiber products of groups

2.

2-Structure i n F i n i t e G r o u p s 7. Involutions 8. Extraspecial groups

3.

Algebras, Codes, and Forms 9. Forms and algebras 10. Codes 11. Derived forms

4.

Syrnplectic 2-Loops 12. Symplectic 2-loops 13. Moufang symplectic Zloops 14. Constructing a 2-local from a loop

46 47 54 57

5.

T h e Discovery, Existence, a n d Uniqueness of the Sporadics 15. History and discovery 16. Existence of the sporadics 17. Uniqueness of the sporadics

65 65 70

A catalog record for this book is available from the British Library. ISBN 0-521-42049-0 hardback

6.

The M a t h i e u Groups, T h e i r Steiner Systems, a n d t h e Golay C o d e 18. Steiner systems for the Mathieu groups 19. The Golay and Todd modules

74

Contents

Contents

vi 7.

The Geometry and Structure of MZ4 20. The geometry of M24 21. The local structure of M24

8.

The Conway Groups and the Leech Lattice 22. The Leech lattice and - 0 23. The Leech Iattice mod 2

9.

Subgroups of - 0 24. The groups Co3, Mc, and H S 25. The groups Col, Coz, SUZ,and J2 26. Some local subgroups of Col

10. The Griess Algebra and the Monster 27. The subgroups C and N of the Monster 28. The Griess algebra 29. The action of N on B 30. N preserves the Griess algebra 31. The automorphism group of the Griess algebra 11.

Subgroups of Groups of Monster Type 32. Subgroups of groups of Monster type

PART I11 12.

Coverings of Graphs and Simplicia1 Complexes 33. The fundamental groupoid 34. Triangulation 35. Coverings of graphs and simplicial complexes

13. The Geometry of Amalgams 36. Amalgams 37. Uniqueness systems 38. The uniqueness system of a string geometry 14. The Uniqueness of Groups of Type M24, He, and L5(2) 39. Some 2-local subgroups in L5(2),Mz4, and He 40. Groups of type L5(2),M24, and He 41. Groups of type L5(2) and M24 42. Groups of type He 43. The root 4-group graph for He 44. The uniqueness of groups of type He

15. The Group U4(3) 45. U4(3)

16. Groups of Conway, Suzuki, and Hall-Janko Type 46. Groups of type Col, Suz, J2, and J3 47. Groups of type J2 48. Groups of type Suz 49. Groups of type Col 17.

Subgroups of Prime Order in Five Sporadic Groups 50. Subgroups of Suz of prime order 51. Subgroups of Col of prime order 52. Subgroups of prime order in He

Symbols Bibliography Index

Preface

The classification of the finite simple groups says that each finite simple group is isomorphic to exactly one of the following:

A group of prime order An alternating group A, of degree n A group of Lie type One of twenty-six sporadic groups

I I

,

I

I

I

I

As a first step in the classification, each of the simple groups must be shown to exist and to be unique subject to suitable hypotheses, and the most basic properties of the group must be established. The existence of the alternating group An comes for free, while the representation of An on its n-set makes possible a first uniqueness proof and easy proofs of most properties of the group. The situation with the groups of Lie type is more difficult, but while groups of Lie rank 1 and 2 cause some problems, Lie theory provides proofs of the existence, uniqueness, and basic structure of the groups of Lie type in terms of their Lie algebras and buildings. However, the situation with the sporadic groups is less satisfactory. Much of the existing treatment of the sporadic groups remains unpublished and the mathematics which does appear in print lacks uniformity, is spread over many papers, and often depends upon machine calculation. Sporadic Groups represents the first step in a program to provide a uniform, self-contained treatment of the foundational material on the sporadic groups. More precisely our eventual aim is to provide complete proofs of the existence and uniqueness of the twenty-six sporadic groups subject to appropriate hypotheses, and to derive the most basic structure of the sporadics, such as the group order and the normalizers of subgroups of prime order. While much of this program is necessarily technical and specialized, other parts are accessible to mathematicians with only a basic knowledge of finite group theory. Moreover some of the sporadic groups are the automorphism groups of combinatorial objects of independent interest, so it is desirable to make this part of the program available to as large an audience as possible. For example, the Mathieu groups are the automorphism groups of Steiner systems and Golay codes while the largest Conway group is the automorphism group of the Leech lattice.

x

Preface

Preface Sporadic Groups begins the treatment of the foundations of the sporadic groups by concentrating on the most accessible chapters of the subject. It is our hope that large parts of the book can be read by the nonspecialist and provide a good picture of the structure of the sporadics and the methods for studying these groups. At the same time the book provides the basis for a complete treatment of the sporadics. The book is divided into three parts: Part I, introductory material (Chapters 1-5); Part 11, existence theorems (Chapters 6-11); and Part 111, uniqueness theorems (Chapters 12-17). The goal of the existence treatment is to construct the largest sporadic group (the Monster) as the group of automorphisms of the Griess algebra. Twenty of the twenty-six sporadic groups are sections of the Monster. We establish the existence of these groups via these embeddings. To construct the Griess algebra one must first construct the Leech lattice and the Conway groups, and to construct the Leech lattice one must first construct the Mathieu groups, their Steiner systems, and the binary Golay code. There are many constructions of the Mathieu groups. Our treatment proceeds by constructing the Steiner systems for the Mathieu groups as a tower of extensions of the projective plane of order 4. This method has the advantage of supplying the extremely detailed information about the Mathieu groups, their Steiner systems, and the Golay code module and Todd module necessary both for the construction of the Leech lattice and the Griess algebra, and for the proof of the uniqueness of various sporadics. The construction given here of the Leech lattice and the subgroups stabilizing various sublattices is the standard one due to Conway in [Col] The construction of the Griess algebra combines aspects of and [CO~]. and Tits [T2], plus the treatments due to Griess [Gr2], Conway [CO~], a few extra wrinkles. The basis of the construction is Parker's loop and Conway's construction via the Parker loop of the normalizer N of a certain Csubgroup of the Monster. Chapter 4 contains a discussion of a general class of loops which includes the Parker loop. This discussion contains much material not needed to construct the Parker loop or the Griess algebra, but the extra discussion provides a context which hopefully makes the Parker loop and Conway's construction of N more natural. The majority of the sporadic groups contain a large extraspecial 2subgroup. Such subgroups provide one of the unifying features of our treatment. The basic theory of large extraspecial subgroups is developed

xi

in Chapter 2. The theory is used to recognize and establish the simplicity of the sporadics contained in the Monster that are not symmetry groups of any nice structure. The eventual object of the uniqueness treatment is to prove each sporadic is unique subject to suitable hypotheses. Here is a typical hypothesis; let w be a positive integer and L a group. (See Chapter 2 for terminology and notation.) Hypothesis Z(w, L): G is a finite gmup containing an involution z such that F*(CG(z))= Q is an extraspecial &subgroup of order 22W+1, CG(z)/Q L, and z is not weakly closed in Q with respect to G .

I

1 \

,

I

For example, Hypothesis 7f(12,Col) characterizes the Monster. Hypotheses of this sort are the appropriate ones for characterizing the sporadic~for purposes of the classification. Sporadic Groups lays the foundation for a proof of the uniqueness of each of the sporadics and supplies actual uniqueness proofs for five of the sporadic groups: M24, He, J2, Suz, and Col. Our approach to the uniqueness problem follows Aschbacher and Segev in [ASl]. Namely given a group theoretic hypothesis 31 we associate t o each group G satisfying 7f a coset graph A defined by some family F of subgroups of G. We prove the amalgam A of 3 is determined up t o isomorphism by 31 independently of G, and form the free amalgamated product G of A and its coset graph A. Now there exists a covering d :A A of graphs. To complete the proof we show A is simply connetted so d is an isomorphism and hence G = is determined up t o isomorphism by H. After developing the most basic part of the conceptual base for our treatment of the sporadic groups in Part I, Chapter 5 closes the first part of the book with an overview of the sporadic groups including the hypotheses by which we expect each group to be characterized, the approach for constructing each of the twenty sporadics involved in the Monster, and a number of historical remarks. While Sporadic Groups concentrates on some of the most accessible and least technical aspects of the study of the sporadic groups, a complete treatment of even this material sometimes requires some difficult specialized arguments. The reader wishing to minimize contact with such arguments can do so as follows. As a general rule the book becomes progressively more difficult in the later chapters. Thus most of the material in Part I should cause little difficulty. A possible exception is Chapter 4, containing the discussion of loops. However, much of this material is not

-

xii

Preface

needed in the rest of the book, and none is needed outside of Chapter 10, where the Griess algebra is constructed. As Chapter 10 is the most technical part of Part 11, some readers may wish to skip both Chapter 4 and Chapter 10. Part I1 contains constructions df the Mathieu groups, the Conway group Col and its sporadic sections, and the Monster and its sporadic sections. Two chapters are devoted to the Mathieu groups and two to the Conway groups. In each case the second of the two chapters is the most technical. Thus the reader may wish to read Chapters 6 and 8, while skipping or skimming Chapters 7 and 9. As suggested in the previous paragraph, dilettantes should skip the construction in Chapter 10 of the Griess algebra and the Monster. The existence proofs for the sporadic sections of the Monster not contained in Col appear in the very short Chapter 11. The Steiner systems and Golay codes associated to the Mathieu groups and the Leech lattice associated to the Conway groups are beautiful and natural objects. Most of the discussion of these objects appears in Chapters 6 and 8. There is some evidence that the Griess algebra is also natural, in that it is the 0-graded submodule of a conformal field theory preserved by the Monster (cf. [FLM]). However, the construction of the Griess algebra in Chapter 10 is not particularly natural or edifying. The first two chapters of Part I11 provide the conceptual base for proving the uniqueness of the sporadic groups. These chapters are fairly elementary. Sections 39 through 41 establishing the uniqueness of M24 and L5(2)probably provide the easiest example of how to apply this machinery to establish uniqueness. On the other hand the proofs of the uniqueness of He,J2, Suz, and Col, while more difficult, are also more representative of the complexity involved in proving the uniqueness of the sporadic groups. The book closes with tables describing the basic structure of the five sporadic groups considered in detail in Sporadic Groups: M 2 4 7 He, J2, Suz, and Gol. These tabIes enumerate the subgroups of prime order of each group G and the normalizers of these subgroups. Much of this information comes out during the proof of the uniqueness of G, but some of the loose ends are tied up in Chapter 17.

PART I

Chapter 1 Preliminary Results

We take a s our starting point the text Finite Group Theory [FGT], although we need only a fraction of the material in that text. Requently quoted results from [FGT]will be recorded in this chapter and in other of the introductory chapters. Chapters 1 and 2 record some of the most basic terminology and notation we will be using plus some elementary results. The reader should consult [FGT] for other basic group theoretic terminology and notation, although we will try to recall such notation when it is first used, or at least give a specific reference to [FGT] at that point. There is a "List of Symbols" at the end of [FGT] which can be used to help hunt down notation. We begin in Section 1 with a brief discussion of abstract representations of groups. Then in Section 2 we specialize to permutation representations. In Section 3 we consider graphs and in Section 4 geometries (in the sense of J. Tits) and geometric complexes. In the last few sections of the chapter we record a few basic facts about the general linear group and fiber products of groups.

1. Abstract representations Let C be a category. For X an object in C, we write Aut(X) for the group of automorphisms of X under the operation of composition in C (cf. Section 2 in [FGT]).A ~presentatzonof a group G in the category C is a group homomorphism ?r; G -+ Aut(X). For example, a permutation representation is a representation in the category of sets and a linear

2

Chapter 1 Preliminary Results representation is a representation in the category of vector spaces and linear maps. If a :A -+ B is an isomorphism of objects in C then a induces a map

2. Permutation representations

3

Let n : G -+ S y m ( X ) be a permutation representation of a group G on X. Usually we suppress n and write xg for the image x(gn) of a point x E X under the permutation gn, g E G. For S C G, we write Fix(S) = Fixx(S) for the set of fixed points of S on X . For Y X,

c

and a* restricts to an isomorphism a* : Aut(A) -+ Aut(B). Thus in particular if A r B then Aut(A) E Aut(B). A representation n : G -t Aut(A) is faithful if n is injective. Two representations n : G -, Aut(A) and u : G + Aut(B) in C are equivalent if there exists an isomorphism a : A --, B such that a = na* is the composition of n with a*. Equivalently for all g E G, (gn)a= a(ga). Similarly if Ti : Gi -+ Aut(Ai), i = 1,2, are representations of groups Gi on objects Ai in C, then nl is said to be quasiequivalent to n2 if there exists a group isomorphism f l : GI -, G2 and an isomorphism cr : Al -+ A2 such that 1r2 = p-lnlat. Observe that we have a permutation representation of Aut(G) on the equivalence classes of representations of G via cr : n H a n with the orbits the quasiequivalence classes. Write Aut(G), for the stabilizer of the equivalence class of n under this representation. The following result is Exercise 1.7 in [FGT]:

Lemma 1.1: Let n,u : G -+ Aut(A) be faithful representations. Then

is quasiequivalent to a if and only if Gn is conjugate to Ga in Aut (A). (2) Aut,,t(,) (Gn)2 Aut(G),. (1)

R

If H < G then write AutG(H) = Na(H)/CG(H)for the group of automorphiims of H induced by G. Also

CG(H)= ( c E G :ch=hcfor all h~ H ) is the centralizer in G of H and NG(H) is the normalizer in G of H, that is, the largest subgroup of G in which H is normal.

2. Permutation representations In this section X is a set. We refer the reader to Section 5 of [FGT] for our notational conventions involving permutation groups, although we record a few of the most frequently used conventions here. In particular we write S y m ( X ) for the symmetric group on X and if X is finite we write Alt(X) for the alternating group on X . Further S,, A, denote the symmetric and alternating groups of degree n; that is, Sn = S y m ( X ) and A, = Alt(X) for X of order n.

G y = { g E G : yg = y for all y E Y) is the pointwise stabilizer of Y in G,

is the global stabilizer of Y in G, and G~ = G ( Y ) / G y is the image of G ( Y ) in Sym(Y) under the restriction map. In particular Gp denotes the stabilizer of a point y E X. Recall the orbit of x E X under G is XG = {xg : g E G ) and G is transitive on X if G has just one orbit on X. If G is transitive on X then our representation n is equivalent to the representation of G by right multiplication on the coset space G/G, via the map Gzg t+ xg (cf. 5.9 in [FGT]). A subgroup K of G is a regular normal subgroup of G if K G and K is regular on X; that is, K is transitive on X and Kx = 1 for x E X. Recall a transitive permutation group G is primitive on X if G preserves no nontrivial partition on X . Further G is primitive on X if and only if Gx is maximal in G (cf. 5.19 in [FGT]).

Lemma 2.1: Let G be transitive on X , x

E X,

and K < G. Then

(1) K is transitive on X if and only if G = GZK.

If 1 # K q G and G is primitive on X then K is transitive on X . (3) If K is a regular normal subgroup of G then the representations of Gx on X and on K by conjugation are equivalent.

(2)

Proof: These are all well known; see, for example, 5.20, 15.15, and 15.11 in [FGT].

Recall that G is t-transitive on X if G is transitive on ordered t-tuples of distinct points of X . In Chapter 6 we will find that the Mathieu group Mm+t is t-transitive on m+t points for m = 19 and t = 3,4,5 and m = 7 and t = 4,5.

Lemma 2.2: Let G be t-transitive on a finite set X with t 2 2, x E X, and 1 # K G. Then

(1) G is primitive on X . (2) K is transitive on X and G = G,K.

3. Graphs

Chapter 1 Pmliminary Results

(3) If K is regular on X then 1KI = 1x1 = pe is a power of some prime p, and if t > 2 then p = 2. (4) If t = 3 < 1x1 and IG : KI = 2 then K is 2-transitive on X . Proof: Again these are well-known facts. See, for example, 15.14 and 15.13 in [FGT]for (1) and (3), respectively. Part ( 2 ) follows from ( 1 ) and 1.1. Part (4) is left as Exercise 1.1.

3. Graphs A graph A = (A,*) consists of a set A of vertices (or objects or points) together with a symmetric relation * called adjacency (or incidence or something else). The ordered pairs in the relation are called the edges of the graph. We write u * v to indicate two vertices are related via * and say u is adjacent to v. Denote by A ( u ) the set of vertices adjacent to u and distinct from u and define uL = A ( u ) LJ{u). A path of length n from u to v is a sequence of vertices u = uo, u l , . ., u,= v such that ui+l E uf for each i. Denote by d(u,v ) the minimal length of a path from u to v. If no such path exists set d(u,v) = oo. d(u,v ) is the distance from u to v. The relation on A defined by u v if and only if d(u, v ) < OCI is an equivalence relation on A. The equivalence classes of this relation are called the connected components of the graph. The graph is connected if it has just one connected component. Equivalently there is a path between any pair of vertices. A morphism of graphs is a function a : A -+ At from the vertex set of A to the vertex set of At which preserves adjacency; that is, uLa C (ua)' for each u E A. A group G of automorphisms of A is edge transitive on A if G is transitive on A and on the edges of A. Representations of groups on graphs play a big role in this book. For example, we prove the uniqueness of some of the sporadics G by considering a representation of G on a suitable graph. The following construction supplies us with such graphs. Let G be a transitive permutation group on a finite set A. Recall the orbitals of G on A are the orbits of G on the set product A2 = A x A. The permutation rank of G is the number of orbitals of G; recall this is also the number of orbits of Gz on A for x E A. Given an orbital R of G , the paired orbital Rp of R is

.

-

N

Stp = { ( Y ,x ) : (2,y)

E R).

5

Evidently RP is an orbital of G with (Rp)P = R. The orbital R is said to be self-paired if RP = 0. For example, the diagonal orbital { ( x , x ) : x E A ) is a self-paired orbital.

Lemma 3.1: (1) A nondiagonal orbital ( x ,y)G of G is self-paired i f and only i f ( x ,y) is a cycle i n some g E G. (2) If G is finite then G possesses a nondiagonal self-paired orbital i f and only i f G Gs of even order. (3) If G is of even order and pennutation mnk 3 then all orbitals of G are self-paired. Proof:See 16.1 in [FGT].

Lemma 3.2: (1) Let R be a self-paired orbital o f G . Then R is a symmetric relation on A, so A = (A,51) is a graph and G is an edge transitive group of automorphisms of A. (2) Conversely i f H is an edge transitive group of automorphisms of a graph A = ( A , *) then the set * of edges of A is a self-paired orbital of G on A, and A is the graph determined by this orbital. Many of the sporadics have representations as rank 3 permutation groups. Indeed some were discovered via such representations; see Chap ter 5 for a discussion of the sporadics discovered this way. See also Exercise 16.5, which considers the rank 3 representation of J2, and Lemmas 24.6, 24.7, and 24.11, which establish the existence of rank 3 representations of Mc, U4(3),and H S . In the remainder of this section assume G is of even order and permutation rank 3 on a set X . Hence G has two nondiagonal orbitals A and I' and by 3.1, each is self-paired. Further for x E X , G z has two orbits A ( x ) and r ( x ) on X - { x ) , where A ( x ) = { y E X : (2,y) E A } and r ( x ) = { z E X : ( x , z ) E I?}. By 3.2, X = ( X , A ) is a graph and G is an edge transitive group of automorphisms of X. Notice A ( x ) = X ( x ) in our old notation. The following notation is standard for rank 3 groups and their graphs: k = lA(x)l, 1 = Ir(x)l, X = l A ( x ) n A(y)l for y E A ( x ) , and p = I A ( x ) n A ( z )( for z E r ( x ) . The integers k , 1, A, p are the parameters of the rank 3 group G. Also let n = 1x1 be the degree of the representation.

Lemma 3.3: Let G be a mnk 3 permutation group of even order on a finite set of order n with parameters k , 1, A, p. Then

6

Chapter 1 Preliminary Results If p # 0 or k then G is primitive and the graph B of G is connected. (4) Assume G is primitive. Then either (a) k = 1 and p = X + 1 = k/2, or (b) d = (A - p)2 4(k - p) is a square and setting D = 2k + (A - p)(k l), d1f2 divides D and 2d1I2 divides D i f and only i f n is odd. (3)

+

+

Proof: See Section 16 of [FGT].

4. Geometries and complexes In this book we adopt a notion of geometry due to J. Tits in [TI]. Let I be a finite set. For J I, let J' = I - J be the complement of J in I. A geometry over I is a triple ( l ? , ~*), where I' is a set of objects, 7 : I' + I is a surjective type function, and * is a symmetric incidence relation on I' such that objects u and v of the same type are incident if and only if u = v. We call ~ ( uthe ) type of the object u. Notice (I?,*) is a graph. We usually write I'for the geometry (I?, 7, *) and ri for the set of objects of I? of type i. The rank of the geometry l? is the cardinality of I. A flag of I? is a subset T of I' such that each pair of objects in T is incident. Notice our one axiom insures that if T is a flag then the type function T : T -t I is injective. Define the type of T to be T(T) and the rank of T to be the cardinality of T. The chambers of I? are the flags of type I. A morphism a : I' -+ I" of geometries is a function a : I' -+ I" of the associated object sets which preserves type and incidence; that is, ) ~ ' ( u a )and ua *' var. A group G of if u, v E I? with u * v then ~ ( u = automorphisms of I' is edge transitive if G is transitive on flags of type J for each subset J of I of order at most 2. Similarly G is flag transitive on I? if G is transitive on flags of type J for all J E I. Representations of groups on geometries also play an important role in Sporadic Groups. For example, the Steiner systems in Chapter 6 are rank 2 geometries whose automorphism groups are the Mathieu groups. Here are some other examples: Examples (1) Let V be an n-dimensional vector space over a field F. We associate a geometry PG(V) to V called the projective geometry of V. The objects of PG(V1 are the proper nonzero subspaces of V, with incidence defined by inclusion. The type of U is T(U) = dim(U). Thus

4.

Geometries and complexes

7

PG(V) is of rank n - 1. The projective general linear group on V is a flag transitive group of automorphism of PG(V). (2) A projective plane is a rank 2 geometry I' whose two types of objects are called points and lines and such that: (PP1) Each pair of distinct points is incident with a unique line. (PP2) Each pair of distinct lines is incident with a unique point. (PP3) There exist four points no three of which are on a common line.

Remarks. (1) Rank 2 projective geometries are projective planes. (2) If I'is a finite projective plane then there exists an integer q such that each point is incident with exactly q + 1 l i i , each lime is incident with exactly q + 1 points, and I' has q2 + q + 1 points and lines. Examples (3) If f is a sesquilinear or quadratic form on V then the totally singular subspaces of V are the subspaces U such that f is trivial on U.The set of such subspaces forms a subgeometry of the projective geometry. See, for example, page 99 in [FGT]. (4) Let G be a group and 3 = (Gi : i E I ) a family of subgroups of G. Define I'(G,3) to be the geometry whose set of objects of type i is the coset space GIGi and with objects Gix and Gjy incident if Gix n Gjy # 0.Observe: Lemma 4.1: (1) G is represented as an edge transitive group of automorphisms of r(G, 3)via right multiplication and r(G, 3 ) possesses a chamber. (2) Conversely if H is an edge transitive group of automorphisms of a geometry I' and I' possesses a chamber C, then I' r I'(H,3), where 3 = ( H , : c E C).

The construction of 4.1 allows us to represent each group G on various geometries. The construction is used in Chapter 13 as part of our machine for establishing the uniqueness of groups. Further the construction associates to each sporadic group G various geometries which can be used to study the subgroup structure of G. The latter point of view is not explored to any extent in Sporadic Groups;see instead [A21 or [RS] where such geometries are discussed. We do use the 2-local geometry of M24 to study that group in Chapter 7. Define the direct sum of geometries Pi on Ii,i = 1,2, to be the geometry rl @r2over the disjoint union I of Il and I2whose object set is the disjoint union of rl and I'2, whose type function is TI U 72, and whose incidence is inherited from rl and I'2 with each object in incident with each object in I'2.

8

4.

Chapter 1 Preliminary Results

Example (5) A generalized digon is a rank 2 geometry which is the direct sum of rank 1 geometries. That is, each element of type 1 is incident with each element of type 2.

Lemma 4.2: Let G be a group and 3 = (GI, G2) a pair of subgroups of G . Then I'(G,F) is a generalized digon if and only if G = G1G2. Proof: As G is edge transitive on I', I? is a generalized digon if and only if G2 is transitive on I'l if and only if G = GIG2 by 2.1.1.

Given a flag T, let r ( T ) consist of all v E I' - T such that v * t for all t E T. We regard I'(T) as a geometry over I - T(T). The geometry r ( T ) is called the residue of T.

Example (6) Let I' = PG(V) be the projective geometry of an ndimensional vector space. Then for U E I', the residue r ( U ) of the object U is isomorphic t o PG(U) $ PG(V/U). The category of geometries is not large enough; we must also consider either the category of chamber systems or the category of geometric complexes. A chamber system over I is a set X together with a collection of equivalence relations y, i E I. For J E I and x E X , let N J be the equivalence relation generated by the relations -j, j E J, and [ x ] the ~ equivalence class of J containing x. Define X t o be nondegenerate if for ~ [x]j = n i E j t [ x ] iA. morphism each x E X , and j E I , {XI = n i [ x ] iand of chamber systems over I is a map preserving each equivalence relation. The notion of "chamber system" was introduced by J. Tits in [TI]. Recall that a simplicial complex K consists of a set X of vertices together with a distinguished set of nonempty subsets of X called the simplices of K such that each nonempty subset of simplex is a simplex. The morphiims of simplicial complexes are the simplicial maps; that is, a simplicial map f : K --, K' is a map f : X --,X' of vertices such that f (s) is a simplex of K' for each simplex s of K.

-

Example (7) If A is a graph then the clique complex K(A) is the simplicial complex whose vertices are the vertices of A and whose simplices are the finite cliques of A. Recall a clique of A is a set Y of vertices such that y E xL for each x, y E Y . Conversely if K is a simplicial complex then the gmph of K is the graph A = A(K) whose vertices are the vertices of K and with x * y if { x , y) is a simplex of K. Observe K is a subcomplex of K(A(K)). Given a simplicial complex K and a simplex s of K , define the star of s to be the subcomplex s t K ( s ) consisting of the simplices t of K such that

,

Geometries and complexes

9

s U t is a simplex of K. Define the link LinkK(s) t o be the subcomplex of s t K ( s ) consisting of the simplices t of s t K ( s ) such that t n s = 0. A geometric complex over I is a geometry I? over I together with a collection C of distinguished chambers of I' such that each flag of rank 1 or 2 is contained in a member of C. The simplices of the complex are the subflags of members of C. A morphism cu :C -,C' of complexes over I is a morphism of geometries with Ca G C'. Notice a geometric complex is just a simplicial complex together with a type function on vertices that is injective on simplices.

Example (8) The flag complex of a geometry I? is the simplicial complex on I? in which all chambers are distinguished. Notice the flag complex is a geometric complex if and only if each flag of rank a t most 2 is contained in a chamber. Further as a simplicial complex, the flag complex is just the clique complex of I'regarded as a graph. Many theorems about geometries are best established in the larger categories of geometric complexes or chamber systems. Theorem 4.11 is an example of such a result. We find in a moment in Lemma 4.3 below that the category of nondegenerate chamber systems is isomorphic t o the category of geometric complexes. I find the latter category more intuitive and so work with complexes rather than chamber systems. But others prefer chamber systems and there is a growing literature on the subject. Given a chamber system X define rx to be the geometry whose objects of type i are the equivalence classes of the relation - i t with A* B if and only if A n B # 0.For x E X let C, be the set of equivalence classes containing x; thus C, is a chamber in I?x. Define CX to be the set of chambers C,, x E X , of I'x. If a : X --, X' is a morphism of chamber systems define ac : CX 4 CXt t o be the morphism of complexes such that ac : A H A' for A a -it equivalence class of X and A' the equivalence class containing Aa. Conversely given a geometric complex C over I let -i be the equivrtlence relation on C defined by A -i B if A and B have the same subflag of type it. Then we have a chamber system XCwith chamber set C and k r t h e r if a : C -+ C' is a morphism of comequivalence relations plexes let ax : XC--, Xct be the morphism of chamber systems defined by the induced map on chambers.

Lemma 4.3: The categoy of nondegenemte chamber systems over I is isomorphic to the category of geometric complexes over I via the maps X w C x andC-XC.

10

Chapter 1 Preliminary Results

4.

Example (9) Let G be a group and 3 = (Gi : i E I ) a family of subgroups of I. For J 5 I and x E G define SJ,Z = {Gjx :j E J ) . Thus Sj,, is a flag of the geometry I'(G, 3)of type J . Observe that the Gj. Define stabilizer of the flag SJ = SJllis the subgroup G j = C(G,3) to be the geometric complex over I with geometry I'(G,3) and distinguished chambers Sz,,, x E G. Then C(G,3) is a geometric complex with simplices Sj,,, J E I, x f G, and G acts as an edge transitive group of automorphisms of C(G, 3 ) via right multiplication, and transitively on C(G, 3 ) . Indeed:

njE

Lemma 4.4: Assume C is a geometric complex over I and G is an edge transitive group of automorphisms with C = CG for some C E C. Let Gi = G,,, where xi E C is of type i, and let 3= (Gi : i E I). Then the Gig is an isomorphism of C with C(G, 3 ) . map x,g Further we have a chamber system X(G,3) whose chamber set is GIGz and with Gzx ~i Gzy if and only if xy-' E Gil. Observe that the map GIs I+ S1,, defines an isomorphim of the chamber systems X(Gl 3 ) and XC(G,F). The construction of 4.4 allows us to represent a group G on many complexes. We make use of this construction in Chapter 13 as part of our uniqueness machine. Let C = (I?, C) be a geometric complex over I. Given a simplex S of type J, regard the link Linkc(S) of S to be a geometric complex over J'; thus the objects of Linkc(S) of type i E J' are those u E ri such that S U {v) is a simplex and with v * u if S U {u, v) is a simplex, and the chamber set C(S) of Linkc(S) consists of the simplices C - S with S E C E C. For example, C = Linkc(@) is the link of the empty simplex. Notice that if all flags are simplices then the geometry of Linkc($) is the residue r(S) of S in the geometry r. We say C is residually connected if the link of each simplex of corank at least two (including 0 if 111 1 2) is connected. A geometry 'I is residually connected if each flag is contained in a chamber and the flag complex of I? is residually connected.

Lemma 4.5: Let 3= (Gi : i E I ) be a family of subgroups of G. Then (1) I?(G,3 ) is connected if and only if G = (3). (2) Linkc ( Sj ) C(Gj,3 j ) for each J E I, where

11

Geometries and complexes

(3) C(G,3) is residually connected if and only if GJ = ( 3 j ) for all

J E I. Proof: Notice (1) and (2) imply (3) so it remains to prove (1) and (2). As 3is a chamber, the connected component A of G, in I' is the same for each i, and H = ( 3 ) acts on A. Conversely as Gi is transitive on rj(Gi) for each j, A A' = Uj GjH, so A = A' and H is transitive on ri n A for each i . Thus as G is transitive on I?,, I' is connected if and only if H is transitive on I'i for each i, and as Gi < H this holds if and only if G = H. Thus (1) is established. In (2) the desired isomorphim is Gkx w SK,, for x E G j , K = J U {k).

Lemma 4.6: Assume C is a residually connected geometric complex over I, J C_ I with IJI 2 2, and x, y E I?. Then there exists a path x = vo, vm = y in I? with 7(vi) E J for all 0 < i < m.

...

Proof: Choose x, y to be a counterexample with d = d(x, y) minimal. As the residue l? of the simplex 0 is connected, d is finite, and clearly d > 1. Let x = vo vd = y be a path. By minimality of d there is a path vl = ug . .um = y with 7(ui) E J for 0 < i < m. Thus if r(vl) E J then xu0 % is the desired path, so assume 7(vl) 4 3. We also induct on the rank of C; if the rank is 2 the lemma is trivial, so our induction is anchored. Now Linkc(vl) is a residually connected complex and x,u1 E Linkc(ul), SO by induction on the rank of C, there is a path x = wo-..wk = u1 with 7(wi) E J for 0 < i < k. NOW ~ the job. x = w ~ " ' w ~ u ~="y 'does Given geometric complexes C over J and over 3 define C €D to be the geometric complex over the disjoint union I of 3 and whose geometry is I? @ and with chamber set {C U : C E C, C E The basic diagram for a geometric complex C over I is the graph on I obtained by joining distinct i, j in I if for some simplex T of type {i,j}' (including 0 if III = 2), Linkc(T) is not a generalized digon. The basic diagram of a geometry is the basic diagram of its flag complex. Diagrams containing more information can also be associated to each geometry or geometric complex. The study of such diagrams was hitiated b y ' ~ i t s[TI] and Buekenout [Bu]. A graph on I is a string if we can order I = (1,. , n ) so that the edges of I are {i,i 11, 1 i < n. Such an ordering will be termed a string ordering. A string geometry is a geometry whose basic diagram is a string. Most of the geometries considered in Sporadic Groups are string geometries; for example:

. ..

c

+

<

c}.

..

12

Chapter 1 Preliminary Results

Example (10) The basic diagram of projective geometry is a string. Lemma 4.7: Assume C is a ~esidudlyconnected geometric complex such that I = I1 I2 is a partition of I such that Il and I2 are unions of connected components of the basic diagmnt of I . Then C = c1 $ C2, where Ci consists of the simplices of type Ii.

+

Proof: We may assume Ii # 0 for i = 1,2. By definition of the bwic diagram, the lemma holds if I' is of rank 2. Thus we may assume Il has rank at least 2. Let xi E ri;by 4.6 there cxists a path xl = vg . .vm = zz with T ( v ~E) I1 for i < m . Choose this path with m minimal; if m = 1 for each choice of Xi we are done, so choose xi such that m is minimal subject to m > 1. Then of course m = 2, so xi E Linkc(vl). But by induction on the rank of I?, xl is incident with x2 in Linkc(vl), and hence also in

5. The general linear group and its projective geometry

.

Theorem 4.11: Let G be a group, I = ( 1 , . . ,n), and 3 = (Gi : i E I ) a family of subgroups of G. Assume 3)is residually connected; that is, G j = ( 3 j )for all J C_ I. (a) C(G, (b) The diagram of C(G,3)is a union of strings; that is, (GillGjl) = GrGy for all i, j E I with li - jl > 1. Then -

(1) G is flag transitive on r ( G ,3). (2) r ( S j ) S r ( G j , F ~ ) f o r a l l J C I . Proof: This follows from 4.9 and 4.10. Use 4.5 to see that the conditions of (a) are equivalent and 4.2 to see that the conditions of (b) are equivalent.

r.

5. The general linear group and its projective geometry

The proof of the following result is trivial:

Lemma 4.8: If C is a geometric complez then the following are equivalent: ( I ) All flags of I? are simplices. (2) Linkc(S) = r ( S ) for each simplex S of C.

Lemma 4.9: Assume C is a residually connected geometric complex such that the connected components of the basic diagram of C are strings. Then all flags of C are simplices. Proof: Assume not and let T be a flag of minimal rank m which is not a simplex. As C is a geometric complex, m > 2. Pick a string ordering for I and let T = { X I , .. ,xm) with r ( x i ) < r(xi+i). Let x = x2. By minimality of m , { x i ,X )and (22,. . ,~ mare ) simplices. Further by 4.7, Linkc(x) = where Ci is the subgeometry on Ii7 Il = {I), and 12 = (3, ,n}. Thus { x i ,x3,. . ,xm} is a simplex in LinkC(x),so T is a simplex of C.

.

. ..

$c2,

.

.

Lemma 4.10: Let G be a group and 3 = (Gi : i E I ) cs family of subgroups of G, and assume C = C(G,F)is residually connected. Then the following are equivalent: (1) G is flag transitive on r(G,F).

(2) Each flag of r(G,3)is a simplex. (3) r ( S J )= Linkc(SJ) 2 I?(GJ,FJ) for each J E I . Proof: By 4.5.2 and 4.8, (2) and (3) are equivalent. As G is transitive on simplices of C of type J for each J E I, (1) and (2) are equivalent.

13

In this section F is a field, n is a positive integer, and V is an ndimensional vector space over F. Recall that the group of vector space automorphisms of V is the general linear group GL(V).We assume the reader is familiar with basic facts about GL(V),such as can be found in Section 13 of [FGT]. For example, as the isomorphism type of V depends only on n and F , the same is true for GL(V),so we can also write GLn(F) for GL(V). Recall that from Section 13 in [FGT] that each ordered basis X = ( x l , . . ,x,) of V determines an isomorphism Mx of GL(V) with the group of all nonsingular n-by-n matrices over F defined by Mx(g) = (gij), where for g E GL(V),gij E F is defined by Xi9 = C jgijXj:j. Thus we will sometimes view GL(V) as this matrix group. We write S L ( V ) or SLn(F) for the subgroup of matrices in G L ( V )of determinant 1. Thus SLn(F) is the special linear group. As the kernel of the determinant map, SLn(F) is a normal subgroup of GL,(F). A semilinear transfornation of V is a bijection g : V --+ V that pr* serves addition and such that there exists u(g) E Aut(F) such that for each a E F and v E V , (av)g = ao(g)v. Define I' = r ( V ) to be the set of all semilinear transformations of V . Notice the map (T : I' -+ Aut(F) is a surjective group homomorphism with kernel GL(V) and r(V)is the split extension of GL(V) by the group { f a : a E Aut(F)) 3 Aut(F) of field automorphisms determined by the basis X of V , where

.

14

Chapter 1 Prelimina y Results

Notice also that r ( V ) permutes the points of the projective geometry PG(V) and this action induces a representation of r ( V ) as a group of automorphisms of PG(V) with kernel the scalar matrices. Thus the image PI'(V) is a group of automorphiims of PG(V) which is the split extension of PGL(V) by the group of field automorphisms. If F = GF(q)is the finite field of order q we write GLn(q)for GLn(F), SLn(q) for SLn(F), PGLn(q) for PGLn(F), and Ln(q) = PSLn(q) for PSL,(F). See Section 13 in [FGT] for the definition of the transvections in GL(V) and properties of transvections.

Lemma 5.1: Let G = PGL(V), S = PSL(V), and H the stabilizer in G of a point p of PG(V). Assume n 2 2. Then (1) H is the split extension of the group Q of all transvections of V with center p by the stabilizer L of p and a hyperplane U of V complementing p. (2) Q 2 U,L r GL(U), and the action of L by conjugation on Q is equivalent to the action of L on U. (3) Q is the unique minimal normal subgroup of H n L. Proof:Let G = GL(V)and regard G as a group of matrices relative to a basis X for V such that p = ( x l ) .Then the preimage H of H in G consists of all matrices

for our construction in Chapter 6 of the Mathieu groups and their Steiner systems. In particular we will need the following result: Lemma 5.2: Let F be the field of order 4. Then (1) The group of automorphisms of the projective plane over F is PI'3(F). (2) A field automorphism f fies exactly 'sevenpoints of PG(V)

.

Proof: Let X = { x l , x ~ , x 3be ) a basis of V . First a proof of (2):A typical point of V is of the form p = (Ciaixi), with aj = 1 for some j. Then p f = (Cia f x i ) ,since the automorphism of F of order 2 defining f is a w a2. Thus pf = p if and only if there exists b E F# with bai = a: for each i. It follows that ai = 0 or b for all i. But as aj = 1, also b = 1. Hence f fixes p if and only if all coefficients ai are in GF(2). So there are precisely seven choices for p. Next let M = Aut(PG(V)) and p = (xl).Then 'I = PI'(V) I M . As I' is transitive on the points of PG(V), M = I' Mp,SO it remains to show Mp 5 rp= H . Let A be the set of five lines through p. Then HA = QB, where Q E1fj is the subgroup of 5.1 and B = (P) 2 Z3, where p = diag(a,1 , l ) and (a) = F#. Further H* = Sym(A), so Mp = HMA and it remains to show MA = QB. Now Q is regular on the sixteen lines not through p, SO M A = QD, where D is the subgroup of MA fixing the line k = (x2,x3).We must

.

S ~ O WD

with a(g) E F#, a(g) a row matrix, and A(g) E GL(U). Moreover Q consists of the matrices g with a(g) = 1 and A(g) = I, while 2/ consists of all matrices h with a(h) = 0. Further gh E Q with a(gh) = a ( h ) ~ ( h ) - l a ( g In ) . particular H is the split extension of Q by L, and Q 2 U is abelian. Further 2= Lo x K , where K is the group of scalar matrices and Lo consists of those h E 2with a(h) = 1. Thus the image L of 2/ in G is isomorphic to Lo 2 GL(U), and the action of L by conjugation on H is equivalent to the action of L 2 Lo on U 2 Q. So (1) and (2) are established. Finally as the action of L on Q is equivalent to its action on U,L (and even LnS) is faithful and irreducible on Q, so Q is minimal normal in H. Now if M is a second minimal normal subgroup of H , then ( M , Q ) = M x Q, so M 5 CH(Q)and M n Q = 1. But as H = LQ with L faithful on Q, Q = CH(Q),contradicting MnQ=l. The projective plane over the field of order 4 will be the starting point

15

6. Fiber products of groups

5 r.

First D fixes k n m for each m E A , so D fixes each point of k. Suppose d E D fixes a point T on m distinct from p and k n m. Then for each point t not on m , d fixes r ( ( T t ) n k ) = r t and then also fixes t = ( r t ) n ( p t). But then d fixes each point not on m, so d = 1. We have shown D is regular on the three points of m not on k and distinct from p, for each m E A. Hence D = B 5 G, completing the proof.

+

+

+ +

+

6. Fiber products of groups We will need the notion of the fiber product of groups at several points. For example, the notion is used in the proof of Lemma 8.17 and in the construction in Section 27 of the centralizer of an involution in the Monster. Let cYi : Ai -, &, i = 1,2, be group homomorphisms and consider the fiber product

Chapter 1 Preliminary Results

Exercises

Let pi : A -, Ai, i = 1,2, be the ith projection, and observe that we have a commutative diagram:

we have (vi @ v2)@ ( P i @ P 2 ) ) = the hypotheses of (2), for b E B, vi E 6, vl(bP1)@ vz(P2) = vl(bhp1) 63 vz(bhp2) = (vi 8 vz)(bh(pl@p2), so (2) holds.

'

Moreover the fiber product satisfies the following universal property: Whenever we have a commutative diagram

17

Remarks. The material on rank 3 permutation groups in Section 3 comes from D. Higman [Hi]. Most of the discussion on geometries, complexes, and chamber systems in Section 4 is due to Tits in [TI], with the last few lemmas from Section 4 appearing in [All. Our discussion of diagrams associated to geometries and geometric complexes has been restricted to the basic diagram. However, there is a much more extensive theory of diagrams in the literature begun by Tits , [Tl] and Buekenout [Bu]. See also [A21 and [RS].

Exercises then there exists a unique map h : B diagram commutes:

-, A

such that the following

B

.

A2

A1 a

Jaz

A0 We record this as:

Lemma 6.1: Let Pi : B -+ Ai, i = 1,2, be group homomorphisms with Plal = Pza2. Then there exists a unique group homomorphism h :B -+ A = A1 XA, A2 such that (1) (2)

hpi = P i fori =1,2. ker(pi) = ((1,az) : a2 G ker(a2)).

Lemma 6.2: Assume Ai IGL(Q),for F-spaces Q, i = 1,2. Then (I) (2)

pi @ p z : A -r GL(Q @ b)is an FA-representation. Under the hypotheses of 6.1, h(p1 @ p2) = & @ P2 is an FBrepresentation on V1@ V2.

Proof:First pi @ ps : A1 x A2 -t GL(Vl 8 V2) is a representation of A1 x A2 which restricts to a representation of A, so (1) holds. Next under

1. Prove Lemma 2.2.4. G be a 4-transitive subgroup of S6. Prove G = S6 or A6, and if the stabilizer in G of three points is of order 3 then G = A6. 3. Let I' be the projective plane over the field of order 4, L = L3(4) 5 Aut(r), and g E Aut(r) an involution fixing exactly three points on some line of I'. Let G = ( g , L) and prove G is L extended by a field automorphism and if x is a point of I' fixed by g and A the set of lines of I' through x then G$ = S5. 4. Let G be a primitive rank 3 group of even order on a set X of finite order n and let x E X. Prove (1) If K is a regular normal subgroup of G then n = pe is the power of a prime and K r Epe. (2) If n is not a prime power and G, is simple then G is simple. 5. Let I' be a string geometry on a string ordered set I = (1,. .. ,n). Prove that if ai * aj * ak with i < j < k then ai * ak. 2. Let

7. Involutions Given t E S E G and h E G, we write (S) for the subgroup of G generated by S, th = h-lth and sh= {sh : s E S) for the conjugates of t and S under h, and sG= {Sg :g E G) for the conjugacy class of S under G .

Chapter 2

Lemma 7.2 (Thompson Order Formula): Assume G has k 1 2 conjugacy classes of involutions with representatives xi, 1 5 i 5 k , and define ni to be the number of ordered pairs (u,v ) with u E xf, v E x?, and xi E (uv).Then

2-Structure in Finite Groups Proof: This is 45.6 in [FGT]. The proof is an easy counting argument. Lemma 7.3: Let I be a G-invariant collection of involutions of G and H 5 G . Then the following are equivalent: In this chapter we record some facts about the 2-subgroups of finite groups. In particular in Section 7 we recall some standard facts about involutions; that is, elements of order 2. Then in Section 8 we consider so-called large extraspecial 2-subg~0upsof a finite group G. Most of the sporadic groups contain such subgroups. They will serve as an important tool both in analyzing the structure of the sporadic groups and as part of the hypotheses under which we characterize many of the sporadics. See Chapter 5 for an idea of how this goes.

7 . Involutions In this section G is a finite group. Recall that an involution in G is an element of order 2. The following elementary result appears as 45.2 in [FGT]:

Lemma 7.1: Let x and y be distinct involutions in G , n = Ixyl, and D = (x, y) Then

.

(1) D is a dihedral group D2, of onler 2n. (2) Each element in D (xy) is an involution. (9) If n is odd then D is transitive on its involutions, so in particular x is conjugate to y in D.

-

(4) If n is even then each involution in G is conjugate to exactly one of x, y, or z, where z is the unique involution in (xy). Further z E Z(D). (5) If n is even and z is the involution in (xy) then xz is conjugate to x in D i f and only if n r 0 mod 4.

<

(1) xG fl H = xH and CG(x) H for each x E I fl H . (2) H n H g n I is empty f o r g ~ G - H . (3) The members of H n I fix a unique point in the permutation representation of G on G/H.

Proof: See 46.1 in [FGT]. Again the proof is easy. An involution z is 2-central in G if z is in the center of a Sylow 2subgroup of G .

Lemma 7.4: Assume z is a ¢ral involution in G and H 5 G such that H is the unique point of GJH @ed b y z. Then (1)

IG : HI is odd.

(2)

V x is a %element of H then xG fl H = xH.

Proof: By 7.3, CG(z) 5 H. As z is Zcentral, CG(z) contains a Sylow 2-subgroup T of G. So T 5 H and hence (1) holds. Let x be a 2-element and g E G with x,xg H. We must show xg = xh for some h E H. Conjugating in H we may take x,x9 E T. Then z E C(xg), so zg-I E K = CG(x). Let R,S be Sylow 2-subgroups of K containing z and zge1, respectively. As H is the unique point of G / H fixed by t and R is nilpotent, H is the unique point of G / H fixed by R (cf. Exercise 2.5). Similarly H ~ is-the~ unique point fixed by S. By Sylow's Theorem there is k E K with R~ = S. Then {Hg-') = Fix(S) = Fix(R)k = {Hk), so kg E H. Then xg = xkg with kg E H, completing the proof.

19

20

Chapter 2 2-Structure in Finite Gmups Lemma 7.5: Let I and J be distinct conjugacy classes of involutions of G and H a subgroup of G such that:

(a) Each member of I U J &es a unique point of G / H . ( b ) For all a E I n H and b E ( I U J ) n H with ab = ba, we have CG(ab)5 H . Then G = H .

Proof: Assume the lemma is false and let x E H n I and y E H n J. If J H then for all g E G , { H g ) = Fix(y)g = Fix(yg) = { H ) . But then H = G , as desired. So let u E J - H and D = (x,u). Then u $ xD so by 7.1, x u has even order m d x z E I U J , where z is the involution in (xu).But then x z E CG(x) 5 H , so by hypothesis (b), u E CG(z) 5 H, a contradiction. A subgroup H of a group G of even order is strongly embedded in G if H is a proper subgroup of G and H n H9 is of odd order for each g f G - H. Notice that by 7.3, this is equivalent to the assertion that H is proper in G and each nontrivial 2-element in G fixes a unique point of G / H . Strongly embedded subgroups arise in this book as follows. Let X be some subgroup of G , M = C G ( X ) ,and H 5 M. We wish to show H = M, given that C G ( ( X , t ) )5 H for enough involutions t E H . We use this information to show that if H # M then H is strongly embedded in M; then we obtain a contradiction from Lemma 7.6. For example, we may show that some 2-central involution of H fixes a unique point of M / H and then use Exercise 2.10 to show H is strongly embedded in M.

Lemma 7.6: Let H be a stmngly embedded subgroup of G. Then there exists a subgroup of H of odd order transitive on the involutions of H .

Proof: This is essentially contained in Exercise 16.5 in [FGT]; here are the details. Let I be the set of involutions in G , t E I n H , u E I - H , and K = H n Hu. Then K is the subgroup of G fixing the points H and H u of G / H and L = K ( u ) is the global stabilizer of (H, Hu). Also as each nontrivial 2-element of G fixes a unique point of G / H , we have: (a) K is of odd order. Let J = ( k E K : kU = k-'1. By (a), uv is of odd order for each v E I n L, so by 7.1, u J = I n L and K is transitive on u J . We claim:

(b) C H ( j ) is of odd order for each 1 # j E J . For if t E C H ( j ) then t is not conjugate to u in X = C G ( j ) ( u )as CG(j)a X. On the other hand X n H is strongly embedded in X , so 7.5 supplies a contradiction. Thus (b) is established. Notice (b) implies:

8. Extraspecial groups (c) Distinct involutions in L are in distinct cosets of C H ( t ) , and ( I nL( 5 IH : CH(t)l. Namely the maps v I-+ uvCH(t) and v t - i vCH(t) are injections of I n L into H / C H ( t ) and G / C H ( t ) ,respectively. For if v , w are distinct in I n L then 1 # wv E J so wv $ CG(t).Next we claim: (d) I J I = l I n H I . For let m = / I n HI and consider the set S of triples ( i , x ,y) such that i E I and ( x ,y) is a cycle of i on G J H . Observe that III = mn, where n = IG : HI, and i has (n- l ) J 2 cycles of length 2. Hence IS1 = m n ( n - 1)/2. But also

where Mz,y is the number of involutions with cycle ( x ,y) and M is the maximum of M,,,. Observe that by 7.5, G is transitive on I , so . by (c), by 7.3, H is transitive on I n H. Thus m = JW : C H ( t ) JBut IH : CH(t)l 2 M , so M < m. It follows that MXly= m for all x , y; that is, (d) holds. Finally observe: (e) Distinct elements of J are in distinct cosets of CH(t). For if j, k E J with k-l j E C H ( t )then ju, ku E I n L with ( u k ) - l ( u j ) = k-luuj = k-I j E C H ( t ) ,contrary to (c). It follows from (d) and (e) that IK : CK(t)l 1 IJI = 11 n HI. But IK : CK(t)l = ItK/ and tK C_ I n H , so tK = I n H . That is, K is transitive on I n H, completing the proof. Let H and S be subgroups of G. We say H controls fusion in S if sG n S = sH for each s E S. We say S is weakly closed in H with respect n H = { S ) . Part (1) of the following lemma appears as 37.6 to G if in [FGT]; part (2) is easy.

sG

Lemma 7.7: Let p be a prime and T E SylP(G). Then (1) If W is a weakly closed subgmup of G then N G ( W ) wntmls

fision i n C G ( W ) . (2) If A, B g T and A E B~ then A and B are conjugate i n NG(T). 8. Extraspecial groups The h t t i n i subgmup of a group G is the intersection of all maximal subgroups of G. Write +(G) for the Frattini subgroup of G. Evidently

22

Chapter 2 2-Structure in Finite Groups 9(G) is a characteristic subgroup of G. hrther

Lemma 8.1: If G is a group, X C G, and G = (X, 9(G)) then G = (X). Lemma 8.2: Let G be a jinite p-group and A a group of automorphisms of G of order prime to p. Then (1) @(G) is the smallest normal subgroup H of G such that G/H is elementary abelian. (2) A is faithful on G/@(G).

Proof: See 23.2 and 24.1 in [FGT].

An elementary abelian p-group of order pm is a direct product of m copies of the group Zp of order p and is denoted here by Epm.We can regard such a group as an m-dimensional vector space over GF(p). Thus if G is a pgroup then by 8.2.1 we can regard G/+(G) as a vector space over GF(p), so by 8.2.2, A 5 GL(G/@(G)) for each group A of automorphisms of G of order prime to p. For x, y E G, [x, y] = x-ly-lzy is the commutator of x and y, and for X,Y 5 G, [X,Y] = ([x,y] : x E X, y E Y). Recall that [X,Y] 5 X if and only if Y 5 NG(X) (cf. 8.5.5 in [FGT]). A pgroup E is extraspecial if E is finite with @(E)= Z(E) = [ElEl and Z(E) is cyclic. As a consequence, Z(E) is of order p (cf. 23.7 in [FGTI). Lemma 8.3: Let E be an extrarpecial p-group, Z = Z(E), and E = EIZ. Regard Z as GF(p) and E as a vector space over Z. Define f : E X E - ~ Zby f(f,g)=[x,y]. Then (1) (B,f )_ is a symplectic space over Z. (2) dim(E) = 2n is even. (3) If p = 2 define Q : E -+ Z by Q(Z) = x2. Then Q is a quadratic form on E associated to f , so (E,Q) is an orthogonal space over Z. (4) Let Z 5 U 5 E. Then U is extraspecial or abelian if and only if U is nondegenemte or totally isotropic, respectively. If p = 2 then U is elementary abelian if and only if 0 is totally singular. --

--

,

- . . -Proof: See 23.10 in [FGT]. Also see Section 12.

Remark. (1) See Chapter 7 in [FGT] for a discussion of syrnplectic and quadratic forms. The integer n in (2) is the width of E. We write plf 2n to denote an extraspecial pgroup of order if p is odd we also require that the extraspecial group be of exponent p.

23

8. Extraspecial groups up to isomorphism; see, for example, For odd p this determines 23.13 in [FGT]. In the rest of this section we concentrate on extraspecial 2-groups. Recall Ds and Qg are the dihedral and quaternion groups of order 8, respectively. Notice each is extraspecial of width 1. Given two extraspecial groups El and E2 define the central product El * E2 of El and E2 to be the group (El x E2)/((zlr z2)), where (y)= Z(Ei). Notice El * E2 is also extraspecial. We can extend this construction to form the central product

E1*.*.*Em=(E1*...*Em-l)*Em

---

of an arbitrary number m of extraspecial groups Ei. Thus El * * Em is extraspecial. Write DtQZ(Z for the central product of n copies of D8 with m copies of Qg.

Lemma 8.4: Up to isomorphism D t and D ; - ~ Q ~are the unique extraspecial groups of width n. The .%-'rank of D; is n 1 while D;-~Q* has %rank n; hence the groups are not isomorphic.

+

Proof: See 23.14 in [FGT].

Remark. (2) Given a prime p, the p-rank v ( G ) of a group G is the maximum m such that G has a subgroup isomorphic to Epm. Notice that if E is an extraspecial 2-group then by 8.3.4, m2(E)- 1 is the Witt index of E, so if E = Dj; then E has Witt index n while if E = D:-IQ~ then E has Witt index n - 1 (cf. page 78 in [FGT] for the definition of the Witt index). Lemma 8.5: Let E G D t or D ~ - ' Q ~ Z , = Z(E), E = EIZ, and Q the quadratic form on & induced by E. Let A = Aut(E). Then (1) c A ( E ) = Inn(E) % Ep. (2) A/c~(E) = O(E,Q) r 0Sn(2), where e = +I if E r Dg and e = -1 if E 3 D;-'Q~. Proof:See Exercise 8.5 in [FGT]. Also see Section 12.

Remark. (3) Given a group H, Inn(H) is the group of inner automorphisms of H: That is, those automorphiims of the form i, : h H hx for some x E H. Notice Inn(H) is a normal subgroup of Aut(H). Define a subgroup Q of a finite group G to be a large extraspecial tsubgroup of G if: (Ll) Q is an extraspecial e subgroup of G. (L2) CG(Q) = z(Q) = 2.

24

Chapter 2 bStructure i n Finite Groups

8. Extraspecial groups

(L3) Q 5 CG(Z). (L4) IG(Q, 2') = 1. Here klG(Q,2') is the set of subgroups K of G of odd order such that Q 5 NG(K). Given a group A, let A# = A - (1).

In particulm A = Z(R). Also [R Cl P, R] 5 [Q,Q] n P = Z n P = 1. SO R n P 5 Z(R) n P = Zg. Thus [Cp(Z),R] P n R = Zg. Thus [Cp(Z)*,R] = 1. Also Cp.(Z) = Np(A)/Zg = Cp(Z)/Zg as M 5 CG(Z).SOCp. ( 2 ) = Cp(Z)* = Cp*(R). Let Y = Cp(Z)R. Then Y* = Cp(Z)* x S* with S* 2 S extraspecial. Thus Q(Y*) = cP(S*) = Z*. Therefore Np. (Y*) 5 Cp. (2) Y*. Thus P* 5 Y* (cf. 9.10 in [FGT]) so P 5 CG(Z), contradicting (L2), and completing the proof.

Lemma 8.6: Let E4 Z A 5 G and X

E

MG(A,2'). Then

Proof: See, for example, Exercise 8.1 in [FGT]. Recall that if n is a set of primes then O,(G) is the largest normal subgroup of G whose order is divisible only by primes in n. Thus if p is a prime then Op(G) denotes the largest normal psubgroup of G. Also p' is the set of primes distinct from p and Op!(G) is the largest normal subgroup of order prime to p. O(G) = 021(G) is the largest normal subgroup of odd order. Similarly On(G) is the smallest normal subgroup N of G such that G/H has order divisible only by primes in T. So O*(G) is the subgroup generated by all p'-elements of G.

Lemma 8.7: Assume Q satisfies (L1)-(L3) in G. Then (1) If Z 5 02(G) then Q is large in G. (2) If there exists an involution b in Q- Z with Z I O2(CC(b)) then Q is large in G. (8) Assume the width of Q is at least 2 and g E G - NG(Z) with Zg 5 Q. Then Z 5 Qg, NG(ZZg)/CG(ZZg) Z S3, and Q is large in G.

Proof: Let X E MG(Q,2'). If Z 5 Oz(G) then [Z,X] 5 02(G)n X = 1. Thus X 5 CG(Z), so as Q a CG(Z), [Q,X] 5 Q n X = 1. Hence X 5 CG(Q) = 2, so X = 1. That is, (1) holds. Assume the hypothesis of (2) and let Z = (2). Then A = (z, b) E4. Notice bz = bx for some x E Q, so z E 02(CG(bz)).Thus s E 0 2 ( C ~ ( a ) ) for all a E A#. By 8.6, X = (Cx(a) : a E A#). As Z 5 Oz(CG(a)), [Z, Cx (a)] < Oz(CG(a))n X = 1. Thus X 5 CG(Z), so X = 1 by (1) applied t o CG(Z) in place of G. Finally assume the hypothesis of (3). Here let b = zg and A = (z, b). Again bx = bz for some x E Q. Observe x E M = NG(A) and x induces the transposition (b, bz) on A#. So if y E M - CG(Z) then (y, x ) 5 M induces S3 on A#. In particular we may choose g E M to act as (b, z) on A#. Then Zg 5 Q so Z = 2 g 2 L: Qg, and (3) holds. So assume M 5 CG(Z). Then Z & Qg. Let P = Qg, R = CQ(b), and P* = P/Zg. As Q has width at least 2, R = Zg x S with S extraspecial.

25

<

We now consider some examples. See Section 10 in [FGT] for a discussion of the notions of extension, split extension, and complement.

Lemma 8.8: Let G be the split extension of U % E2n+l, n 2 1, by M = GL(U), and let Z be of o d e r 2 in U. Let Q = 02(CG(Z)).Then Q Z Dg is large in G and CG(Z) is the split extension of Q by Ln(2). Proof: Observe CM(Z) is the split extension of the group W E2n of transvections with center Z by L S Ln(2). Let Q = U W and H = CG(Z).Then Q = 02(H) so (L3) holds. As M is faithful on U, CG(U) U, SO CG(Q) Z(Q) 5 U. Also Z = CV(W), SO Z = Z(Q). Thus (L2) holds. Next [W, U ]= Z, so Z = [Q,Q] = G(Q). Hence (Ll) holds. As Z 5 02(G), Q is large in G by 8.7.1. Now Q is of order 22n+11so Q has width n . Then by 8.4, m(Q) 5 n+l. But m(U) = n 1, so m(Q) = n 1 and by 8.4, Q E Dg.

<

+

+

Lemma 8.9: Let G = L,+2(2), n 2 2, z be a tmnsvection in G, and Q = O2(CG(z)). Then Q E D? is large in G and CG(z) is the split extension of Q by Ln(2). Proof: First G = GL(V) for some n+ 2-dimensional vector space V over GF(2). Let (v) be the center of z and K = CG(v).Then by 5.1, K is the split extension of U r E2n+l by M = GL(U). Further CG(s) 5 K, so CG(z) = CK(2). Therefore Q % Dg satisfies (L1)-(L3) in K , and hence also in G by 8.9. Finally Zg < U < Q for g E K - NG(Q),so Q is large in G by 8.7.3.

Lemma 8.10: Let G = M24 and (X, C) the Steiner system for G (cf. Section 18). Let z be an involution in G with Cx(z) E C. Then Q = 02(CG(z))r D: is large in G and cG(z) is the split extension of Q by L3(2). Proof: Let B E C. By 19.1, NG(B) is the split extension of U E16 by M E L4(2). Further for z E u#,B = FixX(z); for example, if T is a 3-set in B then B - T is a line in the projective plane on X - T and z is a transvection in L3(4) with axis B - T.

27

8. Extraspecial groups

Chapter 2 &Structure in Finite Groups

1

So CG(z) 5 NG(B). Now complete the proof as in 8.9.

Remark. (4) Notice by 8.9 and 8.10, G1 = M24 and G2 = L5(2) are simple groups such that there exist involutions zi E Gi with CG,(zl) E CG, (z2). There is one further simple group G3 possessing an involution z3 with CG, (z3) E CGI(zl): Namely the sporadic group H e of Held. This is the only example of three nonisomorphic simple groups Gi, 1 i 3, possessing involutions zi with CGi(zi) E CGj(zj) for all i,j. However, the classification of the finite simple groups is required to verify this fact. On the other hand the following result has an elementary prooE

< <

Brauer-Fowler Theorem: Let H be a finite group. Then there exist at most a finite number of finite simple groups G with an involution t such that CG(t) 2 H . See, for example, 45.5 in [FGT] for a proof of the Brauer-Fowler Theorem, or the original proof in [BF]. The Brauer-Fowler Theorem supplies the philosophical base for the classification. We find in Chapter 14 that M24, L5(2), and H e are the only simple groups G possessing an involution z with CG(z) isomorphic to the centralizer of a transvection in L5(2). The next two lemmas show that a group with a large extraspecial 2-subgroup is close to being simple.

Lemma 8.11: Let Q be a large extraspecial Zsubgroup of G, Z(Q) = Z = (z), and M = (ZG). Then one of the following holds: (1) Q 3 G. (2) M is abelian and F*(G) = Oz(G). (3) M = F*(G) is a nonabelian simple group. (4) M = L x Lu = F*(G) for some nonabelian simple group L and

u E Q. Further z = zlz; for some zl E L with CL(zl) = R Sy12(L) and @(R) I(zl). Moreover Q # Oz(CG(z)).

E

Remarks. (5) See Section 31 in [FGT] for the definition of the generalized Fitting subgroup F*(G) of G and a discussion of the properties of this subgroup. In particular F*(G) is the product of the Fitting subgroup F(G) of G with the product E(G)of the components of G. F'urther the - components of G are the subnormal quasisimple subgroups of G, while . a group L is quasisimple if L = [L, L] and L/Z(L) is simple. (6) With some extra work it can be shown that in case (4), G E ZzwrAs, G 2 ZzwrL3(2), or G is of index 2 in Z2wrS5. For example, Exercise 2.3 is a beginning in that direction. See Section 11 in FGT] for the definition of the wreath product AwrB of groups A and B.

Now the proof of 8.11. First O(G) E MG(Q,2') = 1 by (L4). Suppose P = 02(G) # 1. Then 1 # Cp(PQ), so' Z = Cp(QP) by (L2). Then Z 5 Z(P) G, so M = (ZG) 5 Z(P). In particular M is abelian. Also by (L2), 02(F*(cG(Z))) = 1. But as Z 02(G), 0 2 ( ~ * ( G )= ) o ~ ( F * ( C ~ ( Z ) )(cf. ) 31.14.2 in [FGT]). Thus F*(G) = 02(G) and (2) holds in this case. So we may assume G has no nontrivial solvable normal subgroup. Hence F*(G) = E(G). Let L be a component of G and Y = (LQ).As O(G) = I, L has even order, so by (L2), Z 5 Y. Now if K is a component of G not in LQ then K 5 CG(Y) CG(Z), so K E(CG(Z)) = 1. Thus Y = F *(G). It remains to show (3) or (4) holds, so we may assume G = YQ but Y # L. Hence as Q is generated by involutions, there is an involution u E Q - NG(L), Let R = CL(z) and zl the projection of z on L. Define a map n : R -, Y by x.rr = [x,u] = x-'xu. As (L, Lu) = L x Lu, n is an injection and the restriction of n to any abelian subgroup of R is a group homomorphism. h r t h e r [x, u]E Q by (L3), so RT G Q. Then for x E R, T : (3) + Q is an injective group homomorphism, so 1x1 divides 4 with ( x T ) ~= z in case 1x1 = 4. Therefore x2 = zl if 1x1 = 4. We conclude R is a 2-group with @(R) 5 (zl). By (L2), Z is in the center of a Sylow 2-group of G and hence zl is in the center of a Sylow 2-group of L. So R = CL(z) = CL(zl) E Sy12(L). As L is simple, IRI > 2 (cf. 39.2 in [FGT]), so there is r E R - (zl) and as [r,u]$ Z, r $ Q. Thus as R 5 02(CG(Z))), Q # 02(CG(Z)). It remains to show Y = LLu. As IRI > 2, Rn contains a subgroup of Q of order at least 4, so for each v E Q, 1# CRn(v). However, r E CR,(v) projects only on L and LU, so v acts on {L, LU). Thus as Y = (LQ), Y = LLU.

<

<

<

Lemma 8.12: Let z be an involution in a finite group G such that (1) F*(CG(z)) = Q is an extraspecial Bgmup of Width at least 2. (2) K 5 G with U = (zK) abelian and CG(z) = ( U ~ G ( Z ) ) .

(3) Either zK n Q # {z) or there exists u E U z E 02(CG(u)). Then G is simple and Q is large i n G.

n Q - (z)

with

Proof: Let H = CG(z). As Q = F*(H), CG(Q) = CH(Q) I Q (cf. 31.13 in FCT]). So Q satisfies (L1)-(L3) in G. Next if zK n Q # {z) then Q is large in G by 8.7.3. Similarly if u E U n Q - (z) with z E 02(CG(u)) then Q is large in G by 8.7.2. So in any event Q is large in G. Let M = (zG). Then U 5 M, so

28

Chapter 2 2-Stmcture i n Finite Groups H = (uH) 5 M. In particular as Q = 02(H), M -$ 02(G). Also as U = (zK) f (I) and (z) = Z(Q), Q is not normal in G. Hence M = F*(G) is simple by 8.11. Finally as H 5 M , Exercise 2.1 says G = M. That is, G is simpIe. The remaining lemmas in this section, while crucial to the analysis in Part 111, are more difficult and technical. The reader may wish to skip or postpone these lemmas. Lemma 8.13: Let z be an involution in G with Q = F*(CG(z)) extraspecial and A a subgroup of odd order in CG(z). Let R = CQ(A) and assume lRI 2 32, O(CG(AR)) 5 A, and g E G with zg E R - Z. Let M = CG(A)A, and M* = MIA. Then AZ = CM(RA)A, R* is a large extraspecial subgroup of M*, and zg is conjugate to z in (R, CQg (A)).

Proof: By Exercise 2.2, R is extraspecial and Q = R * [Q,A]. Thus if we let D = CM(RA) and P E Sy12(D), we have P n Q = Z. Therefore P / Z is faithful on Q/Z and hence by the Thompson A x B Lemma (cf. 24.2 in [FGT]), P / Z is faithful on R/Z. That is, P = Z. So DA = A x Z. Next by 8.7, z E Qg. So Zg # R1 = CQs(A). Then by Exercise 2.2, R1 is extraspecial, so z is fused to tzg under R1. Hence (R, R1) is transitive on (z, zg)#, so we may choose g E (R, R1). As AZ = CM(RA)A, Z* = CM.(R*), so R* satisfies (L1)-(L3) in M*. As g E M with zg E R Z, R* is large in M* by 8.7.

-

Lemma 8.14: Let z be an involution in G, H = CG(a), Q = F*(H) extraspecial of width at least 3, and t an involution in Q n Qg - zG for some g E G with zg E Q - Z. Assume CH (CQ(t)/(t, z)) = Q and let M = CG(~),and M* = M/(t). Then CQ(t)* is a large extmspecial subgroup of M* and zg is conjugate to z in (CQ(t),CQg(t)). Proof: Let P = C Q ( ~and ) observe that P * is extraspecial. As t 4 zG but tz E tQ, z is weakly closed in (z, t) with respect to G, so CM*(zt) = CH((a,t))*. In particular P* 5 CM=(t*)and as CH(P/(t,z)) = Q, CM=(P*)= Z*. That is, P * satisfies (L1)-(L3) in M*. Next t E Q n Qg and as t 4 zG, t # (z, zg) = A, so there exist T E P - C(z9) and s E CQg(t) H. Then (r, s) induces S3 on A so we may take g E (P,CQ9(t)). Therefore P * is large in M* by 8.7.3.

-

Lemma 8.15: Let Q be a large edmpecial bsubgmup of a finite gmup G, IQI = 21+2w With w 2 2, Z = (I) = Z(Q), H = CG(z), g E G - H with zg € Q, V = (2, zg), X = (Q, Qg), R = (Qg n H)(Q n Hg), and

8. Extmspecial groups

E = Q n Qg. Then (1) R = Cx(V) = Oz(X) and X/R 2 S3. (2) V = Z(R), E 5 Zz(R), and R has nilpotence class at most 3. (3) R and R / E are elementary abelian with R / E = (Q n Hg)/E x (Qg n H)/E. (4) [X, El 5 V. (5) NG(V) = XCH (V) with X 5 NG(V) and NG(V)/R = X/R x CH(V)/R. (6) R / E is the tensor product of the Zdimensional irreducible for X/R 2 S3 with the module (Q n Hg)/ E for CH(V)/R. (7) m2(E) < w 1 and in case of equality Q 2 Dg, E / V is dual to (Q n Hg)/E as a CH(V)/ R-module, and (Qg n H ) / E induces the full group of transvections on E / Z with center V/Z. (8) Let m+ 1 = m2(E). Then E22w-m-1 Z RQ/Q CHIQ(V/Z). In particular if E2"-x 2 O2(CHIQ(V/Z)) then m(E) = w 1.

+

+

Proof: As Q is a large extraspecial 2-subgroup of G, Q 9 H, so Qg nH and Q n Hg are normal in R. F'urther R = CQR(V) 5 QR. Also by 8.7, X/Cx(V) 2 S3 is transitive on v#, so we have symmetry between z and zg , and hence Qg < NG(R), so that R 5 X. As 1Q : Q n RI = 2, X/R is dihedral. Indeed as Q is conjugate to Qg in X, X/R 2 D2, with n odd by 7.1. Then QR/R inverts O(X/R) while [Q, Cx(V)] 5 CO(V) 5 R, so Cx(V) = R. Then as X/CX(V) Z S3, (1) is established: Next [Q,El = Z 5 V, so (4) holds and E 5 Zz(R).As [Qg n H, Q n Hg] Q n Q9 = El we conclude R / E = (Q n Hg)/E x (Qg n H ) / E and as @(Q)= Z, R / E is elementary abelian. Similarly @(E)5 Z n Z g = 1, so E is elementary abelian and (2) and (3) are established, except for the statement V = Z(R). But suppose y E Z(R) - V. As V = Z(Q n Hg), @ ~ . ~ i m i l ya&r lQ~g , s o y = a b , a ~ Q n R - Q g a n d b ~ Q g n R - Q . Now [Q n Hg, b] = [Q n Hg, a] = Z, so y* = b* induces a transvection on 0 with axis = (Q n Hg)/Z. This is impossible as the axis of a transvection in O(Q)is the subspace orthogonal to a nonsingular point, whereas is singular. So the proof of (2) is complete. As X induces S3 = Aut(V) on V, NG(V) = XCH(V). AS X = (Qx : x E NG(V)), X a NG(V). Then as R = Cx(V), NG(V)/R = X/R x CH(V)/R. Now Q/R interchanges (Qg n H ) / E and (&st nH ) / E for t E Q - R,

<

vL

v

30

Chapter 2 2-Structure in Finite Groups

8. Extraspecial groups

and as [Q n Hg, t] I Z, (Q nHg)/E = CRIE(t). Hence R/E is the sum of m((Q n Hg)/E) 2-dimensional irreducibles for X/R E S3, and thus (6) holds. Finally as @(E) = 1, m(E) L w 1 by 8.3. By (5), RQIQ a CH/~(V/Z).Also RQ/Q R/(R n Q) = (Qg n H)(R n Q)/(R n Q) r (Q n Hg)/E E E2zw-m-1. Thus (8) holds, Assume m(E) = w 1. Then by 8.4, Q G D?. Further E is a maximal totally singular subspace of Q, so Q/Eis dual to E as an NH(E)-module, and hence as = (Q n Hg)/Z, also (Q f l Hg)/E is dual to E/V as an NH (E)-module.

Lemma 8.17: Let G be a perfect group with F*(G) = Q an extraspecial 2-group and G/Q quasisimple. Let p : G -t G be the universal covering of G, 2 = ker(p), and Q = p - l ( ~ ) .Assume

+

+

vL

Lemma 8.16: Let Q be a large extraspecial 2-subgroup of a finite group G of width w 2 2, Z = (z), H = CG(z), g, k E G with E8 g V = (z, zg,zk) ) Q n Qg n Q~ = U. Let X = ( Q , Q ~Q, ~ ) ,R = CQ(V)CQ#(V)CQ~(V),S = (Q n Qg)(Q n Q ~ ) ( Q nQ ~ ~ I+ ,1 ~= m(Q n Qg), and u = m(U). Then (1) R = Cx (V) = Oz(X) and X/R = GL(V) cx L3(2). (2) V 5 Z(R) and [X, U] 5 V with @(U)= 1. (3) NG(V) = XCH(V) with X NG(V) and NG(V)/R = X/R x

CH(~)/R(4) S/U is the sum of m - u + 1 copies of the dual of V* and R/S is the sum of 2(w - m) u - 3 copies of V as an X/R-module.

+

ProoE The proof is much like that of 8.15. As in 8.15, R = CQR(V) 5 QR, and then R q X. Moreover Q, Qg, Q~ induce the group of transvections on V with center 2, Zg, Zk, respectively, so V 4 X and X/CX(V) = GL(V) s L3(2). Next [Q,Cx(V)] 5 CQ(V) 5 R, so Cx(V)/R 5 Z(X/R). Let Y = (Q,Qg). By 8.15, YR/R C Sq, so by Gaushutz's Theorem (cf. 10.4 in [FGT]) X/R splits over Cx(V)/R. Further Q n Hg < O ~ ( Y ) R 5 XmR and similarly Q n Hk 5 XwR, so Q = (Q n Hg)(Q nH ~ 5) XwR and hence X = XwR and R = Cx(V). Thus (1) is established. The proof of (2) and (3) are as in 8.15. Next [Y,QnQg] 5 ZZg 5 V by 8.15, and S/(Q n Qg) = ( S n Q)(S nQg)/(Q n Qg) is the sum of natural modules for Y/02(Y) by 8.15, with [S,Oq(Y)] < Q n Qg. Similarly (Q, Q ~ acts ) on S, so S a X. Then S/U is the sum of m((Q n Qg)/U) copies of V* by Exercise 2.9. Similarly Qg n H and Q~ n H centralize c Q ( v ) s / s , so H n x = ( ~ ngH , Q ~n H ) centralizes c ~ ( v )= Q n R modulo S. Further by 8.15, R/(Q n R)S G (Qg n R ) ( Q ~ n R)/(Qg n Q ~ ) is the sum of natural modules for (H f l X)/02(H n X), and then (4) holds by Exercise 2.9.

31

(a) Q = Q/Z(G) is an absolutely irreducible GF(2)G-module, and (b) H ~ ( G , Q )=o. Then (1) Q = [ Q , G ] x z ~ ~ ~ ~ [ Q , G ] E Q . (2) The natural map a : G/Q -t G/Q is the universal covering of

G/Q with 2 %! ker(u). (3) H is a finite group with F*(H) 3 Q and H/Z(H) g G/Z(G) if G] in Q. and only if H G/V for some complement V to IQ, (4) If G/Q = R(Q) and cr is a transvection in O(Q) then a lifts to an automorphism of G.

Remark. (7) See Section 33 in [FGT] for a discussion of coverings. Recall O(Q) and R(Q) are the isometry group of the orthogonal space Q and the commutator group of that isometry group, respectively. Now the proof of lemma 8.17. Let R = o~(Q).As G/Z r G is perfect with 2 5 Z(G) and Z(Q) = Z(G), Z(G) = p - l ( ~ ( ~ ) )Let . Z = z(&). Observe Q = RZ. As Q is extraspecial, R is of class 2. Let P = [R,GI. As H ~ ( G , Q )= 0, R/?(R) splits over (Z n R)/@(R)) by 17.12 in [FGT]. Thus P/@(P) E Q and Q = PZ with @(P)= P f l Z. As is of exponent 2 and P is of class 2, @(P)is of exponent 2 (cf. 23.7 in [FGT]). Let U be a hyperplane of @(P).Then P/U is extraspecial so the commutator map induces a nondegenerate bilinear form ( , ) on P/@(P) Q preserved by G/Q G G/Q as described in 8.3. As G is absolutely irreducible on Q, ( , ) is the unique G-invariant nondegenerate bilinear form on Q (cf. Exercise 9.1 in [FGT]). Pick x, y E P with (Z,&) # 0 and let u = [x, y]. If @(P)# (u) we can pick U with u E U,SO 0 = [x, y] mod U, contradicting (Z, 6) # 0. Thus @(P)= (u), so P r Q and Q = P x 2. That is, (1) holds. As G is perfect, so is G and hence also G/P.Thus as G/Q is quasisimple, so is G/P.Let a1 : L -+G/Pbe the universal covering of G/Pand a2 : G -+ G/Pthe natural map. Form the fiber product A = L x,=,/~ G with respect to the maps a1 and crq as in Section 6, and let pl : A -+ L be the projection. By 6.1, ker(pl) = ( ( 1 , ~ ): a E ker(aa)) P. As P = [P, GI, P i s generated by commutators a = [x, b], b E P, x E G. As a 1 is surjective there is y E L with yal = xcrq. Then (y,x) E A and

32

Chapter 2 &Structure in Finite Groups

[(y, x ) , (1, b)] = ([y, 11,[ x ,b]) = (1,a). Therefore ker(pl) _< [A, A]. But as

L is perfect, A = [A,A]ker(pl), so A is perfect. Now A is perfect and p2 : A -, G is a surjection with ker(pz) = {(c,l) : c E ker(a1)). But ker(a1) L Z(L), so k e r h ) 6 Z(A), and hence p2 is a covering. Therefore as p : G + G is universal, p2 is an isomorphism (cf. 33.7 in [FGT]). Thus 1= ker(al) and G/Pr L. Thus (2) is established. Assume the hypotheses of (3). Then H is a perfect central extension of G/Z(G). But as G is the universal covering group of G, it is also the universal covering group of G/Z(G) (cf. 33.7 in [FGT]), so we have a surjection r#~: G + H with ker(4) 2. Then as 4 : P -t 02(H) is an isomorphism, ker(4) is a complement to P in Q, establishing (3). Finally assume the hypotheses of (4). Then a lifts to an automorphism /3 of G/Z(G) as G/Z(G) = Aut(Q) (cf. Exercise 8.5.3 in [FGT] or 12.16). Then as G is the covering group of G/Z(G) and P is a covering of G, /3 lifts to an automorphism of G.

<

Remarks. The Brauer-Fowler Theorem appears in [BF]. See Chapter 5 for more discussion of its importance in the study of simple groups. I believe 7.6 is due to Feit. Phillip Hall introduced the notion of an extraspecial group, and Thompson and Janko did the early work on groups with a large extraspecia1 2-subgroup. Eventually such groups were classified via the efforts of a number of mathematicians, most notably Timmesfeld in [Tm]. Many of the Iater results in Section 8 come from Section 17 of [A2j.

Exercises 1. Let Q be a large extraspecial subgroup of G and Q 5 L q G. Prove

G = LNG(Q). 2. Let Q be a large extraspecial 2-subgroup of G and A a subgroup NG(Q) of odd order. Prove Q = CQ(A) * [Q,A], [Q,A] is extraspecial, and either CQ(A) = Z ( Q ) or CQ(A) is extraspecial. 3. Let L be a nonabelian simple group and z an involution in L such that R = CL(z) E Sylz(L) and @(R)= (z). Prove R D8. (Hint: Use transfer and fusion arguments such as in Section 37 and Exercise 13.1 in [FGT]. In particular prove z is not weakly closed in R with respect to G and exploit this fact.) 4. Assume z is an involution in a finite group G, H = CG(z), Q = F*(H) is extraspecial of width at least 2, z is not weakly closed in Q with

Exercises

33

respect to G,H/& is simple, H is irreducible on Q/(z), and Q is not the weak closure of z in H. Prove G is simple. 5 . Assume G is transitive on X and z E G fixes a unique point of X. Prove each nilpotent subgroup of G containing z fures a unique point of X. 6. Let G be a finite group with F*(G) = Q r ~ ; and f G/Q L?L' Sli(2). Prove there exists Z3 2 A G with CQ(A) = R Q;, and that CG(A)= A x L with R = F*(L)and LIREQg(2). 7. Let A be a nontrivial elementary abelian 2-subgroup of G, M = NG(A), and assume each element of A# fixes a unique point of G/M. Prove either G = M or NM((t,A)) is transitive on cA(t)# for each involution t E M. 8. Let G be a 2-group, H be of index 2 in G, t E G - H , and assume

<

is a series of normal subgroups of G such that Hi+i/Hi is elementary ( tIHiS1/Hil )1~ for all 0 5 i < n. Prove abelian and 1 ~ ~ , + , / ~ , = (1) G is transitive on the involutions in tH. (2) = !HI9. Let G = L3(2), S4 r H G, F the field of order 2, X the permutation module for G on G / H over F, and Y the natural &dimensional module FG-module in which H fixes a point. Prove (1) dim(Cx(H)) = 2. (2) X = [X, G] @ Cx(G) with dim(Cx(G)) = 1, Soc([X, q)% Y*, and X/Soc(X) 2 Y . (3) If V is an FG-module in which [V,02(H)I 5 CV(H)and V/CV(H) is the sum of natural modules for H/02(H) L2(2), then V is the sum of dim(CV(H)) copies of Y as an FG-module. 10. Let G be a finite group, H a proper subgroup of G, and z a %central involution of H such that z fixes a unique point of GIH. Let U be the set of Zsubgroups U of H such that CG(U) is not contained in H. Then (1) G has a normal subgroup M such that M n H = 02(.H). In particular if z E o ~ ( H )then M, o ~ ( H )satisfy our hypotheses. (2) U # 0 and if U is maximal in U,K = CG(U), and K* = K/U then (a) ( H n K)* is strongly embedded in K*. (b) ZG n u = 0.

<

34

Chapter 2 2-Structure in Finite Groups (c) (H n K)* has a subgroup of odd order transitive on the involutions of ( H n K)*. (Hint: Use 37.5 in [FGT]) for (I).) 11. Let H be a finite group with F*(H) = Q an extraspecial 2-group, (z) = Z(Q), H = H/(z), and H* = HI&. Recall by 8.3 that Q is an orthogonal space over GF(2) and H* 5 o(Q). Prove (1) Each involution t* in H* is of type am, bm, or h,where m = m ( [ ~ , t ] )m , is even for type a and c but odd for type b, [Q,t] is totally isotropic but [Q, t] is totally singular if and only if t* is of type a, and finally t* $ R(Q) if and only if t* is of type b. (2) tz E tQ if and only if t is of type b or c. (3) iQ= ~ Q Fwhere , Q ~ / ( z= ) [~,t]. (4) Each involution in tQ is in tQ:, where Q$/(z) = Ca(t). (5) O(Q) is transitive on involutions of type a,, b, and c, for each m, and a ( ~ is) also transitive except that if m = dina(~)/2 then R(Q) has two orbits a& and a%, where for tr E a i , [Q,ti], i = 1,2, are the two classes of maximal totally singular subspaces of Q under R(Q). (Hint: See [ASe].)

i

.

Chapter 3 Algebras, Codes, and Forms

The Griess Algebra is a certain nonassociative, commutative algebra on 196,884 dimensional Euclidean space constructed by R. Griess. Its a u t e morphism group is the largest sporadic group, the Monster. In Section 9 we record a few elementary facts about algebras; in particular we find that a nondegenerate bilinear form 7 on a space V determines an isomorphism between the space of algebras on V and the space of trilinear forms on V. Thus the Griess algebra corresponds to a trilinear form and in Chapter 10 we use both the algebra and the form to construct the Monster. In Section 11 we find that each map P : V -+ F of a vector space V over the field F of order 2 into F with P(0) = 0 determines a sequence of derived forms P, : Vm -t F. This sequence is used in Chapter 4 to study a certain class of loops we call symplectic %loops which are generalizations of 2-groups of symplectic type. Following Conway, we use a certain symplectic 2-loop discovered by Parker to construct a 2local in the Monster and then in Chapter 10 use this 2-local to assist in the construction of the Griess algebra. In Section 10 we briefly recall a few elementary facts about error correcting codes. In Chapter 6 the Steiner system for the Mathieu group M24 is used to construct the Golay code, a doubly even binary code preIn Section 11 we find that each binary code V naturally served by M24. determines a map P and its derived sequence. If the code is doubly even then P3 is a trilinear form. In Chapter 7 we use the trilinear form defined by the Golay code to study the Mathieu groups.

36

9. Forms and algebras

Chapter 3 Algebras, Codes, and Forms 9. Forms and algebras

In this section F is a field and V an F-space. Given vector spaces I/i, 1 i n, denote by M(V1,... ,Vn;V ) the F-space of all functions f : Vl x x Vn -, V where addition and scalar multiplication are defined by

< <

-

.

for a , € M(V1,. . ,Vn;V ) ,xi E I/i, and a E F. Write L(V1,. . . ,Vn;V ) for the subspace of all n-linear maps a ; that is, for all i and each choice of X j E ~ , j # i , t h e m a p z i ~ a ( x l ., ,xn)isalinearmapfrom&toV. .. We write Ln(V) for L(V1,. . . ,Vn;F ) and Mn(V) for M(Vl, ... ,Vn;F ) , when V, = V for each i. Thus Ln(V) is the space of n-linearforms on V . Given a group G represented on each &, we obtain a representation of G on M ( h , . . . ,Vn;V )via (ag)(xl,.. . ,x,) = ,xng-l). In particular we have that GL(V) acts on Ln(V) in this manner. We also have a representation of the symmetric group

. ..

Sn = Sym((1,.. . ,n ) ) on Ln(V) via (as)(xl,... ,x,) = a(xls-1,. .. ,xn,-I). The form a is symmetric if a is fixed by each element of Sn and a is alternating if a s = sgn(s)a for each s E Sn, where sgn(s) = 1if s E An and sgn(s) = -1 if s E Sn -An. Let a E Mn(V). An isomety of (V,a) is some g E GL(V) such that cug = a , or equivalently for all xi E V . Write O(V,a ) for the isomety group of the form. Let X = { x i , . .. ,xm) be a basis for V and write X n for the set of all ordered n-tuples from X. Given a E Ln(V),we write

37

Remark 9.2. If a is symmetric then o = JE13JS3 a J , where r3/s3 is some set of representatives for the orbits of S3 on I3 acting via subject to our convention of displaying only one monomial of a in an orbit of S3.

Lemma 9.3: Assume a is a symmetric trilinear form on V , G 5 O(V,a) is a 2-group of exponent 2 acting on Fx for each x in some basis X for V , and azyzxyz is a monomial of a with azyz # 0. Then

CG((X, Y ) ) = CG(.Z). Ifg E G inverts x and y then zg = z. (2) (1)

Proof: Let g E G, by hypothesis wg = bw , for each w E X and some b, E F. As G is of exponent 2, bw = f l . Now axyz = a ( x ,y,z) = a(xg,yg,zg) = bxbybzaxyz,so as axyz# 0,bxbybz= 1.

An algebra on V is some T E L 2 ( v ;V ) = L(V, V ;V ) . We write u * v for the image r ( u ,v ) of the pair (u,v ) E V x V under T and call u * v the product of u and v. Thus an algebra is a vector space together with a bilinear product.

Lemma 9.4: Assume 7 is a nondegenemte bilinear form on V . Then

There exists an isomorphism by : r I+ by(r)of the space L'(v;V ) of all algebras on V with the space L 3 ( v ) of all trilinear forms on V , where b7(r)(x,y , Z ) = d x * y, z ) . (2) The inverse of this isomorphism is t7 : a -+ t y ( a ) , where t7(a)(x,y ) is the unique element of V such that r(t,(x, y), z ) = a($,y, z ) for all z E V . (3) The stabilizers in O(V,y) of r and by(r) are the same.

(1)

We have the following analogue of 9.1:

eiEl&

..

to indicate that a(yl,... ,yn) = ayl,...,y,. The term ayl,... ,y, y l q yn is called a monomial of a. Observe that as a is n-linear, a is determined by its monomials. Further if a is symmetric then ay ,, - ayla,... , for all s E Sn, so our convention is to write down just one monomial from each orbit of S,.

Lemma 9.1: Let v = Bier& and a E L ~ ( vThen ). a = JEZ3 aJ , where for J = (i,j, k ) E I ~ a5 , € L ( K ,V j ,Vk;F ) is defined by ( Y J ( Y ~ , Y ~ >=Ya~() ~ i , ~ j , for ~ k YT) , E K.

Lemma 9.5: Let V = and T an algebra on V . Then T = J E Z 3 T J , where for J = (i,j, k ) E 13, T J E L(K, %; Vk) i s defined by T J ( Y ~yj) , = ~ k ( r ( y~j )i) , for YT E VT and pk : V Vk the kth projection. Remark 9.6. Assume 7 is a nondegenerate bilinear form on V . Define an algebra r to be symmetric with respect to 7 if the trilinear form b = by(r) of lemma 9.4 is symmetric. Notice that this forces T t o be commutative; that is, r ( x ,y ) = ~ ( yx ), for all x , y E V . Assume in addition that V is the orthogonal direct sum of subspaces &, i E I ; that is, (&, 5)= 0 for i # j. (We write V = V I L IVk to

38

Chapter 9 Algebras, Codes, and F o m

39

9. Forms and algebras

indicate that V is the orthogonal direct sum of Vl,. . . ,Vk.) By Remark 9.2, b = CJEIsIS3 bj , where 13/s3is some set of representatives for the . claim: orbits of S3 on I ~We

Lemma 9.7: bJ = b7(rJ). For let J = (i,j, k ) . Then for xr E

6,

Now r(xiCi, x j ) = CTP T ( T ( X ~ ,x j ) ) = CrTijr(xi1x ~and ) then as our sum is orthogonal1 xi,xj),~ k =) ( ~ i j k ( xxj), i , xk) = bnl(7~)(xi, x j 1x k ) , establishing the claim. It follows from 9.2, 9.4, and 9.7 that T is determined by the maps rJ, J E 13/s3, so we abuse notation and write

Notice that for i,j E I , ri = riii is symmetric with respect to the restriction of 7 to and riij is commutative. Conversely if for each J = {i,j, k) E 13/s3 we are given maps t~ E L(V,, Vj; V k ) such that t~ is symmetric with respect to the restriction of y to V, when i = j = k and t~ is commutative when i = j, then t = CJ E I S I S 3 t~ is a symmetric algebra (subject to our notational conventions) with bt = CJ E I B I S 3 btJ .

Lemma 9.8: Assume X is a basis for V and G 5 GL(V) permutes Y = f X . Let 0 be a symmetric trilinear form on V and

the set of nonzem monomials of

p

on V . Then

(1) G 5 O(V,p) if and only if G permutes M . That is, for all g E G and Y , z) E Y 3 , axyz = axg,yg,zg. (2) Assume H a G, K F G with G = H K , and M o G M such that K permutes Mo, H < O(V,p), and each member of M is H-conjugate to some member of Mo. Then G < O(V,P). (

~

1

ProoE Part (1) follows as /3 is determined by its monomials. Assume the hypothesis of (2).As H < O(V,P)and G = H K , it suffices to show K < O(V,p). Hence by (1) it suffices to show K permutes M. But if axy,xyz E M then by hypothesis there are h E H and a~,p,~3$.i. E Mo with ( t , g , i ) h = ( x ,y, 2). As K acts on Mo, agk,pk,ik = a5,pc for all k E K , and as H < O(V,P), auvw = aug,vg,wgfor all g E H and

Lemma 9.9: Let char(F) # 2, -y a nondegenerate symmetric bilinear form on V , r a symmetric algebm map on V , b = b7(r), G < O(V,y, b), and z an involution in the center of G. Let V+ = Cv(z), V- = [V,z], Q 5 G a 2-group with 9(Q)5 (z),Vs = CV(Q), and & = [V+,Q]. For a E V define

Then (1) v = v - I V , I & . (2) X E Hom(V,End(V)),where X : a H A,. (9) V+ is a subalgebm of V . (4) If U is a G-invariant subspace ofV then G acts on End(U) via 9 : P ++ ~ 9 where , 4 ( u ) = p(ug-l)g. Further A: = Xag for each aEV andg~G. (5) X induces E H m ( U ,End(W)) for W ) = (V+,V-), (Vs,V-), (V,, V-), and (Vs,V,), where A$ ( w ) = Xu(w) fm u€Uandw€W.

El

Proof: As char(F) # 2 and @(Q)5 ( z ) ,we can diagonalize Q on V+. Pick a basis X for V that is the union of bases for V,, V,, and V-, and such that Q is diagonal with respect to X on V+. Let Xi = X il4. Now for x E X+, y E X->y(xly) = y(xz1yz) = T(x,-Y) = - ~ ( x , Y ) , so y(x, y ) = 0 and V = V-IV+. Similarly as Q is diagonal on V+, v+= V S I & . As T : (u,v ) H u * v is bilinear, (2) holds. As z preserves r , (3) holds. = The first statement in (4) is easy. For the second, A:(v) = (a * ~ 9 - l = )~ ag * v = Aag(v),SO XQ = Aag. Suppose V = U I W and b(u,ul,w) = 0 for all u,u' E U, w E W . Then 0 = b(u,ul,W ) = (U * w,ul) = (AU(w),d),SO XU(W)C U I = W , and hence E H m ( U ,End(W)) by restricting (2). In particular b(V+,V+,V-) = 0 by 9.3 applied to G = ( z ) ,so (5) holds for U = V+ and W = V . . Then ( 5 ) also holds for U = Vs and V, by restriction. Similarly b(G,V,, V,) = 0 by 9.3 applied to G = Q / ( z )acting on V+, so (5) holds for (U,W ) = (V,, V,). ~~l~

40

41

Chapter 3 Algebras, Codes, and Forms

11. Derived Forms

10. Codes

In Section 19 we will encounter codes preserved by the Mathieu groups including the binary Golay code which is a perfect (23,12)-code with minimum weight 7 and correcting 3 errors.

A linear binary e m r correcting code is a triple C = (V,X, U), where V is a finite-dimensional vector space over GF(2), X is a basis for V, snd U is a subspace of V. The elements of U are the code words. If d i m ( V ) = n and dim(U) = m then the code is an (m,n)-code. We can view V as the power set of X with addition of two subsets u, v of X defined to be the symmetric difference of u and v. From this point of azx view the basis X consists of the one element subsets and v = EXEX is identified with its support {x E X : ax = 1). The weight of a code word u is the order of its support in the basis X, or equivalently if we regard u as a subset of X by identifying u with its support, then the weight of u is (uI.The Hamming distance d on U is defined by

Thus d counts the number of places in which u and u' differ. The minim u m weight of the code is the minimum weight of a nonzero code word. Notice this is also the minimum distance between distinct code words. The code words can be regarded as the admissible words sent as a string of n zeros and ones. If errors are introduced in transmission we receive a different word, hopefully in V- U.If M is the minimum weight of the code, then the code can correct e errors, where e = [(M- 1)/2]. Namely if we receive a word v then we decode v as the member of U of minimum distance from v. As long as at most e errors have been introduced, there will be a unique code word u at minimum distance from v, and u will have been the word sent. Thus we will have corrected the error introduced in transmission. Given any u E U and positive integer r , we can consider the ball B,(u) = {v E V : d(u, v)

< r)

of radius r around u. A code C is perfect if C has minimum weight 2ef 1 and for each v E V, there is a unique u E U with v E BT(u). That is, the code is e-error correcting and every member of V can be decoded uniquely. The group of automorphisms of the code (V,X, U)is the subgroup of Sym(X) acting on U.Equivalently it is the subgroup of GL(V) acting on X and U. The level of the code is the greatest integer k such that 1 2 ~ 1 = 0 mod for all u E U.If the level of the code is at least 1 the code is said to be even and if the level is at least 2 the code is doubly even. Define the code to be strictly doubly even if the code has level 2.

11. Derived Forms In this section V is a finite-dimensional vector space over F = GF(2) and P : V -+ F is a function with P(0) = 0. For m a positive integer and x = (xi,. . . ,xm) E Vm, write 2m for the power set of (1,. ,m) and for I E 2m let I(x) = xi. Define

..

xiGI

The form Pmis the mth derived form of P.

Lemma 11.1: Let P form of P. Then

E M(V; F) with

P(0) = 0 and Pmthe m t h derived

(1) Pmis symmetric. (2) FOT all x,y,xi E V,

If v = {xl,. .. ,xm) is linearly dependent then Pm(v) = 0. (4) If v = {xl,. ,xm) is linearly independent then Pm(v) = C u E ( v ) P(u).

(3)

..

Proof: Parts (1) and (4) are immediate from the definition of Pm.In part (2) let z = (x2,. .. ,x,). Then by definition of Pm,

so (2) holds. We prove (3) by induction on m. If m = 1 then (3) holds as P(0) = 0. Assume the result for m. Let y = x i and x = Xm+l. We may assume x E (y, x2,. ,xm) and x y E (22,. ,xm). SO by induction on m, Pm(x + y, Z) = 0.Further if (xi,. . ,x,) is linearly dependent then P(x,z) = P(y,z) = 0 by induction, while if (xl,. . . ,xm) is linearly independent then P(x, z) = P(y, z) by (4). Thus Pm+l(x, y, Z) = P,(z, Z) P m ( ~Z), = 0, SO that (3) holds.

..

+

+

.

..

42

Chapter 3 Algebras, Codes, and F o m

11. Derived F o m s

Define deg(P) to be the maximum d such that Pd # 0. By 11.1.2, Pm = 0 for m > dim(V), so deg(P) is well defined and deg(P) 5 dim(V). Observe also that by 11.1:

We now specialize to the case where P is a form of degree 3, and let C = Pz and f = P3. Thus f is a symmetric trilinear form by 11.2. The triple (P,C, f ) induces extra geometric structure on the projective geometry PG(V) of V (cf. Example 1 in Section 4) by allowing us to distinguish subspaces of the same dimension via the restriction of our forms to such subspaces. The next few lemmas introduce some geometric concepts useful in studying this geometry. Recall the radical of a symmetric bilinear form b on V is Rad(b) = V I = {u E V : (u,v) = 0 for all v E V). Similarly a vector v E V is singular with respect to a quadratic form q on V if q(v) = 0. More generally see Chapter 7 in [FGT] for a discussion of bilinear and quadratic forms. In particular the proof of the following lemma follows trividy from the definition of a quadratic form.

Lemma 11.2: Let P E M(V; F) with P(0) = 0 and d = deg(P). Then Pd is a symmetric d-linear form on V. Examples (1) The form P is of degree 1 if and only if P is linear. (2) The form P is of degree 2 if and only if P is a quadratic form on V (cf. Chapter 7 in [FGT]). Moreover in that event Pl is the bilinear form associated with P and O(V, P) is an orthogonal group. Lemma 11.3: If X is a set, V the GF(2) space of all subsets of X under symmetric diffemnce, and u,v, w E V then

Lemma 11.4: Assume (V, X, U) is a binary error correcting code of level 1 and identify each v = CzEX azx E V with its support {x E X : ax = 1). Define P E M(U; F ) by P(u) = Ju1/2' mod 2. Let (xl, ,xm) E Urn. Then

...

(1) I n g l x i l 5 0 mod 21-m+1 f o r m < l + l . (2) Pm(x1, . . . , ~ ~ ) = l x l n . ~ . n x ~ l /mod2. 2'+~-~ (3) deg(P) S 1 1.

+

Proof: We first prove (1) and (2) by induction on m. Notice when m = 1, (1) holds as our code is of level 1 while (2) holds by definition of P. So the induction is anchored. Next let x = XI, y = x2, z = (xg,. . . ,xrn), and v = xi. Then by induction on m and 11.3,

nZ3

43

Lemma 11.5: For x E V let f, E L ~ ( v )and Cx E M ~ ( v ) be defined by fsc(ylz) = f (x, y, Z) and Cx(y) = C(x, y). Then CZ is a quadratic form on V with associated bilinear form fx.

nUEU

Define R(x) = Rad(f,) and for U 2 V define R(U) = R(u) and UB = {x E V : f, = 0 on U). Observe that as f is trilinear, UB is a subspace of V. Define U to be subhyperbolic if UB is a hyperplane of V. Lemma 11.6: Let U be a subhyperbolic subspace and define fv E M ~ ( u ) by fv(x, y) = f (z, x, y) for z E V - UB. Then (1) fu is independent of the choice of z V U8. (2) fv is a symmetric bilinear form on U. (3) For A 5 U,R(A) n U is the subspace of U orthogonal to A with respect to fu.

-

Proof: Parts (1) and (2) are trivial. For u E U and v E UB, f (A, u, v) = 0, so u E R(A) if and only if fu(A, u) = 0. Thus (3) holds.

=Prn-1(x,z)+Pm-l(y,z)-l~nynvl/2 mod 2. Thus as Pm-l(u, z) E Z for u = x, y,x + y, we conclude lx n y n vl = 0

Define a U 5 V to be singular if P is trivial on U and V = UB. Define a line 1 of V t o be hyperbolic if all points of 1 are singular but 1 is not singular.

mod 21-m+1 and by 11.1.2, Pm(x, y, z) = lx n y n ~ 1 / 2 l - ~ + 'mod 2. That is, (1) and (2) are established. Finally when m > 1 1, (2) says -P~(X y,, z) = 0, so (3) holds.

Lemma 11.7: Let S V with P(s) = 0 for all s E S. Then (S) is singular if and only if S G R(s) for each s E S.

Example (3) Let X be a basis for the GF(2)-space V and U the core of V with respect to X; that is, U consists of the vectors of even weight. Then (V,X, U) is of level 1, so the form P of 11.4 is of degree 2. That is, P is a quadratic form on U.

Proof: Let U = (S)and x E V. Certainly the condition on S is necessary for fx t o be trivial on U for all x E V. Conversely given the condition, U is generated by a set of pairwise orthogonal singular points with respect to the quadratic form C,,so C, is trivial on U.Finally let u = CKTt for some T C S; we prove P(u) = 0 by induction on )TI.Namely u = s v,

+

+

44

+

where v = Ctf t and by induction P(v) = 0, so P(u) = P(v) P(s) C(s,v ) = 0.

+

Lemma 11.8: Assume (V,X , U ) is a strictly doubly even code and define P E M'(v), C E M ~ ( V )and , f E M 3 ( v ) by P ( x ) = Ixl/4 mod 2; C ( x ,y ) = 1x n y1/2 mod 2;

f ( x ,y, z) = 1x n y n zl mod 2. Then (1) C = P2 and f = P3 are the derived forms of P. (2) deg(P) 5 3 so f is a symmetric trilinear form on V with f ( x , x ,y) = 0 for all x , y E V . (3) If dim(V) r 0 mod 8 then there are induced forms on V / ( X ) .

Proof: This follows from earlier lemmas. Define a 3-form on V to be a triple F = (T,Q, f ) such that T : V -, F , Q : v2-* F, Q is linear in the first variable, and for all x, y, z E F:

(F1) T ( x + Y ) = T ( x )+ T ( Y )+ Q(x,Y ) + Q(Y,2). (F2) f is a symmetric trilinear font on V . (F3) Qx is a quadratic fonn with associated bilinear fonn f,; that is, QX(Y+ 4 = Q X ( y + ) Q x ( ~+ ) f d y 14, where QZ : Y Q(x,Y ) and fz : ( Y , z ) H f (2,Y , 4. Thus T is a cubic form and Q is a linear family of quadratic forms.

Lemma 11.9: Let X be an ordered basis for V , and f a symmetric trilinear form on V with f ( x ,x , y) = 0 for all x , y E X . Then (1) 3 = (T,Q, f ) is a 3-form, where

(2) 3 is unique subject to T ( x )= Q(v,x ) = 0 for all v E V , x E X . (3) If G O(V,f) and for all x E X , u E xG, and v E V , T ( u ) = Q(v,u ) = 0, then G 5 O(V,3).

<

Proof: Exercise 3.1.

45

Exercises

Chapter 3 Algebras, Codes, and F o m

Remarks. The material in Section 11 comes from [A41 but part of that material is a rediscovery of earlier work of H. Ward. In particular the notion of a derived form is a special case of what Ward calls combinatorial polarization in [Wa2],while the coding theoretic construction of Lemma 11.4 appears first in [Wa3].

Exercises 1. Prove Lemma 11.9. 2. Assume the hypotheses of Lemma 11.8 and define €:U-tv* by cJ(v)= I J nvl mod 2. Prove: (1) E is a well-defined surjective linear map with V 5 ker(e). (2) If dim(V) = dim(U)/2 then e induces an isomorphiim of U/V and V*. (3) For u,v,w E V , cunw(v)= f(u,w,v). 3. Let V be a GF(2)-space and P : V -+ GF(2)a form. Suppose w 5 V with P(w) = 1 for all w E w#. Prove: (1) If dim(W) = 3 then P3(x,y, z) = 1 for each basis ( x ,y, z ) of V . (2) If P has degree at most 3 then dim(W) 3.

<

12. Symplectic 2-loops

Chapter 4 Symplectic 2-Loops

Recall that a loop is a set L together with a binary operation possessing an identity, and such that for each x E L, the left and right translation maps a H xu and a H ax are permutations of L. Notice that the associative loops are precisely the groups. A loop is Moufang if it satisfies a certain weak form of associativity (cf. Section 12). For example, by Moufang's Theorem (cf. 12.2) each 2-generator subloop of a Moufang loop is a group. In Section 12 we study loops L which are extensions of the additive group of the field F of order 2 by a finite-dimensional vector space V over F. We term such loops symplectic &loops, as the groups of this form are the direct product of an elementary abelian 2-group with a 2-group of symplectic type (cf. page 109 in [FGT]).In particular the extraspecial Zgroups of Section 8 are examples of symplectic 2-loops. Each symplectic 2-loop L is determined by a cocycle 0 : V x V -,F and comes equipped with a power map P, a commutator C, and an associator A. weview asforms forms P : v-+F, C : v2-+F, and^: -+ F. In Section 1 2 we find that the central isomorphism type of L is determined by the equivalence class of 8 modulo the coboundaries, and that this class is determined in turn by the parameters par(L) = (P,C,A). Then in Section 13 we find that L is Moufang if and only if (in the language of Section 11) P has degree at most 3 and C = P2 and A = P3 are the derived forms of P. Moreover if L is Moufang then there exists a cocycle 8 defining L such that (P,$,A) is a 3-form in the sense of Section 11.

v3

47

Finally in Section 14 we see that if L is a Moufang symplectic %loop of order 2nf l such that the radical of the associator A of L is 1-dimensional, then a construction of Conway suppIies a group N containing a large extraspecial 2-group Q 2 D i such that IN : NN(Q)I = 3. In Chapter 7 we will construct a form P of degree 3 on a 12-dimensiona1 F-space preserved by the Mathieu group M24. The corresponding syrnplectic 2-loop is the Parker loop. When Conway's construction is applied to the Parker loop we obtain the normalizer N of a 4group in the Monster. This group is used in Chapter 10 to construct the Griess algebra and the Monster. This chapter is used elsewhere in the book only in Chapter 10 in the construction of the Griess algebra and the Monster. Thus readers skipping that chapter can skip this one too. Also much of the discussion in Sections 12 and 13 is not necessary for the construction of the Monster. The Remarks at the end of this chapter indicate which lemmas are necessary. On the other hand the material in these sections places the Conway construction in a larger context which hopefully makes the construction more natural and hence easier to understand. Further it answers questions which will probably occur to the reader as the construction unfolds. 12. Symplectic 2-loops We begin by recalling some generalities about loops which can be found in [B]. A loop is a set L together with a binary operation (x, y) I+ xy on L such that (Ll) L has an identity 1. (L2) For each x E L the maps a I+ xa and a I-+ m defined via left and right multiplication by x are permutations of L. In the remainder of this section assume L is a loop. The loop L is said to be Moufang if L satisfies the Moufang condition (MF): (MF) For all x, y,z E L, (xy)(zx) = (x(yz))x. Further L is diassociative if the subloop generated by each pair of elements of L is a group.

Lemma 12.1: Assume L is Moufang and let x, y E L. Then

48

Chapter 4 Symptectic 2-Loops Proof: Specialize y to 1 in ( M F ) to get (1). Then specializing z to x in (MI?)and using (1)we get (xy)x2= (x(yx))x= ((xy)x)x.But by (L2), each u E L can be written in the form u = XY for some y E L, so ux2 = ( u x ) ~giving , (2).Similarly (3)follows by specializing y to x in (MF). Theorem 12.2 (Moufang's Theorem): Moufang loops are diassociative. Proof: See Chapter VII, Section 4 of [B]. We require Moufang's Theorem only for syrnplectic 2-loops, where the result is easy to prove using 12.1 and 12.3; see, for example, Exercise 4.5.

Define L to be a symplectic 2-loop if L is the extension of the additive group Z2 of the field F = GF(2) by a finihdimensional vector space V over F. That is, here exists a surjective morphiim q5 : L -+ V of loops such that ker(q5)E Z2. In the remainder of this section assume L is a symplectic 2-loop with defining morphism q5 : L -+ V and ( 1 , ~ = ) ker(q5). Let n = dim(V). Observe

Id Symplectic 2-loops (2) There exist 8 E 8 and an isomorphism a! : L -+ L(8) such that = (1,O).

Ira!

Proof: Exercise 4.1. Remark 12.5. From 12.4 we have a bijection 8 I-+ L(8) between the F-space 8 ( n )and the set of ail symplectic 2-loops defined on F x V.

Because of 12.4 and 12.5, we may take L = L(8) for some cocycle 0 and n = (1,O). For x, y, z E L, define P(x) = x2, C(x,Y ) = ( X Y ) ( Y X ) - ~ ~ 4 x 1 Y,z) = (x(yz))((xy)z)--l.

Thus P is the power map, C ( x ,y ) is the commutator of x and y, and A(x, y, z ) is the associator of x, y , z. The parameters of L are the triple par ( L )= (P,C,A). Write par(8) for par (L(8)). Notice that P(xn) = P ( x ),

Lemma 12.3: (1) For x, y E L, 4 ( x ) = $ ( y ) if and only if y = x or nx. (2) n i s in the center of L. That is, for all x , y E L, xn = nx and (xy)n = x(yn) = (xn)y. (3) For each x E L - (n),the subloop ( x ,n ) generated by x and n is a group of order 4, and the inverse x-I for x in ( x ,n) is the unique left, right inverse for x in L.

and

Proof: As $(L) = V is a group of exponent 2, d ( x ) = $(y) if and only if 1 = 4(x)q5(y)= $(xy) if and only if xy E ker(4) = (1,n). In particular, by (L2) there are exactly two elements y E L with 4 ( x ) = +(y),and these must be x and xn = T X . So (1)holds. Then as +(xy)= q5(x(yn))= #((xn)y),(1) implies (2). Finally (2) implies (3).

for all x, y,z E L by 12.3. Further as Lq5 = V is an elementary abelian 2-group, P(x),C ( x ,y), A(x,y , z ) E (n). Thus we can and will regard the parameters P, C,A as maps from V n into F. Further we write 8(x,y) for e(q5(z),+(Y)), 8(x,Y z) for 0 ( 4 ( ~ ) , 4 ( y )d(z)),etc. Subject to these conventions we calculate:

The usual construction from homological algebra for studying extensions of abelian groups can be used to analyze our loops. Define a cocycle on V to be a map 8 : V x V F such that 8(v,0) = 8(0,v) = 0 for all u E V. Write 8 = 8(n)for the space of all cocycles on a space V of dimension n. Given a cocycle 8 define L(8) = F x V and define a product on L(8) via

c(x7ri,y d )= C ( x ,y),

+

Lemma 12.6: For all x,y, z E V: (1) P ( X )= e(x,X I . (2) C ( x ,y ) = B(z,y) 8(y,x), so C is symmetric. (3) 4 3 ,Yl z ) = @(x, Y ) O(Y, z) + '4x9 Y + z ) + e(x Y 1 4 . Write FV for the set of all maps E : V -+ F such that ~ ( 0=) 0. For E F~ denote by the cocycle defined by

+

-+

(a,u)(b,v ) = (a + b

+

E

+

+

eE

+ O(U, v ) ,u + v).

Lemma 12.4: (1) For each 8 E 8, L(8) is a loop and the projection (a,u) w u is a surjective morphism of loops whose kernel ((0, O ) , (1,O)) is in the center of L(0) and isomorphic to Z2.

Such cocycles will be termed coboundaries. Evidently the set of coboundaries forms a subspace of the space of cocycles. Define an equivalence relation on @ via 8 8' if 8' 8 is a coboundary. N

N

+

51

Chapter 4 Symplectic 2-Loops

12. Symplectic 2-loops

Define Map(L) to be the set of functions a : L -t L permuting the cosets of (T), acting as the identity on ( x ) , and such that the induced map on V is linear. We also write a for this induced map; thus cr E GL(V).Observe that there exists E, E FVsuch that

Lemma 12.10: Assume V = (x, y), par(8) = par(O1), and 8(x, y) = #(x,Y). Then 8 = a'.

50

&(a,u) = (a + ea(u), a(u))Conversely given r E FVand a E GL(V) we get a E Map(L) defined by a(a, u) = (a r(u), a(u)). Given Ole1E 8, a centml isomorphism cr : L(8) 4 L(8') is an isomorphism acting as the identity on (T) and such that the induced map on V is the identity. Thus if a is a central isomorphiim then a E Map(L) with the induced map a on V the identity.

+

Lemma 12.7: Let 8,B'

E €3

and a E Map(L). Then

+

+

(1) a : L(8) -+ L(0') is an isomorphism if and only if 8 8' o a es, = 0. (2) a is a centml autornorphisrn of L(8) if and only if the induced map is the identity on V and e, f H m F ( V ,F). Proof: Part (1) is an easy calculation, while (2) is a consequence of (I). Lemma 12.8: Let U be a hyperplane of V and x E V - U. Assume 8,8/ E 8 with par(8) = and 8 = 8' on U and on all lines of V amugh X. Then 8 = 8'. Proof: The proof is by induction on n. If n = 1 the result is trivial as 8(x,x) = P(x) = B'(x,x). Similarly if n = 2 then V is a line through x, so the lemma holds. Hence n 2 3. If n > 3 and y,z E V then W = (x, y, z) # V, so by induction on n, 8 = I3' on W. In particular 8(y, Z) = el(ylZ) and as this holds for all y, z E V, 8 = 8'. So let n = 3. Let U = (y,z). Now as 8 = 8' on U, 8(y, z) = 8'(y, z) and as 8 = 8' on all lines through x, 8(x, y) = B1(x,y) and B(x, y + z) = 8'(x, y 3). Hence by 12.6.3, 8(x y, t ) = 8'(x y, z). That is,

+

+

+

+

+

(a) 8(x u, v) = B1(x U,v) for a11 U, v E U. Next substituting z x for z in 12.6.3, we conclude:

+

+

+

+

+

Proof: First by 12.6.2, 8(y, x) = B(x, y) C(X,y) = 8'(y, x). Next specializing y to x and z to y in 12.6.3, respectively, we obtain: Lemma 12.11: A(x, x, z) = B(x, x)+B(x, z)+B(x, x+z) and A(x, y, y) = e(x, y) B(Y,Y) 8 ( ~Y,Y)for all X,Y, z E V. We conclude from 12.11 that 8(x, x+ y) = A(%,x, y) +P(x)+O(x, y) = B1(x,x + y) and similarly B(y, x Y) = #(Y, x y). So indeed 8 = O', completing the proof of 12.10.

+

+ +

+

+

Let I = { l , ... , n ) , X = { x i : i € I) beabasisforv, and for k c Z let A(k) = {J C I : IJI 2 k). For J E A(l), let m(J) = max(j : j E J), 3 = J - {m(J)], and XJ = xj. Define r j E FVby r j ( ~ j = ) 1 and Q(V) = 0 for v # XJ.

xjer

Lemma 12.12: Let d E 9 ( n ) and a : A(2) -+ F. Then there exists a unique 8 E 8(n) with par(8) = par(bl) and 8(xjlxm( J)) = a(J) for all J E A(2). h r t h e r 8 13'.

-

Proof: Let U(m) = (xi :i 5 m). Proceeding by induction on m, we pro)) duce 8, = 6' N 8' with par(elU(,)) = pa~(8i&,)) and ~ ( x ~ , x , ( ~= a(J) for all J E A(2) with m(J) < m. Further OIU(,) is unique. If m = 1 then 8 = 8' works. Assume the result for m. Replacing 8' by dm, we may assume 8'(xJ, x,( J)) = a(J) for all J E A(2) with m(J) < m. For J E A(2) with m(J) = m 1, let b(J) = e1(xj1x,+~) a(J), and set E = CJ b(J)€j, where the sum is over those J E A(2) with m(J) = m + 1. Then 8 = 8' + 8, satisfies S(XJ,X,+~) = a(J) for all J with m(J) < m + 1. Further by induction, BlU(,) is unique, so the uniqueness of 81u(,+l) follows from 12.8 and 12.10.

+

+

Theorem 12.13: Let 8, d E 8.Then the following are equivalent: (1)

eNel.

(2) L(8) is centmlly isomorphic to ~ ( 8 ' ) . (3) par(8) = par(8').

+

Now (a) and 12.9 imply 8(x y, x z) = B1(x y,x 2). That is, (b) 8 ( ~ + ~ , ~ + ~ ) = 8 ' ( ~ + u , x + v ) f o r a l l ~ , v ~ U . Notice that (a), (b), and the fact that 8 = 8' on U complete the proof of 12.8.

Proof: By 12.7.1, (1) and (2) are equivalent. Further if L(8) is centrally isomorphic to L(0'), certainly the two loops have the same parameters. Finally if par(8) = par(B1)then by 12.12 there exists 8* with 8 N 8* N 8', so (3) implies (1).

Remark 12.14. Let eO(n)be the space of all coboundaries and

Chapter 4 Symplectic 2-Loops

12. Symplectic bloops

Then by 12.13 the map iL(8)]I+ 8+80(n)is a bijection between the set of central isomorphism classes of symplectic 2-loops of order 2"+l and eO(n).Thus we may regard e O ( n )as the space of all symplectic &loops of order 2"+l. Notice that dim(^') = 2n - 1 and the kernel of the = 2" - n - 1. map c H Be is Hom(V, F) of dimension n, so dirn(eO(n)) Therefore as dim(6(n))= (2n - I ) ~ we , conclude:

if 6' o a 8 8,- = 0. So L(8) is isomorphic to L(8') if and only if e + e O ( n )is conjugate to 8' e O ( n )under GL(V),and by 12.13 this is equivalent to par(8') conjugate to par(8) under GL(V).That is, (1) holds. Also O(par(8))is the stabilizer of the central equivalence class of el SO (2)holds.

52

L e m m a 12.15: dim(eO(n)) = 2(2" - 1)(2"-l

- 1) + n.

Write O(par(8))= O(P,C,A) for the subgroup of GL(V) fixing P, C , and A. Theorem 12.16: Let G = A t ~ t ( L ( 8 ) )Then ~.

E

= CG(L(8)/(?r)) &' V * = Hom(V,F ) ,

GIE 2 O(par(B)),and these isomorphisms are equivariant with respect to the representations of GIE by conjugation on E and the dual action of O(par(0))on V*. Proof: Identify V with L / ( x ) via r$ and for g E G, let g be the map induced by g on V . For c E V* let a , be the central isomorphism induced by E as in 12.7.2, and let E = {a, : e E V*). By 12.7, E = CG(V)and the map e t,a , is an isomorphism of V * with E. Thus E is the kernel of the map g I+ g. Further g fixes P, C,A, so d 5 O(pur(8)).Conversely given d E O(par(B)),we have par(8od) = par(B), so by 12.13, there.exists 6 E pV with 8 o d 8,5 = 8. Let g E Map(L) with g = d and eg = 6 . By 12.7.1, g E G, so as g = d , G =OOCpaO)). Finally (u)= ~ 9 ( g - ~ ( uso )),

+

g f f ~ (ua) ,= ga,g-l(a, u ) = gcu,(a

+ 6(g-'(u)), g-l

(u))

+

= g(a + GI-~(u)) + e(g-l(u)),g-l(u))= (a e(g-'(u)),~), so ga, = aEOg-1= ag(r).Thus the proof is complete. Theorem 12.17: (1) L(8) is isomorphic to L(O1)if and only ifpar(8) is conjugate to ~ a r ( 8 'under ) GL(V) i;f and only if 8 e O ( n )is conjugate to 8' e O ( n )under the action of GL(V) on e O ( n ) . (2) IGL(V) : O(par(8))I is the number of central isomorphism classes in the n-stable isomorphism class of L(8).

+

+

Proof: Notice that (*) is the intersection of the maximal subloops of ~ ( 8unless ) par(8) = 0 and 15 E~.+I. In this case (1) is trivial, So We may wr(') f O hence each isomorphism @ : L(8) -t L(81) n' Thus by 12'7, a : L(8) L(O') is an if and only

+ +

53

+

We say that 8 E 8 is diassociative if 0 is bilinear on each Zdimensional subspace of V ; that is, for all z, y E V , Lemma 12.18: Let par(8) = (P,C ,A). Then the following are equivalent: (1) L(8) is diassociative. (2) A(x, y, z) = 0 for all linearly dependent subsets ( x ,y, z ) of V . (3) 8 is diassociative. Moreover if8 is diassociative then C(x,y) = C(x,x+ y ) and C(x,y) = P(x) P(y) P(x y) for all x, y E V .

+

+

+

Proof: The equivalence of ( 1 ) and (2) is just the definition of diassociativity. Notice (2) implies 8 is diassociative by 12.11. Finally assume (3). Summing the equations for diassociativity of 8, we get C(x,y ) = C(x,x y). Then summing the three images of 12.11 under the permutation (2,z, x t ) ,we get 0 = P(x) P(z) P(x z ) C(x,z ) C ( ~ , x + z ) + C ( z , x + z so ) , asC(x,x+z) = C ( z , s + z ) , also C(x,z) = P(x) P(z) P(x 2). Specializing z to s in 12.6.3, we get:

+

+

+

+

+

+

+ +

+

+

Lemma 12.19: A(x, y,x) = C ( x ,y)

+ C ( x , x + y ) for all x , y E V.

So as C ( x ,y ) = C ( x ,x + y), A(x,y, x ) = 0. Also diassociativity of 8 and 12.11 shows A(x,x,y) = A(x, y, y ) = 0. Thus it remains t o show A(x, y, x y) = 0. Specializing z to x y in 12.6.3 yields:

+

+

Lemma 12.20: A(x,y, x for all x, y E V .

+ y ) = 8(xly ) + 0(y,x + Y ) + P(x)+ P(x + Y )

Then by 12.20, diassociativity of 8, and 12.6, A(x,Y , X+Y) = C(X,Y)+ P(y)+ P ( x )+ P(x y). Hence A(x,y, x y) = 0 by an earlier remark.

+

+

Remark 12.21. Write e l ( n ) for the space of diassociative members = el(n)/eo(n). B~ 12.18 and m a r k 12.14, of e(n)and let have a bijection bekeen the set of central isomorphism dasses of diassociative symplectic 2-loops of order 2"+' and e l ( n ) .Thus we view el(,) the space diassopjative dymplectic2-loops of oder T+'.

,

55

Chapter 4 Symplectic 2-Loops

13. Moufang symplectic 2-loops

Notice that given x, y E v#,diassociativity of 8 and the elements 8(x,y), P ( x ) ,P(y),and P(x y) determine 8 on ( x ,y), so dim(Ql (n))= (2n 1)(2"'l- 1)/ 3 2n - 1. Hence as dim(eo( n ) )= 2n -n - 1, we conclude:

C(u,v ) = Q(u,v ) Q(v,u ) for all u, v E V . Further f = P3 is the third derived f o m of P.

54

+

+

+

Lemma 12.22: dim(el(n))= (2n

ProoE Let v

- 1 ) ( 2 ~ --l 1)/3+ n.

E V.

Then v = EXEX axx. Hence from 11.9,

13. Moufang symplectic 2-loops

=

Lemma 13.1: Let par(8) = (P,C,A). Then (1) If L is Moufang then C = Pz is the second derived fonn of P. (2) L is Moufang if and only if A = P3 is the third derived f o m

of P. Proof: Part (1)follows from 12.2 and 12.18. Let x , y , z E V and assume L is diassociative. Then

1 azq,azf ( x ,Y , 4 = P(v)

x
In this section L = L(8) is a symplectic 2-loop. We continue the notation of Section 12. We also use the terminology of Section 11.

by symmetry of f and as f (x,x,y) = 0 for all x, y E X . That is, P(v) = Q(v1v). By 12.6.2, C(u,v ) = Q(u,v ) Q(v,u).By (F3) in the definition of a 3-form,

+

,

+

Q(v,21 w ) = Q(v,4 + Q(v,w) + f (v,21, w ) and as Q is linear in its first variable, Q(v + u, w ) = Q(v,w ) Therefore f ( v ,u, W ) = Q(v,u )

+ Q(u, + Q(v, W)

So by 12.6, par(L(Q))= (P,C1f Finally by ( F l ) ,

21

+ Q(u,w).

+ W ) + Q(v+ 21, w).

1.

=xz+P(y)+A(x,y,y+z)+C(x+y,y+z) by diassociativity. Similarly

+ P(y) + P ( z )+ C(x,y) + a x , + C ( Y , ~+)f (xlylz) by (F3).But by ( F l ) , C(x,y) = P ( x )+ P(y) + P(x + y), so f ( x ,Y , z ) = = P(x)

by diassociativity. Therefore (y(zx))y= (yz)(xy)if and only if

by diassociativity and 12.18. That is, if L is diassociative then L is Moufang if and only if A = P3. But if L is Moufang then by 12.2, L is diassociative, so A = P3. Conversely if A = P3 then A(%,x, z ) = A(x,y, y ) = 0, so by 12.11 and 12.18, L is diassociative and hence L is Moufang.

Lemma 13.2: Assume X is a basis for V and f is a symmetric trilinear form on V with f ( x ,x, y) = 0 for all x, y E X . Let 3 = (P,Q, f ) be the unique 3-form with P ( x ) = Q(v,x) = 0 for all x E X , v E V . Then L(Q) is a Moufang symplectic &loop with par(&) = (P,C,f), where

2)

P3(x,y, z). Hence L(Q) is Moufang by 13.1. Example 13.3 Let n = 3, X = { x ,y, z ) a basis for V , and f the symmetric trilinear form on V with one monomial xyz in the basis X. Form the 3-form 3= (P,Q, f ) as in 13.2 and let L = L(Q).Then setting s = x + y + z, we have P(s) = 1 and P(v) = 0 for v E V - (s).Further for distinct a, b E X , Q = 0 on (a,b), while for W = ( x f y,x z ) , Q(a,b) = 1 for all distinct a, b E w#. For the purposes of Lemma 13.5 it will be more convenient to consider a different cocycle 8 with 8 Q. Namely define E f F~ by E = 1 on ~#andr=OonV-W,andlet8=Q+8,.Observethat8=OonW while 8(u,v ) = 1 for all distinct u,v E (a,b)# and all distinct a, b E X . By 12.12,B is the unique cocycle such that par(8) = (P,C,f ) , 8 = 0 on W , and 8(s,w ) = 1 for all w E w#.

+

N

Chapter 4 Symplectic 2-Loops

14. Constructing a 2-local from a loop

For 0 5 m E Z , define F; to consist of those P E FV such that deg(P) 5 m , where deg(P) is defined in Section 11. Then

power map of the Moufang loop L(@. Hence for any P E vF- F:, 3 = P P for some P E F? and 8 = 8 8 is a cocycle such that L(8) has power map p. So without loss of generality P ( x ) = 1 for a unique x E V. Let U be a complement to ( 5 ) in V . Pick a basis Y = (xl,x2,x3) for U and let X = Y U {x). By 12.12 we may assume @(xi, xj) = 0 = 8(x3,xl x2) for all 1 5 j < i 2 3 and B(x,u) = 1 for all u E u#.Then as P = 0 on U,0 = O on U by the uniqueness statement in 12.12. Let u ,v be distinct members of U# and V = (u,v, x). Then 8 = 0 on W = U n V while B(z,w ) = 1 for w E w#, so 8 satisfies the defining relations of the second cocycle of example 13.3 on V . Hence from the discussion of that example, 8(u,v x ) = 1. Let Z = ( x l ,2 2 , x3 x). Then P = 0 on Z so all associators on Z are 0. Hence

56

Lemma 13.4: (1) 0 = F: 5 F,V 5 ... 5 F: = FV is a GL(V)invariant filtration of FV. vlFmv 1 ) = (g) (2) dim(Fm (3) For each P E F? there exists a wcycle 8 such that L(8) is a Moufang symplectic &loop with par(L(8)) = (P,P2,P3) and (P,8, P3) is a 3-form.

-

Proof: For (1) and (2) observe that the map 11, : P H Pm is a linear map of F: into the space Rm(V) of all symmetric m-linear forms f on V such that

.

FL-~,

SO (1) holds. Also for all X I , . . ,Xn-1 E V . F'urther ker(11,) = dim(Rm(V)) = Hence dim(F:) = Cgl d(i) with d(i) = v 5 dim(Fiv /E-l) So as CL1d(i) = 2"- 1 = dim(FV),(2) follows. Thus we may take P E F: and it remains to prove (3). Observe first that if B is a cocycle with par(8) = (P,P2,P3) then (P,8, P3) is a 3-form if and only if 8 is linear in its first variable. This follows from 12.6. We term such 8 1-linear. Hence it remains to show there exists a 1-linear cocycle 8 with par(L(8))= (P,P2, P3). Next by 13.2 there exists P' E F: and a 1-linear cocycle 8' such that Pi = P3 and par(B1)= (P', 5,pi). Thus P = P P' E F:, so if 8 is a 1-linear cocycle for P, then 8 = 8' 8 is a 1-linear cocycle for P. So we may assume P E Notice this means P is a quadratic form on V and P2 is the bilinear form defined by P. Then V = Vo$ Vl$. $ V,, with the summands orthogonal with respect to P2, Vo = Rad(P2),and d i m ( 5 ) = 2 for i > 0. There is a bilinear cocycle 80 for P on Vo and each cocycle Bi for P on 5 is diassociative for i > 0 by 12.18, and Bi is bilinear with B(x,x ) = P ( x ) and hence bilinear. Therefore 8 = Ci 8(x,y) 8(y,x ) = P2(x1y) for all x , y E V . Bilinearity of 8 shows 12.6.3 is satisfied, so by 12.6, 8 is a cocycIe for P , completing the proof.

(E).

+

57

+

+

+

+

(7).

FT.

+

+

+

Lemma 13.5: Let P E F'. Then there &ts a Moufang symplectic 2-loop L with par(L) = (P,P2, P3) if and only $ deg(P) 5 3. Proof: If deg(P) 5 3 then L exists by 13.4. Conversely assume L = L(8) is a symplectic 2-loop with par(8) = (P,C,A) and A = P3, but deg(p) > 3. Then restricting P to a 4-dimensional subspace of V on which P4 # 0, we may assume n = 4. Now by 13.4, F? is a hyperplane of FV such that for all P E v:, there exists a cocycle # with P the

But by the discussion above, the first term in this sum is 0, while the last three terms are 1, so 1 = A(xl,x2,x3 x), a contradiction. This completes the proof of the lemma.

+

Remark 13.6. Write 8 * ( n ) for the space of cocycles 8 such that par(8) = (P,P2,23) for some P E FV. By 13.1, L(8) is Moufang if and only if B E 0 * ( n ) .Thus 8 * ( n )= 0 * ( n ) / e 0 ( n )is the space of Moufang symplectic 2-loops of order 2n+1. -+ e * ( n ) such Theorem 13.7: (1) There is an isomorphism E : that (P,P2,P3) = par(L(E(P))for each P E (2) dim(0*(n))= n(n2$ 5 ) / 6 . (3) L(<(P))r L(<(P'))i f and only if P is conjugate to P' under GL(V). (4) IGL(V) : O(P)I is the number of central isomorphism classes in the n-stable isomorphism class of L(<(P)).

FY.

Proof: Part (1) is a consequence of 12.13, 13.5, and 13.6. Then part (2) foliows from 13.4.2. Parts (3) and (4) follow from 12.17.

14. C o n s t r u c t i n g a 2-local f r o m a loop In this section L is a Moufangsymplectic2-loop with parameters (P,C,A). We continue the notation of Section 13. In particular L consists of all

Chapter 4 Symplectic &Loops

58

14. Constructing a &local from a loop

pairs (a,u ) with a E F, u E V , where multiplication is defined by

+ + B(u, v),u +

(a,u)(b,v ) = (a b

We write $i(a) for $,(a,O), where (a,0) E 1;. Let S3 act on R3 via

V)

with respect to some cocycle 8 such that (P,8, A) is a 3-form. We write ?r for the generator (1,O) of Z(L). We let E = CAUt(L)(V) and recall that by 12.16, Aut(L)/E = O(V,P ) and E consists of d maps a,, E E V*,defined by a, : (a,v) H (a e(u),u). In addition let I'o 5 O(V,P ) and assume vg E V# such that fixes vo and is faithful on V/(vo),vo is in the radical of A, and P(vo)= C(v,vo) = 0 for all v E V . Let E I' Aut(L) with I'/E = and pick s E L such that #(a) = vo,where q5 :(a,u) H u is the projection of L onto V . As vo is in the radical of A and C(v,vo) = 0 for each v E V , s is in the center of L. As P(vo) = 0, s2 = 1. We term a E I' to be even if a 6xes s. Notice the even automorphisms form a subgroup of I' of index 2. Define E E V * to be even if a , is even; that is, if vo E k e r ( ~ ) .

+

ro

< <

ro,

Remark 14.1. Each pair of elements u, w E V determines a unique E ( = , ~ ) E V *via q,,,) ( v ) = A(u, w, v). As A is symmetric and A(x,x, y) = 0 for all x, y E V , this definitionis independent of the generators u , w for (u,w). Write a(,,,) for the corresponding element a,,(, E E. Similarly if d,e E L, write ~ ( d , ~a(d,,) ) , for the maps E ( , + ( ~ ) , ~ ( ~ ) ) , a(,+(e),+(d))l respectively. Observe that a(4e. is even and if

then a(d,e)= 1. Let R = L U ( 0 ) and decree that 0 . d = d 0 = 0 for all d E L. Write fi3 for the set of all ordered %tuples from R. We now associate to each d E L three permutations $is i = 1,2,3, of R3 by decreeing that (a,b, c) is mapped by ~,+(d) for i = 1,2,3 to

.

(d-lad-', $b, cd), (ad,d-'bd-l, dc}, (da,bd, d-led-l), -

respectively. Next we associate three permutations &(a)of R3 to each a E r. If a is even we define (a,b, c)$i(a) = (cia,b a l m ) for each i = 1,2,3. If cr is odd we define the image of (a,b, c) under $i(a) to be

for a E: S3. Define N to be the subgroup of syrn(a3) generated by the maps $i(d),$i(i(a), d E L, a E I ', and i = 1,2,3. Define N+ to be the sub&(a), d E L, a E I' even. Set ki = group generated by the maps $+(d), $i'i-l(s)$i+l(sr) (where the indices are read modulo 3),K = ( k l ,k2,k3), Zi =$i(r), and (zl,z2,~3).

z=

Lemma 14.2: (1) For each a E A3 and x € L U r, $:(x) = $$,(x). (2) +i : I' -t sym(f13)is a faithful permutation representation. (3) $i(d)2 = lli(d2) = &(P(d)) and $i(d)-l = &(d-l) for each d€L. (4) For each a E I' and d E L, ~ ~ ( d ) * = ' ($i(da), ~) $i(dcu)-l, for even, odd, respectively. (5) If d E L and a E I' is odd then $l(d)*2(a) = +3(da)-'. (6) zi centmlizes Qj(d)for all d E L and all i, j . (7) For e E L,+l(d)$l(e)= 'dJl(de)$3(C(e, d))l(ll(a(e,d))(8) For d E L and a , E E , [ k ( d ) ,$i (a,)] = $ i ( ~ ( d ) $i(P(d) ), +~ ( d ) ) for E even, odd, respectively. (9) [$~i(e),+i(d)] = +i([e,dl)= A(C(e,d)). (10) For d , e E L, $1 ( d ) * ~ ( ~=)$1 (d)$3(C(e1d))$i ( ( ~ ( ~ , d ) ) . Proof: Parts (1) and (2) are straightforward, as are (3)-(6) once we recall that L is diassociative. Also, in (8)

( d ) ,+i( a L )= ] $i(d)-'$i(d)'i(')

(c-'a, b-'a, a-'a),

for i = 1,2,3, respectively.

(b-'a, a-'a, c-'a),

= $i (d)-'&(d

+~ ( d ) ) ' ~ ~ " )

by (4).Hence (8) follows from ( I ) , (3), and (7). Similarly in (9),

by repeated applications of (7) and keeping in mind that q3(r)and +l(a(d,e))centralize each other and ql(x) for each x E (e,d). So it remains to prove (7)and (10).We prove (7);the proof of (10) is similar and left as Exercise 4.2. F i s t $1

(a-'a, c-la, b-'a),

59

(d)+l(e): (a,b, c) H (e-l(d-'ad-l)e-',

e(db),(cd)e),

while

&(de) : (a,b, c) H ((de)-'a(de)-l, (de)b,c(de)).

Chapter 4 Symplectic &-Loops

60 Next

14. Constructing a 2-local from a loop For g E N, let

= Kg and adopt the bar convention. Define

A 2 = {(O,d,O) : d E L), A3 = {(0,0,d) : d

Also as A is symmetric trilinear with A(x,x, y) = 0 for all x,y E V, we have A(e, d f a, d e) = A(d, el d a). Therefore $1 (de) maps (a,b, c) to

+

+

+ C(d,e) + A(d,el a), e(db) + C(d,e) + A(d, el b),(cd)e+ A(d, el c ) ) .

(e-l(d-lad-l)e-l

Now (7) follows.

Lemma 14.3: (1) zlz2z3 = 1 = $l(d)$2(d)$~~(d) for all d E L. (2) Z and K are normal 4-subgmt~psof N with centralizer N+. (3) N/N+ 2 s3.

Proof:Part (1) is an elementary calculation. By 14.2.6, Z is centralized by .&(d) for all d E L and all i = 1,2,3. Similarly as s is in the nucleus of L, K is also centralized by these elements. By 14.2.4, KZ is centralized by $i(a) for each even a E I?. So K Z 5 Z(N+). Next let a E I' be odd. By 14.2.4, $ l ( a ) centralizes zl and [$l(s), $i(a)]= 21, while by 14.2.5, [zi,$2(a)]= s and [$i(s),$2(a)l = $2(s)z3. Therefore Z a N , and N / C N ( Z ) S3. Moreover by 14.2, N+ a N and $i(a)2E N+, while we calculate that & ( a ) * ~ = ( ~$3(a), ) so also N/N+ E S3.Therefore (3) holds and N+ = C N ( Z ) . Finally from the commutators in the previous paragraph,

is centralized by +l(a)and [kl,&(a)] = k2, completing the proof of (2).

E L},

and A = A 2 u A 3 . Lemma 14.4: (1) Q1 Nl = C N ( z l ) and kl € Z(N1). (2) Q1 Dg is edmspecial and c R I ( Q 1 )= (zl). (3) E2"+1 2 E+ 9 N . (4) Each g E Nl can be written uniquely in the fonn f o r d , e ~ L a, E r ; a n d g ~ Q 1i f and only i l e € ( n ) a n d a E ~. (5) K n Q1 = ( k l ) . (6) Q2nQ1= E f and N1/Q1 is the split extension of (Q2nNl)/E+Z v by $1 (r)/$l(El 2 ro. Proof: As vo is r-invariant, E+ is a $(I?)-invariant subgroup of N by 14.2.2 and 14.3.2. By 14.2.8, &(d) acts on E+ for each i = 1,2,3 and d E L. Hence E+ a N . Next by 14.2.2, 14.2.8, and 14.2.9, [Q1,Q1] ( z l ) . Then by 14.2.2, 14.2.7, and 14.3.1, each g E Q1 can be written as g = $l(d)l12(f)$l(a), with d E L, f E ( n ) , and a E E. For uniqueness observe that ES is the kernel of the action of Q1 on the space A / ( z l ) of orbits of (21) on A and Q 1 / ~ is + regular on A / ( z l ) . Similarly E+ is faithful on A of order 2n+1. So the uniqueness statement in (4) is established for elements of Q1. By 14.3, Nl = N+(llrl(a)) for any odd a and kl E Z ( N l ) . Next Q1 is the kernel of the action of N1 on A l / ( k l ) 2 V with (Q2n N+)/E+ acting by translation and qbl(I') acting naturally. This shows Q1 N1, establishes (6),and shows k2 Q1, so (5) holds. Now (6) gives the existence of an expression for g E Nl as in (4), so to complete the proof of (4) suppose g = g' = $l(d')$2(e')$l(cr') is a second expression. Then as Q1 is the kernel of the action on V , 7+b2(e)$l(a)5 $2(e')$l(a') mod Q1. Then by (6) and the regular action of (Q2n N+)/E+ on V , we have 4(e) = r$(et)and a z a' mod E. Then

<

Chapter 4 Symplectic &Loops

Exercises

multiplying by $1 (~t)-l$~(e)-l on the right and appealing to 14.2.2 and 14.2.7, we may assume g,g' E Q1. But we handled that case in the previous paragraph, so the proof of (4) is complete. We have seen that [Q1,Q1] 5 (21). Further E+ is an abelian subgroup of Q1 and for each d E L - (n) there exists e E V* with e(d) # 0, so by 14.2.9, [$l(d), I+bl(cu,)] = zl. Hence Z(Q1) 5 E+. Similarly for 0 # e E V* there is d E L with ~ ( d # ) 0 and hence [$i(d), I+bl(cud)]= 21. Thus Q1 is extraspecial of order 22nS1 and as Ef 5 Q1 is isomorphic to E2n+l, Q1 % Dg. Finally by 14.2.4, (Q2nN+)/3+ acts as the full group of transvections of Ef/(zl) with center 2/(zl). Also by the hypothesis of this section, $(I')/$(E) 2 I'0 is faithful on E+/z E V/(vo). So from (6), Q1 is the kernel of the action of on E+/(E~).This completes the proof of (2) and the lemma.

and then an easy calculation shows the kernel is J = ($2(~n),$3(s)). By construction zl acts as -1, so Q1 acts as Q1/(JnQ1) = Q1/Kl % Q1 S D;. In particular a s the minimal dimension of a faithful representation of an extraspecial group of order 22n+1 is 2n (cf. 34.9 in [FGT]), Q1 acts irreducibly.

62

Lemma 14.5: Let R = (Q1 n N2)(Q2 n Nl). Then ( I ) R q N. (2) N/R = (N-/R) x (Nf/R) with N-/R E S3 and N+/R E ro. n (3) R is class at most 3 with Z(R) = 2 % E4, Z2(R) 2 Ef = Q2 r E2n+1,and R/E+ = ((Q1 nN2)/E+) x ((Q2n Nl)/Ef) S E2s("-I).

(4)

N- induces S3on 2 and centralizes E+/Z, while Nf centralizes

Z and E+/z E Hom(V/(vo), F) as a module for N+/R

S

rO.

(5) R/E+ is isomorphic to the tensor product of the &dimensional irreducible module for R+/R E S3 with the module V/(vo) for N+/R 2 Po.

Proof: This follows from 14.4.2, 14.3, and 8.15. Lemma 14.6: There &ts a monomial representation of Nl of dimension 2n over R with kernel ($2(sn), $3(s)) in which Q1 acts imducibly as Dg and z1 acts as -1.

Proof: Recall the set A defined earlier. Then Kl = (A1) has orbits _

and d3 = ((0, Old), (40, dsn)), d E L, on A. h r t h e r z1 is fued point free on A/Kl. So by Exercise 4.3, the monomial representation of N over R with basis A/K1 subject to the constraint that 4 z l = -di is of dimension 2n = tA/K11/2. As 2.1 is faithful, the kernel of our representation is contained in K(zl) by 14.4.2,

63

Remarks. In [CO~], Conway uses the Parker loop to construct the normalizer of a 4subgroup of the Monster and uses this normaluer to construct the Griess algebra and the Monster. In the terminology of Section 12, the Parker loop is the Moufang symplectic 2-loop L(0) with parameters (P,C,A), where V is the 12dimensional Golay code module of Section 19 for M24 and (P,C, A) = (P, P2,P3)is the series of derived forms on V defined in Section 20 using 11.8, and (P,O,A) is a 3-form. By 13.4.3, such a cocycle exists, and by 13.1, L(0) is Moufang. By 12.13, the isomorphism type of L(8) depends only on P, not on 0. The construction of the group N in Section 14 is essentially the same as Conway's construction in [CO~].From 14.4 and 14.5, N has a large extraspecial Zsubgroup Q1 and a 4subgroup 2 = (fl, Z2) 5 Q1 with (2,) = Z(Ql), 2 9 N, and N/Cm(Z) % S3. For the Parker loop L(B), Q1 turns out to be a large extraspecial subgroup of the Monster and fi is the normalizer of 2 in the Monster. Only a small part of the discussion in Sections 12 and 13is necessary to construct # and make the calculations needed to construct the Monster. We use 13.2 and the first few lemmas in Section 12 to construct the Parker loop L. We need that L is diassociative, which is easily verified using arguments in 12.18. We need that Aut(L) is large enough, which follows from 12.16, which depends in turn on some of the earlier results in Section 12. This information is sufficient to apply the construction of Section 14 to construct N from L.

Exercises 1. Prove Lemma 12.4. 2. Prove Lemma 14.2.10. 3. Let g : G -t S y m ( X ) be a permutation representation of a group G such that za is fixed point free on X for some involution z with [z,GI ker(0). Prove there exists a monomial ZG-representation p such that (1) zp acts as -1 on the module V for p. (2) The permutation representation of G on fY is equivalent to o for some basis Y of V.

<

64

Chapter 4 Symplectic %Loops (3) G p preserves the bilinear form on V with orthonormal basis Y. 4. Let Q be an extraspecial 2-group and q the quadratic form of 8.3.3 on Q = Q / Z ( Q ) . Let p : Q -+ SL(V) = G be the unique faithful irreducible RQ-representation and M = NG(Qp). Prove M/Qp % 0(Q, 9). (Hint: Use 12.16 and 1.1.) 5. Prove that if L is a Moufang symplectic 2-loop then L is diassociative, without using Moufang's Theorem. 6. Assume the hypotheses of Section 14, let D = V x V* as a set, and define addition on D by Define q : D -, GF(2) by q(u,P) = (1 P : D x D * G F ( ~ ) by

,

+ s(P))P(u) + P(u) and

where s(P) = 0 , l for ,f3 even, odd, respectively. Let Q1 = Q 1 / ( ~ l ) and define P : QI * D by P : $i(d)$2(b)+(a4 ( d ~ ( d+ ) t(b)~o,~), where t(b) = 0 , l for b = 1,n, respectively. Prove (1) D is a GF(2)-space under this definition of addition. (2) q is a quadratic form on D with bilinear form p. (3) p is a group homomorphism with kernel (21, kl) inducing an isometry p : 0 1 -, D, where ~1 is an orthogonal space via 8.3. (4) p induces a n equivalence of the actions of Nl on Q1 and Dl when we define $l(e) = 1 on D and $l(a) and &(e) on D, for a E I' andeEL,by '

(Hint: See the proofs of 27.2 and 23.10.) 7. Assume the hypotheses of Section 14 with L associative. Prove GI : L -, Q1 is an injective group homomorphism and $ l ( ~ )a fi with Q~ = E + $(L) ~ and E+ n (L) = 2.

Chapter 5

The Discovery, Existence, and Uniqueness of the Sporadics

In this chapter we discuss the history of how the sporadic groups were discovered and were first shown to exist uniquely. Then we provide a general outline of how Sporadic Groups proves the existence of the twenty sporadics which are realized as sections of the largest sporadic group, the Monster. Finally we outline our approach for proving the uniqueness of the sporadics. Chapter 5 includes several tables. At the end of Section 15 is a table listing the twenty-six sporadic groups with the notation and name used in Sporadic Groups to label the group. This table also lists the group order. The table at the end of Section 16 summarizes the existence results proved in Sporadic Groups.

15. History and discovery Group theory had its beginnings in the early nineteenth century, with Galois making some of the most important early contributions. For example, it was Galois who first defined the notion of a simple group in 1832. During most of the nineteenth century the term "group" meant 'Lpermutationgroup." Thus it is natural that the first sporadic groups were discovered by Mathieu as multiply transitive permutation groups; papers describing this work appeared in 1860 and 1861 [MI] [M2]. More than a century passed before the discovery of the next spcradic group. This is perhaps not too surprising, since with a few notable

Chapter 5 The Sporadics exceptions in the work of mathematicians such as Burnside, Frobenius, Brauer, and Phillip Hall, there was little progress in the theory of finite groups during the first half of the twentieth century. However, in the mid fifties three crucial events occurred in finite group theory. First, at the International Congress of Mathematicians in Amsterdam in 1954, Brauer proposed that finite simple groups might be characterized by the centralizers of involutions. Brauer's program was inspired by the Brauer-Fowler Theorem [BF] (cf. Section 8), which says that the order of a finite simple group G is bounded by the order of CG(z) for z an involution in G. Brauer's program lead to the discovery of many of the sporadic groups and eventually to the classification. The second major event in finite group theory in the fifties was Chevalley's Tohoku paper of 1955 [Ch] in which he gave a uniform construction of Chevalley groups over all fields, and began modern work on the finite groups of Lie type. The Lie theoretic point of view provided a uniform perspective from which to view most of the simple groups known in the fifties, and thus gave hope that the nonabelian finite simple groups might consist of the alternating groups, the groups of Lie type, plus the five "sporadic groups" of Mathieu. The third major event was Thompson's thesis of 1959 establishing the conjecture of Frobenius that Frobenius kernels are nilpotent. Thomp son's thesis marked the beginnings of serious local group theory: the study of finite groups G from the point of view of normalizers of p subgroups of G, soon termed p-local subgroups. All three of these events contributed to the discovery of the next sporadic group by Janko in a paper appearing in 1965. First, Brauer's program had taken hold and group theorists were busy characterizing simple groups via the centralizers of involutions. One class of groups for which such a characterization seemed difficult was the Ree groups 2~2(32n+1), one of the families of groups of Lie type. The problem studied by Janko, Thompson, and Ward was to determine all finite simple groups G possessing an involution z such that CG(z) E 2 2 x L2(q), q odd, and with Sylow 2-subgroups of G abelian. Using a mixture of character theory and local group theory, Janko, Thompson, and Ward [JT],[Wa]showed that under these hypotheses, q is an odd power of 3, or q = 5. In [Jl], Janko showed that there exists a unique group in the last case: the sporadic group now denoted by J1. The next two sporadic groups were also discovered by Janko. Again Janko studied groups with a certain involution centralizer. This time the centralizer satisfied Hypothesis E(2, A5), in the notation of the Preface.

15. Histol~land discovery In a paper appearing in 1968 [J2], Janko showed that if G satisfies Hypothesis X(2, As), then G has one of two possible structures. In particular in each of the two cases, Janko showed that IGI is determined, as is the local structure of G. On the other hand Janko did not show that a group existed in either case, nor did he show that such a group was unique up to isomorphism. The existence and uniqueness of the Janko groups J2and J3 were established by M. Hall and D. Wales in the first and by G. Higman and J. MacKay [HM] in the second. case This was the first instance of a fairly typical pattern. Namely many sporadic groups were discovered via considering groups with a certain centralizer, often satisfying Hypothesis E(w, L) for suitable w and L. Further the mathematician "discovering" the group often only showed that the order and local structure of a group satisfying his hypotheses was determined, but did not demonstrate the existence or uniqueness of the group. HaH and Wales constructed the group J2 as a rank 3 permutation group (cf. Section 3). Very soon after the Hall-Wales construction, three more sporadic groups were discovered as rank 3 groups: HS discovered by D. Higman and C. Sirns [HS]; Mc discovered by J. McLauglin [Mc]; and Suz discovered by M. Suzuki [Suz]. Indeed Higman and Sirns constructed their group in just a few hours, shortly after hearing Hall lecture on the construction of Jz. Somewhat later, Rudvaliis discovered the group Ru as a rank 3 group and Conway and Wales constructed a 2-fold cover of the group as a subgroup of GL2*(C) [CW]. J. Conway discovered the three Conway groups Col, Co2, and Cog as automorphism groups of the Leech lattice; this work appears in his 1969 paper [Co2] (cf. Chapter 8). Also visible as stabilizers of sublattices of the Leech lattice or local subgroups of Col were J2,HS, Mc, and Suz. Thus if Conway had become interested in the Leech lattice a little earlier, he would have discovered a slew of sporadic groups. In any event these embeddings supplied convenient existence proofs for the groups. Bernd Ficher discovered the three Fischer groups by studying groups generated by %transpositions. This work appears in his Warwick preprint, the first few sections of which make up the 1971 paper [F]. As defined by Fischer, a set of 3-transpositions of a group G is a G-invariant set D of involutions such that for all a,b E Dl lab1 5 3. In addition to the three Fischer groups, the symmetric groups and several families of groups of Lie type are generated by 3-transpositions. The theory of groups generated by %transpositions is beautiful and useful. Indeed it is

[HW,

68

15. History and discovery

Chapter 5 The Sporadics probably the best way to study and prove the uniqueness of the Fischer groups. The theory is not considered here, as it does not fit well with the main topics of this book. But the author is planning a book on 3-transposition groups to cover the topic. Fischer was also lead t o the Fischer group F2 (also known as the Baby Monster) in 1973 via the study of groups generated by (3,4)transpositions. The existence and uniqueness of the Baby Monster were established later by Sims and Leon using extensive machine calculation discussed in [LS]. The remaining sporadics were discovered via the centralizer of involution approach. For example, Held discovered He. In particular in his 1969 paper [He], Held studied groups satisfying Hypothesis H ( 3 , L3(2)) and showed each such simple group is isomorphic to L s ( 2 ) or M24, or has a uniquely determined order and local structure. Soon after, G. Higman and MacKay showed there exists a unique group He in this last case. In Chapter 14 we give a simplified treatment of groups satisfying Hypothesis 'H(3,L3(2)). This is perhaps the most accessible illustration in Sporadic Groups of how to prove the uniqueness and derive the basic structure of a sporadic group. The group L y was discovered by Lyons (with the paper [Ly] describing this work appearing in 1972) and shown t o exist uniquely by Sims [Si2] using the machine. O'Nan discovered O'N; his work appeared in 1976 in [ON]. Fischer and R. Griess independently were led to consider groups satisfying Hypothesis 3C(12,Col). One quickly sees that if G is a group satisfying this hypothesis then G has a second class of involutions whose centralizer is the covering group of the Baby Monster. During one of his annual meetings in Bielefeld, Fischer discussed the possibilities of such a group with mathematicians then a t Cambridge, including Conway, Thompson, K. Harada, and S. Norton. The Cambridge mathematicians began to study the potential group, which they dubbed the Monster. They showed that if the Monster existed then the centralizers of certain elements of order 3 and 5 had to contain sections which were new sporadic groups. In Griess' notation these sporadics are F3 and F5. Harada [Ha] concentrated on the Harada group F5, pinning down its order and local structure. By some oversight, no uniqueness proof for the group was generated until Y. Segev [Se] took up the problem in 1989. Thompson studied the Thompson group F3, generating the group order, local structure, and even the character table. With the character

69

Table 1 The sporadic groups Notation MII M12 M22 M23 Mz4 51 Jz

27.36.5.17.19 221 . 3 3 a 5 . 7 - 1 1 3 * 2 3 . 2 9 -31 .37 43

J4

LY He

Ru O'N cog COZ

Higman-Sims McLaughlin Suzuki Lyons Held Rudvalis O'Nan

Conway

'

29 . 3 2 . 5 3 . 7 . 1 1 27.36.53.7.11 213.37-52.7-11.13 28.37.56-7-11.31-37.67 210.33 .52 .73 .17 214 33 53 7 . 1 3 .29 29 .34 5 + 7 3- 11 1 9 . 3 1 210.37-53a7.11.23 21sa3"53.7.11.23 2" .3' 54 * 7' 11 ~ 1 323 217.39.52.7.11.13 2'' 313 - 5' . 7 11 13 17 . 2 3 2" . 316 * 5' . 73 11 . 13 .17 2 3 . 2 9 215 .31° . 53 . 7 2 . 1 3 . 1 9 . 3 1

-

(301

M(22) M(23) M (24)' F3 F5 F2 Fl

* 27.32.5.7.11 27.32.5.7-11.23 21° . 3 3 . 5 . 7 . 1 1 . 2 3 23.3.5.7-11.19 27 .33 -52 - 7

Janko

53

HS Mc Suz

Order 24.32.5.11 26.33.5.11

Name

Mathieu

Fischer

- -

a

Thompson Harada Baby Monster Monster

a

214 .36 . s 6 . 7 . i i . 19 241 313 56 7' 11 13 17.19 ~ 2 331 47 226 3" .5' 76 .112 1 3 ~17 .19 - 23 .29.31.41.47.59.71

-

-

a

table, Thompson had a proof of the existence of an irreducible representation of degree 248. This representation makes possible a one page proof of the uniqueness of the Thompson group. Using machine calculation done by P. Smith, Thompson [Thl]showed F3to be a subgroup of the exceptional group E8(3) of Lie type, hence also establishing the existence of the group.

70

71

Chapter 5 The Sporadics

16. Existence of the sporadics

Thompson [Th2] also showed that if the Monster Fl possesses an irreducible representation of degree 196,883, and several other fairly weak constraints are satisfied, then the Monster is unique. Norton went on to verify the existence of this representation assuming certain other fairly weak properties, but he did not publish this work. Finally in 1989, Griess, Meierfrankenfeld, and Segev [GMS] published the first complete uniqueness proof for the Monster. In the meantime, working independently from the Cambridge group, Griess had been generating properties of the Monster too. In 1980 he was able to construct the Monster as the group of automorphisms of a real algebra in 196,883 dimensions [Grl]. Later in [Gr2], Griess worked with the Griess algebra, a 196,884dimensional real algebra with an identity. Griess' existence proof established a t the same time the existence of the twenty sporadic groups which are sections of the Monster. Conway [Co3] and Tits [T2] later produced simplifications of portions of Griess' arguments. It is fitting that Janko, who discovered the first of the modern sporadic groups, also discovered the last of the sporadics, J4.Janko [J4] discovered J q in 1975 while studying groups satisfying Hypothesis U(6, Aut(M22IZ3)). (Our notational convention is that Hl/H2/. . /Hn denotes a group with normal series

systems provide a geometric point of view for studying multiply transitive groups. Witt was the first t o study M24 via its Steiner system in hi 1938 paper [Wl]. His approach is rather different from the one used here, which proceeds by constructing a tower of extensions of Steiner systems beginning with the projective plane of order 4. The five Mathieu groups are then realized as stabilizers of subsystems of the Steiner system for M24. This gives a constructive existence proof for the Mathieu groups, which makes it easy to establish various properties of the groups such as their order and local structure. In later instances, some of our existence proofs are nonconstructive, so we get relatively little information about the group. In particular in many cases we do not calculate the group order. Table 2, on page 74, indicates which sporadics have their order calculated in Sporadic Groups; for each of these groups, either our existence proof is constructive, or we calculate the group order as part of the uniqueness proof. Next the Steiner system is used to construct two 11-dimensional modules for M24 over the field of order 2, which in turn provide a realization for the 2-local geometry of M24. One of these modules is associated t o an error correcting code (cf. Section 10) for M24, the binary Golay code. The Steiner system and Golay code for M24 are then used in Chapter 8 to construct the Leech lattice A. The Leech lattice is a certain 2 4 dimensional Z-submodule of 24di~nensionalEuclidean space. Its group of automorphiims is the covering group 0 of the largest Conway group Col. The sporadic groups C m , C03, HS, and Mc are shown to exist as stabilizers of certain sublattices of A. Again these existence proofs are constructive and allow us to calculate the group orders. On the other hand we show the existence of J2and Suz as sections of Col by nonconstructive methods. Namely we prove that G = Col has a large extraspecial 2-subgroup Q (cf. Section 8). Then we produce subgroups Ap of NG(Q) of order p = 3,5 and use 8.13 to show that CQ(Ap)* is a large extraspecial subgroup of CG(Ap)* = CG(Ap)/Ap. Then we appeal to 8.12 or Exercise 2.4 to see that CG(Ap)* is a simple group satisfying Hypothesis X(w, L), where w is the width of CQ(Ap) and L = (NG(Q) f? NQ(Ap))/CQ(Ap)Ap. Thus, for example, we prove there exist simple groups satisfying Hypothesis Z(3, Ri(2)) and Hypothesis X(2,A5),living as sections of Col, and define these sections t o be Suz and J2, respectively. We postpone to Chapter 16 the problem of proving the uniqueness and deriving the structure of such groups. The theory of large extraspecial subgroups is also used t o show that Coi and Co2 are simple. This theory is used in an analogous manner in

~ = G ~ A 9G ...A ~ -G o~= G with Gi-l/Gi 2 Hi.) As usual he determined the order and local structure of such a group. Norton [N] established the existence and uniqueness of J4 using the machine. Table 1, on page 69, lists the twenty-six sporadic groups and their orders.

16. Existence of the sporadics In this section we give a very broad outline of how Sporadic Groups proves the existence of the twenty sporadics realized as sections of the Monster. We also speculate briefly on how best to establish the existence of the remaining six sporadics. Our first step is to construct the Mathieu group M24. Thii is accomplished in Chapter 6 by realizing M24 as the group of automorphisms of its Steiner system S(24,8,5). A Steiner system S(v, k,t) is a rank 2 geometry (cf. Section 4), whose objects are called points and blocks, such that the geometry has v points, each block is incident with exactly k points, and each set o f t points is incident with a unique block. Steiner

Chapter 5 The Sporadics Chapter 11t o establish the existence and simplicity of sporadic sections of the Monster. The Monster is constructed as the group of automorphisms of the Griess algebra. Unfortunately this object is not so natural, so it gives us little information about the Monster and its subgroups. Following Conway, we begin by constructing the Parker loop and using it to construct a group N which will turn out to be the normalizer of a certain Cstlbgroup of the Monster. The necessary disctission of loops and of Conway's construction of fl from the Park.jr loop is contained in Chapter 4. The discussion in Chapter 4 gives a general method for constructing Moufang symplectic 2-loops given a form of degree 3 over the field of order 2 (cf. Section 11). The Parker loop is constructed from the form defined by the Golay code. Our next step is to construct a group C which will turn out to be the centralizer of an involution z in the Monster G. The group C is constructed t o have a normal extraspecial 2-subgroup Q of order 21+24 with C/Q Col and Q/(z) 2 A/2A. Then the Griess algebra is constructed so as to admit C and &? with C n fl of index 3 in N. Finally using arguments of Tits we show G = (C,fl) is finite and C = CG(z). Thus G is constructed so as to satisfy Hypothesis R(12, Col), so in particular G is simple. We define G to be the Monster; it is constructed to satisfy Hypothesis 'H(12,Col, ). In [FLM],Frenkel, Lepowski, and Meurman use an alternate approach t o construct the Monster as the symmetry group of a vertex operator algebra. Vertex operator algebras are infinite-dimensional graded Lie algebras together with a family of 'bertex operators." They play a role in string theory but are also of independent interest to mathematicians. The Frenkel, Lepowski, and Meurman construction has several advantages. For one thing it can be viewed as fairly natural within the context of the theory of vertex operator algebras. For another it "explains" the connection between the Monster and modular functions of genus 0, discussed in [CN]. The reader is directed to the introduction of [FLM] for a nice discussion of these very interesting properties of the Monster. On the other hand the Frenkel-Lepowski-Meurman construction is quite lengthy and complicated. Thus we have opted here for the shorter, more group theoretic approach of Griess, Conway, and Tits. The six sporadic groups F2, F3, F5,He, and M(24) are shown t o exists as sections of CG(A) for suitable subgroups A of NG(Q) using the same approach we used to prove the existence of J2 and S u t in Col. Namely we show the existence of a group satisfying Hypothesis

16. Existence of the sporadics

73

x ( w , L) for suitable w, L. In most cases we do not obtain any further information about the structure of such groups. However, in Chapter 14 we do investigate groups satisfying Hypothesis x(3,L3(2)). In the process we derive the order of He and much of its local structure. This illustrates how the structure of a sporadic can be derived beginning only from Hypothesis X(w, L). In Chapters 16 and 17 we determine the group order and structure of J2,SUZ,and Col, and prove the uniqueness of these groups. The group M(24) is the largest of Fischer's sporadic 3-transposition groups. We leave till later the problem of showing the subgroup NG(A) of the Monster of type M(24) is generated by 3-transpositions. Once that is accomplished, the theory of 3-transpositions can be used t o prove the existence of the M(22) and M(23) sections of M(24) and t o determine the order and local structure of these groups. This is our outline of the existence proofs for the twenty sporadic groups which live as sections of the Monster. We add a few words of speculation as to the best means for demonstrating the existence of the remaining six sporadics. The groups J1, J3, and Ru can be shown t o exist as subgroups of the exceptional groups of Lie type G2(ll), E6(4), and E7(5).The existence proofs for these groups via these embeddings are fairly elegant and satisfactory. I have no good ideas to advance for establishing the existence of O'N, Ly, and Jq. The present existence proofs are highly machine dependent. We close this section with Table 2 summarizing the existence results established in Sporadic Groups. Column 1lists the sporadic group. Column 2 lists the hypothesis 7-1 under which the group is shown to exist; that is, we show that there exists a simple group G satisfying Hypothesis 3t. However, often very little information is obtained about the group other than its simplicity; for example, sometimes the group order is not calculated. Column 3 indicates if the order of at least one group satisfying Hypothesis 3-1 is calculated in Sporadic Groups. Column 4 indicates the page where existence is established. The question marks in column 2 and asterisks in column 4 for J1, J3, J4, Ly, Ru, and O'N indicate that the existence question for these groups is not addressed in Sporadic Groups. Similarly the asterisk in column 4 for M(22) and M(23) indicates that the proof of existence of these groups is postponed until a later book, where the normalizer in the Monster of the subgroup of order 3 described in 32.4.3 is shown to be a 3-transposition group, and 3-transposition theory is used t o demonstrate the existence of the smaller Fischer groups. Sometimes more

Chapter

5 The Sporadics

17. Uniqueness of the sporadics

Table 2 Existence of the sporadic groups Group

ExistenceHypothesis

Ml1 M12 M22 M23

Multiply transitive group of automorphisms of a Steiner system

M 2 4

?-t(3,L3(2)

51 J2 53

H(2, A5)

54

HS Mc SUZ

? ? ?

rank 3 group and stabilizer of a sublattice of the Leech lattice x(4,s~6(2))

LY

?

He Ru O'N

71(3,L3(2)) ? ?

Stabilizer of a sublattice of the Leech lattice c01 a(4, fG(2)) M(22) Will exist as a subgroup of M(24) using M(23) 3-transposition theory once a suitable subgroup M(24)' of the Monster is shown to be a 3-transposition group F3 H(4, As) Fs w(4, As UJT Z2) F2 H(11, (302) Fl %(I%Col) c 0 3

co2

O ~ d e r ?Page Y e s 89 Yes 89 Y e s 82 Yes 84 Yes 85 No * Y e s 135 No * No * Y e s 119 Y e s 118 Ye3 135 No * Y e s 173 No * No * Y e s 116 Y e s 116 Y e s 116 No * No * No No No No No

x;

173 173 173 173 169

, than one existence hypothesis is obtained. For example, we show there exists a simple 5-transitive group of automorphiirns of a Steiner system S(24,8,5) and we also calculate that such a group satisfies Hypothesis w 3 , L3(2)).

17. Uniqueness of the sporadics The present treatment of the uniqueness of the sporadics is ad hoc. In many cases the uniqueness proofs are machine aided. One goal of

Spomdic Groups is to correct this situation by providing a uniform a p preach to uniqueness which will work for almost all the sporadics. Our approach is the one introduced by the author and Yoav Segev in [ASl]. Conceptually the approach is as follows: We consider groups G satisfying a suitable group theoretic hypothesis for example, 7-l will usually be W(w, L) for suitable w and L. We show that any such G possesses a suitable family F(G) = (Gi : i E I ) of subgroups. Associated to 3 is a simplicial complex K and the topological space IKI of K. We form the free amalgamated product G of the amalgam defined by the family F(G) and the topological space I K Iof the complex K defined by 3(@. The universal property of G translates into a covering I K (-t ( K Jof topological spaces. On the other hand we prove (KI is simply connected, so 1 ~ =1 IKI and hence G = G. As this holds for any G satisfying I-I, and as the isomorphism type of the amalgam of 3 ( G ) depends only on 3-1 and not on G, we have our uniqueness proof. While this is conceptually one way of viewing what goes on, in practice we introduce no topology, but instead carry out our proof at a combinatorial level. Thus we consider the coset geometry I' = r(G, 3 ) of the family 3 as defined in Section 4 and'form the collinearity graph A on G/G1 defined by .'I Our simplicia1 complex is the clique complex K = K ( A ) of the graph A (cf. Section 4). Similarly we have the coset geometry I;= ~(G,F(G) and its collinearity graph d and clique complex K = K(&).The covering of topological spaces corresponds to a covering of clique complexes which corresponds in turn to a covering d : -+ A of graphs. Here d : A -t A is a covering of graphs if d is a surjective local isomorphism of graphs (cf. Section 35). Then we can define A to be simply connected if it possesses no proper connected coverings, and this turns out to be equivalent to IKJbeing simply connected. We work only with this combinatorial definition of graph covering. Part I11 of Sporadic Groups puts in place the machinery for i m p k menting this approach for proving the uniqueness of the sporadics. Thus in Chapter 12 we develop the basic theory of coverings of graphs, including various means for verifying that a graph is simply connected. Then in Chapter 13 we record the theory of group amalgams we will need, including - results which will allow us to prove that our group theoretic hypothesis H determines the isomorphism type of our amalgam 3 ( G ) independently of the group G satisfying 'H. The discussion in Section 4 and Chapters 12 and 13 is more extensive than necessary simply to prove the uniqueness of the sporadic groups.

76

PART

Chapter 5 The Spomdics

However, the extra material is useful in studying geometries and simplicial complexes associated t o finite groups, such as the Quillen complexes [Q]. Finally the last few chapters of the book illustrate how the machinery can be used by proving the uniqueness of various sporadics. In particular in Chapter 14 we prove Ls(2), M24, and H e are the only simple groups satisfying the Hypothesis N(3, L3(2)).In Chapter 15 we prove the uniqueness of U4(3). While U4(3) is not a sporadic group it does exhibit sporadic behavior, and its characterization is crucial to proving the uniqueness of various sporadics. In Chapter 16 we prove J2, Suz, and Col are unique. In the process we also derive the group order of J2 and Suz and much of the subgroup structure of these groups. Finally Chapter 17 closes the book with a list of the subgroups of prime order and their normalizers in the five sporadic groups M24, He, J2, SUZ, and Col considered here in detail. Most of this information is generated while proving the uniqueness of the groups; the remainder is established in Chapter 17,

Chapter 6 The Mathieu Groups, Their Steiner Systems, and the Golay Code

The first five sporadic groups were discovered by Mathieu as multiply transitive permutation groups around 1860. There are by now many constructions of the Mathieu groups. Our approach is to construct the groups as automorphism groups of their Steiner systems. We recall that a Steiner system S(v, k, t) is a rank 2 geometry (whose objects are called points and blocks) with v points, such that each block contains exactly k points and each set oft points is incident with a unique block. For example, the projective plane of order 4 is a S(21,5,2). In Section 18 we construct a tower

I

I

of Steiner systems admitting the corresponding Mathieu groups. At the same time we establish many of the properties of these Steiner systems needed t o analyze the structure of the Mathieu groups and construct the Leech lattice, the Griess algebra, the Conway groups, and the Monster. The blocks in the Steiner system S(24,8,5) are called octads. In Section 19 we construct certain subgroups of the largest Mathieu group M24 and determine the action of these subgroups on the octads. Further we study two 11-dimensional GF(2)-modules for M24, the Golay code module and the Todd module. These modules are sections of the

78

Chapter 6 The Mathieu Groups

18. Steiner systems for the Mathieu groups

permutation module for M24 on the points of its Steiner system. The Golay code module is an image of a (24,12)-code module.

Write I,(X) for the set of all independent subsets of X of order rn. Define an extension subset of X to be a subset C of Ik+l such that each member of It+lis contained in a unique member of C.

18. Steiner systems for the Mathieu groups

Lemma 18.2: Let t > 1, x E X , C(x) the set of blocks of X not containing x, and Y = D(X,x) the residual design of X at x. Then C(x) is an extension subset of Y.

A Steiner system S(v, k, t) or t-design is a rank 2 geometry (X,U) (cf. Section 4) whose objects are a set X of points and a collection U of k-subsets of X called blocks such that each t-subset of X is contained in a unique block. Of course incidence in this geometry is inclusion. Example (1) Recall the definition of a projective plane from Exarnple 2 in Section 4. Each projective plane of order q is a Steiner system s(q2 + q + l , q + l , 2 ) . Let (X, U) be a Steiner system S(v, k, t). Given a point x E X define the residual design D(X, x) of X at x to be the geometry (X(x), U(x)), where X(x) =X

- {x),

U(x) ={B- {x) : x E B €23).

79

Lemma 18.3: Let C be an extension subset of X. Then (1) There exists an extension Z of X such that C is the extension subset C(z) induced by Z and the point z in Z - X. (2) The restriction map a I-+ alx defines an isomolphism of Aut(Z)Z with NAU~(X) Proof: The blocks of Z are the members of C together with the blocks {z)uB, B E U. Remark 18.4. If Z is an extension of X at some point z we identify Aut(Z), with NAUt(x)(C(z)) via the isomorphism of 18.3.2. For example, this convention is used in the statement of the following hypothesis: Extension Hypothesis: Y is a Steiner system S(v, k,t) with t y E Y, and A A U ~ ( Y )Further ~.

<

Observe: Lemma 18.1: If t > 1 then the residual design D(X,x) of X at x is a Steiner system S(v - 1, k - 1,t - 1).

(Exl)

Next define an extension of X to be a Steiner system (2, A) such that (X, 23) = D(Z, z) is the residual design of Z at some point z E Z. Notice if Z is an extension of X then by 18.1, Z has parameters (v+l, k+1, t+l). Our object is to construct a tower

(Ex2) (ExJ)

of extensions of Steiner systems beginning with the projective plane of order 4, such that the Mathieu group Mv is an automorphism group of the Steiner system S(v, k, t). The remainder of this section is devoted to this construction and to the generation of properties of these Steiner systems that we will need to analyze the Mathieu groups. -Define a subset I of X to be independent if no (t 1)-subset of I is contained in a block of X .

+

Example (2) If X is the projective plane of a vector space V, then the independent subsets are just the sets O of points such that each triple of points in 0 is linearly independent in V in the usual sense.

(Ex4)

> 1,

Aut(Y) is transitive on the extension subsets of Y invariant under some Aut(Y)-conjugate of A, and there exists such an extension subset. I f A S A ' < A U ~ ( Y then ) ~ A=A'. If Y' is an extension of D(Y, y) with A 5 Aut(Y1) then there exists an isomorphism ?r :Y -t Y' acting on D(Y, y). NAUt(Y)(C)is t-transitive on Y for each A-invariant extension subset C of Y.

Lemma 18.5: Let Y be a Steiner system S(v, k, t) and A Then

< Aut(Y).

(1) If A,Y satisfies (Exl) then, up to isomorphism, there exists a unique extension X of Y with A 5 NAut(x)(Y). Moreover NAut(x)(Y) = NAut(Y)(C), where C is the extension subset of Y induced by X. (2) Assume t > 1 and A,Y satisfies the Extension Hypothesis with respect to some y E Y. Then Aut(X) is (t 1)-transitive on X and transitive on the blocks of X.

+

+

Proof: For i = 1,2, let Xi be (t 1)-designs, xi E Xi, & = D(Xi, xi), and Ci the extension subset of Yi induced by Xi, and assume ai :Y -+ &

80

Chapter 6 The Mathieu Groups

18. Steiner systems for the Mathieu groups

c

<

is an isomorphism with Aaf NAut(Xi)(q).Notice a = a ~ : Yi~+ a Y2 is an isomorphism. Then C l a is an extension subset of Y2 invariant = Aa;. SO by (Exl),there exists 0 E Aut(Y2) with under (A+* Clap = C2. Let y = a@,and extend 7 to X1 by xi7 = x2. Then C17 = C2. Hence 7 : X1 + X2 is an isomorphism. Therefore the uniqueness statement in (1) is established while the other parts of (1) follow from 18.3. So assume the hypothesis of (2) and let X = Y U { x ) be the unique extension of Y with A < Aut(X), supplied by (1). Let xl = x, x2 = y, and define % and Ci as in the previous paragraph with respect to X = X1 = X2. Then Y2 is an extension of Z = D(Y,y ) with A Aut(Y2), so by (Ex3) there exists an isomorphism a2 : Y 4 Y2 with ya2 = x and Zaz = Z. Then Aa?j 2 A 5 Aut(Y2), so by EX^), A = Aa;. Thus Aa?j = A 5 N A U ~ ( X ) Let a1 : Y 4 Y be the identity map. Then we have achieved the hypothesis of paragraph one, so by that paragraph there exists an automorphism 7 of X with xy = y. Hence as NA2ct(Y)(C) = NAut(X)(Y) is t-transitive on Y , Aut(X) is (t 1)-transitive on X. Then as each (t 1)-subset of X is in a unique block, Aut(X) is also transitive on blocks.

~ then b = 1. So all orbits of B on C are regular. However, by a counting argument, ICI = 16 = IBI, so B is regular on C.Thus NA(m) = E is a complement to B in A and C = mB. Further A, and hence also El is transitive on A and therefore also on m = { mnK : k f A}. So m = qE is the orbit of E on Y containing any point q 6 m. Pick q E 1. Observe that Aq = BqDq with Dq = ND(k), where q E k E A. Similarly Aq = BqEq with Eq = NE(k). Now there exists an automorphism a of A centralizing B and A / B with Da = E (cf. 17.3 in [FGT]). As a centralizes B , (Bq)a= Bq. Also as a centralizes A / B , NA(k)a= NA(k).SO (Dq)a= Da Cl NA(k)a = E n NA(k) = Ep.Thus (Aq)a= (Bq)a(Dq)a= Aq. Define a on Y by (qa)a = q(aa) for a E A. As Aqa = Aq, this action is well defined. Then la = (qD)a = q(Da) = qE = m. As a commutes with B and A is the set of orbits of B on Y, a permutes A. So a E Aut(Y). Similarly Qa = (1B)a = 1aB = m B = C. Thus (Exl) is satisfied, and the proof is complete.

<

(%)a

+

+

We now begin to construct the Steiner systems for the Mathieu groups. This requires a detailed analysis of the independent subsets in the plane of order 4.

L e m m a 18.6: Let F = GF(4),X the projective plane over F, p a point of X , and A the stabilizer of p in L3(F). Then, up to isomorphism, X is the unique extension of D(X,p) admitting A. Proof: By Lemma 18.5, it suffices to establish (Exl) for the pair Y, A, where Y = D(X,p). Let V be a vector space over F with basis { X Ix2, , x3) and take X = PG(V) and p = ( X I ) . Let B be the group of transvections with center p and D the stabilizer of p and the line 1 = (x2,x3). From 5.1, B S EI6, D 2 L2(4)S A5, and A is the split extension of B by D. Let A be the set of lines of X through p and R the remaining lines. Let A = {k - ( p ) : k E A). Then A is the set of five blocks of the residual design Y and 0 is the extension subset induced by X . Observe that the members of A are the 5 orbits of B on Y of length 4 and that no member of B# fixes a point in each orbit. Also B is regular on R and D is the stabilizer in A of 1 E 0. Let C be an A-invariant extension subset of Y and m E C. As lml = 5 = 1dl and each pair of points in m is independent, it follows that m n k is a point for each k E A. If b E B fixes m then b fixes the point m n k for each k E A. But

81

Lemma 18.7: Let F = GF(4), V be a vector space over F with basis {xl,x2,x3), Y = PG(V), M = PGL(V), G = PSL(V), H = NG((xl)),and I' = P r ( V ) . Then M is transitive on I4 as a set of ordered .&tuples. G is tmnsitive on 13, and has three orbits on 4 as a set of unordered 4-tuples. Each S E 14 is contained in a unique I ( S ) E 16. (3) (4) For S = { ( x i ) (, x z ) ,( ~ 3( 2)1 + ~ 2 2 ~ 3 )E) 14,

(I) (2) . .

+

+ + a-'x3), ( x i + a-'x2 + ax3)),

I ( S ) = S U { ( x i ax2 1

where F# = (a). (5) M is tmnsitive on I6 and G has three orbits 1 i 3, on 16 (6) Each S 6 I3 is contained in a unique member li(s)of I: for each i = 1,2,3. I:, 1 < i < 3, are the H-invariant extension subsets of Y and (7) H has two orbits on I$. (8) For I E 16, NG(I) A6, Nr(I) 2 196, and Nr(I) is faithful on I.

-

i

;

I

16, < <

82

Proof: Let pi = (xi). Clearly each member of I3 is conjugate to T3 = (p1,m,p3) under G. Let = ( X I 2 2 x3), pi = ( X I 2 2 ax3), and = (xl +z2 + a-lz3). Then NG(T3) = (g),where g = diag(l,a,adl). So 1 5 i 5 3, are representatives for the orbits of NG(T3) on points p of Y with T3U { p ) E Iq. Hence T:, 1 i 5 3, are representatives for the action of G on I4 as a set of ordered 4tuples, where = T3 U { p i ) . Also N M (T3) = (g,h ) , where h = diag(1, 1, a). So N M (T3)is transitive on members of I4 through T3, and hence (1) holds. To prove (3) and (4) it suffices by (1) to prove I ( T ~ is) the unique member of Is containing But if x = xl rx2 sx3 E V with T: U { ( x ) ) E I5 then, as x l x2 tx3 E ( ~ 3 , ~ :for ) all t E F#, r E {a, adl). Similarly s = r - l , so (3) and (4) are established. . (1) and (3), N M ( I ) is 4transitive on I . As (g) = Let I = I ( T ~ )By N M ( I ) * ~N , M ( I ) is faithful on I and isomorphic to A6 by Exercise 1.2. Then N M ( I ) = NM(I)= 5 G, so N M ( I ) = NG(I). Also the field automorphism f determined by T3 induces a transposition on I , so &(I) S6. Thus (8) holds. By (8), N G ( I ) is transitive on I3 r l I , so NG(T3) is transitive on the members of I G through T3. Also as N G ( I ) is 4transitive on I , q is not G-conjugate to a subset of I for i = 2,3. Therefore ( 2 ) holds. Thus I ( q ) G = I:, 1 5 i 6 3, are the orbits of G on 16. Of course M is transitive on I6 by (1) and (3). F'urther (6) holds with l i ( ~ 3= ) I(q). As N G ( I ) is transitive on the points of I, H is transitive on the set A, of members of I: through pl. Also if J E I6 with pl 4 J , then J is conjugate under H to a &set containing T = {p2,p3,(x1+x2)).Hence as T is in a unique member of I:, H is transitive on the set B, of members of I; not through pl. Notice by (6) that I; is a G-invariant extension subset of Y for each i. Conversely suppose C is an H-invariant extension set. We may assume I E C , and it remains to show C = 161. But A1 = I H E C so if C # 161 there is J E C n I: for j = 2 or 3 with pl $ J. Then ~j = J H % C . But if T E 13 r l I with pl 4 T , then T C J, E I;. Also I = I ( T U {pl)), so J, E B ~Thus . T is in two members of C , a contradiction. Hence ( 7 ) is established.

pi

pi

+ +

pi,

+ +

<

~i.

+ +

+

83

18. Steiner systems for the Mathieu groups

Chapter 6 The Mathieu Groups

+

Lemma 18.8: Let F = GF(4), Y the projective plane over F , p a point in Y , and H the stabilizer in L 3 ( F ) of p. Then

<

(1) There exists a unique extension X of Y with H A u t ( X ) . (2) Let M = A u t ( X ) and G = M fA ll t ( X ) = Mzz. Then IM : GI = 2, N G ( Y ) = L3(F), N M ( Y ) is N G ( Y ) extended by a

(3)

(4) (5)

(6)

(7)

field automorphism, G is 3-transitive on X , and G is transitive on B ( X ) . M is tmnsitive on h ( X ) and G hos two orbits I$(x), i = 1,2, on I7 ( X ). G is tmnsitive on 1 4 ( X ) , M is transitive on 1 5 ( X ) , and G has two orbits on 15(X). Each member of 1 5 ( X )is contained i n a unique member of 17(X) and each member of 1 4 ( X ) is contained i n a unique member of I$(x), i = 1,2. For I E 1 7 ( X ) ,N G ( I ) Gi A7 is faithful on I . i = 1,2, are the unique NG(Y)-invariant extension subsets of X .

q(~),

Proof: We claim the pair (Y,H) satisfies the Extension Hypothesis. First ( E x l ) and (Ex4) are satisfied by 18.7.5 and 18.7.7. By 5.1 and 5.2, H = A u t ( Y ) F , so (Ex2) holds. Finally (Ex3) holds by 18.6. So the claim is established. Now the claim and 18.5 implies ( I ) , and we may take the extension subset Cx of Y induced by X to be I ~ ( Y )Then . by 18.5 and 18.7, NM(Y)= NAUt(Y)(CX)is L3(F) extended by a field automorphism f . Notice f interchanges 1; ( Y )and I ~ ( Y )Further . by 5.2.2, f fixes exactly seven points of Y . So f has seven orbits of length 2 on X , and hence f @ G. Therefore N G ( Y ) = L3(F) and lM : GI = 2. By 18.5, M is 3-transitive on X , so G is transitive on X . Then as L3(F) is 2-transitive on Y , G is also 3-transitive on X . So G is transitive on B ( X ) . Thus (2) holds. Let q be the point in X - Y and suppose I E 17(X). Conjugating in G , wemay takeq E I . Then I. = I - { q ) E 16(Y)and as lion BI 5 3 ) i = 1 or 2. for each B E C X ,IO E I ~ ( Yfor Conversely let J = { q ) U Ji, Ji E I:(Y), i = 1 or 2. As J, E 16(Y), no 4-subset of J through q is in a block of X . As i < 2, IJi n BI 5 3 for each B E CX by 18.7.3, so no Csubset of Ji is in a block of X . Hence J E 17(X).That is, the members of 17(X) through q are the sets Ji U {q), J, E I:(Y), i = l , 2 . If U E 1 5 ( X ) , conjugating in G we may t&e q E U. Then Uo = U - {q) E 14(Y),so by 18.7.3, Uo is in a unique member Eo of 16(Y). As U E 1 5 ( X ) ,Eo E I i ( Y ) for i 5 2. So by the previous paragraph, E = {q) U Eo is the unique member of 1 7 ( X ) containing U. Similarly if q E W E 1 4 ( X )then Wo = W' - {q) is in a unique member Ji of for i = 1,2.

G(Y),

84

Chapter 6 The Mathieu Groups

19. The Golay and Todd modules

Further by 18.7.1 and 18.7.2, G is transitive on 14(X), G has a t most two orbits on 15(X), and M is transitive on 15(X) as f interchanges the two G-orbits on Iq(Y) whose members are not in a member of I~(Y). As M is transitive on 15(X) and each member of 15(X) is in a unique member of 17(X), M is transitive on 17(X) and NM(I) is &transitive on I E 17(X). As NG(Y) n N(I) = NM(Y) n N(I) = A6 is faithful on I , NG(I) = NM(I) = A7 is faithful on I. (I.e., (6) holds.) In particular NG(I) is transitive on I, so as { q ) J1 ~ = Il is not conjugate t o { q ) J2 ~= I 2 in Gq, I 2 4 IIG. Thus G has two orbits I$(x) on 17(X) and (3) is established. Indeed the members of l i ( x ) containing q are of the form {q)UKi, Ki E I~(Y). Similarly as NG(U) is transitive on U E 15(X) but NG(q) has two orbits on members of 15(X) through q, G has two orbits on 15(X). This completes the proof of (4). Also each member of 14(X) through q is in a unique member of I+(x), so as NG(W) is transitive on W, the proof of (5) is complete. Finally by (51, I~(x), i = 1,2, are G-invariant extension subsets of X . Conversely suppose C is an NG(Y)-invariant extension subset of X. Let Cq = {I E C : q E I) and C: = {I- {q) : I E CQ1. Then C: is an NG(Y)-inmriant extension subset of Y, so by 18.7, Ci = I~(Y). As Cx = I:(Y), i 2. Now NG(Y) is transitive on Cq. Similarly, for y E Y, N G ( Y ) ~is transitive on {S E C : y E S, q 4 S) from the penultimate paragraph of the proof of 18.7. Thus as NG(Y) is transitive on Y, NG(Y) is transitive on C - Cq.Now an argument in the last paragraph of the proof of 18.7 completes the proof of (7).

Lemma 18.10: Let Y be the 4-design of M23 and H the stabilizer Mz2 in M23 of a point of Y . Then (1) Them exists a unique extension X of Y with H 5 Aut(X). (2) Let G = Aut(X) = M24. Then NG(Y) = M23, G is 5-transitive on X , and G i.9 transitive on B(X). (3) I7(X) is empty. (4) G is transitive on Is(X). Notice that by 18.10.3, the M24-design has IIO extension subsets, and hence the M24-design cannot be extended.

Lemma 18.11: (1) The groups M22, M23, and M24 are simple. (2) lMz21= 27 . 32 5 . 7 - 11. (3) lM231 = 2 7 . 3 2 . 5 - 7 . 1 1 . 2 3 . (4) IMZ41= 2'' .33 . 5 7 . 11.23.

-

-

Proof: Let G be M22, M23, or M2*, and let X be the design of G. Then G is t-transitive on X for t = 3,4, or 5. Suppose 1 # K 9 G, let p E X , and let H = Gp. By 2.2.2, G = K H . By 2.2.3, H n K # 1. But if G = MZ2 then H = L3(F) is simple, while if G is M23 or M24 then or M23 is simple by induction. Then as H n K <1 H, we have H= H=HnK
<

19. The Golay and Todd modules In this section (X,C) is the Steiner system S(24,8,5) for M24 and G = Aut(X,C) is M24, as discussed in Section 18. Let V be the binary permutation module for G on X and proceeding as in Section 10, identify V with the power set of X by identifying v E V with its support. The members of C will be called octads. As (X,C) is a 5-design, each 5-subset U of X is contained in a unique octad; denote this octad by B(U). So there are (?)I(;)= 23.11 . 3 = 759 octads. Recall also from 18.11 that G has order 21° 33 - 5 - 7.11.23. We first prove two lemmas giving fairly complete information about the action of G on the octads.

The following two lemmas can be proved along the lines of the proof of 18.8:

Lemma 18.9: Let Y be the 3-design of Mz2 and H the stabilizer L3(4) in M22 of a point of Y. Then

-

<

(1) There exists a unique extension X of Y with H Aut(X). (2) Let G = Aut(X) = M23. Then NG(Y) = M22, G is 4-transitive on X , and G is transitive on B(X). (3) G is transitive on 18(X). (4) G is transitive on Im(X) for m = 5 and 6, and each member of Im(X) is contained in a unique member of 18(X). (5) For I E I8(X), NG(I) = As is faithful on I. (6) 18(X) is the unique extension subset of X .

Lemma 19.1: Let B be an octad, M = NG(B), Q = GB, and z E X - B . Then 1

i

(1) M is the split extension of Q % El6 by Mz &' As. (2) Q is regular on X - B . (3) M, acts faithfully as Alt(l3) on B.

86

Chapter 6 The Mathieu Groups (4) Mz = GL(Q).I n particular As 2 L4(2). (5) M is 3-transitive o n X-B, with the representation of Mx on XB equivalent to the representation of Mz on Q via conjugation. Proof: Let Y = D(X,x).Then B E Is(Y),so by 18.9.5, M, = NGz( B )= Alt(B) = As. LetUbeabsubsetofB,H=Gu,Y=X-U,andl=B-U.Then Y is a projective plane, 1 is a line in Y , H = L3(4),and Q = Hl. So Q S El6, H n Mx = L2(4)is faithful on Q, and Q is regular on X - B by the dual of 5.1. Next as Q is regular on X - B, M = QMx is the split extension of Q by Mx % As. As Mx is simple and CM=(Q) s Mx, CM=(Q) = 1 or Mz. The latter is impossible as H n Mz is faithful on Q. Thus Mx is faithful on Q, so Mx 5 GL(Q)2 L4(2). As IL4(2)1 = IA81, Mx = GL(Q)and L4(2) G As. By 2.1, the representation of M, on X - B is equivalent to the r e p resentation of Mx on Q by conjugation. So as L4(2)is 2-transitive on Q#,M is Stransitive on X - B. For v E V denote by C , ( v ) the set of all octads B such that IvnBl = n.

Lemma 19.2: Let A, B be octads, M = NG(B),Q = GB,and if A # B l e t x E A - B . Then (1) lAnBl = 0, 2, 4, or 8. (2) M is transitive on Cn(B)for each n. (3) ICn(B)I = 30, 448, 280, 1, for n = 0, 2, 4, 8, respectively.

(4) If A E C4(B)then NM(A)is the split eztension of NQ(A)r E4 by NM(A)x = NM=( An B ) z2/(A4 x 444). (5) If A € C2(B)then NM( A ) is a complement to Q i n NM( An B ) , 8 0 NM( A ) s6. (6) If A E Co(B)then NQ(A) Eg is a hyperplane of Q and NM(A) is the split eztension of NQ(A)by NMm (NQ(A))% L3(2)/Es. (7) If A E Co(B) then X A B = X - ( AU B ) is a n octad. (8) If A E C4(B) then A B is an octad.

+ + +

Proof: Let A be distinct from B and x E A - B. Notice NM(A)5 NM( An B) = QNMz( Ar l B ) , and NQ(A)is semiregular on A - ( An B), SO INQ(A)Ii ( A - ( A n B ) ( 2= [ A nBI2. As each 5-subset is in a unique octad, IA n BI = n 2 4. Also if U is a Csubset of B, then there exists a unique octad B(U,x) containing U and x; further U = B(U,x ) n B. Counting the set R of pairs U,x in two ways we get 16(:) = 41C4(B)Ilso IC4(B)(= 280. As Q is transitive on

87

19. The Golay and Todd modules

X - B and Mx is transitive on the 4-subsets of B, M is transitive on R,

and hence on C4(B).So I M : NM(A)I = 280 for A E C4(B). Next NMs( AilB ) Z Z2/(A4X A*). Thus NMz( An B)! = INM(An B)/Ql 2 INM(A)Q/&I= INM(A)/NQ(A)I2 INM(A)1/4= INMZ(An B ) [ ,as INM(A)I = (M1/280 = 41NM= ( An B)I. SO d inequalities are equalities and hence NQ(A) E4 is regular on A - B, so NM(A)= NQ(A)NM=( A )with NMs( A )= NMz( An B). Thus (4)holds. Also there exists t E Mz with (AnB)t= B A. Indeed t E NM(NQ(A)), so ( A - B)t = (xNQ(A))t=xNQ(A)= A - B. Thus At = ( B - A ) + ( A - B ) = A + B. So A + B = At is an octad, establishing (8). If U is a 3-subset of B then Y = X - U is a projective plane and B - U is a line in Y . Then if U 5 A, A - U is another line, so ( B- U )n ( A- U) is a point. In particular n # 3. Counting the set of pairs (U,C) with U a 2-subset of C E C in two ways, we get 759(:) = (\4) N ~where , N2 is the number of octads through a iixed 2-subset U of B. Hence Nz = 77. hrther there are &subsets W of B through U, and 4 octads in C4(B)through W, so there are 60 members of C4(B)through U. This leaves 77 - (60 1) = 16 octads in C2(B)through U. Let A E C2(B)with A n B = U. If g E NQ(A)# then U = FixA(g), so g induces an odd permutation on A. This is impossible as N G ( A ) = ~ Alt(A). So NQ(A)= 1. Thus Q is regular on the sixteen members of C2(B)through U. So NM(A)is a complement to Q in NM(U),and hence NM(A)2 Ss and (5) is established. Also as Q is transitive on the members of C2(B)through U and M is transitive on 2-subsets of B, M is transitive on C2(B).Then IC2 (B)I = IM : NM(A)J= 448. We are left with 759 - (1 280 448) = 30 octads in C1(B)UCo(B).If A E C1 ( B )then NQ( A )= 1 as IA- BI is odd. firther NM(A) NM( A n B),so IAM] = I M : NM(A)I2 JM:NM(AnB)I-1627.16>30.S0 C1(B)is empty and Co(B)is of order 30. Thus (1) and (3)hold. Let A E Co(B).As Q is regular on X - B, Q # NQ(A).On the other hand as (Co(B)I-= 2 mod 4, we may choose A so that 1 AM1 is not divisible by 4. Hence a Sylow 2-subgroup of NM(A)is of index 2 in a Sylow 2-group of M and NQ(A)is regular on A. Indeed A and A+ B + X are the two orbits of NQ(A)on X - B, and g E Q - NQ(A)interchanges . these orbits, so A f B X = Ag is also an octad, establishing (7). Finally as NQ(A)is regular on A, NM(A)= NQ(A)NM=(A). Also NM=(A) L N M (NQ(A)), ~ so

-

(62)

+

+ +

+

<

88

Chapter 6 T h e Mathieu Groups

19. T h e Golay and Todd modules

so all inequalities are equalities. That is, NM=(A) = NMz(NQ(A)) and M is transitive on Co(B). Therefore (2) and (6) hold, and the lemma is established.

A n D E B(D), so (5) holds. Now (5) and the transitivity of K on B(D) show K is 5-transitive on D, completing the proof of (2). Finally as K is 5-transitive on D, IK1 = 12 11.10.9.8. IKuI, for U a 5-subset of A n D. But K U = NK(A n D)u = 1, so (4) holds.

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+

Define a dodecad of X to be a subset of the form A B, where B is an octad and A E C2(B). Thus each dodecad is of order 12.

Given a dodecad D and subsets Di of D of order i, 0 5 i 2 4, denote by Mi2.4 the pointwise stabilizer in NG(D) of Di. Thus M12 = NG(D). The groups M24, M23, M22, Mi2, Mi1 are the Mathieu groups. Notice that each Mathieu group is t-transitive on its t-design for t = 3, 4, or 5. It turns out that the Mathieu groups, the symmetric groups, and the alternating groups are the only Ctransitive groups. So far this has only been proved using the classification of the finite simple groups. We have already seen that the first three Mathieu groups are simple. By Exercise 6.1, M12 and Mll are also simple, while Mlo has a unique minimal normal subgroup, and that subgroup is isomorphic to A6.

+

Lemma 19.3: Let D = A B be a dodecad, with B an octad and A E CZ(B). Let C be an octad. Then

(1) IC n Dl is even. (2) D contains no octads. (3) For each 5-subset U of Dl B(U) E C6(D) and B(U) octad.

+ D is an

Proofi L e t r = I C n A n B I , s = IAnCnDI, a n d t = ( B n C n D I . T h e n r+s=IAnCIandr+t=IBnCIareevenby19.2.1.Thuss=rrt mod 2, so s t = IC n Dl r 0 mod 2, and (1) is established. SupposeC C D. T h e n r = 0 and s + t = 8. By 19.2.1, s , t 5 4, so s = t = 4. By 19.2.8, C' = C B is an octad. But IC' f l A1 = 6, contradicting 19.2.1. Hence (2) holds. Notice (1) and (2) imply the first part of (3). Let C = B(U), so that s + t = 6. Then s or t is a t least 3, say t. So by 19.2.1, either C =,B or IB n C ( = 4, and we may assume the latter. Then B C is an octad by 19.2.8. Further by inspection, IAn(B+C)I = 4, so also A+(B+C) is an octad. But addition in .V is associative, so A + (B + C) = (A +B ) C = D C. Hence the proof of (3) is complete.

+

Lemma 19.5: Let D be a dodecad. Then

+

(1) D + X is a dodecad. There exists g E G interchanging D and D

(2)

Proof: By definition D = A + B for suitable octads A, B. Let C E Co(B) and C' = X + B + C . Then C' is anoctad by 19.2.7. Now D n C = A n C is of even order, as is A n C'. Also (An C) + ( A n C') = A n D. So either A n C o r A n C 1 isoforder 4, say A n C . T h u s A + C = C* is anoctad. Finally D + X = ( A + B ) + ( B + C + C i ) = A + C + C 1 = C * + C ' , so D + X is a dodecad. By 19.4.3 there exists g E G with Dg = D X,Then (D X)g = Dg+X=D+X+X=D,soginterchangesDandD+X.

+

+

+

+

Given a dodecad D define O(D) = {B(U) n D : U is a 5-subset of D). (D, L?(D)) is a Steiner system S(12,6,5). K is transitive o n B(D) and 5-transitive and faithful on D. G is transitive on dodecads. IKI = 1 2 - 1 1 . 1 0 - 9 . 8 = 2 6 - 3 3 - 5 . 1 1 . (5) For S E B(D), NK(S) = S6 is faithful o n S and D S.

+

Proof: By 19.3.3, each 5-subset of D is in a unique member of B(D), so (D, B(D)) is a Steiner system S(12,6,5). By 19.2, G is transitive on pairs A, B of octads with A B a dodecad, so (3) holds and K is transitive on O(D). Further if A + B = D then by 19.2.5, NK(A) acts faithfully as the symmetric group on A n D and on B n D = D + ( A n D). Of course

+

+

Lemma 19.6: Let D be a dodecad, K = NG(D), and B an octad. Then

Lemma 19.4: Let D be a dodecad and K = NG(D).

(1) (2) (3) (4)

+X .

( I ) I D n B I = 2, 4, or 6. (2) K is transitive on C,(D) for each n. (3) If B G C6(D) then NK(B) E S6 is faithful on B nD and D - B. Also B + D is an octad. (4) If B E C2(D) then NK(B) 2! S6 is transitive o n B f l D and faithful and &-transitive on D - B. h r t h e r B D X is an octad. (5) If B E C4(D)then NK ( B )= NK (BnD) and B+ D is a dodecad. (6) Cn (D) is of order 132, 495, 132, for n = 2,4,6, respectively.

+ +

I

I

/

I

Proof: As K is transitive on B(D), K is transitive on C6(D). Further IC6(D)I = (l52)/6 = 132. Also (3) holds by 19.4.5.

+

an octad, or a dodecad. If u = 0 then v = v, is an octad. If u is an octad then by 19.2, u vn E S. Finally if u is a dodecad then by 19.6, u Vn E S.

Let D' = D X. Then C2(D) = C6(D1), so (C2(D)I = 132 and NK(B) r S6 for B E C2(D). Also D + B X = D' B is an octad. By 19.2.5, NK(B) is transitive on B - D' = B n D. As K is sharply 5-transitive on D, KsnD is btransitive on D - B and IKBnDI = 10 9 8 = (NK(B)l. Thus NK(B)BnD is of index 2 in KBnD and hence is 2-transitive on D - B by 2.2.4. So (4) is established. By 19.3, I B n Dl is even for each octad B, but no octad is contained in D or Dr. So (1) holds. Thus as ICz(D)I = (C6(Dt)I= 132 and ICI = 759, IC4(D)I = 495. But 495 = (I:), SO the map B I-+ B n D is a bijection of C4(D) with the set of Csubsets of D. Hence as K is Ctransitive on D, K is transitive on C4(D). Thii completes the proof of (2) and (6) and shows NK(B) = N K ( B n D ) for B fC4(D). Further B n D I A n D for someA~C~(D).NowC=A+DisanoctadandAnB=BnDisof order 4, so A B = B' is an octad. Notice IB' n CI = 2, so B' C is adodecad. Then B + D = B + ( A + C ) = ( B + A ) + C = B 1 + C is a dodecad.

+

+

Observe that (V,X, Vc) is a (24,12)-code with minimum weight 8. Thus the code corrects 3 errors. As G = Aut(X,C), G is the group of automorphiims of the code. Pick x E X and let X' = X {x), V1 = ( X I ) , and Vi the projection of VC on V' with respect to the decomposition V = (2) CB V'. Then (V', X', Vi) is a (23,12)-code with minimum weight 7 which corrects 3 errors. This code is the binary Golay code. It turns out the binary Golay code is perfect (in the sense of The proof of these remarks Section 10) with automorphiim group is left as an exercise. Set = V/(X). Define Vc to be the 11-dimensional Golay code module for G = M24.

-

~t:

Lemma 19.8: (1) G has czactly two orbits on the images of the octads, of length 759, and the images of the dodecads, of length 1,288. (2) If B is an octad then NG(B) = NG(B) E As/Els. (3) If D D a dodecad then NG(D) = NG((D, D X)) Z2/M12. (4) Vc is an irreducible GF(2)G-module.

We now use the detailed knowledge of the action of G on its octads and dodecads which we have developed to construct and study several GF(2)G-modules. These modules can be studied using the theory of codes and forms in Chapter 3, which in turn endow the modules with a geometric structure in the sense of Section 4. The resulting algebraic and geometric structure is interesting in its own right, but is also an important tool in studying the Mathieu groups. Denote by Vc the subspace of V generated by C. The error correcting I+) is the extended binary Golay code. As G code (cf. Section 10) (KX, permutes C, G acts on Vc, so Vc is a GF(2)G-module.

+

v~X

+

+

+

ProoE Rom 19.7.2, each member of is of the form X v, with v = 0, an octad, or a dodecad. So each element of V# is the image of an octad or a dodecad. As G is transitive on octads and dodecads, their If B is an octad then B X is not an images are the orbits of G on octad, so NG(B) = NG({B, B X)) = NG(B). But if D is a dodecad, 19.5 says NG(D) is of index 2 in NG(D). If 0 # a is a G-invariant subspace of Vc then a# is a union of orbits of G. So if 0 Vc, then o# is the set of images of octads or dodecads, and hence of order 759 or 1,288, respectively. But then 101is not a power of 2.

~t. +

+

+

Lemma 19.7: (1) Vc is of dimension 12. (2) G has 5 orbits on the vectors of Vc: 0, X , the octads, the dodecads, and the complements of the octads.

ProoE We know G is transitive on octads and on dodecads, so the five subsets of V listed in (2) are orbits of G on Vc. Further there are 759 octads and hence 759 complements. By 19.5, the complement of a dodecad is a dodecad. Also there are IG : M121 = 2,576 dodecads. ." Thus the union S of the five orbits of (2) is of order 4,096 = 212. So to complete the proof, it remains to show each element of VC is in S. Let v E VC. Then v = vl ..+ v, with vi E C. We prove v E S by induction on n. As 0 E S we may take n > 1. Then v = u vn with u E S by induction on n. Indeed u or u + X is 0, an octad, or a dodecad, so as v E S if and only if v X E S, without loss of generality u is 0,

+

+

+

+

+ +

91

19. The Golay and Todd modules

Chapter 6 The Mathieu Groups

90

Define a sextet to be a partition A of X into six Csubsets such that for each pair of distinct S,T from A, S + T is an octad. I

I

Lemma 19.9: Let S be a 4-subset of X. Then (1) S is contained in a unique sextet A(S). (2) For each 3-subset T of S, A(S) = {S,L-{s) : L E t), where C is the set of lines through s E S - T in the projective plane X - T.

92

Chapter 6 The Mathieu Groups

19. The Golay and Todd modules

(3) G is transitive on sextets. (4) If A is a sextet then N ~ ( A ) * = S6 and N G ( A ) ~is the split extension of E64 by a self-centralizing subgmup of order 3. (5) There are exactly (?)/6 = 1,771 sextets.

93

v#,

Lemma 19.10: (1) G has four orbits Vnl 1 5 n < 4, on consisting of the images of the n-subsets of V. for n = 1,2,3. and IV41 = (7116 = 1,771. (2) IVnl = (9) Each coset in Vn contains a unique n-set for n < 3, while the &sets in a coset of V4 form a sextet. (4) Let U = wre(V); then = V2 u Vq.

(z)

Proof: Let T be a %subset of S, s E S - T, and C the set of five lines through s in the projective plane T on X - T. Then T 1 E C for each 1 E t by construction of C. Define A(S) as in (2). Then for Proof: Let B E 8# with v of weight n 5 4. Then for u E v$, (u vl S f R E A ( ~ ) , R + s = ~ ( R ) E C , S O S + R = T + ~ ( R ) ~ ~ ~ ~ ~lulC- ~IvI ~>~4. with A ~ equality SO if and only if u E C, n = 4, and v C u. This if R' E A(S) - {R, S) then R R' is the sum of the octads S R and proves that if n I: 3 then v is the unique element of weight at most 4 S+ R' with ( S R) n ( S + R') = S of order 4. Thus R R' is an octad in the coset C = v Vc. Further for n = 4, the set of elements in 6 of by 19.2.8. Hence A = A(S) is a sextet. weight at most 4 are v plus those of the form u + v, u E C, v cl: u. But Suppose A' is a sextet through S. For x E X - S there are unique 4 by 19.9.3, these elements are just the members of A(v). So (3) holds. sets R E A and R' E A' containing x. Then S R = B(S+x) = S+R', we also have (2). By (2), Further as the number of n-sets is so R = R'. Hence A = A'. This completes the proof of (1) and (2). Moreover (1) and Ctransitivity of G on X imply G is transitive on pairs (S, A) such that S is a 4-set and A a sextet through S. In particular (3) holds and NG(A) is transitive on A. so (1) holds. As the members of core(V) are the vectors of even weight, By (1) there are (244)/6 = 1,771 sextets; that is, (5) holds. (4) holds. Let T = {tl, t2, t3) and H = G s . Notice GT = L3(4) acts faithfully on n with H the split extension of E = HA E E16by A5 = Alt(A - {S)). Observe that 19.10 shows each vector v in V is of the form u w for ) ~A8, so there is an involution As S R = B is an octad, N ~ ( B = some w of weight at most 4 and some u E VC. Thus d(u,v) = lu vl = g E NG(B) acting as (tl, t2) on T and fixing exactly 2 points of R. Then lwl 5 4. So each vector in V is in a sphere of diameter 4 from some code Tg = T, so g acts on n and fixes exactly 3 points on the line {s, R). word in the extended Golay code. So by Exercise 1.3, (g, GT)*is L3(4) extended by a field automorphism Lemma 19.11: Let A be a sextet, L = NG(A), and K = LA. Then ~ = Sym(A - {S)). Thus as NG(A) is transitive on A, and ( H , ~ =) Sg N ~ ( A= ) Sym(A) ~ = Ss. (1) Ci(A), 1 5 i < 3, are the orbits of L on C. Let K = N G ( A ) ~be the kernel of the action of NG(A) on A. Then (2) Ci(A) is of order 15, 360, 384, for n = 1, 2, 3, respectively. Kg = EB Z E4, so by Ztransitivity of NG(A) on A, KA E E4, where (3) C1(A) consists of those B with B = S T for some S,T E A. A = R R', R' E A - {S,R). As E is faithful on A, KA n E = 1. Thus NL (B) = NL({S,T)) . Also KA and E are normal in K , so M = (KA,E) = KA x E % E64 (4) C2(A) consists of those B with lBn Rl = 2 for all R f A -{S, T) and M K. But by ( 5 ) , ING(A)I = 3 26 IS61,SO as NG(A)/K S6, for some S, T E A. NL(B) is the split extension of NK(B) FZ E8 1K1 = 3 26. Hence M E Syl2(K) and K is the split extension of M by by N L ( B ) ~= NL({S,TI)~. P of order 3. By Ctransitivity of G, NG(S)~= S4. Also M~ % E4,so (5) C3(A) consists of those B such that IB n SI = 3 for some S E A M~ E4 by symmetry. Thus ( P M ) ~= K~ 2 A4 since E q q K~ S4. andIBnTI = 1f o r T E A-{S). NK(B) Z Z 3 andIVL(~)*= Thus CM(P)< Ks. Then by transitivity of NG(A) on A, CM(P) = 1. NL(S)*.

o#

+

+

+

+

+ >

+

+

(z),

+

+ +

+

+

+ -

<

Set

v = V I E . The module

is the 12-dimensional Todd module for Notice that if U = core(V) is the subspace of vectors of V of even order then 0 is an 11-dimensional GF(2)G-submodule of called the 11-dimensional Todd module for M24.

G=

v

Proof: By 19.9, C1(A) is an orbit of L of length 15 and (3) holds. Let S,T E A and A = S + T E C1(A). Then Co(A) is of order 30 by 19.2 and hence the set C2(A,A) of octads in &(A) but not in Cl(A) is of order 24. As IB n C J is even for each B E C2(A,A) and C E Cl(A),

94

Exercises

Chapter 6 The Mathieu Groups

Exercises

we conclude lR n BI = 2 for each R E A - {S,T). Thus Cz(A>=

U

Cz(A1A)

AECi (A)

is of order 15.24 = 360. Let Si E A intersect A trivially, 1 5 i 5 3, and let P = 02(K). Then NK(B) acts on B r l Si of order 2 so NK(B) 5 Np(B n Si) and IP : Np(B n Si)l = 2. Then NK(B) Np(B nSi), which is of index 24 in K. So IK : NK(B)I 24 = lC2(A,A)I, and hence K is transitive on C2(A,A). Thus as L is transitive on Cl (A), L is transitive on C2(A) and (4) holds. Finally let xi E S, and B = B(Sl +XI 2 2 +x3). As IBn (Sl R)) is even for all R E A, we conclude B n R is a point for each R E A - (Sl) and S1 + X I = B n S1 is of order 3. That is, B E C3(A). Also K is transitive on triples (xl,x2, 23) with xi E Siand hence transitive on

>

< ni

+

95

+

of order 64. So INK(B)I = IKI/IRI = 3, and hence as L is transitive on A, L is transitive on C3(A) and (5) holds. Finally CilC,(A)l = 759 = )CI, so (1) holds, completing the proof.

Remarks. The Mathieu groups made their first appearance in two papers of Mathieu in 1860 [MI] and 1861 [M2]. However, M24 was mentioned only briefly and Mathieu did not supply details about the group until 1873 [M3]. Even then there remained uncertainty about the group. For example, in 1898 [Mil], G. A. Miiler published what puQorted to be a proof that M24 did not exist, although a year later in [Mi21 he had realized his mistake. In 1938, Witt published two papers [Wl], [W2]studying the Mathieu groups from the point of view of their Steiner systems. This may have been the first fairly rigorous existence proof for M24. J. Todd's papers in 1959 [Toll and 1966 [To21 seem to represent the first serious attempt to study the Steiner systems of the Mathieu groups. In particular Todd begins the study of the Todd module, and indirectly the Golay code module, as the latter module is the dual of the Todd module. It is in these papers that the terminology of octads, sextets, etc. is introduced and most of the results of Section 19 first proved. The existence of the invariant submodule Vc of the 24-dimensional binary permutation module was recognized at least implicitly by Paige [Pa] in 1957, two years before Todd's first paper. The Golay code had been discovered in 1949 by Golay in [GI,although he did not realize it was invariant under the Mathieu group M23.

1. The Mathieu groups Mi1 and M12 are simple while F*(Mlo) E A6. 2. Assume the hypotheses and notation of Lemma 18.7 and let T E 13. Prove (1) There exist ~ , T ' , T *E l3with I ~ ( T )= T + T', I3(T) = T 'ii, and I3(TI) = TI T*. (2) T T* E I& SO r2(T) = 'ii T*.

+

+

+

+

3. Let (X, C) be the Steiner system S(24,8,5) for Mz4 and let x and y be distinct points of x. Prove (1) There are exactly 330 octads missing x and y. (2) The number of members of Vc containing exactly one of x and y is 211. 4. Prove the binary Golay code is a perfect (23,12)-code with minimum weight 7 which corrects three errors. Its automorphism group is M23. 5. Let G be a group t-transitive on a set X of order n, T a t-subset of X , and B a k-subset of X with T 2 B such that G(T) 5 G ( B ) and G(B) is t-transitive on B. Let B = {Bg : g E G).Prove (X,B) is a Steiner system S(n, k, t). 6. Let Xg = (Xg, D9) be the affine plane over GF(3). Prove (1) X9 is a Steiner system S(9,3,2). (2) Aut(Ug) = HE is a split extension of E by H, where E E Eg is the translation group and H S GL2(3). (3) Let Q8 S A 5 H. Prove Xg, A satisfies the Extension Hypothesis. (4) Let Xlo be the projective plane over GF(9) and 2310 the collection of translates of the projective line B = {m, O,1, -1) over GF(3) under M = PI'L2(9). Prove Xlo = (Xlo, Blo) is a Steiner system S(10,4,3), M = Aut(Xlo), and Xlo is the unique extension X of x 9 with A I N~,,t(x)(Xg). (5) Prove Xlo,AE satisfy the Extension Hypothesis and there exists a unique extension Xll = (Xll,Ull) of X10 admitting AE. Moreover Aut(Xll) = M11. (6) Prove Xl1, Mll satisfy the Extension Hypothesis and there exists a unique extension X12 = (X12,B12)of X11 admitting Mil. Moreover Aut(X12) = M12. (7) Mil, Mi2 are the unique 4,5 transitive groups with point stabilizer Mlo, Mll, respectively. (Hint: To prove (7) use (5) and (6) plus Exercise 6.5.)

20. The geometry of Mz4 of forms P, C, f defined on Vc via

Moreover by 11.8:

Chapter 7

Lemma 20.1: (1) f is a symmetric trilinear form with f (x,x, y) = 0 for all x , y E VC. (2) Rad(Vc) = (X), so we have induced forms i n which we also write as P,C ,f .

vc

The Geometry and Structure of M2,

In Lemma 11.8 we saw that if (V,X, U) is a strictly doubly even binary code then there exists a triple (P, C, f ) of forms on U inducing a geometric structure on U.In this chapter we study the geometry induced on the Golay code module Vc in this fashion. In particular we find that the octads are the singular points in this module with the trios corresponding to the singular lines and the sextets to maximal hyperbolic subspaces. The tlocal geometry for M24 is the geometry of octads, trios, and sextets with incidence defined by inclusion, subject to the identification of these objects with subspaces of Vc just described. In Section 20 we derive various properties of this geometry which will be used in Section 21 t o investigate the basic local structure of M24, and later t o establish the uniqueness of M24 and other sporadics.

3

,

Lemma 20.2: (1) G is transitive on the 3,795 = 23.11 5 . 3 trios. (2) The stabilizer L of a trio {A,B, A B X ) is the split extension of E64 by S3 X L3(2)(3) Let A be a sextet such that A, B E C1(A). Then L = NL(B)NL(A) with lL : NL(B)I = 3 and IL : NL(A)I = 7.

+ +

+ +

20. The geometry of M24

In this section we continue the hypothesis and notation of Sections 18 and 19. In particular G is the Mathieu group M24 and (X,C) is the Steiner system for G. Recall V is the power set of X regarded as a GF(2)G-module via the operation of symmetric difference. Also Vc is the 12-dimensional Golay code submodule of V generated by the octads and Vc = Vc/(X) is the 11-dimensional Golay code module. By 19.7, (V,X, Vc) is a doubly even code so by 11.8 there is a triple

For x E Vc, fx is the bilinear form fx(y,z) = f(x,y,z), Cz is the quadratic form Cx(y) = C(x,y) (which has fx as its associated bilinear form) and R(x) = Rad(fx). In Section 11 we saw how this algebraic structure on Vc can be used t o define a geometric structure preserved by the isometry group of P. In particular M24 preserves this geometry, which is an excellent tool for studying the group. For example, recall from Section 11that a subspace U of Vc is singular if P is trivial on U and U is singular with respect to f , for all x E V. Further given S C_ Vc, SO consists of those v E Vc such that f,, = 0 on S. Recall from the discussion in Section 11 that SO is a subspace of Vc. Further a subspace U of Vc is subhyperbolic if U8 is a hyperplane of Vc and P is trivial on U. Define a trio to be a triple {B1, B2, B3) of octads such that X is the disjoint union of B1, B2, and B3. Thus Bi E Co(Bj) for each i # j and B3=B1+Bz+X.

I

Proof: Let 7 = {A, B, A B X) be a trio. Part (1) follows as G is transitive on its 759 octads and Co(B) is an orbit for M = NG(B) of length 30. Also this shows IL : L n MI = 3 and L~ = S3. By 19.2.6, LT is the split extension of Q7 Ea by L3(2)/E8, so as L~ = S3, 02(LT) = QT x 9% E E64 for g E L - NL(B). Then (2) holds. Finally NL(A) induces S3 o n 7 and (NG(A): NL(A)I = 15by 19.9.4, so (3) holds.

Lemma 20.3: (1) The octads are the singular points of dodecads are the nonsingular points.

Vc and the

98

Chapter 7 The Geometry and Structure of M24

20. The geometry of

(2) G has three orbits on lines of Vc generated by octads: the singular lines, the hyperbolic lines, and the lines (A,B ) with A E C2(B). (3) The singular lines of & are the lines 1 with 1# the octads in a trio. (4) The hyperbolic lines of Vc are the lines (A, B} with A E C4(B).

v?:

Proof: By 19.8, G has two orbits on The octads and the dodecads. So as Ivl = 0,2 mod 4 for v an octad, dodecad, respectively, (1) holds. Next by 19.2, G has three orbits on lines of Vc generated by singular points, and thoseprbits have representatives (B,Ai), where Ai E Ci(B),

i=0,2,4.NowB+A:!isadodecadso~(~+~q)=1,whie~+~~is an octad for i = 0,4. By 19.11 there exists an octad C with IBnA4nCI odd, so f(B,A4,C) = 1 and hence (B,A4) is hyperbolic. Finally as B n Ao = 0,IC n B nAol is even for all C E C, so (B, Ao) is singular.

Lemma 20.4: Let A be a sextet. Then (1) {B : B E C1(A)) = U# for some 4-dimensional subspace U =

U(A) of &. (2) U is subhyperbolic. (3) NG(U) is irreducible on U, U8/U, and Vc/UB of dimension 4,6,1, respectively. (4) Let fu = fv for v E V - U8. Then (U, fu) is a 4-dimensional symplectic space over F and NG(U) induces the symplectic group O(U, f,) Z Sp4(2) on U. (5) For A 5 U, R(A) n U is the subspace of U orthogonal to A under

fun (6) If P E Sy13(CG(U)) then NG(P) 5 NG(U) and CG(P) A6/Z3 is quasisimple.

+

+ +

E

Proof: For distinct A, B E Cl(A), A B or A B X is in Cl(A), so as lCl(A)l = 15, (1) holds. Let L = NG(U) and K = LA a s i n 19.9, and take P E Sy13(K). By 19.9.4, CL(P) S A6/Z3 and 02(K) G% E64- Thus 02(L) is a faithful 3-dimensional GF(4)CL(P)-module, so a Sylow 3-subgroup R of CL(P) is--Sylow in SL3(4), and hence is isomorphic to 31+2. Therefore CL(P) is quasisimp!e. As K fixes each member of Cl(A), K = CG(U) and K acts on the hyperplane (A, B)B of Vc for (A, B) a hyperbolic line in U. Thus dim(Cvc(P)) dim(U) 1 = 5, so dim([Vc, PI) 2 6. So as 6 is the minimal dimension of a faithful GF(2)R-module, dim([Vc, PI) = 6 and

>

+

M24

+

CL(P) is irreducible on ([Vc, P] U)/U of dimension 6. In particular [Vc, PI U is generated by octads. Now by 19.11, if C E C3(A) then IAnBnCl is odd, so f (A, B, C) = 1 and C 4 UB, while if C 4 C3(A) then f (A, B, C) = 0, so C UB. Thus as [Vc, P] U is a hyperplane generated by octads, U6 = [Vc, P] U, so (2) and (3) hold. Then (4) and (5) hold by 11.6 and 19.9. We also see that dim(Cvc (P)) = 5 and U is the unique subhyperbolic hyperplane of Cvc(P),so NG(P) IL, completing the proof of (6).

+

+

<

+

Using 20.3 and 20.4, we identify the octads, trios, and sextets with subspaces of Vc of dimension 1,2,4, respectively. Proceeding as in Section 4, let 'I be the geometry on I = {1,2,3) whose objects of type i are the octads, trios, sextets, for i = 1,2,3, respectively, and with incidence equal to inclusion, subject to our identification of the objects of I" with subspaces of Vc. The geometry I'is the 2-local geometry of M24, since the stabilizers of objects in a chamber are the maximal 2-local subgroups of M24 containing a Sylow Zsubgroup of M24 fixing the chamber. The identification of I' with subspaces of Vc gives an injection of geometries l? -t PG(Vc) embedding I' in the projective geometry of Vc. The algebraic structure supplied by the forms P, C, f allows us to distinguish subspaces of Vc and hence use the embedding of r in pG(VC) effectively. We say a pair of octads are collinear if they are incident with a common trio and octads or trios are coplanar if they are incident with a common sextet. See Section 4 for definitions of geometric terminology such a s "residue," etc. See Chapter 7 in [FGT] for a discussion of syrnplectic and orthogonal spaces.

Lemma 20.5: (1) r is a residually connected string geometry. (2) G is flag tmnsitive on .'I (3) The residue of an octad is isomorphic to the points and lines of 4-dimensional projective space over GF(2). (4) The residue of a sextet is isomorphic to the singular points and lines of 4-dimensional symplectic space over GF(2). .(5)-The-e are three octads and seven sextets incident with each trio. Proof: Part (4) follows from 20.4.4 and 20.4.5 and part (3) follows from 20.6.1 and 20.6.3 below. Moreover these lemmas show the stabilizer of an octad or sextet is flag transitive on its residue. Notice (3) and (4) say the residues of octads and sextets are connected. By 20.2.3 part (5) holds and the residue of a trio is a generalized digon with the stabilizer

Chapter 7 The Geometry and Structure of M24

20. The geometry of M2*

of the trio flag transitive on the residue. Thus to complete the proof it suffices to prove I? is connected. But from 19.2, G is primitive on octads, so I' is connected.

and NG(U) = NG(UB), U is the unique sextet containing h. By 20.5, U n R(h) = k is the line orthogonal to h in U, so U = h @ k. h r t h e r if x E R(h) then x h W E 1'3 by (3), so by uniqueness of U,W = U and hence x E k.

100

+ <

Lemma 20.6: If x is an octad then (1) dim(R(x)) = 5, G, is the stabilizer of x in GL(R(x)), and R(x) is the union of the trios through x. (2) (Vc/R(x), C,) is a 6-dimensional orthogonal space of sign +1, Gz acts as Ri(2) on Vc/~(x),and for U E r3(x), U+R(x)/R(x) is a singular point of Vc/R(x) and UB/R(x) is the subspace orthogonal to that point. (3) The map U -+ R(x) nU is a bijection between r3(x) and the set of 3-subspaces of R(x) containing x. (4) Each pair of distinct coplaner trios is contained in a unique sextet. (5) If U E F3(x) then G,,u has three orbits on r3(x): {U), the eighteen sextets W in UB with U n W E r2, and the sixteen sextets Z not contained in UB with Z n U = (x). (6) Each hyperbolic line h is contained in a unique sextet U, and u = h $ R(h).

Proof:Adopt the notation of 19.2 with B = x and (x, I, U) a chamber. Thus, for example, M = Gx, and by 19.2, M/Q r L4(2), while by 19.2.6, Ml/Q is the stabilizer of a point in the natural module for MI&. In particular M is therefore 2-transitive on r2(x). But by 20.5.4 and 20.5.5, A C is an octwl if 1 = (B, A) and k = (B, 6 )are trios in U,so A C is an octad for all distinct A, C E Co(B). Hence as ICo(B)I = 30, (1) holds. Next by 20.4, R(x) n U is the hyperplane of U orthogonal to x under fU, so 0 is a point in Vc = VC/R(U).Further (vc, C,) is a nondegenerate orthogonal space of dimension 6 and Gx induces Rgf(2) on Vc, so the space is of sign +l. By definition of C,, U is a singular point in PC and 8 0 = O1.Then as Mu is the parabolic subgroup of M stabilizing R(x) n U and 0, (3) and (5) hold. If 1 and m are trios in U then either 1 m = U or 1n m is an octad which we take to be x. In the first case clearly U is the unique sextet containing 1 and m. In the second 1 m = R(x) n U, so U is unique by (3). Thus (4) is established. I t remains to prove (6). As each sextet contains a hyperbolic line h and G is transitive on hyperbolic lines, we may take h 5 U. Then UB 5 he, so a s UB is a hyperplane, UB = he. Then as G is transitive on sextets

+

101

Given a trio 1 let ((1) = (R(x) : x

E 1).

L e m m a 20.7: Let 1 be a trio and x, y E 1 distinct octads. Then

+

( I ) dim(C(1)) = 8 and c(1) = R(x) R(y) with R(x) n R(y) = 1. (2) ((1)/1 is the tensor product of natural modules for the factors of G1/02(Gr) L3(2) x L2(2). (3) G~ acts as ~3 (2) on Vc/c(l). (4) Each octad in ((1) - 1 is in a unique R(u), u E z#.

Proof: From 20.6, Gz,y acts irreducibly on R(x)/l as L3(2), so R(x) n R(y) = 1. Thus dim(R(x) R(y)) = 8. Let Q(z) = 02(Gz) for z = x, y. By 20.6, [Q(z), VC] L R(z), so S = (Q(x), Q(Y)) acts on R(x) + R(Y). Thus as GI = GZIyS,it follows that ((1) = R(x) + R(y) is of dimension 8, and (2) holds. As R(x) nR(y) = 1, each octad in ((1) -1 is in at most one R(u), u E 1#. By (2), GI has two orbits on vectors in <(I) - I, and as one of these contains nonsingular points, each such octad is in at least one R(u). Thus (4) is established. Finally dim(Vc/C(l)) = 3, so from the action of Gx,y on Vc obtained from 20.6, (3) holds.

+

i

Lemma 20.8: The stabilizer GI of a trio 1 has five orbits fii(l), 0 5 i 5 4, on I'2 as follows:

+

60(1) = {ll. S1(l)= {k:O# k n l #1) is of order42. S2(1) = {k : k + 1 E r3)is of order 56. (4) s3(l) = {k : dim(k n c(1)) = 1) is of order 1,008. (5) S4(1) = {k : k n ((1) = 0) is of order 2,688.

(1)

(2) (3)

+

+

i

Proof: First Gl is transitive on the three octads on 1 and taking x E 1, by 20.6, R(x) contains fifteen trios through x and G, acts Ztransitively as L4(2) on these trios, so there are 3 . 14 = 42 trios in Sl(1) and they form an orbit under G1. Second GI is transitive on the seven sextets incident with 1 and if U is such a sextet, then from 20.5, Gu,1 is transitive on the eight trios of U intersecting 1 trivially, so there are 7 8 = 56 trios in S2(1) and they form an orbit under Gr. Third by 20.6, GI,, is transitive on the 28 octads z in R(x) - 1. By 20.6.3, z + I is contained in a unique sextet U and by 20.6.3, 20.6.6,

Chapter 7 The Geometry and Structure of M24

102

and 20.7.4, ( ( 1 ) n R ( z ) = U n R ( z ) is a 3-subspace of R ( z ) with Gz,u = Gz,UnR(z).Now CG(U) < G,,l,u is transitive on the 12 trios in R ( z ) <(1). Thus the 3.28 12 = 1,008 trios in b3(l) form an orbit. This leaves 2,688 trios k with k n ( ( 1 ) = 0. Checking the permutation character of G on GIGl, we find G is rank 5 on lines, and hence conclude that &i4(1) forms the fifth orbit.

Lemma 20.9: The stabilizer GU of a sextet U has orbits Ei(U), 0 i 5 3, on sextets, where:

<

(1) = {U}. (2) Z1(U) = { W : W n U E r 2 ) is of order 90. (3) Z 2 ( U ) = { W : W n U E r l ) is of order 240. (4) E3(U) = { W : U n W = 0) is of order 1,440.

Moreover members of E;(U) are not contained i n U 8 for i = 2,3. Proof: As G u is transitive on the fifteen trios 1 E r 2 ( U ) and Gl is 2-transitive on the seven sextets in r3(1), Z1(U) is an orbit of length 1 5 . 6 = 90. As G u is transitive on the fifteen octads x E r l ( U ) and by 20.6.5, Gut, is transitive on the sixteen sextets W E r 3 ( x ) with ( x ) = U n W , B2(U) is an orbit of length 15 16 = 240. Further by 20.6.5, W $ UO. This leaves 1,440 sextets W with W n U = 0. 'From the character table of G, G is rank 4 on GIGU, so E3(U) is the fourth orbit of G u on planes. Lemma 20.10: Let U E r3,W Then((1)nU = W n U ~ r 2

E Z l ( U ) , and

1 E r z ( W ) with I n U = 0.

.

Proof: If x E U n ( ( l ) # then by 20.7.4, x E R ( y ) for some y E 1#. Further if ( x , x ' ) is a trio in U then as U = ( R ( x )n U ) + ( R ( x f )n U ) , ( R ( x ) U )n ( R ( x / n ) U ) = U ( R ( x ) n R(x')) = U , SO ~ ( yn )u = ( 2 ) . Thus by 20.7.7,

+

103

21. The local structure of M24

+

Lemma 20.11: For each sextet U , GU is transitive on the 384 octads and 640 nonsingular points i n PC- U0, and if x is such an octad then -. (U,Cx) is an orthogonal space of sign -1 and GU,, r Z3 x S5 induces the isometry group of Cx on U with kernel of o d e r 3. Proof: Let x E Vc - UB be an octad. Now 0 2 ( G u )= E is the group of transvections with axis UB/U on V/U, so E is regular on V/U - UB/U. Thus H = Gu,u+, is a complement to E in Gu, so H Z S6/Z3. Now H

+

has two orbits xiH on x U of order 6,10, respectively. So as 27 does not divide the order of Gx, we conclude x = x i , G u is transitive on the 64 6 octads in V - UB, and Gu,, is as claimed.

Lemma 20.12: The stabilizer of a sextet U has orbits y ( U ) , 0 < i 5 3, on r2, where: (1) no(U) = r 2 ( U ) is of order 15. (2) tcl(U) = {I : 1 n U E r l ) is of order 180. (3) tc2(U) = (1 : 0 = 1 fl U and 1 5 W E E 2 ( U ) ) w of order 720. (4) rc3(U) = ( 1 : 1 $ UO) is of order 2,880. Proof: Of course q ( U ) is an orbit of length 15. Next Gu is transitive on the 15 octads x E r l ( U ) and by 20.6, GXluis transitive on the twelve trios 1 with U n 1 = ( x ) , so tcl(U) is an orbit of length 15 - 12 = 180. Now W E E1(U) contains eight trios 1 with 1 n U = 0 and by 20.10, W n U = ((1) n U . Then W = ( { ( l )n U ) + 1 is uniquely determined by I, so tc2(U)is of order 90.8 = 720. This leaves 2,880 trios. From 20.11, Gu is transitive on the 384 octads x E V - UO, and G,,u is transitive on the fifteen trios through x , so Gu is transitive on the 384.15/2 = 2,880 trios in tc3(U). Lemma 20.13: W o s 1, m are coplanar if and only if m i 5 2.

E

&(l) for

Proof: This follows as G is flag transitive on I? and if 1 is a trio in a sextet U then the orbits of Gl,u on r 2 ( U ) are bi(l) n U for i 2.

<

21. The local structure of

M24

In this section (X, C) is the Steiner system for G = M24, V is the binary permutation module for'G on X, Vc is the Golay code submodule, and Vc = Vc/(X) is the 11-dimensional Golay code module. We also use the rest of the notation and terminology of Chapter 6 and Section 20. Our object in this section is to determine the conjugacy classes of subgroups of G of prime order and the normalizers of such subgroups.

Lemma 21.1: (1) G has two classes of involutions with representatives z and t . (2) F i x X ( z ) E C and t has no fixed points o n X. (3) F*(CG(z)) = Q % 0;is a large extmspecial subgroup of G and C G ( ~is) the split extension of Q by L3(2). (4) CG(t) 5 NG ( U ) for some sextet U , and CG(t) is the extension of E64 by 5 5 -

Chapter 7 The Geometq and Structure of

104

21. The local structure of Mz4

M24

dim(Cv(d)) is the number of orbits of d on X. This gives the dimensions in (3). We use this same argument below without further comment. Notice also that if g of order 3 fixes an octad B it fixes a point of B , so the final remark in (3) holds too. Next we saw during the proof of 20.4 that a Sylow 3-subgroup P of G is 31+2 and we may take (d) = Z(P).Then from (4), NG(P) has two orbits on elements on P - (d). One of these orbits is fused to d in L3(4), leaving the other fixed point free on X and completing the proof of (2). Let 1be a trio. By 20.7 there is a subgroup Y of order 3 with NG, (Y) L3(2) x S3 a complement to 02(Gl) in Gl and [Vc, Y] = c(1). Then by 21.3, NG(Y) 5 GI, completing the proof of the lemma.

Proof: Let i be an involution in G. As G is simple, (Fixx(i)l 1x15 0 mod 4. But the stabilizer L of three points of X is L3(4), which has one class of involutions with representative z such that FixX(z) E C. Further (3) holds by 8.10. Thus we may assume i has no fixed points on X. Now 40.6 and 40.4.2 complete the proof.

Lemma 21.2: Let v be a dodecad. Then (1) Gu is irreducible on Vc/(8). R(a) = (8).

(2)

Proof: Recall from 19.4 that Gv 2 M12 has order divisible by 11. But 10 is the minimal dimension of a faithful GF(2)Z11-module, so as dim&) = 11, (1) holds. Then as f, # 0 is preserved by Gv, we conclude from (1) that (Vc, f,) is nondegenerate and hence (2) holds.

Lemma 21.5: (1) G is transitive on subgroups of o d e r p for p =5, 7, 11, and 23. (2) NG(A) is h b e n i u s of order 11 10, 2 3 . 11, for A of order 11, 23, respectively. (3) If B is of order 5 then JFixx(B)l = 4, dim(Cvc(B)) = 3, CG(B) 2 Z5 x A4, ING(B) : CG(B)I = 4, and NG(B) &es a seztet. (4) If E is of order 7 then IFixx(E)( = 3, dim(Cvc(E)) = 2, NG(E) 2 Z3/Z7 x S3,and NG(E) jixes a trio.

-

Lemma 21.3: If 1 is a trio (regarded as a singular line of Vc) then 1 is the radical off restricted to C(1). Proof: From 20.6 and 20.7, <(Z)/R(x) is a totally singular %space in Vc/R(x) with respect to C, for each octad x E 1. Thus 1 < R = R ~ d ( f ~ < ( On ~ ) )the . other hand by 20.7, G1 is irreducible on c(1)/1, so if R # I then f is trivial on <(I). But there is a dodecad v with 8 f [ ( l ) , so if f is trivial on c(1) then [(1)/(5) is a totally singular subspace of (G/(G),f ) of dimension 6, whereas by 21.2.2, that space is nondegenerate of dimension 10 and hence has no such subspace.

Proof: For p > 3 a prime divisor of IGI, p divides IG/ to the first power, so (1) follows from Sylow's Theorem. Similarly if p = 11 or 23 then Sylow gives us ING(A) : A1 modulo p and from 21.1 and 21.4, ICG(A)[ is prime to 6. We conclude (2) holds. Next the stabilizer L3(4) of three points contains subgroups B, E of order 5,7, respectively, allowing us to calculate the number of fixed points of our element on X and Vc. In particular from 20.7, CVc(E) = 1 is a trio and then (4) holds. Also Fixx(B) is of order 4 and hence contained in a unique sextet, and this observation together with 20.4 gives (3).

Lemma 21.4: (1) G has two classes of elements of order 3 with representatives y and d. (2) I Fixx (d) 1 = 6 and y has no fixed points on X. (3) dim(C&(d)) = 5 and dim(Cvc(y)) = 3. Moreover y centralizes no octads in Vc. (4) CG(d) S AslZ3 is quasisimple, NG((d))/(d) S Ss, and NG((d)) is a complement to O2(NG(U)) in NG(U) for some sextet U < Cvc (d). (5) NG((y)) 2 53 x L3(2) and NG(Y) is a complement to O2(NG(1)) in NG(l) for some trio 1. Proof: First the stabilizer L3(4) of three points of X has one class of elements of order 3, so G is transitive on elements of order 3 fixing a point of X. Now by 20.4 if U is a sextet then there is an element d of order 3 fixing each block in the sextet, fixing six points of X , and satisfying (4). Notice as V/VCis dual to I+, dim(Cv(d)) = 2dim(Cv,(d)) = 2(dim(CVc(d)) 1) and as V is the permutation module for G on X,

+

105

,

Remarks. I believe Ronan and Smith were the first to discuss the 2local geometry r for M24 in [RS]. Ronan and Smith also point out the representation of r as points, lines, and 4-subspaces of the Golay code module VC. The use of the derived forms (PIC,f ) on Vc to induce and study the structure of I' and G comes from [AS2]. Some of the facts about I? established in Section 20 first appeared in Todd's paper TO^]. Others appear in R. Curtis's thesis [Cu] and still others in [AS2]. The facts about the local structure of M24 fall in the realm of the "well known."

106

Chapter 7 The Geometry and Structure of M24

I

The reader may wonder why the 2-local geometry r of M24 is discussed here in such excruciating detail. It is because l' appears as a residue of the 2-local geometries of Col, Jq, M(24), and Fl. Thus the uniqueness proofs for these groups require detailed knowledge of I?.

Exercises 1. Let A be the collinearity graph of the 2-local geometry I' of M24 and let x E A. Prove (1) A(x) = Co(x), a 2 ( x ) = C4(x), and h3(x) = C2(x), SO A is of diameter 3. (2) xL = R(x) n A and A 1 2 ( ~= ) {y E A : Cx(y) = 0). (3) The collinearity graph of the residue of a plane is the graph of singular points in a Cdimensional symplectic space over GF(2). (4) Each pair of singular lines through x is incident with a plane. (5) If z E A2(x) then x and z are incident with a unique plane. (6) For each line k of there is a unique point y of k with d(x, y) = 4 x 1 k)(7) If U is a plane determined by a sextet S then d(x, U)= 1 if and only if x E C2(S). (8) If d(x, y) = 3 then G=,v 2 Sp4(2) with R(x)/x the 4dimensional symplectic module for Gx,v and U E r3(z) is at distance 1from y if and only if (U 17R(x))/x is a singular line in that space. Hence the graph on r2(x) with 1 * k if and only if d(n(1, k), y) = 1 is , is the unique plane through 1 and k. connected, where ~ ( 1 k) 2. Let G = M24. (1) Let P be the binary permutation module for G on the set C of octads and Q the submodule of P generated by all elements A B C such that {A,B, 6 ) is a singular or hyperbolic line in the Golay code module Vc. Prove P/Q is GF(2)G-isomorphic t o (2) Let M be the stabilizer in G of an octad and V a faithful GF(2)Gmodule such that (i) M stabilizes some z E V with V = (zG). (ii) M is faithful on some U 5 V with 2 E U, dim(U) = 5, and U# E zG. Prove V is GF(2)G-isomorphic to Vc. 3. Let Vc be the Golay code module and W 5 Vc with dim(W) > 3. Prove W contains a singular point. (Hint: Use Exercise 3.3.)

+

+

vc.

Exercises 4. Let Vc be the 11-dimensional Golay code module for G = M24. Prove that for each elementary abelian 2-subgroup A of G, IVc : Cvc (A)I > IAI. 5. Let G = Mu, v = 22,23, or 24, and (X, B) be the Steiner system for G. Prove Aut(G) = Aut(X, B), so Aut(M24) = M24, A~t(M23)= M23, and IAut(Mz2) : M221 = 2.

107

22. The Leech lattice and 0 of M, so for

C, a,

and

Exbx in R

Chapter 8 The Conway Groups and the Leech Lattice

~ ~ , \

/

For v E R~~define q(v) = (v,v)/16. Thus q is a positive definite X , define e y = CvEY y ER ~ ~ . ~ .Y quadratic form on R ~ Given For x E X let Az = e x - 4x. The Leech lattice is the set A of vectors v = C, ass E R~~such that: (Al) (A2) (A3) (A4)

a, E Z for all x E X.

(Ex

m(v) = ax)/4 E Z. a, = m(v) mod 2 for all x E X. C(v) = {x E X : a, f m(v) mod 4) E Vc.

Lemma 22.1: Let u, v E A. Then ( I ) m ( u + v ) =m(u) +m(v) andm(-u) = -m(u). (2) C(U+V)=C(u) +C(v) andC(-u) =C(u).

The Leech lattice is a certain 24dimensional Z-submodule of 24dimensional Euclidean space R~~discovered by John Leech. John Conway showed that the group -0of automorphisms of the Leech lattice is a quasisimple group. The central factor group of S O is the Conway group Col. Further other sporadic groups are the stabilizers of sublattices of the Leech lattice; for example, Conway discovered the Conway groups Co2 and Co3 in this way. In Section 22 we construct the Leech lattice A and the Conway groups, and establish various properties of A and the Conway groups. In Section 23, we consider = A/2A, the Leech lattice mod 2. The Leech lattice mod 2 supplies another tool for studying the Conway groups. For example, it allows us t o construct McLaughlin7sgroup Mc and the Higrnan-Sims group HS.

22. The Leech lattice and - 0 In this section M = M24 and (X,C) is the Steiner system S(24,8,5) for M. Let V be the permutation module over GF(2) for M with basis X and I+ the Golay code submodule. Let R~~be the permutation module over the reals for M with basis X and let ( , ) be the symmetric bilinear form on R~~for which X is an orthonormal basis. Thus R~~together with ( , ) is just 24dimensional Euclidean space admitting the action

(3) A is a 2-submodule 0 . f ~ ~ ~ .

Proof: Parts (1) and (2) are easy. Further (1) and (2) together with the fact that Vc is a subspace of V, imply (3). Let A. denote the set of vectors v E A such that m(v) E 0 mod 4. Lemma 22.2: (1) 4eJ E A for each subset J of X of even order. (2) 2e, E A for each v E VC. (3) 8x E A for all x E X. (4) A. is the Z-submodule spanned by {2eB : B E C).

Proof: Parts (1)-(3) are straightforward calculations, keeping in mind that Ivl r 0 mod 4 for all v E VC. By 22.1.1, A. is a Z-submodule of A. By (2), 2eB E A. for B E C, so the Z-submodule L generated by {2eB : B E C) is contained in Ao. Hence to complete the proof of (4), it remains t o show A. 5 L. We claim 4(x - y) and 162 E L for all x, y E X. Namely if T is a 4subset of X then from 19.9.1, T is contained in a unique sextet A(T). Then for R, S E A(T) - {T), 4eT = 2eq-+s

+ 2 e T + ~- 2 e ~ + Es L.

Then choosing U to be a 5-subset of X containing x, y,

I

110

Chapter 8 The Conway Groups and the Leech Lattice as U + x and U

+ y are Csubsets. Finally

+

tET

where T is a Csubset with x $ T. Thus the claim is established. Now let v = Ex axx E ha;we must show v E L. As m ( v ) 0 mod 4, ax is even for all x E X by (A3),and C(v) = { x X : a, G 2 mod 4 ) E I+ by (A4). But C(v) = CBES B for some S E C, so replacing v by v+CBES 2eB1we may assume C(v) = 0. Now taking X = ( X I , ... ,224) and bk = Cislea,, we have

=

v=

C b i ( ~-i xi+l) + b24~24-

i<24

But a, 0 mod 4 for d l i,so bi 0 mod 4, and hence b i ( ~ i - ~ i + El )L by the claim. Similarly b24 = 4m(v) = 0 mod 16, so b 2 4 ~ 2E4 L by the claim.

Lemma 22.3: Let xo E X . Then A is the Z-submodule generated by Ao and A,, . Proof: Let X = Ax,. Evidently X E A with m(A) = 5. Conversely let v = Ex a,x E A; we must show v is in the Z-span of A. and A. If m ( v ) is odd then as m ( v A) = m ( v ) m(X), replacing v by v X we may take m ( v ) even. Similarly replacing v by v + 2X if necessary, we may assume m ( v ) = 0 mod 4. But then v E Ao, as desired.

+

+

Lemma 22.4: (1) ( x ,y ) 0 mod 8 for all x,y (2) q(x) E Z for all x E A.

111

(1) F o r g e S y m ( X ) , ( e y ) g = e y g and (cy)g=cyg. (2) The map Y H ~y is an isomorphism of V with P as abelian groups which commutes with the action of M I where M acts on P by conjugation. (3) P S y m ( X ) o ( R ~ and ~N) = M Q 5 G .

-

E h.

Write o ( R ~ for~the ) subgroup of G L ( R ~preserving ~) the bilinear form ( , ), or equivalently preserving the quadratic form q. Let G be the subgroup of O ( R 2 4 )acting on A. The group G is the automorphism group of the Leech lattice; it is often denoted by - 0.For Y X , write cy for the element of G L ( R ~such ~ ) that

-

<

Proof: Part (1) is straightforward and, together with the observation that eyez = cy+z, implies (2). As S y m ( X ) permutes X and X is an ~). P o(R~~). orthonormal basis for ( , ), S y m ( X ) 5 o ( R ~Similarly As M preserves VC1 M acts on A. So M 5 G. Let B E C and e = eB. It remains to show Q 5 G, and for this it suffices by 22.2 and 22.3 to show the images 2eAc and Axe of 2eA and Ax under c are in A for each = and 2eB are in - 2 e and ~ A E C and X E X.But A by 22.2. Similarly Axe = Ax - 2eB 82 if x E B, and Axe = A, - 2eg if x $ B , But by 22.2, 2 e and ~ 8x are in A, so the lemma holds.

<

+

In the remainder of this section N will be the subgroup of lemma 22.5.3. Given a positive integer n , write An for the set of vectors v in A with q(v) = n. By 22.4.2,

+

Proof: By 22.2 and 22.3 it suffices to prove the remarks for x = 2eB or A, and y = 2eA, where A, B E C , z E X. But (2eA,2eB) = 41An BI G 0 mod 8 by 19.2.1. Also (2eB,A,) = 8 or 16 for z E B , z !$ B , respectively. Finally q(X,) = 2.

c

I

Lemma 22.5: Let P = {ey : Y E V ) , Q = {ey : Y E VC). Then

4(x - t ) 4 e E~ L,

162 =

22. The Leech lattice and 0

i

A= For v =

UA.. n

Exax%E A and i a nonnegative integer, let Si(v) = { x E X : la,/ = i),

.

and define the shape of v to be (On,, lnl, . .), where ni = ISi(v)). Let A$ be the set of vectors in A of shape (28,016), A$ the vectors in A of shape (3,123),and A: the vectors in A of shape (42,022).

Lemma 22.6: (1) A;, 1 5 i < 3, are the orbits of N on A2. (2) 1~?j1= 27 759, I A ; = ~ 212 - 24, and ) ~ f t = l 22 . (224). (3) lA21 = 196,560 = 2* . 33 .5 7 13. Proof: F i s t observe that by Exercise 8.1, a vector v = C, a,x E R~~ of shape (28,016) is in A if and only if S2(v) E C and T ( v ) = {X E X : a, = 2 ) is of even order. Thus there are 759 choices for S2(v),and givenSz(v) there are 27 choices of signs for the coefficients ax, x E S2(v),so that IT(v)l is even. So (2) holds in this case. Further if u E A: then there is g E M with S2(v) = S2(u)g = S2(ug), so in proving transitivity of N on A$, we may take Sz(u) = S2(v). Next for each subset R of S2(v) of order 2, there exists B E C with

112

2.2. The Leech lattice and

Chapter 8 The Conway Groups and the Leech Lattice

.0

113

S2(v)n B = R by 19.2. Now veg differs from v only by a sign change on R, so adjusting by such elements eg, we can map v to u , since T ( u ) and T ( v ) are each of even order. Therefore we have shown that N is transitive on A;. Similarly by Exercise 8.2, N is transitive on 1\32, while by Exercise 8.1, if v E R~~is of shape (3, 123),then v E A if and only if S = { x E X : ax r 1 mod 4 ) E I+. So there are 212 choices for S , and given S there are twenty-four choices for the place S3(v).Hence (2) holds in this case. Notice Exercise 8.3 handles the case of A!. Finally suppose v = axx E A2. We must show v has one of the three shapes above. As 32 = 16q(v) = C,a%, we have laxl 5 5 for all x. Indeed by (A3), if some az is odd, then all are odd, and hence a$ 5 32 - 23 = 9. So laxl i 3, and if lazl = 3 then the inequality is an equality, so v has shape (3,123),as desired. On the other hand if v has shape 124 then q(v) # 2. So we may take m ( v ) = m to be even and laxl = 0,2, or 4 for all x. By (A4), S z ( v ) E 6. But 41S2(v)I 5 32, so ISz(v)l 5 8. Now if S2(v) is empty then as q(v) = 2, v has shape (42,o ~ ~Thus ) . we may assume S z ( v ) # 0,so as IS2(v)j 5 8, S2(v)E C and v has shape (28,016).

by (A4),Bh E C. Therefore h E NSv,(x)(C) = M ,so we may assume g = cy. Similarly for u E A:, R(u) = ( x E X : a, r 1 mod 4 ) E and R(ug) = R ( u ) Y , so R(u) Y E Vc. As R(u) E VC and Vc is a subspace of V , we conclude Y E Vc, so g E N .

Lemma 22.7: If p is a prime diwisor of IGI then p 5 23.

Proof: For x , y E X , (xq,y ~ =) ( x - eT(z)/2,y - e ~ ( ~ ) = / 2()x ,y). SO q E 0(R2*), and hence as ry- E o ( R ~also ~ )E ,= ET E o(R").Thus by 22.2 and 22.3, it remains to show 2eBt and A,< are in A for each B E C and some x E X. Notice ET = ER~R+Tfor each R E A - {T), so as ~R+TE G, we have complete freedom in our choice of T , independently of B and x. By 19.11, B E Ci(A),i = 1,2,3. If B E C l ( A )then B = R+S for some R, S f A, and by the previous paragraph we may assume R # T # S. Then 2eB( = 2(eg - 2es - 2eR)cT = -2eg E A. If B E C2(A)then B n Si is of order 2 for four Si E A. Let A be the sum of the remaining two 4-sets in A. Then A E C with A n B = 0, so A + B + X = C E C. We may assume T C A. Then 2egt = 2(eB eSi)eT = - 2 e ~E A. Finally if B E C3(A)then B n Si is a point for five Si f A and B n S is of order 3 for the sixth S in A. We may assume T = S . Then

Ex

Proof: Let g E G be of order p, let w be a primitive pth root of 1, and let mi be the multiplicity of wi as an eigenvalue of g for 0 5 i < p. Then mjwi. But as g acts on A, the matrix of g the trace of g is T r ( g ) = is integral, so T r ( g ) is also integral. Thus T r ( g ) is invariant under each wi of Q(w), SO as {wi : 0 5 i < p) is a basis for automorphism w Q(w) over Q , mi = mj for all i,j > 0. In particular as mi > 0 for some i > 0, 24 = dim(^^^) p - 1.

xi

>

Lemma 22.8: If g then g E N .

E

G and z E X with (zg) = ( w ) for some w

E

X,

Proof: Let 2 = X U -X and assume g E G, z E X , with (zg) = ( w ) for some w E X . Then zg zg 2 since q(z) = q(zg). For x E x let S(x)= ( v E R

~ ( v~, x ):= O and 4 ( v + ~ x E ) A2 for e = f l ) .

Observe that using 22.6, S ( x ) = x - { A x ) . As G acts on A2, S(zg) = S(z)g. Thus as zg E X, xg= ( f zg) U S(z)g = {fzg) U S(zg) = X . So it remains to show N ~ ( x = ) N . But N ~ ( ~ ~ =~P). S( yxm ()X ) , where P is defined in 22.5. Thus g = hey for some h E S y m ( X ) and Y E V . Let B E C and v = 2eg. Then v E A and supp(vg) = Bh. Hence

+

Lemma 22.9: N

+

=N~(A!).

<

Proof:By 22.6, N N ~ ( A ! = ) H. Let y, z E X and v = 4(z+ y). Thus v E A;. Notice N(,,) has two orbits I1 and I2 on the set I = A: n vL: namely I1 = { f 4(x - 9 ) ) and I2 = At n xL n yl. So either H(,) acts on Il or H(,) is transitive on I. However III = 926 = 2 463 and 463 is a prime, so the latter case is out by 22.7. Hence H(,)acts on Il and hence also on { ( x ) ,(y)} = {(r+v) :r E II). Therefore H(,) 5 N by 22.8. Then as N is transitive on A;, H = NH@)= N . Lemma 22.10: Let A be a sextet of X and for x E X let T ( x ) be the 4-set in A containing x. Define q to be the element of G L ( R ~ * with ) xq = x - eT(,)/2 for each x E X . Then ET = q e E ~ G - N for each TEA.

xi

is the element of A$ such that a, = 3 for x E S

- B and

Lemma 22.11: If N < H

115

22. The Leech lattice and - 0

Chapter 8 The Conway Groups and the Leech Lattice

114

< G then

lbble 3

(1) H is transitive on A2. (2) Hu is transitive on A2 n uL for each u E A2. (3) IA2 nuLl = 93,150 = 2.34 .52 .23 for u E A2. Proof: By 22.9, H does not act on A;. But A ~ His the union of orbits of N on A2, so by 22.6, either H is transitive on A2 or A ~ H= A$ U for i = 2 or 3. In the latter case by 22.6, A ~ Hhas order 24 .3 - 23.89 or 2 4 , 3.19.109. This contradicts 22.7, as 89 and 109 are primes. So ( 1 ) is established. Next let xo E X and A = A,,. Then there is g E M of order 23 fixing xo and transitive on X - {xo}. So if v = a x x E CA(g) then a, = a is independent of x # xo. Thus if v E L(X) = A2 n X L , then v = fA, contradicting (A,A) # 0. So all orbits of HA on L(A) have length divisible by 23. As H is transitive on A2 and X E A2, the same holds for each u E A2. Let u = 4(xo X I ) for some xo, x l E X. Then u E A2 and we find that the orbits of N(,) on L ( u ) are:

Ex

Orbit Shape

A:

A;

A;

(212,012) (33,l2') (4,28,0'')

(5, 123)

Length 211 .2,576

Orbit

212

(y)

-

2' ,759 16 212 - 24

AZ A: A:+ Shape (216,0') (216,0') (3') 119) (44,020) (4*,2', 014) 2' 759 (126) Length 211 ,759 211 ,759-15 212 (2t)24 )2:( A:+

A:-

.

Orbit Shape

A:(4, 212,0")

Length 212 .2,576.12

-

4

A:

(5,3', 12') 212

(234)

+

A: (6,2', 016) (8,023)

- 3 2' .'759 .8

48

+

{4(x0 - x l ) , 4(x1 - s o ) ) of order 2. { w E A; :xo,xl 4 supp(w)) of order 4(:). { w E A; :so, x l $ supp(w)) of order 330 Z7. { w E A; : xo,xl E supp(w)) of order 77.26. { w = E xalx E A: : xo E S l ( w ) and a,, = -axo) of order 22 - 211. For example, we calculated during the proof of 19.2 that there are exactly 77 octads through { x o , x l ) ; this gives the order of orbit (d). Similarly by Exercise 6.3, there are 330 octads missing both xo and X I , which gives the order of orbit (c). Finally by Exercise 6.3, the number of members of 5 containing exactly one of xo and x1 is 2''. But w in orbit (e) is determined by such a set plus the element x in S3(w). So as xo E S l ( w ) and a,, = -azo, there are 22 - 211 elements in orbit (e). Now the orbit lengths modulo 23 are: 2,4,12,6,22. But no proper sum of these is congruent to 0 modulo 23, so H(,) is transitive on L(u). Then Hu is transitive on L ( u ) as H(,,) = (eB)Hu with B E C fixing w in orbit (b). (a)

(b) (c) (d) (e)

-"

.

-

Lemma 22.12: (1) G is transitive on A2. (2) N is maximal in G. (3) IGJ = 222 -3'- 5 4 - 72 11 - 13.23. (4) If u E A2 and v E A2 n uL, then G,,, = Nu,, 2 Mz2/E210. (5) If u = 4 ( x y) and v = 4(x - y) for x, y E X then GU,,= Nu,,.

-

+

Proof: By 22.10, N is proper in G, so G is transitive on A2 and on S = { ( u , ~ )u: E A2, v E A 2 n u L ) by 22.ll..Similarly if N < H 5 G then H is transitive on S , so G = HGU,,, where u = 4 ( x y ) and v = 4 ( x - y). Now Gu,, fixes (u v ) / 8 = x, so by 22.8, GU,, 5 N . Thus G = H N = H. So ( 2 ) is established. Further Gut, = N,,zr = M22/E210 as the subspace of elements of VC not projecting on x or y is of codimension 2. So

+

+

Lemma 22.13: The orbits of N on A3 and A4 are listed i n Table 3, along with the length and shape of each orbit. The members of A? are the N-conjugates of v,, where v+ has sixteen weficients equal to 2 and v- has two weficients equal to 2. Proof: This is Exercise 8.4.

Lemma 22.14: (1) G is transitive om As and A4. (2) /A3[= 212(212 - 1 ) = 212 . 32 - 5 7 - 13. (3) lA41 = 398,034,000 = 24 37 . 53 7 13.

- -

Proof: This is Exercise 8.5.

116

Chapter 8 The Conway Groups and the Leech Lattice

23. The Leech lattice mod 2

Notice EX is the scalar map on R~~determined by -1, and hence is in the center of G. Denote by Col or .1 the factor group G/(eX). Denote by Co2 or .2 the stabilizer of a vector in A2 and denote by Co3 or -3 the stabilizer of a vector in A3. The groups Col, Coal and Cog are the Conway groups. We will see later that each of the Conway groups is simple.

(3) The members of A4 in a coset in ii4forn a coordinate frame. (4) dimCF(2)(A) = dimz(A) = 24. Proof: Let L = A2 U A3 U A4 and suppose u, v E L with (u) # (v) but G = fi. Then u+v = w E 2A, so q(w) = 4q(w/2) 2 8, with equality if and only if u/2 E hz by Exercise 8.8. Next replacing u by -u if necessary, we may assume (v, u) 5 0. Now q(w) = q(u v) = q(v) q(u) (v, u)/8. But (v, u) 0 and q(v), q(u) 4 as u, v E L, so q(w) 5 8 with quality precisely when u and v are in A4, (u, v) = 0, and (u v)/2 E A2. Hence by 23.1.3, u and v are in the same coordinate frame. Conversely if u, v E A! then u = 82, v = 8y, and u + v = 2(4(x+y)) E 2A by 22.2. So A: ii. Thus (2) and (3) are established. Further by (2) and (3):

Lemma 22.15: (1) lCozl = 218 36 . 5 . 7 - 1 1 - 2 3 . (2) ICo3( = 2'' 37 s3 7 11 ~ 2 3 .

<

. .

Proof: This follows from 22.14 as iContl = IGf/JAml.

23. The Leech lattice mod 2

+

<

+

+

+

In this section we continue the hypothesis and notation of the previous section. In addition, for v E A let A,(v, i) denote the set of u E An such I ~ =I IA21/2 lh31/2 1h41/48 = 224 - 1. that (v, u) = 8i. Thus 24 dirnGF(2)(d) 5 dimz(A). We will show d@(A) 5 24 t o Let 2A = (2v : v E A). Then 2A is a G-invariant Z-submodule of A, so complete the proof. G acts on the factor module = A/2A. The module is the Leech lattice m o d 2 . ~ o r v ~ ~ l e t 8 = v + 2 ~ a n d f o r ~ ~ ~ l e t ~ = { ~ : s Indeed ~ ~ ) the . Zspan K of X is a 24-dimensional free Z-module, so as A is a 2-submodule of K , dimz(A) 5 dimZ(K) = 24. By construction 2G = 0 for all v E A, so is an elementary abelian 2-group which we may view as a GF(2)G-module. Also e x is trivial on Lemma 23.3: is a faithful irreducible ~ ~ ( 2 ) e - m o d u l e . A, so is also a GF(2)-module for G = G/(ex) E! Col. Proof: If 0 # U is a G-submodule of A then U# is the union of orbits Define the coordinate frames of A to be the G-conjugates of A .! Ail i E {2,3,4). But also U# has order 2n - 1 for some n, which forces Lemma 23.1: (1) N = NG(A:).

+

+

<

u = ii.

(2) The coordinate fmmes form a system of imprimitivity for G on A4 of onler 8,292,375 = 36 53 - 7 13. (3) If S is a coordinate frame, u E S, and v E A4(u, 0) with (u+v)/2 E A2 then v E S.

-

Proof: Let u E A! and g E G with ug E A.: Then u = 8%or -82 and ug = 8y or -8y for some x, y E X , and hence g E N by 22.8. This implies (1)and shows the coordinate frames are a system of imprimitivity for G on A4. Then as IA!~ = 48 by 22.13, we complete the proof of (2) using 22.14.3. Assume the hypothesis of (3). Conjugating in G, we may take u = 82. As v E uL, the projection of v on x is trivial and hence the projection of w = (u v)/2 on x is 4. Then as w E A2, 22.6 says w has shape (42, o ~ ~Hence ) . v E A:, as desired.

+

L e m m a 23.2: (1) A;, i = 2,3,4, are the orbits of G on A#. (2) For i = 2,3, each coset in Ai contains exactly two members u and -u of Aij and no member of Aj for i # j E (2,3,4).

We have already observed that e x is in the kernel K of G on A. Conversely K N by 23.1.1. Also K fixes .It for each v = 2eB, B E C, and hence acts on { f v ) by 23.2.2. So K = (EX).

<

Define a bilinear form ( , ) and quadratic form q on

by

(6, G) = (u,v)/8 mod 2 and q(Q) = q(u) mod 2. These maps are well defined since by 22.4, (u,v) r 0 mod 8 for all u, v E A. Notice G preserves the bilinear form ( , ) and quadratic form q on A. Recall a subspace U of A is singular if q is trivial on U.

'

I

is the set Lemma 23.4: A2uX4 is the set of singular points of i\ and of nonsingular points, with respect to the quadratic form q. The bilinear form ( , ) is nondegenerate. Proof: The first two remarks are evident. For the third, G acts on the radical R of the form, so by 23.3, R = A or 0. As q # 0, R # A.

Chapter 8 The Conway Groups and the Leech Lattice

118

Lemma 23.5: Let v E As. Then

+

+

(1) v = u 2w with u E A3, W , u w E A2, (u,W ) = -24, and ( w , u + w ) = 8. (2) There is a unique way to un-ite v as the sum of two elements from A2: namely v = w (u w). (3) GV = NG({w,21 + w ) ) N G ( ( ~ ) ) . (4) G is transitive on A5. (5) lAsl = 214. 33 5.7.13 - 23. (6) [Gu,w~ = 27 3' . 53 7.11 and IGvl = 2(Gu,,l.

+ + <

-

-

Proof: By 23.2, v = u+2w for some u E Ail i = 2,3,4, and some w E A. Then 5 = q(v) = q(u) 4q(w) (u,w ) / 4 = q(u) mod 2. Thus u E A3. Now (u,w ) = 8 16n, where w E A,,. But by the Schwarz inequality, (u,w ) 5~ (u,u)(w,w) = 1 6 ~3n, so 4n2 - 16n 1 5 0. Hence n 5 3. However, if n = 3 then (u,w ) = -40, so q(u + w ) = 1, contrary to Exercise 8.8. Thus n = 2, so (u,w ) = -24 and (w,u + w) = 8. Next by 23.2.2, v = u 2w = -u 2(u w ) are the unique ways to write u as the sum of an element from A3 and 2A. So Gv 5 NG((u))n NG({w,u + w ) ) . Conversely NG({w,u + w ) ) acts on ( u + w ) + w = v. Thus (3) holds. A l s o i f v = s + t withs,t E A2 then ( s , t ) = 8 , s o q ( s - t ) = 3 . Then as v = ( s - t ) + 2t, {s, t ) = {w,u + w ) by the previous paragraph. So (2) holds. By Exercise 8.6, Gw is transitive on A2(w,1) and JA2(w,1))= 211 -23. Hence by (2),G is transitive on As and

+

-

+

+

+

+ +

119

23. The Leech lattice mod 2 Proof: This is Exercise 8.9. The proof is much like that of 23.5 and uses Exercise 8.7. The group Gv of 23.6.6 is the Higman-Sims group HS. We will see later that the Higman-Sims group is simple.

Lemma 23.7: G has 3 orbits on 2-dimensional subspaces (G,9) of with6,o E x 2 . They consist of: (1) Totally singular lines with ii + C E &. C&((ii,ii)) G M22/E21~ for such a line. (2) Totally singular lines with C ii E A2. (3) Nondegenerate lines. CC((ii,6 ) ) 2 Mc for such a line.

+

Proof: Let G E A2. By Exercise 8.6, G5 has 3 orbits on A2 - (5): namely A2(v,i), i = 0, -1, -2. Then choosing u E A2(v,i), q(u v) = 4,3,2, respectively. That is, G + C E &, &, i2, respectively. Then by 23.4, (6,d) is totally singular in the first and third cases and nondegenerate in the second. Further if ii + G E A4 then ( I ) holds by 22.12.4, while if ii + 6 E i3, then (3) holds by 23.5.

+

In the last few lemmas in this section we work toward a description of A as a GF(2)N-module. For J E V define

Lemma 23.8: For all J, K E V,

so (5) holds. Now lGvl = (Gl/jA5Jis as claimed. The group G,,w of 23.5.6 is the McLaughlin group Mc. We will see later that the McLaughlin group is simple.

Proof: Parts (1) and (2) are straightforward calculations and imply (3).

Lemma 23.6: Let v E A7. Then

&ernma23.9: The map f : V homomorphism with kernel Vc

-

(I) -

(2) (3)

(4) (5) (6)

There is a unique way to write v as a sum v = u+w with u E A2 and w E A3. u - w E A ~ ,v = ( u - w ) + 2 w , and(u,w)=16. Gv = Gu%w. G is transitive on A7. /A7(= 213 37. 5 7.13.23. lGvl=29.32.53.7.11.

-

.

-,

A

defined by f : J

I-+

+

f (J) 2A is a

Proof: Observe first that f ( J ) E A. I f ) JI is even this follows from 22.2.1 and 22.2.2, while if I JI is odd then f ( J ) = f ( K ) - A, for x E J and K = J + x of even order. This also shows f ( x ) = -As. Notice next that Vc 5 k e r ( f ) since if J E VC then 15112 is even and fJ E A by 22.2.2. Thus it remains to show f is linear. But this follows from 23.8 as f ( J n K ) E A.

120

Chapter 8 The Conway Groups and the Leech Lattice

Lemma 23.10: Define P,C, f as in Section 20 and make Vc x V/Vc into an F-space via (v, VC

121

23. The Leech lattice mod 2 Further for each v E El f, E A by 22.2.2, so 4 maps D into check that ) is linear amounts to a verification that

A. To

+ J ) + (u, VC + K) = (v + u + C(U,v)X, Vc + J + K + (Vn u)),

a n d d e f f n e ) : l / c x ~ / & + h b y ) : ( v , l / c + J ) ~ f v + f(J)+2A. Then

+

for all u, v E VC. But by 22.2.2 and 23.8.2, fv+, z fv fx for v E V, and then (*) follows from 23.8.2. Therefore 4 : D 4 k is linear. We observed during the proof of 23.9 that f (x) = -Ax. Thus ixE #(D)for each s E X, so as A is spanned by A,, x E X , ) is a surjection. Then as dim(D) = 24 = dim(A), ) is an isomorphism. Hence (1) is established. If v E VC and J E V then q(4(v)) = lv(/4 = P(v) and ()(v), 4(J)) = (jv,f (J)) = Iv n JI - IJllvl/4 mod 2. That is, (3) holds. It is straightforward to see ,i3 acts linearly on D and commutes with more elaborate calculation shows r y is linear. Further ) : (0, x) I-Ax and we check -Axey = )((O,x)ey) for each x, so as & = (-Ax : x € X), 4 commutes with ey. Thus ) is N-equivariant and (4) is established. Let Dl = ((X,O)), D2 = D l W+, W+ = ((0,Vc J ) : s(J) = 01, and D3 = (02, (v,O) : v E V). Set Li = $(Di). To prove (5) given (4), it remains to observe that the map

(1) ) is an isomorphism of F-spaces. (2) )(W) is totally singular, where W = ((0, Vc J) : J E V). (3) 9(4(v,3))=P(v) and(4(v,0),4(0,i'c+J)) =jJnvl+IJllv)/4. (4) N acts on Vc x V/Vc via

+

ey : (v, J

+ I+)

H

+ s(J)Y + ((1+ s(J))C(v, Y) + eJ(Y))X, Vc + J + (1 + s(J))(v n Y))

(v

q5.p

so as to make ) N-equivariant, where s(J) = 0 , l for I JI even, odd, wspectively, ,8 E M, and Y E Vc. (5) N preserves a filtration

+

of A, where L1 = (A:) is a point and L3 = L; is a hyperplane of and L2/L1, L3/L2 are isomorphic to the 11-dimensional Todd module, 11-dimensional Golay code module for M, wspectively. Further 0 2 ( N ) = Q acts on LP as the group of transvections with center L1, and L4/L2 is dual to L2 as an N-module.

A,

Proof: Let D = Vc x V/Vc and notice first that this construction is a special case of the construction of Exercise 4.6; in particular D is an F-space by Exercise 4.6.6. However, we sketch a proof of that fact in this case. Check first that the definition of addition on D is associative. This follows from a straightforward calculation once we observe that C(v,u)

+ C(v + u, w) = P(v) + P(u) + P(w) + P(v + u + w),

that V

~

+U((v + u ) n W)= (Vn u ) u (Vn W ) u (Un w ) ,

that C ( X ,v) = 0 for all v E VC, and that

Next by 23.9,) : W -t is a well-defined linear map. Write )(v),)(J) for )(v, 0),)(0, V+J), respectively, and observe that q()(J)) = 1 Jl(l Jl+1) r 0 mod 2, so (2) holds.

+

is an M-isomorphism of D2/D1 with the Todd module, while the map

1

I

I

1

I

II

is an M-isomorphism of D3/D2 with the Golay code module Vc. Similarly acts on Dz as the group of transvections with center D l and then, as L2 is a maximal totally singular subspace of is dual to Lz as an N-module.

A,NLZ

Remarks. The Leech lattice A was discovered by John Leech in ILe2] extending his earlier construction of the sublattice A. in [Lel]. In [CO~], Conway determined the group .0 of automorphisms of A, proved that - 0 is quasisimple, determined its order, and determined the stabilizers of various sublattices, hence also discovering the Conway groups Co2 and Co3 and finding M c and HS as subgroups of -0. Our treatment of A and .0 follows Conway's treatment in [Col] and [CO~].

122

Exercises

Chapter 8 The Conway Groups and the Leech Lattice

(2) . The map q5 : D -t A of Lemma 23.10 is an isometry; that is, q(d(d)) = q(d) for all d E D. (3) For X # Y E V, (v, J) E CD(ry) if and only if s ( J ) = O = r j(Y) .and V n Y E VC. (4) dim(cA(rv)) = 16.12 for Y an oetad, dodecad, respectively. (Hint: See Exercise 4.6.)

Exercises

,

1. Assume the hypotheses of Section 22 and let v = CXGx ax E R~~ with a, E Z. Then (1) If ay is odd for some y E X then v E A if and only if the following hold: (a) a, is odd for all x E X. (b) {x E X : a, 5 1 mod 4) E VC. (c) {x E X :a, r f3 mod 8) is of odd order. (2) If ay is even for some y E X then v E A if and only if the following hold: (a) ax is even for all x E X. (b) { x E X : a, E 2 mod 4) E Vc. (c) {x E X :a, E 4 or 6 mod 8) is of even order. 2. Let v E A with m(v) odd and ISj(v)J 2 20 for some j,Then N is transitive on the set of elements of shape v. 3. For each k 5 5 and r = 28 _> 4, N is transitive on the set Rk of vectors of shape Further Rk C A and lRkl = (2k4)2k. 4. Prove Lemma 22.13. 5. Prove Lemma 22.14. 6. Let u, v E A2. Prove (1) (v, u) = 8i with lil = 0,1,2, or 4. (2) fA2(v,i)l = 47,104 = 211 -23or 4,600 = 23 52 -23 for JiJ= 1 or 2, respectively, while A2(v,4) = {v) and A2(v, -4) = {-v). (3) Gv is transitive on A2(v, i) for each i. 7. Prove that for v E 112, Gv is transitive on A3(v, 2) and IA3(v, 2)1 = 2'. 34 23. 8. q(v) 2 2 for all v E A#. 9. Prove Lemma 23.6. 10. Define the F-space D = Vc x V/Vc as in 23.10. For v E Vc, J E V, define E~(v) = (vn J ( mod 2 as in Exercise 3.2. Define q : D + F by

-

andr: D x D - t Fby

Prove (1) q is a quadratic form on D with bilinear form 7.

1 I I

1

i I

125

24. The groups Co3, Mc, and H S 24. The groups Co3, Mc, and H S In this section we continue the hypothesis and notation of Chapter 8. In addition let XI,x2 E X,

'

Chapter 9

Subgroups of 0 Let 2, = A2(vl,2) n A2(v2,2), and define

In this chapter we use the machinery developed in Chapter 8 to establish the existence of various subgroups of S O and t o establish various properties of these subgroups. For example, in Section 23 we defined the stabilizer of a nonsingular vector G of to be the Conway group Co3.In Section 24 we prove that Co3 is 2-transitive on the 276 lines of through G generated by points in d2,with the stabilizer in Co3 of such a line isomorphic to ZzlMc. Further we prove Mc is a primitive rank 3 group on the remaining 275 lines. Similarly we show HS is a primitive rank 3 group of degree 100 with point stabilizer Mzz. These representations allow us to prove that Co3, Mc, and HS are simple. In Section 25 we prove that the groups Col and Co2 have large extraspecial 2-subgroups. Similarly we find subgroups A of G = Col of order 3,5, such that CG(A)/Ahas a large extraspecial 2-subgroup; CG(A)/Ais Suz or J2, in the respective case. Our theory of large extraspecial subgroups developed in Section 9 then allows us to prove that Col, Co2, SUZ,and J2 are simple. Finally in Section 26 we establish various facts about the local structure of Col which will be used later to construct various sporadic subgroups of the Monster. Chapters 16 and 17 contain much more information about Col, Suz, and J2.

Then vl,vq E A;, v2 E A;, and v3 E A:. Thus

G3

Cog

as G3is the stabilizer of a vector in A3. Also v2 E A2(vl, 1),so

by 23.5. Similarly vq E A2(v3,2 ) , so

by 23.6. ). Gq fixes v l v4 = 8x1 and Finally let G4 = C G ( ( v l , v 2 , v 4 )Then vl - vq = 8x2, so G4 < N by 22.8. Therefore G4 = C N ( ( x lx2, , v3)) = Mx,,z2Z M22. That is,

+

For w E A and x

w = CXEX wxx.

E

<

X let w, be the coefficient of x in the expansion

<

Notice ( G I ,G2) G3 and Gq G2. We will discover in 24.6 that G2 r Mc acts as a rank 3 group on B of degree 275 with stabilizer G7 = CG2(v5).Further by 24.7 and Exercise 9.5, G7 E U4(3).

126

Chapter 9 Subgroups of - 0 Lemma 24.1: G4 has three orbits En, n = 2,3,4, on 8,where

(1) Z2 consists of those w E A; such that w,, = w,, = 2 and w, = -2 for all x G S2(w)- { x l , x 2 ) . I E ~ ~ = 77. (2) z3 consists of those w E such that w,, = 3, w,, = 1, and C ( W ) = { X E x : w, = 1) E C. 1531 = 176. (3) z4 consists of those w E A%with w,, = 4, w,, = 0, and w, = -4 for X I # x E Sq(w). 1z4(= 22.

~g

Proof: From the proof of Exercise 8.6, w E A2 is in A2(v1,2) if and only if (wx,wy)= (2,2), (3,l) or (1,3), (4,O) or (0,4) for w in A:, A:, A;, respectively. Then we find w E A2(v1,2) is in A2(v2,2) if and only if w is described in (1)-(3). . . . . Further, in (1) w is determined by S2(w),and from the proof of 19.2 there are 77 octads through { x l , x 2 ) with G4 transitive on these octads. is an orbit of length 77 under G4. We can complete the proofs of So (2) and (3) similarly.

z2

Lemma 24.2: (1) A2(v3,3) is an orbit under G3 of length 2 276, with

a system of imprimitivity of order 276. (2) ((63,G) : v E A2(V3,3)) is the set of lines of through 83 generated by elements of A:!. There are 276 such lines and they fonn an orbit under G3. (3)~2(v3,3)=(vl,-V2,Vl-W1W-V2:WE~).

Proof: From 23.5, G is transitive on the set S of pairs ( u ,v ) with u E A3, v E A2(2~,3).F'urther IS1 = 21A5l. So (Az(v3,3)1= 21A51/1A31 = 2 . 276. Thus (1) is established. By 23.7, (1) implies (2). Finally for w E 2, (v3,w ) = (vl - v2, w) = 0. Thus (v3,vl w ) = (v3,vl) = 24 = (v3,-v2) = (v3,w - 212). SOthe set R on the right in (3) is contained in A2(V3,3). B Uby ~ 24.i11al= 275, so IRI= 2.276. T ~ U SIRI= I A Z ( V ~31,1 by ( I ) , so (3) holds.

-

Lemma 24.3: Let u E A2, v E A2(u,2 ) ) and w E A2(v,2) n A2(u,2). Then

(1) IGil,,,l is not divisible by 25. (2) IGu,v,wl is not divisible by 27. ProoE By Exercise 8.6, Gu is transitive on A2(u,2) with Gu,v of order

127

24. The groups C03, Mc, and HS

So (1) holds. Similarly z = v - w E A2(u,0) and by 22.12 Gu,z E Mz2/E21o,so IGulzI is not divisible by 27. But of c o ~ r s e ~ G ~5, ,GUlz. ,,~ Let x3 E X - {xl,xZ), vg = 4(x1 - x3), and Gg = CG4(x3)= MxIlz, 2 L3(4). Let G7 = CG(v5)and notice G5 5 G7. That is, G5 2 L3(4), and G5

< G7.

By 24.1, v5 E 9. Let Ri = E n A2(v5,i ) and 52r = En n A2(v5,i) for i = 1,2, n = 2,3,4. Notice G5 5 Gq 5 G2 and G5 fixes v5.

Lemma 24.4: The orbits of G5 on Z - {v5) are Zr, i = 1,2, n = 2,3,4, is empty and ( n ,i ) # ( 4 , l ) . Moreover

(1) 52: consists of those w E E2 with x3 f S2(w),so 152:l = 56. (2) 52; consists of those w E z3 with 23 E C(w), so 152al = 56. (3) 52; consists of those w E s2with xg E S2(w),so I52;l = 21. (4) consists of those w E E3 with 23 $ C(w), so 152; 1 = 120. 52; = E~ - {us) is of order 21. (5) Proof: These are easy calculations. Notice in (1) that the collection of &sets of the form S ( w ) - {x1,x2),x3 $ S2(w),is one of the three Gginvariant collections of independent &sets of the projective plane on X - { x l , x ~231, , defined in 18.7. Thus this collection is of order 56 by 18.7.8, and therefore 52; is also of order 56. Similarly in (2),the collection of &sets C(w)- (12,231, w E Qf, is an I; with j # i.

Lemma 24.5: (1) G2 is transitive on 3. (2) G7 = CGa(v5) is of order 27 36 - 5 . 7 and is transitive on $21 of order 112 and on 522 of order 162. (3) If u, w E Q1 with u E A2(w,2), then the order of the stabilizer in G7 of u and w is not divisible by 27. Proof: Assume (1) is false. Then from 24.1, Zn is an orbit of G2 on 3 for some n. As the length of En is not divisible by 5, a Sylow 5-group of Gz fixes some w E Zn. But then as 53 divides the order of G2 r Mc (cf. 23.5.6), G,,,,, has order divisible by 53, contradicting 24.3.1. Thus (1) holds, so IG71 = lG21/121= 27 36. 5 7. Suppose G7 is not transitive on 522. Then from 24.4, R$ is an orbit of G7 on Q2 for some n. So as the length of 529 is not divisible by 9, the stabilizer H in G7 of w E 52$ has order divisible by 35. But H Gva,v6,w, contradicting 24.3.2. Indeed the same argument establishes (3);namely the stabilizer in G7 of u and w also fixes vl and by 24.3.2, IGvl,u,wI is not divisible by 27.

-

-

<

130

Chapter 9 Subgroups of .O

24. The groups Cog, Mc, and HS

Thii shows R* = {l(T) : T E 13(r)) C_ St0 and if w E 1 E Ro -St* then 1 C_ R?. But then by an earlier remark, 111 5 2. Further the map T I+ 1(T) of 13(r)onto R* has fibers of order 4, so IR*I = 113(r))/4= 280. Also the set R*(w) of members of R* incident with w is of order (!)I2 = 10. As each member of Q*(w) has a unique member of Cn(w), (9) shows each member of Cn(w) is in a unique member of R*(w) and G6 is transitive on R*(w). This together with previous remarks shows a* = $20 and establishes (1)-(4). Notice we have also shown IC(w)l = 30, which is the parameter X of the McLaughlin graph of 24.6. Next if u E I'(w), then by 24.5.3, 27 does not divide the order of the stabilizer in G7 of w and u. But lC~,(w)l = (G71/112 = 23 36 . 5 = (A61. 34, so as Ir(w)j = 27, H = CG7(w)is transitive on I'(w). Similarly by (3) and Exercise 9.1, H = G6P, where P is the kernel of the action of H on n*(w) and P is of order 81. Now as G6 is simple, either [G6,P ] = 1 or an element g E G6 of order 5 is faithful on P. In P] = 1 the latter case P E E8l and g is irreducible on P. So either [G6, or (7) holds, and we may assume the former. Then P acts on the orbits of G6 on r(w) of length 10 and 20 and hence fixes each point of C(w). But then 24.5.3 supplies a contradiction. So (7) is established. Next P is transitive on 1 - {w) for 1 E R*(w). For if not, P fixes each point of 1, and then by transitivity of G6 on R*(w), P fixes each point of C(w). We observed that thii is not the case in the preceding paragraph. Now H is transitive on C(w) and r(w), so G7 is rank 3 on R with k = IC(w)l = 30 and 1 = IJ?(w)l = 81. Further for u E C(w), C(u) n C(w) = 1- {u, w), where 1 is the line through u and w. Therefore X = 2. Then p = k(k - X - 1)/1 = 10 by 3.3.2. G is primitive on R by 3.3.3. Thus (6) is established. Finally suppose 1 # K a G7. Then as 112 is not a prime power, 1 # H n K by Exercise 1.4.1. But by (7), P is the unique minimal normal subgroup of H , so P K . Further for v E C(w), P(u), $ P, so as HIP is simple, H K. Thus G7 = K is simple.

24.6.1, GQis doubly transitive on C with G2 of index 2 in the stabilizer on C - (1) H of some 1 E C.Indeed G2 is simple by 24.8 and by 24.6.2, so G2 is the unique minimal normal subgroup of H. Thus as It(= 276 is not a prime power, G2 5 K by 2.2.3. So JG3: KI 5 IG2 : G2n Kl 2. Thus we may assume IG3 : KI = 2. Let P E Sy123(K). By a Frattini argument, NG, (P)has even order. This contradicts Exercise 9.2.

Lemma 24.8: McLaughlinls group is simple.

is a GI-invariant partition of I: of order 176 and by 24.10 and 23.2 the (G3, G4,G) is a bijection of T with S. So it remains map {v, v3 - v) to show G1 is Btransitive on T. By 24.1, Gq < GI is transitive on T , so it remains to show H = NG, (a) is transitive on T' = T - {a), for a = {v, v3 - v). Let v = vl - z, z E z3,and L = CG(z). By Exercise 9.3, L has two orbits on T' of length 70 and 105. Thus if H is not transitive on T', these

<

<

Proof: By 24.6, G2 is primitive of rank 3 on B while by 24.7 the stabilizer G7 in G2 of a point of is simple. So as 131= 275 is not a prime power,

G2 is simple by Exercise 1.4.2.

Lemma 24.9: The third Conway group Co3 is simple. Proof: Let 1 # K
131

<

Lemma 24.10: (1) Let 6 = A2(v3,3) nA2(v4,0). Then B = 8l uo2ue4, ~ h e r e e ~ = { v ~ ) , 8 ~ = { va ~n d- @ ~ =: {~ w~- ~v ~~ : )w,E ~ ) . (2) L e t r = A2(~3,3)nA2(v4,-l).ThenI: = {vl-w, w-v2 :w E z3). Proof: Thii is left as Exercise 9.4; use 23.6.

Lemma 24.11: G1 r HS is a primitive rank 3 group on 0 of degree 100 with parameters k = 22, 1 = 77, X = 0, p = 6. The stabilizer in G1 ofvl E e is G~ M ~ ~ .

=

Proof: By 24.1 and 24.10, Gq S Mz2 has orbits en, n = 1,2,4, on 8 of length 1, 77, 22, respectively. So if G1 is not transitive on 8 then G1

fixes vl or has an orbit of length 23 or 78 = 2 .3.13. But G1 HS has order 'J9 . 32 - 53 7.11 by 23.6, which is not divisible by 23 or 13. Further the stabilizer in G1 of vl is CG((~,V4,Vl))= G4 S M22. So indeed G1 is a rank 3 group with k = 22 and 1 = 77. Now 84 E A2(vl, 1), whereas (w, v) = 16 for distinct v, w E 84. Thus X = 0, and then p = k(k - X 1)/1= 6.

-

-

Lemma 24.12: The Higrnan-Sims group is simple. Proof: This follows from 24.11 and the simplicity of M22 via Exercise

1.4.2.

Lemma 24.13: G1 2 HS is doubly tmnsitive on the set S of 3-dimensional subspaces (CQ,64, 5) of through (C3, G4) such that Proof: Adopt the notation of 24.10. Let T be the set of pairs {v, v3 -v) as v ranges over I:. Notice Irl = 2 - 176 by 24.10.2 and 24.1.2. Then T I-,

i

Chapter 9 Subgroups of - 0

132

are also the orbits of H. In particular a subgroup P of index 5 in a Sylow $group of H fixes a point u E 'I with ( u ,v3 - u ) in the orbit of length 105. Then by Exercise 9.3, u E AZ(v,2). But IHI = IG11/176 = 25.32-53.7 by 23.6.6, so IPI = 25, contradicting 24.3.1.

25. The groups C o l , Co2, S u z , and J2 In this section we continue the hypotheses and notation of Chapter 8, except we write E for (eU : U E Vc). Pick an octad B E C and let B = X B. Let A(B) be the sublattice of A consisting of all elements of A whose support lies in B. Let QE be the subgroup of G generated by all elements EA, A E CO(B),and QM the subgroup of M k i n g B pointwise. Let Q = QEQM and z = eg. Finally let G = G / ( e X ) .Thus G 2 col. We will show that 0 2 Q S 21+8 is a large extraspecial 2-subgroup . Chapter 16 we prove that of G and indeed 6 satisfies ' ~ ( 4 , 0 : ( 2 ) )In Col is the unique group satisfying this hypothesis and use the existence of the extraspecial 2-subgroup to study the structure of Col. Similarly we will show that Co2 satisfies X(4, Sp6(2)),Suz satisfies H(3,R<(2)), and J2 satisfies X(2, Rq(2)). In Chapter 16 we characterize Suz and J2 by these hypotheses. On the other hand the existence of Suz and J2follows from work in this section.

+

Lemma 25.1: (1) QE Z E32, z E QE, and QE - ( z ) consists of all e ~ , A E Co(B). (2) QM E16(3) Q D: is extraspecial with Z(Q) = ( z ) . (4) N N ( Q )= N M ( B ) Eand N N ( & ) / Q ( C X )AalE64.

Proof: Part (1) follows from 22.5.2 and 20.6.1. Next (2) holds by 19.1.1. Of course N M ( B ) 5 N M ( Q E ) ,SO Q = QMQE is a subgroup of G of order z9. For g E QM and A f Co(B), = C A =~ e~ or ~ + =ge ~ e gby 22.5, so [Q,Q] 5 ( 2 ) . Also IQM : NQM( A ) [= 2 by 19.2.6. Hence QM induces the group of all transvections of QE with center ( z ) ,so (3) holds. Next NN(Q) 5 C N ( z )= E N M ( B ) .Further N M ( B )acts on QE and QM and hence also on Q. Finally for g E Q and A E C,[eA,g] = ~ A + EA ~ QE because g fixes B pointwise and hence the support of A Ag is contained in B. So E < NN(Q) and NN(Q) = N M ( B ) E .Finally by 19.1, N M ( B ) / Q M2 A8, while by (1) and 22.5.2, E/QE(ex) 2 E64, so ( 4 ) holds.

~i

+

1

25. The groups C o l , Co2, S u z , and J2

!

:

Lemma 25.2: A(B) = CA(z),Q = CG(A(B)),and Q = c&(B)). Thu~&gc~(~).

i

Proof: Evidently A(B) = CA(z).Next as 4(x f y) E A(B) for x, y E B, CG(A(B))and cG(A(B)) are contained in N by 22.8 and 23.2.2. Indeed C N ( A ( B ) )k e s 4(x y) for each x, y E B , so Q = C N ( A ( B ) = ) CG(A(B)).Similarly cG(K(B)) = Q(ex).

133

+

Lemma 25.3: NG(Q) $ N . .

Proof: Let A be a sextet with B = T + S for some S,T E A. Define = tT as in 22.10. Then for x E B, xi$ = ( x - eT(%)/2)eT, where x E T ( x ) E A. So T ( x ) = S or T. Therefore t acts on A(B) and hence on Q by 25.2. Lemma 25.4: Suppose 0 is a totally singular subspace of A. Then qu is a quadratic form on 0 with bilinear form ( , ),y, where qu(ii) = q(u)/2 mod 2 and (6,.CI)u= (u,v)/16 mod 2.

Proof: As 0 is totally singular, q(u) is even and (u,v) is divisible by 16 for all u,v E U. So qu and ( , )u are well defined and visibly are a quadratic form and associated bilinear form. Lemma 25.5: (1) A4(B) is of order 16 . 135 and NG(Q) is transitive on A(B)4. (2) Each coordinate frame containing a member o f A ( B ) contains exactly sixteen members of A(B). (3) A(B)2 is of order 240.

(4) A(B)#

=A

2 ( ~U)A l ( ~ ) .

(5) dimz(A(B))= dimGFcz,(6(~)) = 8. (6) qg is an NG(Q)-invariant quadratic form on A(B) of sign +1, where qB(.CI)= q(v)/2 mod 2 for v E A(B). (7) Q 2 Q = F*(Ce(Z)) with c ~ ( z ) /n8+(2). Q

Proof: Using 22.13, we calculate that A4(B) consists of three orbits under CN(z): (a) A ~ ( Bof ) order 16. (b) A ~ ( Bof) order 16 26. (c) A ~ ( Bof) order (): ~ 1 = 6 70 16. Next the element 6 E CG(z)= H , produced during the proof of 25.3, moves orbit (a) into orbit (c). As 71 does not divide the order of G, it follows that (1)holds. h r t h e r orbit (a) is the intersection of a coordinate frame with A4(B),so (1) implies (2).

.

1

.

Chapter 9 Subgraups of 0

134

26. Some local subgroups of Col

135

Similarly we find A 2 ( B ) falls into two C N ( z ) orbits: A;(B) of order 2' and A!~(B)of order 112. So (3) holds. Now A ( B ) is a sublattice of the &dimensional lattice spanned by B ,

Proof;By Exercise 2.6, A exists and ce(A(z))= A x i l with fi = F*(Z) and E/R r R i ( 2 ) simple. Thus (2) holds and by 25.6 and 8.13, ( 1 ) and (4) hold. Finally (3) holds by Exercise 2.4.

SO

The simple group M* of 25.9 is the sporadic Suzuki group Suz. The same proof shows:

d i m ~ ~ ( 2 , ( A ( B5)d) i m z ( A ( B ) ) 1 8.

-

On the other hand by (1)-(3) and 23.2, A2(B)u A ~ ( Bis) of order 28 1, so (4) and (5) hold. Then ( 6 ) follows from 25.4. Osf(2). Indeed if v = By 25.2 and (6), H / Q o ( ~ ( B q)B, ) 8x, x E B , then the stabilizer in H of 3 is CR(E) and C ~ ( E ) / Q = N M ( B ) E / Q ( e x ) E A8/Es4 by 25.1.4. So as A8/Es4 is the stabilizer of 5 in @(2), (1) and (6) say @/Q S ~ t ( 2 )In. particular H / Q is simple, so either Q = F*(H) or H = QC&(Q). The latter is impossible as C H ~ N ( &=) ( Z , EX).

<

Lemma 25.6: All involutions in Q are fused to 2 i n G. Proof: All involutions in QE are fused to E in N , while by 25.5.7, Ce(Z) is transitive on noncentral involutions of Q. Lemma 25.7: 6 2 Col is simple and Q is a large extraspecial subgroup of G. Proof: This follows from 8.12 with N in the role of "K." Lemma 25.8: Let v

E A2(B) and

H = CG(v). Then

(1) CH( z ) I Q g Sp6(2). (2) Q is a large extmspecial %subgroup of H . (3) H r Co2 is simple.

Proof: CG(z) acts as Q(qB, A(B)) on L(B) with C H ( z ) the stabilizer of the nonsingular vector fi of X(B). Hence (1) holds by 22.5 in [FGT]. In particular C H ( v ) / Qis simple with Q = F*(CH(v))by 25.5.7. v = 4(x f y), u = 4(x - y), x , y E B , and let K = Nv,u. By 22.12.4, ( z K ) is abelian with (zK) $ Q. By 25.6 and 8.7, for each involution u E Q - 2, z E 02(CH(21)),SO (2) and (3) hold by 5.12. Lemma 25.9: Let A be of order 3 i n CG(z) with R = CQ(A) Q:. Let M = CG(A) and M* = MIA. Then (1) R* is a large extraspecial subg~0upof M*. (2) R* = Fr(CM.(z*)) vith CM*( z t ) / R * E Rg ( 2 ) .

(3) M* is simple. (4) All involutions in R* are fused to z* i n M*.

Lemma 25.10: Let A be of order 5 i n CG(z) with R = CQ(A)2 D8Qs. Let M = Cc(A) and M* = MIA. Then (1) (2) (3) (4)

R* is a large extraspecial subgroup of M*.

R* = F*(CM*( z * ) ) with CM-(z*)/R* E Qq (2) S A5. M* is simple. All involutions i n R* are fused to z* i n M*.

The simple group M* of 25.10 is the Hall-Janko gmup J2 or H J .

26. Some local subgroups of C o l In this section we continue the hypotheses and notation of Section 25. We determine the normalizers of certain subgroups of G of prime order. We restrict attention to results needed to establish the existence of certain sporadic groups as sections of the Monster in Chapter 11. Complete results appear in Chapter 17.

Lemma 26.1: Let B S N be cyclic of odd order, v(2) = 2ec, v(3) = A,, and v(4) = 4ezy, with C E C and x , y E X . Then (1) dim(CA(B))= c, where c is the number of cycles of B on X . (2) C j i ( B ) is a nondegenerate subspace of the orthogonal space k. (3) ICA, ( B )U C L (B)I = ( 2 ~ -1e~) ( 2 ~ / ~e), where e is the sign of the orthogonal space C A ( B ) . (4) If B~ n GG =*'B for some zi E Ai, then N G ( B ) is transitive 0" C,ii(B>. (5) If gN n N,(i) = BNw(;)for i = 2,3,4, then NN(B)is tranINN(B) : sitive on Ch:(B) for each i and ICA2(B)I = N N (B)o(i)1 (6) MG(,),E'oii) aw isomorphic to N M ( C ) 2 A8/E16, E2s; MZ % M23, 22; M ( { x ,y ) ) 2/M22, E211for i = 2; 3;4, respectively. Proof: First c = dim(CV(B)) as V is the permutation module on X. Then as V/VCis the dual of Vc,c = 2dim(CL2(B))= 2dim(CVIY,(B)), so 23.10.5 completes the proof of (1). Next A = [A,B] @ C A ( B ) as lBl is odd, so (2) follows from 22.1 in [FGT].

+

zL2

136

Chapter 9 Subgmups of . O

26. Some local subgroups of Col

Part (3) follows as CA2(B) U CA4(B)is the set of singular points in Cii(B). Parts (4) and (5) follow from a standard argument (cf. 5.21 in [FGT]). Part (6) is an easy calculation.

S2(v), S2(v) US6(v) are octads in the second and third cases, and hence intersect D nontrivially by 19.6.1, a contradiction. Finally if u is in case 1then the coordinate frame A of u consists of those v in case 1such that Sz(v) is in the same sextet as S2(u) and v has the same parity of signs as u. In particular by 19.6.1, S2(u) is the unique member of A contained in X + Dl so e fixes only eight members of the coordicate frame of u. We have shown A! is the unique coordinate frame containing 24 fixed vectors of e, so the lemma holds by 23.2.3.

Remark. If B is of prime order p in a group H and K 5 H with IK), 5 p, then by Sylow's Theorem, B~ n K = B ~In. particular if IBI = 7 then /G5I7 = 7 for 6 E Ai, i = 2,4, by 22.15 and 23.1.2. So by 26.1.4, NG(B) is transitive on Cii,(B) for i = 2,4. Similarly if IBI = 5 then Ne(B) is transitive on CA4(B).Finally for p = 5,7, IGvo 1, = p by 26.1.6, so we can calculate ICA,(B)I via 26.1.5. To do so we obtain the structure of NN(B) from Section 21, while NN(B),(~)is easily calculated from 26.1.6. Lemma 26.2: Let B be a subgmup of N of order 7. Then C@(B) 2 Z7 x L3(2) and ING(B) : CG(B)I = 6 with an involution-inverting B inducing an outer automorphism on E(Ce(B)). Further Cji(B) is of dimension 6 and sign +1 and ICA4(B)I= 14. Proof: By 21.5.4, CNIE(B) 2 Z7xS3, CB(B) 2 E4 with B the kernel of the action of CNIE(B) on CB(B), and INN (B) : CN(B)1 = 3. Therefore C,+(B) S Z7 x S4. Next by 26.1.1, W = CA(B) is of dimension 6 and by 26.1.2, W is nondegenerate. From the structure of CN(B) described in the previous paragraph and the Remark above, NG(B) is transitive on CAk(B)for k = 2,4, and B fixes 3,12,6, points of for i = 2,3,4, respectively, and hence ]CA2(B)I = 21. In particular an element of order 7 is induced on W in CG(B), so the orthogonal space W has sign +l. Hence by 26.1.3, ICA4(B)I = 14, so I NG(B)I = 14 I NR(B) I = 24 . 32 . 72. Thus the nontrivial vectors in two irreducibles Wi, i = 1,2, for a Sylow 7-group of W make up the fourteen vectors of w n h 4 , and these are the irreducibles for E(CR(B)) Z L3(2). From 25.5.7, B is inverted in C$(z) for z E CB, (B) by an involution t with dim(CA(B(t))) = 3. So IAutG(B)I = 6 and as dim(Cji(B(t))) = 3, t induces an outer automorphism on E(CG(B)).

-

Lemma 26.3: Let D E Vc be a dodecad and t = ZD N is the split extension of E E E211 by Z2/M12.

E E.

Then CE(t) 5

Proof: First CR(t) is the split extension of E by NM(D) e!Z2/Mlz by 22.5 and 19.9.3. Thus it remains to show CG(t) I N. Now e = ED fixes 24 of the 48 members of A! and inverts the rest. Similarly if v E % is fixed by e then the support S(v) of v has order at most 12 and hence ). by 22.13, u has shape (44,020),(42,28,014), (6,27,~16),or ( 8 , 0 ~ ~But

137

Lemma 26.4: Let Y = (y) be a non-¢ral subgroup of N . Then Ne(Y) E S3 x Ag, dim(CA(Y)) = 8, and Ne(Y) is transitive on the 135 singular vectors of Ci(Y), each of which is in K4. Proof: By 21.4, Y has no fixed points on X , so the second statement of the lemma follows from 26.1.1. Similarly by 21.4, Nfi(Y) S3 x (L3(2)/E8), with O2(NR(Y)) = A = C&(Y). By 21.4.3, Y stabilizes no octad, so each element in A# is of the form ED for some dodecad D. Hence by 26.3, AY/Y is a TI-set in H = Ce (Y)/Y with K = Cfi(Y) = NH(A), subject to the convention of identifying A with its image AY/Y in H. As Y has no fixed points on X, Y stabilizes no member of h2, so all ~ H N is transitive singular points in CA(Y) are in A4 and as Y ~ = yN, on these points. Thus by 26.1.3, there are 135 such points, as 17 does not divide IGI. So IH( = 135 IKI = 26. 34. 5 7. Represent H on W = C'(Y). Then W is an &dimensional orthogonal space of sign fl and K is the stabilizer of a singular point v of W. Bom 23.10, N stabilizes a maximal totally singular subspace L(v) of A and L(v)/v is the Todd module for NIE. By 21.4, dim(CL(,)lv(Y)) = 3, so L(v) n W = W(v) is a maximal totally singular subspace of W. Further by 23.10, E induces the group of transvections on L(v) with center v, so by the Thompson A x B Lemma (cf. 24.2 in [FGT]) A induces the group of transvections on W(v) with center v. Next by Exercise 8.10.4, for a f A#, dim(CA(a)) = 12, so CA(a) = [A, a]. Also as a induces a transvection on L(v), CL(,)(a) is a hyperp!ane of L(v), so Cw(,)(a) is a hyperplane of Cw(a) = [W,a]. In particular

(*) W(v)/v = CwIv(B) for each hyperplane B of A. Let v # w be a point of W(v) and h E H with vh = w . Then B = CA(w) is a hyperplane of A. As A is a TI-set weakly closed in K = NH(A), CAh(B) is a hyperplane of A ~ which , we may take to . z E B#. Then B~ I CK(z), and R = F*(CK(z)) Z 0 8 2 be B ~ Let

; I

138

Chapter 9 Subgroups of .0

~ i t h C ~ ( z ) / R % S ~ . H e n c e ~ ~ n R # l . h r t h e r mI.( l~o~r n Ri ) n R#) 5 K ~and , then (z) = @(R) A ~ a, j else R = (CR(a) : a E i contradiction. So we may take (zh) = B h n and ~ similarly ~n R~ = (z). ' By 8.15.7, R n R~ = D % E8. But c R ( t h ) Z2 x Dg has two Es-subgroups, so as A n R~ = (z), D is uniquely determined as the i E8-subgroup distinct from cA(zh) and DA = R. Now a subgroup F of i order 3 in CK (z) acts faithfully on D and (R, R ~F), = KO < NH(D) induces GL(D) L3(2) on D. We claim zh induces a transvection on W(v). For if not, R = AAo = DAo, where A. is the group of transvections of W(v) with axis CW(,)(z). Thus CWtVI(D) = (z), so as CW(,,)(z) is a hyperplane of CW(z), dim(Cw(D)) 5 2. Hence as Ko/D L3(2), KO I CH(Cw(D)) 6 CH(v) = K, a contradiction. So zh induces a transvection on W(v) with center w. Then as NK(B) is transitive on B#, B induces transvections on W(v) with center w, so ) GL(W(v)) = L4(2) by (*), W(v) = W(w). Thus L = (K, K ~ induces on W(v) and as CH(W(v)) = CK(W(v)) = 1, L G L4(2). Then as ' ]HI = IAgl, IH : LI = 9, so we have a faithful permutation representation of H on H/L of degree 9. Therefore as L % L4(2) G As, H % Ag. That is, the lemma holds.

<

,

,

Lemma 26.5: Let B E Sy15(N). Then dim(CA(B)) = 8, C6(B) Zg x (A5wrZ2), ICA4(B)I= 75, and 4 = IAut&(B)I.

%

Proof:The first statement of the lemma follows from 26.1.1. Using the Remark above, we conclude NG(B) is transitive on CAa(B) and check that B fixes 32, 64, 24 elements of A$ for i = 2,3,4, reipectively. So by 26.1.5, lCA2(B)I= 60. Also as B E Sy15(N) but B is not Sylow in Co2, tCA,(B)I r 0 mod 5, so the number of singular points in CA(B) is divisible by 5, and hence by 26.1.3, that space has sign f l and contains 135 singular points. Thus ICA4(B)I = 75. Therefore ING(B)I = 75. INfi(B)I = 27 32 . 53 by 21.5.3. Let H = Ce(B) and H* = H/B. By21.5.3, IAute(B)I = 4 and c ~ ( B ) /B C (B) ~ 3 A4 is faithful on C - (B) % E8 and stabilizes a hyperplane Z of C3(B). Thus IH*I = 25 .3f- 52 by the previous paragraph. Also a Sylow 2-group T of Cfi(B) is isomorphic to EqwrZ2. Moreover elements in CB(B) - Z are conjugate to i: while those in Z# are of the Ch(B) for each s E Z# by 26.3. form ED, D a dodecad, so CH(s) Therefore T E Sy12(H). Let J = J ( T ) % El6. By the previous paragraph we may assume i E T- J. Next CQ(B) = (z), so from 25.1, B is determined up to conjugacy

<

26. Some local subgroups of Gol

139

in CG(z) and CG(B(z)) Zlo x A5. In particular ICH(z')12 = 8, so gH n J = 0,and hence by Thompson transfer H has a subgroup K of index 2 with i # K . As J = [J, R] for R E Sy13(CR(B)), J E Sy12(K). Let I = NK(J) and recall that CK*(s) = J * for all s E z#. Further by 7.7, I controls fusion in J, so by 7.3, s fixes a unique point of K/I. Also I # K as CK(z) I. Hence by Exercise 2.10, there exists U 5 J such that CI(U)/U is strongly embedded in CK(U)/U, U n Z = 1, and CI(U) is transitive on (J/I)#. As 7 does not divide [HI, transitivity of Cr(U) on (J/I)# implies IUI # 2, so as U n Z = 1, either IUI = 4 and J = Z x U, or U = 1. But in the latter case I is transitive on J#, so as CK.(s*) = J*, Exercise 16.6 in [FGT] says K* % L2(16). This is impossible as 17 does not divide I HI. Thus J = U x Z, so by Exercise 16.6 in [FGT], CK(U)/BU % L2(4). Let J1 = U and R1 E Sy13(CK(Jl) f l NK(J)) be z-invariant. Then RRl E Sy12(K) by an order argument and (z)RR1 has two orbits on J# of length 6 and 9, with elements of in the orbit of length 6. In particular J = J1 x J2, where J2= J f and Jl is RR1-invariant. Now K1 = RCK(J1) is of index 5 in K , so we have a permutation representation a : K* --+ S5.We conclude Kf = CK(J1)* is the kernel of this representation, so K* = KT x K;, where K; = KT. That is, H* r A5wrZ2.

J?

Lemma 26.6: Let Y be a 3-central subgroup of N of order 3. Then dim(CA(Y)) = 12, IAutG(Y)I = 2, ICG(Y) : E(C,+(Y))I = 2, and E(Ce(Y)) U4(3)lE9.

=

Proof: As usual 21.4 and 26.1.1 imply the first two remarks. Next by 21.4, N has two classes of elements of order 3 and from 26.4 the dimensions of the fixed point spaces on A are different for the two Thus we can apply 26.1.4. As in the classes, so they are not fused in Remark above, we check that Y fixes 96, 30 vectors in A; for i = 3,4, and NR(B) has two orbits of length 96 and 60 on A;. SO1CA2(Y)1= 378. Sz Next Y centralizes a subgroup Yl of order 3 such that CG(Yl)/Yl ~ By by 25.9. By 46.6, K = CG(YY1) is quasisimple with K / F 2~U4(3). Exercise 9.6, Yl has no fixed points on A, so CA(Y)has sign fl, and then by 26.1.3, ICA4(Y)]= 35.7 and ING(Y)I = 3 5 . 7 - I ~ R ( ~=) 29-38*5.7. i SO as IU4(3)1 = 2' -36.5.7 and l A ~ t ~ ( Y ) 3, l =we conclude K = E(NG(Y) is of index 2 in CE(Y) and I NG(Y) : KI = 4.

e.

Remarks. Conway was the first to find Co2, Cog, Mc, and HS as subSome of our arguments are roughly the Same as groups of Col in [CO~].

140

Chapter 9 Subgroups of

.0

his, although Conway felt free to use classification theorems in the literature t o identify various subgroups, whereas here we keep our treatment self-contained. This will cause small problems later too. The McLaughlin group Mc and the Higman-Sims group H S were discovered as rank 3 permutation groups by McLaughlin [Mc] and D. Higman and C. Sims [HS], respectively. Conway showed that these rank 3 representations could be reaIized on vectors in the Leech lattice as in 24.6 and 24.11. After Higman and Sims discovered HS, G. Higrnan happened upon H S as a 2-transitive group [HiG]. Sims proved the group discovered by G. Higman was isomorphic to H S [Sill. Conway showed this representation could be realized on the Leech lattice as in 24.13. Conway attributes the calculation of various local subgroups In [CO~], of Col to Thompson. This work was never published, but a fairly complete description of the local structure of Col was derived using different methods by Nick Patterson in his thesis [PI. In many cases we follow Patterson's treatment, although again Patterson felt free to quote the literature, whereas we do not. Also when possible, we use the theory of large extraspecial subgroups in Chapter 2 t o replace Patterson's arguments.

Exercises 1. Let X be a set of order 10 and As r A 5 H 5 Sym(X) with A transitive on X and IH I = IAl 3i. Prove H = A. 2. Let G = Col and P E Sy123(G). Prove ING(P)I = 23 - 11. 3. Adopt the notation of Section 24 and let v = vl - z for some z E z3. Let L = CG,(z), = (vl - w : w E and y = I'l - (v). Prove L has two orbits 7%and 72 on 7, where Ti = Az(v, i). F'urther lyll = 70 and 1721 = 105. 4. Prove Lemma 24.10. Use 23.6. 5. Let R = (R1,RO) be a rank 2 geometry satisfying the properties of 24.7 and G a group of automorphisrns of R satisfying the properties of G7 in 24.7. Denote by A the collinearity graph of R. Let (x, 1) be a flag of R, z E ~ ' ( x ) , (y) = A(z) fl R(Z), and P = P(x) the kernel of the action of Gx on the set Ro(x) of lines through x. Prove (1) Gx is faithful on A(x). (2) P is regular on A2(x) and Gx,z is a complement to P in G,. (3) Gx,, preserves an orthogonal space structure of sign -1 on P and GXs,l is the stabilizer in G x , of a singular point W(1) of P.

rl

z3),

!

I

Exercises

(4) The kernel &(I) of the action of GI on R(1) is of order 2 . 35 with P(1) = Os(Q(l)) the unique normal 31+4 subgroup of GZll and with W(1)= Z(P(1)). (5) P n ~ ( y =) W (I). ( 6 ) For h E Gx and P E P1 G X , = ~ Gx1z,1(Pn P(1)); yh E A(Y) if and only if h E G1 = G,,,,JP; and zp E A(y) if and only if p E P f l P(1). (7) For p E P#, zp E A(z) if and only if p E W(k) for some k E Ro(x). (8) G U4(3). (Hint: See the proof of 45.11 and use Exercise 15.3.) 6. Assume the hypothesis of Lemma 25.9. Prove CA(A) = 0. (Hint: Prove ha n CIAVzI(A) = 0,and use this to show A is fked point free on [A, z] and hence also on h/cA(z). Conclude CA(A) 5 CA(CG(A)). Finally observe ICG(A)I2 does not divide ICo31, and hence Cc(A) h e s no nonsingular point of h.)

141

27. The subgroups C and N of the Monster Table 4 List of symbols in Chapter 10

Symbol

Chapter 10

The Griess Algebra and the Monster

-

In this chapter we construct the Griess algebra and its automorphiim group, which is the largest sporadic group: the Monster. We begin in Section 27 by specializing the construction of Section 14 to the Parker loop L. The subgroup N = N/K supplied by this construction is the normalizer of the Cgroup Z = (zl, zz) in the Monster. We saw in Section 14 that N contains a large extraspecial 2-subgroup Q 0i2. In Section 27 we construct a group C with F*(C) = Q and C/Q E Col. The group C is the centralizer of the involution z~ in the Monster and Q is a large extraspecial subgroup of the Monster. In Section 27 we construct a 196,884dimensional RC-module B for C admitting a C-invariant bilinear form 7. Next in Section 28, we define an algebra map T on B preserved by C. The algebra (B,T) is the Griess algebra first constructed by Griess. From Section 9, the algebra map T is equivalent to a C-invariant trilinear form 0. The construction follows Conway in [Co3)and uses the Parker loop L. Next Nl = Cfijzl) is a subgroup of C. In Section 29 we extend the representatio~lof % to N in such a way that N preserves y. Thus Go=(C,~)
V,F, U, P(v), C(u, v), 4 7 4 ,v, w) A,E,E+, n,4 , ~ c , v oX11 r1ro,Q11Clil ziLslki1 Z1K ~,~+,~,~i,~ilfi~f(~),e~,&l,SlrS ~i ,Bj(Ai) 6 ( d ) 6: ( 4 , Lf(x) 1 Bi* BL,B-,x@~~(~),C,P~,P~,Y CltC2r ~ 2 Xz1 1 ex1 c01 P Bs, XY, BR, Br;'% ~ ( r ) B Z , B ~ B&, , 73', uxurvxyrW& B,T,T~~Ts~,P,Y~~T,*~TT~TsTr- I Pi B;,$,R,BE xi,B& BO X

B;,

BE

The Remarks at the end of this chapter contain a brief diicussion of the approaches of Griess, Conway, and Tits to constructing the Monster, and a comparison of those approaches to the treatment given here. Since this chapter is replete with specialized notation, we close this introduction with a list of notation used in Chapter 10. Column 1 of Table 4 lists the symbol and column 2 the page where the symbol is defined.

27. The subgroups C and N of the Monster In this section we assume the hypotheses of Section 14 with V = V , the Golay code module for the Steiner system (X,C) and P(v) = lvl/4 mod 2 as discussed in Sections 18, 19, and 20. Write U for the power set of X regarded as a vector space over the field F of order 2 under symmetric difference; thus V 5 U.Notice this differs slightly from the notational conventions of Sections 19 and 20. Recall from Section 14 that L is a Moufang syrnplectic 2-loop with parameters (P,C, A), where from 11.8 the commutator map C : V x V 4 F is given by C(u, v) =

IU

n vl/2 mod 2

144

27. The subgroups C and N of the Monster

Chapter 10 The Griess Algebra and the Monster

A

,fl : Q1 -,

and the associator map A : V x V x V -,F is given by

+

t 1 :$l(d)'@2(b)$l(aJ) r-' f+(d)+t(b)~ f ( J )

A(u, v , w ) = lu n v n wl mod 2. The loop L is the Parker loop. Recall also from Section 14 that the center of L is (n)= (1, n ) and q5 : L -, V is a surjection of loops with kernel (n). Further the centralizer E in Aut(L) of L / ( n ) consists of the maps a,, e E V * , with a,(d) = dnE@) for d E L. For d E L and f = 0 or 1 in GF(2) we sometimes write d f for d r f . So a,(d) = d ~ ( dunder ) this convention. We choose the distinguished element vo of Section 14 to be vo = X and take ro = M24. Recall E J? 5 Aut(L) with r / E = J?o. In particular the natural map -,o'I gives us a permutation representation of I'on X with kernel E and for a E I' we write x a for the image of x E X under a via this representation. Hence, for example, for each E E V * , a, is in the kernel of the action of r on X. Recall from Section 14 that fl = L U ( 0 ) with 0 . d = d . 0 = 0 for all d E L, and to each d E L and a E there are associated permutations qi(d), $i(a), i = 1,2,3, of a3.See Section 14 for the definition of these permutations and lemma 14.2 for an extensive list of properties of these permutations and how they multiply. Recall N is the subgroup of ~ ~ r n ( i 'generated 3~) by these permutations. Further zj = Ilri(r),s E L with #(s) = X , ki = $ i - l ( ~ ) h + l ( ~ ~ ) , Z = (21, z2), and K = ( k l ,k2). By 14.3, 1 = $ ~ ( d ) $ ~ ( d ) $ ~ for (d) each d E L, so in particular 212223 = klk2k3 = 1. Also by 14.2, Z and K are normal in N . Further N+ = CN(Z)= C N ( K ) and N/N+ S3 by 14.3. We write N = N I K . Recall Ni = C N ( q ) and Qi = ($i(d),&(a) : d E L , a E E ) . By 14.4, Qi a Ni, ~ iki , E Z ( N i ) , and Qi/(ki) g Qi S is extraspecial of width 12. For J E U,define the elements f J and f ( J ) of the Leech lattice A as in Section 23; thus

+

+

with kernel ( z l , k l ) defined by

<

0i2

By Exercise 3.2.1 we have a surjection J H E J of U onto V * with kernel V , where E J ( v ) = I J n vl mod 2. Write a J for a,, E El and observe using Exercise 3.2 and Remark 14.1 that:

Lemma 27.1: (1) The map J H a~ is an isomorphism of U / V with E. (2) all d, e E L, a(d,e) = a+(d)n4(e). Lemma 27.2: Let = Q l / ( z l ,k l ) and regard Q I as an orthogonal space over F . Then there exists a surjective group homomorphism

+ 2&

where t(b) = 0 , l for b = 1, n , respectively, which induces an isometry

i:$il+A. Proof:Let D = V x U / V and regard D as an F-space as in 23.10 with quadratic form q defined in Exercise 8.10. Then 23.10 and Exercise 8.10 give us an isometry L : D -+ ii defined by L

:(v,V+

J)

H

fv

+ f ( J ) + 2A.

Also Exercise 4.6 gives us a group homomorphism p : Q1 -+ D inducing an isometry ,ii: $il --+ A defined by P : *l(d)$2(b)$l(aJ)

( 4 ( d )+ t(b)Xl V

+J).

Then El = PL is our homomorphism inducing the isometry ( = f i b . Exercises 4.6 and 8.10 apply in a general setting; we give a proof now that p is a homomorphism and an isometry in this special case. First the map p is well defined by 14.4.4. By 14.2.7,

i

$l(d)$l(e) = $1(de)$3(C(~id))$l(a(e,d))* By 27-1-27$ l ( a ( e , d ) ) = '@l(a+(d)n+(e)). Also for b E [$1 ( Q J ) ,$l(d)l

=-- [$I ( Q J ) , $2(b)l

=1

(T)?

mod

(4

by 14.2.4 and 14.2.5, since $2(b)$3(b) = $l(b). Therefore

Then as zl is in the center of Q1 and the kernel of p, the check that p is a homomorphism is a straightforward calculation. So : Q1 -, is an isomorphism of F-spaces. To check that ( is an isometry, we first observe that a5. $1 ( E ) is elementary abelian, (1 ( E ) is totally ) ) singular. Thus totally singular. Further by 23.10.2, i ( $ l (is~ it suffices to show (a) g2 = 1 if and only if q ( t ( 3 ) )= 0 for g E $ i ( L ) , and (b) [g,h] = 1 if and only if (&), f(h)) = 0 for g E $1(L), h E $1(E). But if d E L then $l(d)2 = $ i ( P ( d ) ) ,so (a) holds by 23.10.3. Similarly [$l(d),$l(aJ)]= $ l ( t J ( d ) + lJIP(d))= $l(IJndl

+ IJIld1/4)

by 14.2.8, so 23.10.3 also implies (b). We next construct several R-modules for Nl and N+.

146

Chapter 10 The Griess Algebra and the Monster

Recall from Section 14 that Ai consists of those members of 523 with ith entry in L and 0 elsewhere. For d E L, write di(d) for the element of hi determined by d. For i # j let Bj(Ai) be the R-space with basis the orbit space Ai/(kj) subject to the convention that 6(d)l(rj(r) = -6(d), where 6(d) = $(d) denotes the orbit under (kj) of 4(d). Observe that ~ a) 211-dimensional monomial module by 27.3 and Exercise 4.3, B ~ ( A is for N+ with respect to the basis 6(d), d E L, via g : 6(d) +-+ S(d)g. Regard Bj(Ai) as an orthogonal space with orthonormal basis 6(d), d varying over a suitable transversal. By Exercise 4.3, N+ preserves this inner product. Also an easy calculation shows: Lemma 27.3: For all b, d E L and a E I' (1) Ill ( 4 : 62(b) (2) $2(d) : 62(b)

62(db). 62(b + P(d) + C(b1d)). (3) 1)2(a) : 62(b) 62(ba) if a is even. (4) $2(4 : 62(b) 1-t 62(ba+P(b)) and $l(ff) : 62(b) 63(ba+P(b)) if a is odd. (5) ($l(s),$2(7r), &(s)) is the kernel of the action of N+ on B1(A2). (6) ($l(s7r),$2(n), g2(s)) is the kernel of the action of N+ on ~ ~ ( ~ 2 1 .

[ B

Next by 27.3, B1(Ai), i = 2,3, are monomial N+-modules with respect to the bases bi(d), d E L, with B ' ( A ~ ) $ ~ ( = ~ )B1(A3) for a odd, so B: = B1(A2) @ B1(A3) is a monomial Nl-module. Indeed B; is the Nl-module constructed in 14.6. Notice that by 14.6, zl acts as -1 on B?. Let B- = Bi4 @ B,: regarded as an Nl-module. Also B- is an orthogonal space whose form is the tensor product of the forms on the factors. ) 6:(d) of B-. Observe We write x @J bi(d) for the basis vector L ~ ( x @ that a suitable subset of these vectors forms an orthonormal basis for the I), form on B-. Notice also that -6: (d) = 6: (d)zl = 6;(d?r) = 6:(d so L'+€(x) @J $(d) = Le(x) @ 6: (d 1) = -(Le(x) @ 6:(d)).

1

+-+

(I) h(d) frzes L:(x) and Ilj (d) : Li (z) I-+ L~+'~('(~))(X) for j # i. (2) ?,bi(&): LZ(x) 1-t Li(xa). (3) $ ~ ( a ): L:(x) ++ Ls(xa) if a is odd. -- .(4) Qi is the kernel of the action of Ni on Bi4 and kj inverts Bi4 for j # i. Proof: These are easy computations. For example, (a, 0,0)$2(d) = (ad, 0,O) and €,(+(ad)) = ez(4(a) + +(dl) = ~ ~ ( 4 ( a+) )6~(4(d)),so e+e= (4 L! (4$2(d) = L1 (a).

Lemma 2'7.5: ( I ) B- is a monomial module for N1 of dimension 212.24 with respect to the basis x 8 bi(d), x E X, d E L, i = 2,3, and Nl preserves the form on B-. (2) zl acts as -1 on B-. (3) K is the kernel of the action of N1 on B-. (4) I12(s?r) is trivial on B: and inverts B;~.

Bid,

Lf (x) = {bi(d) : d E L and eZ(4(d))= e )

Lemma 27.4: For each x E X I d E L, a E I?, and e E {0,1):

+

+

Proof: We have observed that B- is a monomial Nl-module, so (1) holds. As zl centralizes Bi4 and inverts B,: (2) holds. From 14.6, ($2(~7r),$3(~))is the kernel of the action'of Nl on B;. In particular together with 27.4.4 this means that kl is trivial on B; and Bi4 and hence also on B-. Further k2 = $2(sn)z1 inverts B; and so k2 is trivial on B- and (4) holds. That is, K is contained in the kernel J of the action of Nl on B-. By 27.4, [Q1, Bi4] = 0, so as (~h(s?r),$3(s)) is the kernel of the action of Q1 on B!, Q1 n ($2(~7r),7,h3(s)) = (kl) is the kernel of the action of Q1 on B1. Thus [Q1, J ] 5 Q1 n J = (kl). Then J 5 CN,(Q~K/K= ) (21, K) by 14.4.2, so J = K.

Next for x E X , e. E {O,l), define and using Exercise 4.3, define Bi4 to be the 24dimensional R-space with basis Li(x), x E X , E E {0,1}, subject to the convention that ~ f '(x) + = -LZ(x). Make Bi4 into an orthogonal space by decreeing that L:(X), x E X, be an orthonormal basis. By Exercise 4.3 and 27.4, Bi4 is a monomial module for N, and Ni preserves the form on Bi4 as Ni permutes the vectors Li(x) via right multiplication; indeed:

27. The subgroups C and N of the Monster

!

I1

I

We next construct the centralizer C of a 2-central involution in the Monster and an amalgam C

t

Nl

-t

N.

Let p1 : Nl + SL(B?) be the representation of Nl on B1. Then by 14.6, Q1pl is extraspecial and irreducible on B;, so N1p1 5 Ml the normaliier in SL(BI) of Qlpl. In particular by Exercise 4.4, MlIQlp1 is the isometry group of the quadratic form on Q1 = QIK/K(zl). We use the isomorphism of 27.2 to give ~1 the structure of and let Cl be the subgroup of MI preserving this structure with Cl/Qlpl Col. We will see that Nlpl < Cl. Let 4 : Nlpl o ( Q ~ ) be the natural map with kernel Q1pl. Similarly let p2 : Nl -+ S L ( B ~ ~be) the representation of Nl on Bi4. -+

148

~i~

The map L: (x) I-+ fiex defines an isomorphism of with R @zA in which N1p2 Stabilizes the inverse image of A, so identifying this sublattice with A via the isomorphism, we have N1p2 5 C2 the stabilizer in S L ( B ? ~ ~ ) of A. The map A --, A Z Q1 induces a surjection vz : C2 + Col. We claim plvl = p2v2; this will show Nlpl Cl as mentioned earlier. Our claim amounts to the assertion that the map of 27.2 is Nlequivariant. By construction during the proof of 27.2, = DL, while by 23.10.4 and Exercise 4.6, the maps j2 and L are Nl-equivariant, establishing the claim. This proof depends on the general Exercise 4.6; we prove the equivariance directly in our special case now. It suffices to show gplvl = gp2v2 for g = $l(P), P E M24, and g = $72(e), $(e) E C, as these elements together with Q1 generate Nl. Similarly as the elements ix, x E X, generate A S ~ 1it ,suffices to check that gplvl and g p p 2 agree on Xx for each x E X. If p E M24 then by 27.4, L:lx)llrl(P) = L~(XP),so as L;(X) ++ d e , , we conclude ~ x $ 1 ( ~ ) p 2 v= z Xxp. On the other hand <(qbl(crx)) = ix (cf. 27.8.3) and &(aX)$l(fl = $l(cxzo) by 14.2.4, so ~ x ~ l ( ~ ) p=l ivx lp too. Next for $(e) E C, $ ~ ( c u , ) @ ~=( ~&(e. ) ~ ~ ~ ( ~ ) ) $ l ( (cf. c r , 29.6.2) ) so Xx$2(e)nu1 = -Az + ffmce,+C=(e),+ 2A. Finally by 27.4, ~ : ( y ) d d e )= L:""('(~))(~), so &(e) changes the signs of the entries q, in -h for y E $(e), and hence indeed XX$2(e)p2v2 = ~ x ~ ( e ) p l vestablishing l, the claim. Finally form the fiber product Co = C1 x~,, C2 with respect to the diagram

<

i

c1-% CO, 2c2 as in Section 6. As plvl = p2v2, 6.1 gives us a map p : Nl -+ Co with ppi = pi for i = f,2, and 6.2 sayspl@pz : Co + O(B-), where pi : Co + .Ci is the ith projection, and p(pl @pz)= p l @p2 : Nl + O(B-). Let C denote the image of Co under p l @n, so that p1@pa : Nl C 5 O(B-). -+

Lemma 27.6: (1) F*(C) = Q1(pl 8 p2) Z (2) C/F*(C) 2 Col.

0i2.

(3) k - 4 4 €4 ~ 2 =) K . (4) Qi = QiK/K(zl) is isomorphic to as a C/Fe(C)-module via the isomorphism of 27.2. Further ((22) = 8ex 2A and Nl/QIK r

A

+

M24/E211 is the stabilizer in C/F*(C) of 22 with E+K/Kz isomorphic to the 11-dimensional Todd module for Nl/O2(N1).

Proof: Part (3) is a restatement of 27.5.3. By (3) and 14.4, Q1(p1@p2) Q1/K D.:'

149

27, The subgroups C and N of the Monster

Chapter 10 The Griess Algebra and the Monster

"

Next by 27.5.4, $2(s7r) E ker(pl) with ($2(sn)p2) = ker(v2). Thus by 6.1, ker(p1) = ($~(ST)P)I (Q1K)p. Then as (QiK)pi = F*(CI) with C1/(QIK)pl E Coi, (Q1K)p = F*(Co) and CoI(Q1K)p 2 Coi. Hence (3) implies (1) and (2). By construction, Q1 A as a C/Q1-module via [, with ( ( i z=)fx 2A = 8ex + 2A. Then as Nl stabilizes 32 and INl : QIKl = (M24/E2~a1 with M24/E211 the stabilizer of 8ex 2A in Colr all but the final reniark in (4) hold. Finally by 14.2.4, $ ~ l ( a ~ ) @ l (=p ql(aJp) ) for J E U ,P E M, so the map

=

+

+

$l(aj)KZ

I+

J

+V

is an N+-equivariant isomorphism of E+ K/KZ with the 11-dimensional Todd module T. Let N = N / K and for g E N set jj = gK. Then p1 @ p2 induces an injection R1 -+ C and we identify fil with its image under this injection and regard fil as a subgroup of C. In particular under this convention, Q1 = F*(C) by 27.6.1, while by 27.6.2, C / Q ~ E Col. Let B, = s 2 ( ~ i 4be ) the symmetric square of regarded as a Cmodule with quadratic form equal to the tensor product of the forms on Thus B, has a basis {xy : x, y E X), where

~h~

~ 1 ~ .

+

xy = L;(z) 8 ~ : ( y ) ~:(y)@ L:(x) if x

#y

' ) 0 if xy # x'y', while and x2 = L:(x) 8 L:(x). In particular ( X ~ , X ' ~= (x2,x2) = 1 and (xz, xz) = 2 for x # z. Further Lemma 27.7: (1) Q1 is the kernel of the action of C on B,. (2) B, is a monomial module for Nl with 0; jixing Rxy for all ~ , EyX . (3) B, = B ~ I B & ,where BR = (x2 : x E X) and B; = (xy : x, y x, x # Y).

Proof: Part (I) follows from 27.4.4 and the fact that 6 / Q 1 is simple. = (q2(d) : d E L)E+. Part (3) follows Part (2) follows from 27.4.1 as from the definition of the quadratic form on B.

~zf

Next identify Q1 = Q1/(zl) with k via the isometry (of 27.2. By construction C1/(QIK)pl Col preserves this structure, so as k e r ( ~ ~ ) (QIK)p, so does C. For g E Q1 or j j E Q1, write j for the image of g in Q1. Write R for the set of r E Q1 such that i E Az. Then C acts on R via conjugation, inducing a monomial R-module B, with basis y(r),

=

<

Chapter 10 The Griess Algebra and the Monster

28. The Griess algebra

r E R, subject to y(rzl) = -y(r), via y(r)g = y(r9) for r E R and g E C. Observe that C preserves the quadratic form on BT for which our basis is orthonormal.

28. The Griess algebra

150

Remark. In the next lemma we associate to each r E R an element ( ( r ) E A2 such that [ ( r )+ 2A = ((r).Recall from 23.2 that there are exactly two A E Az with = [(r).We caution that for r E R2,( ( r )is not well defined in that r(c,J 4(c))= r(c,J ) but ((r(c,J + $(c)))= -((r(c, J)). However, we only encounter E(r) in the guise of some function h(E(r))with the property that h(-t(r)) = h(((r)),so this abuse of notation causes no problems.

+

Lemma 27.8: (1) R = R2U R3U 7Z4, where Ri = {r E R : i ( r )E A;) and [(r)= [ ( r ) 2A, where ( ( r ) is defined below. (2) R2 = {r(c,J ) : c E L, J C_ $(c)E C, 1 JI even), whem

+

4% J ) = 4l(c)dl(aJ)4l(s)' J"2 and M c , J ) ) = f+(c) - 2f.J. (3) R3 = {r(x,d ) : x E X, d

E L),

where

and < ( r ( ~ , d=) )fd+e.(d)~- AX(4) R4 =_ {ru(xy)Pq,rv(xy)~~-: x, y E X , x # y, E = 0,1), where rv(xy) = $l(aXy),T U ( X Y ) = $l(aZy)$l(s),E ( T ~ ( X Y ) ) = 2fxy, and t(rv(xy))= 2(fx - f y ) . (5) BT = B ~ I B ~ I B & L ~ ,w h e ~Bz = (y(r) : r E R2), = ( ~ (: 4r E R3),B& = (wzy : X,Y E X ) , uxy = Y ( ~ , ( x Y ) ) , vxy = y(rv(xy)),w%,= vxy+ uxy, and w&, = vxy - uxy.

~4

Proof: Part (1) is 22.6. Also 22.6 and its proof tell when X E A;; then we use this description and 27.2 to calculate (2)-(4).Part ( 5 ) follows from (2)-(4)and the definition of the quadratic form on Br. We close this section by defining B to be the F-space with quadratic form which is the orthogonal direct sum T h u s C is a group of isometries of B. Observe that zl centralizes Bs by 27.7.1. Also as 2.1 E Z(Q1) and C acts on R by conjugation, zl fixes y(r) for each r E R. Thus zl centralizes BT.Finally by 27.5.2, zl inverts B-. Thus

Lemma 27.9: B- = [B,zl] and Bs

+ BT = CB(zl).

151

In this section we use the results of Section 9 to define an algebra structure on the C-module B defined at the end of Section 27. Let 7 = ( , ) be the bilinear form on B and recall

B =B-IBTIBs is the orthogonal direct sum of three subspaces. We define maps ~i E

L(Bi,Bi;Bi) and rij E L(Bi,B j ;B j ) , which determine a symmetric algebra map r E L ~ ( BB;) by

subject to the conventions of Section 9, and T and 7 in turn determine a symmetric trilinear form b7(r) = @ = ( , , ) by 9.4. Recall Bs = is the symmetric square of B $ and ~ has basis xy, x, y E X . It will sometimes be convenient to view Bs as the space of all 24-by-24 symmetric matrices by identifying Ex,yaxarx y with the ; by matrix (asy).Subject to this identification we define 7,E L 2 ( ~ ,Bs)

s2(~i4)

where AA' is the usual matrix product of the matrices A and A'. Next BT has basis {y(r): r E R ) and we define TST E L(Bs,BT;Br) by

(28.2) A * y(r) = (A,t ( r )8 <(r))y(r), for A f Bs, T E R.

~i~

R Q9z A via x = L:(x) H &ex Here we need to recall that and ( ( r )E A is defined in 27.8. Thus ( ( r )8 E(r) and A are members of B, and the inner product (A,<(r)8 J ( r ) )takes place in Bs. Indeed as ex = x / d , we have: Lemma 28.3: If ( ( r ) = ax/& in Bs.

Exaxex then the inner product

(<(r),x)=

Define T,. E L ~ ( BBr) ~ ;by

(28.4) For r, r' E R, y(r) * y(rl) = y(rr') if rr' E R and O otherwise. Next B- = ~ ?@ jB f~ and we define TS- E L(Bs,B-; B-) by

(28.5) A * (a8 b) = aA Q9 b and b E ~ 1 ,

+ (Tr(A)/8) (aQ9 b) for A E Bs, a E B&,

where aA is the product of the vector a = Ex asx with the matrix A.

152

28. The Griess algebra

Chapter 10 The Griess Algebra and the Monster Finally we define T,-

Lemma 28.9: Tsr 6 commutative and fl,, has monomials

E L(B,, B-; B-) by

(28.6) y(r) *(a@b) = (a - 2(a, J(r))t(r)) @ (br)/8, for r b E BL.

E R,

a

E Bi4,

Here br denotes the image of b under r. Next by 9.4, T, and the restriction Ti of 7 to Bi determine a trilinear form b-,,(~~)= pi on Bi via

and

~ idetermines j

Pij E L(Bi, Bj, Bj; F) via

forx,y€X, r € R .

Proof: The list of monomials is immediate from 28.2 and the fact that y(r) is of norm 1. In particular from the list of monomials we see P,, is invariant under the permutation (2,3), so its algebra map T~~ is commutative. Lemma 28.10: r9- is commutative and PS.- has monomials

Pij(a, b, c) = (7ij(a, b), C ) . In the next few lemmas we determine the monomials of these forms with respect to our standard bases.

Lemma 28.7: PS is symmetric with monomials 4.x2~xy.xy,4-x2.x2.x2, and 4 xy yz zx for distinct x, y, z E X.

- -

' A * ( ~ 8 6 =) x A @ S+ T T ( A ) / ~(X . €36).

Proof: Let A, A' be the matrices corresponding to ab, arb', respectively, where a, a', b, b' E X. Then unless {a, b) n (a', b') # 0 , AA' = A'A = 0, so A * A' = 0 and hence there is no monomial AA'A*. So assume {a, b) n {a', b') # 0. If a = a' and b = b' then A2 = a2+b2, a2, for a # b, a = b, respectively. Hence A * A = 4(a2 b2), 4a2, respectively. So as a2 has norm 1,we get monomials 4 ab ab a2, 4 a2 a2 a2,respectively. If a' = b' = a # b then A * A' = 2ab, so as ab has norm 2 we get monomials 4 . a2 ab ab. Finally if a' = b and 6' = c with a, b, c distinct, then A*A1 = 2ca, contributing a monomial 4.ab.bc.ca. Thus the lemma is established.

+ . .

Lemma 28.8: r, r', TT' E 72,

Proof: Let A be the matrix of ab and 6 = 6(d). Then

PT is

Now TT(A) = 0 unless a = b, where Tr(A) = 1. Further XA = 0, b, a, for x $ {a, b), x = a, x = b, respectively. So as (x @ 6, y @ 6) = (x, y) = the monomials are as claimed, and hence T ~ - is commutative.

Lemma 28.11: 7,- is commutatzve and PT- has monomials

'

. . .

symmetric with monomials y(r)y(rt)y(rrl), for

Proof: By definition, ~(r)*y(r')= 0 unless rr' E R, where y(r)* Y(T') = y(rrr), and we may assume the latter. As y ( r ~ ' )is of norm 1, we get monomials y(r)~ ( r ' ~) ( r r ' ) ,T, r', rr' E: R. So it remains to show the form is symmetric. But as r, T', rr' E R and @(c(s))= 0 for each s E R with an isometry by 27.2, we conclude that s2 = 1 and 0 = (C(T),~(T'))= [r,~']. Further r TT' = T'. So the form is indeed symmetric.

6

f o r x , y ~ X , x # y , r ~ R , d ~wLh e, r e ~ ( r ) = C , a x e x .

Proof: Write 6 for 6(d). Then y ( ~ ) (*~ € 3 6= ) (~-2(x,t(r))<(r))@6r/8, so (y(r),x @ 6,y @ 5') = ( ( 2 , ~) 2(x,t(~))(~,tfr)))(6',6~)/8.

(*)

In particular (*) is 0 unless 6' = fSr, in which case we may normalize and take 6' = 6r. Then as 6' is of norm 1, appealing t o 28.3 we obtain:

Thus the lemma holds.

Lemma 28.12: The algebra map

29. The action of

Chapter 10 The Griess Algebra and the Monster

154

defined via the convention of Remark 9.6 is symmetric as is its trilinear form 0 = b7(r) defined via 9.4. Moreover and

T

and 0 have the same stabilizer i n the isometry group of B .

Proof: The first remark follows from Remark 9.6, since from the lemmas

above each of the summands T; are symmetric and the summands Tij are commutative. The second follows from 9.4.3.

Lemma 28.13: C stabilizes T and P. Proof: By 28.12 it suffices to show C stabilizes T , and for this it suffices

to show C stabilizes each ri and rij. But this is almost immediate from the definitions.

29. The action of N on B We have seen in 27.7 and 27.8 that:

Lemma 29.1: As an N+-module

N

on B

155

we check that D centralizes B D for D = R, E l 2, we have Bo = [ B ,Z ] and (4) holds. As E C ( Z ) ,B z C B ( Z ) .On the other hand

z2

<

Lemma 29.3: CE+(r(c,J)) = Z ( & ( a K ) : eK(c) = o), since [ql(aK),r(c,J ) ] = [?j1(ffK),ql(c)] = 2;"'') by 27.8 and 14.2.8. Thus Bz = [BZ, E+], so (3) holds once we show [D,BD] = 0 for D =

R, E.

+

Next by 27.7.1, B R B& = BS 6 C(Q1).Also Q$ acts on Ra: 6 B:, x E X , by 27.4.1, so Q$ centralizes z 2 and hence R 6 C(BR). Further by 27.4.1, B k = [B;,Q.~],SO B& = [B&,R]. Then 29.4 below completes the proof of 29.2.

<

Lemma 29.4: (1) $l(d) :w:, H (-l)c-u(d)w:y for k = 2,3. (2) $2 ( d ) : wz, r (- l ) e = u w?, and f i e s wz, . (3) $l(a) : w:, I+ w : ~ for , ~ k = 2,3 and a even. (4) E+ centralizes B; for k = 2,3, and $q(az)has cycle (w&,,w&) for z E X . Proof: By 14.2.8, [$i(d),$l(axar)] = $i(%ar(d)) and by 14.2.9 and

where (1) B R = ( x 2 : ~ € X ) S B s . (2) B E = B & L B $ I B ~ , where B& = ( x y : x , y E X , x # y) 5 B , and = (wig : x , y E X ) 5 Br for i = 2,3 are described in 27.8.5. < BT. (3) Bz = ( ~ ( r: r) E z2) (4) Bo = B&LB&~.B~, where B; = ( y ( r ) : r E 5 B , and B- = Bo2-l~; with B ~ = ( X @ ~ ~ ( ~ ) : X EL)Xfori=2,3. , ~ E

BL

z3)

Actually this is the first time we have defined B: and B:. These subspaces are N+-invariant by 27.3. Recall from 14.5 that R = (Q1n N2)(Q2n N l ) is a normal subgroup of N with R of class at most 3 , Z = Z ( R ) , and Z 2 ( R ) 2 E+ = Ql n Q 2 .

Lemma 29.2: (1) B R = C B ( R ) 5 CB(Q1). (2) B E = [CB(E+),R] and B& = cB,(Q:) with [B;, Q1]= 0. (3) B Z = [CB(Z),E+]., (4) Bo = [ B ,Z ] with Bh = CB, (Zi). Proof: As we have already observed in 27.9, B- = [ B ,zl]. By 27.3 and 27.4, B; = CB-(z2), so B; = CB- (z3) as zlz2z3 = 1. For r E 7Z3,

y(r) E Q1 - C ( z 2 ) ,so z2 inverts

BA,

and hence B; = [B;,Z ] . Then once

14.2.10, qi(d) centralizes q l ( s ) for all d E L. Hence (1) holds and as $2(e) = (Ql(s)kl)' r $l(s)' mod K by 14.3, we also have (2). Part (3) is easy. Finally & ( E ) is abelian and E S centralizes $ l ( s ) , so E f cens), = z1 by 14.2.8, so +l(az) tralizes B; for k = 2,3. Also [ $ ~ ~ ($l(aZ)] centralizes vxU and inverts uw, and hence has cycle (w&, w&). We wish to extend the representation of Nl on B to a representation of N that preserves our bilinear and trilinear forms, and hence also the Griess algebra. We see in general from 29.1 and 29.2 how this must be done. Namely B D must be an N-submodule for D = R, E l Z , 0 (as each such D is normal. in N ) , and N / N + must induce S3 on the three summands of B E and Bo. More precisely we have a permutation representation of N on {1,2,3) with kernel N+ such that for g E N, = zig, k: = kigl (Ai)g = A,, etc. In particular, $(dlg = 6ii(dt) and LP(x)g = L$(x') for some d L , xt E X , and 6'. One extension is easy. If B R is to be an N-submodule then as Q1 centralizes BR, so must (Q1,Q2). But by 14.5.2, N / R = (Q1,Q2)/Rx N + / R , so:

Lemma 29.5: BR extends uniquely to an N-module with [(Q1, Q2),BR] = 0. Further N preserves the restriction of our forms to BR.

29. T h e action of N on B

C h a p t e r 10 The Griess Algebra a n d the Monster

156

Lemma 29.6: Let x

EX,

b, e , d E L, a E I ', and r ( x , b) E 7Z3.T h e n

and by 29.7 the map

x : B;

[ r ( x ,b), &(d)l = q i ( ~ x ( d+) P ( d ) + C ( d , b)). (2) r ( x , b)*z(") = r ( x , eb e,(e)). (3) r ( x , b ) + ~ (= ~ ~) ( X Qba), , rtza, ba+%(b) + P ( b ) ) for a even, odd, respectively. (1)

+

ProoE Straightforward calculations using 14.2 and the definition of T ( X , b) in 27.8. For example, by 14.2, [$l(d),$l(a,)] = $q(P(d)+e,(d)), [$l(d),Ilrl(s)l = 1, and [$l(d),$l(b)l = $l(C(d,b)), so ( 1 ) holds. In ( 2 ) use the fact that

+

B;,

x @ 6; (b) = L: ( x ) @ 6; (b) H L; ( x ) @ 64 (b) = x @ 6; (b); x : ~ : + ~ ; , X @ 6; (b) = L: ( x ) @ 6; ( b ) I--+ L; ( x ) @ 6;(b

+ ex@)) = x @ 6; (b)

induces an N+-isomorphism B; -, Bi for i = 2,3, and of course this extends uniquely to an Nl-isomorphism. Next

Lemma 29.8: The map Lemma 29.7: (1) The map Xi :Lidl ( x )

Y(.(.,

6:-I (b) H Lt+l ( x )@ 6:"

(b

+ ex (b))

extends

i s an isomorphism of Biz1 @ B i - l ( a i ) with B;$' @ B i + l ( ~ i ) . (2) The stabilizer in N+ of x @ ~ 5 ~ ( consists 1) of the elements such that x a = x and e,(e)

+ P ( e ) +j = 0

mod 2.

Proof: Applying 14.2.1, we may assume i = 2. Observe first that the map is well defined. That is, 6b(b) = 6!j(bt) if and only if b' = b or bs, since 6z(b)kl = 62(bs). Similarly @(b) = 6$(bt) if and only if b' = b or bsx, since 6z(b)k3 = dz(bs?r). But bs ex(bs) = bsn e,(b), so @(bs eZ(bs)) = 6 : ( b s ~ ex (b)) = 623 (b eX(b)),as desired. So ( 1 ) is established. Next by 14.4 each member of N+ is of the form g = $l(d)$2(e)$l(a) for some d , e E L and even a E J?. Then we calculate

+

+

g : L:(x)@6;(b)

-

+

+

+

L~(xcu)@6~((db)a+e,(e)+~(e)+~(db,e)), (29.7.3)

+

g : L: ( x )@ @(b) H L; ( x a ) @ 6$((db)a e,(ed)

+ P ( e ) + C(db,e ) ) . (29.7.4)

This shows the map

xz is N+-equivariant

and establishes (2).

We write x @ 6: (b) for L: (2)@ 6; (b) and identify B;i1 @ Bi-l ( a i ) with ?B'+'(A~) via the map Xi of 29.7 which identifies x @ 6 f 1 ( b ) with x @ ( b Q (6)). We denote this space by B: and let

Bit1

q+l +

x

b)) I--+ L i ( x ) @ 6!(b)

to a n Nl-isomorphism of B; with B;.

Proof: We check first that the map is a well-defined bijection; that is, r ( x ,b) = ~ ( x 'b') , if and only if x = xt and b' = b or bs if and only if L;(x) @ 6t(b) = L;(x') @ 6!(bt) by 14.4.4. Then we observe that if g = $ ~ ~ ( d ) $ ~ ( e ) $ with ~ ( a )a even then using 29.6,

+ + P ( d ) + C ( b ,d ) ) , ~ ( xb)g , = ~ ( x a(eb)a , + e,(ed) + P ( d ) + C(d,b))

g : L;(x) Q9 6 f ( b ) ++ ~ i ( x a@) 6t((eb)a e,(ed)

so x is N+-equivariant. Similarly using 29.6.3, we check that the map commutes with $1(az) for z E X with qhl(az) : x @ 6;(b) = L;(x) 8 6: (b) +-+ L; ( x ) @ 6 f ( b P(b) exz(b)) L: ( x ) @ (b P(b) ex, (b)) = x @ S?(b P ( b ) exz(b)).

+

+

+

+

+

+

BA

By 29.8, our Nl-isomorphism x : -+ B;,i = 2,3, extends to an Nl-isomorphism x : Bo -+ Bo. We pull back the representation of N on Bo via x to obtain a representation of N on Bo which makes x an equivalence. This is our action of N on Bo. Our standard basis for B;, i = 2,3, consists of the elements x @ Ji(d), x E X , d E L, while the standard basis elements for B; are y ( r ( x , b)), x E X , b E L. We write x@Sl (b) for y ( r ( x , b)). This gives us a symmetric notation for the standard basis elements of Bo and the notation makes sense thanks to 29.8.

Lemma 29.9: N preserves the restriction of our forms to Bo. BO = B ~ I B ; ~=@ (B&@B~(A~))~(B~~@B~(A~))~.(B~~@B~(A~)). Proof: First N preserves 7 as it permutes our standard basis up to sign, Then N is naturally represented on Bo via and that basis is orthonormal. To show N preserves P we appeal to 9.8, so we need information about monomials. (29.7.5) g : L;+,(X) 63 b;+l(b) I--+ L ~ + ~ ( @ X ~) :~+ l ( b ) ~ ,

t

Chapter 10 T h e Griess Algebra and the Monster

29. T h e action of N o n 3

By 9.3 applied to Z in the role of the G of 9.3, and using 29.2.4, if

(2)

m = amlm2m3 is a nonzero monomial in @ on Bo then we may take mi E B;. Hence by 28.11, m is one of

(3)

-azayl32 . y(r) - ( x 8 62(d)) . (Y @ Sz(d)r)1 with r E 'R3 and with x # y in the second case. We seek to determine m up to conjugation under N f , so as N f is transitive on 'R3we may take ml = y(r), where r = T ( Z , 1) = &(az). SO by our notational convention, ml = z 8 & ( I ) . Next by 29.7.3, Q: is transitive on basis = (Z3)C*:(&(a,)) and z3 inverts Bi, each vectors x 8 b2(d),so as Q: monomial is conjugate under N+ to either

.

with x # y in the second case. But x @ b2(l)r = x 8 b3(l) by 27.3.4, and E(r) = -Az by 27.8.3, so by 28.11, a% = 3, -1, for x = z , x # z , respectively. Therefore m is conjugate to

where a = -1/32,3/32, -5132 for x , y, z all distinct, two of x , y, z equal, x = y = z , respectively. Let Mo be this set of monomials and M the set of all nonzero monomials of /3 on Bo.By 9.8.2, it remains to show (&(aw), $2(a,)) permutes M o But &(a,) fixes z 8 6 1 ( l ) and interchanges x @ & ( l ) and x @ & ( l ) , while $2 ( a w ) fixes x @ & ( I ) and interchanges z 8 b1( 1 ) and z 8 63 ( I ) , completing the proof. Write Bk for the subspace ( ( x Y ) ~: x , y E X , x # y) of s ~ ( B ; ~ ) , where ( x Y ) ~= L ~ ( x 8 ) L: ( y ) Li ( y ) @ L:(x). Observe that the map g : Lf ( x ) o L: ( x ) g induces a representation of N on

+

BE = B&IB&IB&.

We use the isomorphism of 29.10 to pull back the action of N on BE to an action on BE. Having defined our representation on BE we prove:

Lemma 29.11: N przsemes the restriction of our forms to BE. Proof: The argument is similar to 29.9. Again it is clear the bilinear form is preserved and we appeal t o 9.8 to handle the trilinear form. , a # 0. So let m = amlm2mg be a monomial in our basis ( x Y ) ~with We apply 9.3 to R / E f , recalling from 29.2 that BE C ( E + ) and = c B E ( Q + ) ,and recalling also that R/E+ is of exponent 2 by 14.5. Thus by 9.3 we may assume either

<

(a) mi E (b) mi E

~b for i = 1,2,3, or

EL for some j and all i.

Let Qi(xy) = (E', &(d) : %y(d)) = 0. Then C R ( ( X Y ) I = ) Q~QZ(XY) and CR((uz)2)= Q$ Q I ( u z ) . Thus in case (a) if ml = ( x y ) l and ma = ( u z )then ~ C R ( ( ~ mz)) I , = Q I ( ~ Z ) Q Z ( X YBut ) . C B ~ ( Q ~ ( U ~ ) Q Z (=X Y ) ) R ( x y ) 3 fl R ( u z ) 3 , so by 9.3, uz = XY and mi = ( ~ 9 ) ~ . Similarly in case (b) if ml = ( x y ) l and m2 = ( u z ) l then whose centralizer on B& is (ml,ma)unless y = u and x # z , where we . by 9.3 either m3 = ml, or m2 or ml = ( X Y ) ~ , get (ml,ma, ( Z X ) ~ ) Hence mz = ( y z ) l ,and m3 = ( Z X ) ~ But . thereexists e E L with Q2(e)inverting ml (and centralizing m2 if { x , y) n{u,z } = 0 )so 9.3.2 says ml = ( x y ) l , m2 = (yz)2, and mg = ( Z X ) ~ . Now in case ( a ) by 28.2 we have monomials where v(1) = v and v(-1) = u. By 27.8, ((rv(,)(xy)) = 2 ( f x - e f y ) so by 28.3,

Lemma 29.10: Let g = $l(d)$2(e)$l(a) E N+. Then

+

BL

(xY,J(~V(,)(XY)) @ E ( r v ( e ) ( ~ ~. )v)()~ ) x .v(e)xy, y

Of course B; 2 B; as an Nl-module. Indeed

(1) 9 : ( x Y ) I~ + ( - ~ ) ' ( ~ ) ( X C N Y C X and )~ g : wig i = 2,3, where ~ ( g=) exy(d) icxy(e).

Proof:We use 27.4 to calculate the action of g on B& and 29.4 to make for i = 2,3. This gives ( I ) , and (2) is obtained the calculation on similarly. Then ( 1 ) and (2) and 29.4.4 imply (3).

BL

-axaar132. ( z 8 6 1 ( 1 ) ) ( x 8 & ( I ) ) (Y 8 6 2 ( l ) r )

--

has cycle ( ( ~ ~ 1( 2~, ~ 1and 3 )fie8 ( x Y ) while ~ $ 3 ( 4 &es ( ~ ~ and 1 3 has cycle ( ( x Y ) ~( x, Y ) ~ ) . The map x y I+ ( x y ) l , w&, H ( x Y ) ~i ,= 2,3, is an Nl-isomorphism of BE with B E .

$1

++

(-1)'(9)w&,,

for

160

29. The action of N on B

Chapter 10 The Griess Algebra and the Monster

so R/CR(BzVc) r 21+12. As dim(Bz,,) = 26 is the minimal dimension of a faithful module for 2'+12, (4) holds. Finally by (1)-(4), Bz is an induced module xN1, where x is the representation of N1,, whose kernel K = Cr(~)CR(BZ,c)is Nc-invariant and such that N1,,/K E ~ ~ / with 2 x~the +unique ~ faithful ~ irreducible representation for F*(Nl,,/K) = 21+12. Then x is Nc-invariant, so it extends to X N ~and then X N l extends to X{c. Thus (5) is established.

Therefore (xy, wZy, w&) = -8, so we have monomials

in case (a). Next in case (b) with i = 1 by 28.7 we have monomials

BL,

161

in while if i = 2 , 3 the relevant monomials in BT are Y(T)Y(T')~(TT') for r,rl,rr' E Further in our case r = rv(,)(xy) and r' = r,,(6)(y~). Now r21(C) (ZY)T,,(~)(YZ) = rv(,6) (zx), SO we get monomials

z4.

We use 29.13.5 to define the action of N on BZ.Thus we have defined

N on each of the summands of B of 29.1, and hence have embedded N in GL(B). We close this section by proving:

Lemma 29.14: N preserves the restriction of our forms to BZ. Hence permutes the monomials in (k*), Finally by 29.10.2, ($l(a,), $J~(cY,)) k = 1,2,3, so 9.8 completes the proof.

Lemma 29.12: Let c E L with 4(c) a E I?+. nLen

E

C, J Z: 4(c), e,d

E

L, and

(1) r(c, ~ ) * l ( = ~ )r ( m , J a ) . (2) r(c, J)*'(~) = r ( c + ~ ~ ( d ) + C ( c , d ) , J ) . (3) r ( c , ~ ) * ~ ( ~ ) = r ( c + C ( e , c ) , J + ( c n e ) ) . Proof: Calculate using 14.2 and the definition of T(C,J) in 27.8. In making the calculation in (3), recall $2 ( T ) = $1 (s) and $3 (T) = $1 (ST). This reduces us to a yerification that I J (4(c) n 4(e))1/2 = 15112 C(c, e) I J n 4(e)l mod 2, which follows as C(c, e) = I4(c) n 4(e)l/2 mod 2 and I$(c) n 4(e) n.JI = I J n 4(e)l since J S $(c).

+

+

Lemma 29.13: (1) B z =

+

eCEc B Z , ~where ,

B z , ~= (y(r(cl, J)) : J G c = +(cl)). (2) B z , ~= CBZ(E:), where E$ = (2, &(a J) : c~ ( c = ) 0). (3) Cr(Bz,c) = Cr(c) and Nr+(c)lCr(c) 2 4 3 . (4) cR(BZ,c) =E$ ($1 ( 4 4 2 (e) :enc, dncE (0, c)) andR/CR(Bz,c) 2'+12 is irreducible on Bz,,. (5) Bz extends to an N-module in which Q~ acts on Bz,, for each CEC. Proof: Part (1) is trivial while 29.12 implies (2) and the first statement of (3) and (4). Then 19.1 implies the second statement in (3). Similarly by 14.5 and the first part of (4), R/CR(Bzc) is of order 213 and the product of the elementary abelian groups Q~CR(BZ,~)/CR(BZ,~) is of order 2', while by 14.2.10, Z(R/CR(BZ,c)) = E+CR(BZ,~)/CR(BZ,~),

Proof: As B is the orthogonal direct sum of the four summands of 29.1 and we have shown N preserves the restriction of 7 to the summands BD,D = R, E, 0, it follows that N preserves +y on B if it preserves its restriction to BZ. We prove N preserves this restriction by proving N+ is irreducible on Bz and hence, up to a scalar multiple, 7 is the unique bilinear form on Bz preserved by N+. Hence N preserves +y up to a scalar, so as N = (N?') and Nl preserves 7, so does N. We use the same argument to show N preserves p on Bz; that is, we show that, up to a scalar, p is the unique N+-invariant trilinear form with a certain property and the image of fi under each g E N has that property. So it remains to establish these claims. Let Y = {y(r) : r E 7Z2) so that Y is a signed basis for B z with y(rzl) = -y(r). Now for r E Z(Cq; (r)) = (r, 2) and from 27.2, i((?, 2))nii; = {r'), so the weight spaces of Qf on Bz are 1-dimensional and hence and are ? = {Ry : y E Y). Thus as N+ is transitive on z2, also on ?, N+ is irreducible on Bz as claimed. Next let q # 0 be an N+-invariant trilinear form on BZ and m = amlm2mg be a nonzero monomial of q in the basis Y. Let mi = y(ri). Then mi E BztCi for some ci E C, and by 29.13.2 and 9.3, ci # c j for i # j. Further by 9.3, ra E Z(CQ: ((rl, r2))) = g(rl, r2), so since we just showed that 7Z2 n Z(ri) =- {_ri,TiE1) and c3 # cl or c2, it follows that r3 E Zrlr2. Then C(f3) E J(ZflP2) n i ; , which implies cl +c2 = c3 and either (i) {cl, c2, c3) is a trio and r'lr'2 = ?3S, or (ii) lcl n c21 = 4 and FlF2 = Y3. By 28.8, ,LIZ = PB, has no monomials of type (i). Further N+ is

z2,

Chapter 10 T h e Griess Algebra and the Monster

30. N preseives the Griess algebra

transitive on triples (rl,r2,r3) with rlr2 = ~ 3 so, up to a scalar PZ is the unique N+-invariant trilinear form such that if xi E B z I G with /3z(xl,x2, x3) # 0 then cl + c2 = c3 and lcl n c21 = 4. But by 29.13.5, N permutes the subspaces Bz,,, c E C , so N permutes such forms, completing the proof.

K, K'. As g preserves 7 and { y ( K ) : K E K ) is orthonormal of order 64, it follows that aK = f118 for all K E EC. That is, (1) is established.

162

30. N preserves the Griess algebra In the last section we constructed a representation of N on B preserving the decomposition of 29.1 and the bilinear form y on B. In this section we prove N preserves the trilinear form p and hence also the Griess algebra. Now by Remark 9.2, JeI3/S3 where I = {R, ElZ,0) and I ~ /isssome ~ set of representatives for the = I x I x I . Further N preserves 3 / if and only if orbits of S3 on it preserves PJ for each J E 1 ~ 1 ~ 3We . have shown that N preserves the restriction of p to BD for each D E I; that is, N preserves pJ for J = (DlDl D ) . I t remains to treat the other projections PJ. Ram the discussion in Section 29, we have a standard basis for B with respect to N. Namely the basis for B R consists of the vectors x2, x E X , the basis for B; consists of the vectors ( x Y ) ~x,, y E X distinct, the basis for B z consists of the vectors y(r), T E 'R2, and the basis for B; consists of the vectors x @I bi(d), x E X , d E L. Our subbases are monomial basis for N+ and, except for B z , even monomial bases for N. We form the monomials for /3 from the standard basis.

Lemma 30.1: For c

E

L with $(c)

E

C, J

$(c) of even order, and

XEX:

(1) y ( r ( ~J ,) ) * ' ( ~ = = ) 1 / 8 -CKGc tK(J)y('(c, K ) ) with t K ( J ) = f1 and the sum over any set K of 64 representatives for the even subsets K of $(c) modulo the elation K K + 4(c). (2) t K ( 0 ) = ( 1 ) ~ (for~ )each K E K . N

Proof: Let y ( J ) = y(r(c, J ) ) , g = &(a,), and x ( J ) = y(J)g. BY 29.13.5, g acts on Bq4(,), so as { R y ( J ) : J E K} is the set of weight spaces of Q: on BZ,@(,) and ( 9 ? ) 9 = Qzf, { x ( J ) : J E K ) is the set of weight spaces for Q$ on BZ,+(~). Next x ( J ) = CKEK a K y ( K ) and as Q; is transitive on the Q: weight l all spaces and permutes (fy ( J ) : J E K ) , we conclude aK = fa ~ for

163

Next by 29.12, R Y ( Q ) = c~~,+(,)(Qi(@)), where

so &;(a) = ( ~ t ) g= ~ , f ( + ~ (:eC)( c ,e ) = 0 ) by 14.2.5, and &(a) = c ~ ~ , + ~ ~ ) ( Q 2But ( 0 )by) .29.12, Y ( K ) $ ~ ( ' )= ( - 1 ) C ( e ' C ) 3 (+ ~ ($('I n e E L with C(c,e) = 0, # ( e ) ) ) ,so t K ( 0 ) = tK+(+(c)n4(e))(0)for since we just showed $2(e) fixes x ( 0 ) for each such e. Now C ( c ,e ) = I$(c) n d(e)1/2,so we conclude that the subgroup ($(c)n $(e) :C ( e ,c ) = 0 ) = { K : K c q5(c),(KI even), and hence that t = tK( 0 )is independent of K . and y = y ( 0 ) . Then by 14.2 and the definition of Let h = T ( C , K ) in 27.8, y ( K ) h = (-1)IK1/2+'z(c)y(~).Therefore yh = by, where b = (-l)'*(~ and ) ygh = t / 8 CKy ( K ) h = t b / 8 . C Kt K y ( K ) , where tK = ( - 1 ) 1 ~ 1 / ~Therefore . ( y ,yg) = t / 8 while

-

as there are 36, 28 choices for K with IKI = 0 , 2 mod 4, respectively. Finally h9 = gh = qh2(aX),so b/8 = (yg,ygh) = ( y ,yhg) = ( y ,ygh) = (yh,yhg) = ( y ,yg) = t / 8 as yh = by. That is, t = b.

Lemma 30.2: N preserves PJ for any J E I ~ / containing s ~ an entry R. Proof: Assume m = amlm2mg is a nonzero monomial in PJ in our standard basis with ml E BR. Then ml = x2 for some x E X. As [R,BR] = 0 we conclude from 9.3 that C R ( m 2 ) = CR(m3) and then conclude from 29.2 that J = (R, Dl D ) for some D. As N preserves the restriction of p to R, we may assume y = ma $ BR. Assume y E BZ.Then y = y ( r ) for some r = r(c, J ) E 'R2, and by 28.9 and 28.3, y = m3 and a = ( x , ~ ( r )=) ~a 3 8 = ~,(c)/2. Let g = $3(a,). Then x2g = x2 and by 30.1, yg = 118. '&t K ( J ) y ( K ) ,so (x2g,yg, yg) = 1 / 6 4 . C K ( X Y (~K,)Y, ( K ) )= 4 4 1 2 = (x2,Y , Y ) . That is, N = (9,N l ) preserves P j , as desired. If i = 1 then y = bc for distinct b, c E X Assume next that y E and by 28.7, y = m 3 and a = 4eZ(bc). On the other hand if i # 1 then Y = ( b ~=) w~i,and by 28.9 and 28.3, P ( x ~ , v ~ = ~ a218 , v ~=~2)~ z ( b c= ) P ( x ~ , u ~ , , u ~ , )SO, a = 4ez(bc). Hence as g permutes ( b ~by ) ~29.10.2, N preserves PJ in this case too. This leaves the case y E B;. Then y = u @ 4 ( d ) for some u E X,

~ k .

164

Chapter 10 The Griess Algebra and the Monster

30. N preserves the Griess algebra

d f L. If i = 1 then from 29.8, y = y(r), where r = r(u,d) € 7Z3 and by 28.3 and 28.9, y = ma and a = ag/8 = 918, 118 for x = u, x # u, respectively. On the other hand if i # 1then we obtain the same answer from 28.10. This time using 27.3 and 27.4, we calculate the action of g on Bo and using the N-equivariant isomorphism x :Bo -+ Bo of Section 29, we see that g permutes our monomials, completing the proof.

Lemma 30.3: N preserves PJ for any J E I

~ /containing s ~ an entry E.

Proof: Again take m = a m l m ~ m gto be a nonzero monomial, this time with m l E Bb. We have handled the case where each mi is in BE, so we may assume m2 4 BE. As m l E BE, [ml, E+] = 0, so by 9.3, CE+ (m2) = CE+(m3). In particular if m2 E BE then so is m3, contrary to the choice of m3. Thus neither m2 nor m3 is in BE. Similarly if ma E BZ so is m3. That is, m2,m3 E BD for D = Z or D = 0. So it remains to show N+ preserves the restriction fl of 0 to BE x BD x BD. Next N = (Nl,g), where g = q3(az) for some z f X, so as Nl preserves p, it remains to show g preserves p. Write for the restriction C2, of p to B$ x BD x BD.It sufficesto show g preserves C3 and C = as p = 6 C. We save the proof that g preserves C3 till the end. Notice Clg E L(B& BD, Bo; F) is N+-invariant and to show g preserves C we must show Clg = (2. In particular we may take ml E B;. Thus ml = xy for some choice of distinct x, y E X, and by 29.10.2, m19 = ( x Y ) = ~ w;~. Let p E L ( B ~BD, , BD; F) be N+-invariant and bmln2n3 be a nonzero monomial in p. Suppose D = Z. Then ni = y(ri) with ri = r ( q , J i ) E 7Z2, and as CE+(n2) = CE+(n3), 4(c2) = 4(c3) by 29.13.2. So replacing n3 by -n3 if necessary, we may assume c2 = c3 = c. Also xy E CCB(~:), so CQ:(n2) = CQ:(n3) and hence by 29.12.2, n2 = ng = y(r), where r = r(c, J). If {x, y) is not contained in 4(c) we can pick e E L with +(e) n $(c) = 0 and #(e) containing exactly one of x, y. Then by 27.4, $JZ(e)inverts xy while by 29.12.3, q2(e) fixes n2, contradicting 9.3. Thus x, y E #(c). We saw during the proof of 30.1 that Q$ is transitive on the Q: weight spaces in Bz,,(,), so as Q: fixes x and Y, we may take J = 0 and p is determined up to a scalar. In particular Clg E R42, so it remains to show a = (xy, n2, n2) = (wzy,nag, n2g). As ri =T(c,D), f(ri) = fd(,) by 27.8. So by28.9and 28.3, a = 1. By 30.1.2, n2g = -1/8.CK r(c, K). Further if we set v(1) = v, vxy(l) = vxy and v(0) = u, vxy(0) = uxyr then using the definitions in 27.8, we calculate

ci

+

+

completing the proof in this case. So take D = 0. By 9.3, CZ(m2) = Cz(rn3), so m2,m3 E ~ g for k the + -$+ where is the restriction of p to same k. Hence S, = 82 x B; x B;. Therefore it suffices to show = for j = 1,2, and = Similarly we get maps E L(B' ,B;,B;: F) associated to p. Suppose j = 1. Then ni = y(ri), ri E,'R and Cq:(n2) = Cq:(n3), so f 2 = Y3 and hence we may take n2 = n3. Fhrther by 29.6, N N + ( k y ) has two orbits on the weight spaces of Q: on B: with representatives q QD S1(l) = y(r(q, I)), where g = x or q # x, y, respectively. Now as r = r(q, 1), c(r) = -Aq by 27.8. Then by 28.9 and 28.3, b = -314, 114, for q = x, q # x, y, respectively. Thus = @ if (w;~,(q 8 S1(l))g, (q @ Sl (1))g) = -314, 114, for the respective q. Now using 27.3 and 27.4 we calculate the action of g on Bo and use the N-equivariant isomorphism x : Bo -+ Bo to check that (q 8 S1(l))9 = q QD &(I). Also by 28.11, (vxy(e),q €3 &(I), q @ &(I)) = (1 - ai/4)/8 = -318, 1/8, for the respective q, and hence (wzy,n29, nzg) = -314, 114, as desired. So take j = 2 or 3. By 27.4.1, +j(e) inverts xy if exy(e) = 1. Then from 29.7.3, there exists e E L such that +j(e) inverts xy and centralizes q @ 6j(d) and q' @ Sj(dl) unless {q,q') = {x, y). Thus each nonzero monomial of is conjugate under NN+(Rzy) to bxy xy (X @ Sj(1)) . (y @ dj(l)), so to show = and <;g = it remains to show ( x ~ , x @ b j ( l~QD6j(l)) ), = (wZy1~ @ 6 k ( ~ ) (Y1@6k(j)(l))1 )1 where k(2) = 1 and k(3) = 3. Now by 28.10, (xy, x 8 Sj(l), y @ Sj (1)) = 1. Further x @ 62(l) = y(r(x, 1))and r(x, l)r(y, 1) = r,(xy), so 28.8 gives the desired equality when j = 2. So take j = 3. Now from 27.3, S3(l)r,(4(xy) = a3(s1-'), so y @ 63(l)rv(4(xy) = (-l)Lfl(y @ 63(1)). Then by 28.11, (vxy(e),x 8 &(I), y 69 b3(1)) = 112, so (w?,, x 8 ~ ~ ( 1y@b3(1)) ), = 1,completing the proof that Clg = <2. Finally we show g preserves c3. Set h = q2(crz) and k = gl(aZ). Then h = gk. As 61 is Nl-invariant and N! = N3, it suffices to show Clh = b. But h = = Clgk = c21c = C3, completing the proof.

el

(tg (i.



8,

el

~ 1 2 ~ ~2 1

6

- -

clgk

Lemma 30.4: N preserves the Griess algebra. Proof: As we observed earlier, to show N preserves the Griess algebra, We~may . assume it suffices to show N preserves p j for each J E I ~ / s

Chapter 10 The Griess Algebra and the Monster

$1. The automorphism group of the Griess algebra

J # (DlDl D), so by 30.2 and 30.3, it remains to consider the cases J = (Z,Z, 0) or (Z,0, 0). In particular we need only consider nonzero monomials m = amlm~mgwith ml E Bz. Then by 9.3, Cz(m2) = Cz(m3), so J = (Z,0,0) and mp, m3 E B& for some i. If we argue as in the proof of 30.3, it suffices to show Cig = 52 and C3g = (3, where Ci is the restriction of p to Bz x B: x B: and g = 1+h3(a,). Let pi E L(Bz, B;, B:; F) be N+-invariant and n = bmln2n3 a nonzero monomial in pi. Conjugating in NS we may take ml = y(r) for r = T(C,0 ) and z 4 #(c). Then mlg = 1/8 CKy(r(c, K)) by 30.1.2. , E R~ for j = Suppose i = 1. Then n j = y(rj) for some r j = T ( x ~bj) 2,3. Now unless r E Z(r2,r3), there exists h E CQ: ((r2,r3))- Cg:(r). But then h fixes n2 and n3 and inverts ml, contrary to 9.3. Therefore r E Z(r2,rg), which forces r E Zrlr2. Recall that if p = C then by 28.8, T = T2T3. Now by Exercise 10.1.6, CNt (ml) has two orbits on pairs (r2,r3) such that rr2 = r3 E 7Z3,with representatives (r(x, l),r(x,c)), x 4 $J(c),and (r(x,d),~(y,Cd)),d E C with I$(c) n $J(d)l = 2 and xy = $(c) n 4(d). By 28.8, (ml, n2,n3) = 1 when rr2 = r3. Similarly if r2r3 = rZ2 then by Exercise 10.1.7, up to conjugation in N + , r2 = T(X,1) and r3 = r(x, cs) with x E #(c). Now Clg is an N f -invariant form so by the previous paragraphs to show 51g = Z2, it sufficesto show p2(mlg,n2g,n3g) = 1, 1, 0 for the three choices of n2,n3 above. Now using 27.3 and 27.4, y(r(x, d))g = (x 8 6f(d))g = x @ 6; (d P(d) ezx (d)). Thus if r2, r3 = r (x, I), r (2, c) with x 4 #(c), then

Finally take r2, r3 = r(x, 1), r(x, cs) with x E $J(c). This time our terms are of the form

166

+

+

-1/8

167

- (y(r(c, K)), x 8 6(l),a: 8 6(c))

since S2(cs) = b2(c). Now as x E $(c) each such term is 0 by 28.11. Thus we have shown that clg = C2, so it remains only to show g and k = $ J ~ ( ( Y Then ~ ) . h = gk. So as 51 preserves 53. Set h = qp(aZ) is Nl-invariant and N: = N3, it suffice to show Clh = 53. But Clh = = Clgk = C2k = 53, completing the proof.

clgk

31. The automorphism group

of the Griess algebra In this section G = O(B, 0, y) denotes the group of isometries of the bilinear form y and the trilinear form P. In particular G is also a group of automorphiims of the Griess algebra. Using the convention of Section 28 for regarding elements of BS as symmetric matrices indexed by X, we write I for the element of BS corresponding to the identity matrix. Observe first from the definition of 7 in Section 28 that:

Lemma 31.1: The element id = 1/4 of B, is the identity for the Griess algebra. In particular G f i e s id and acts on the subspace B = idL orthogonal to id. Observe that B = B- LB,IB,,

where

B ~ = ( X ~ , X ~ - ~ ~ : xZ# y, )~. E X ,

Lemma 31.2: CG(zl) acts on Bi for each i E (r, s, -).

+

and as 6;(b)r(c, K) = 6;(cb eK(cb)) by 27.3, it follows from 28.11 that as x 4 4(c), each term in this sum is 118, so the sum is indeed 1 as desired. Next take r2, r g = r(x, d), r(y, cd). Then this time we get a sum of 64 terms

4 28.11, and as xy = #(c) C!#(d) such a term is equal to ( - 1 ) ~ ( ~ ) / 6by where s(K) = 1 ezz(c) ~ ~ ( c d eK(xy). ) Further s(K) = 0, since x f $J(c), z 4 $J(c), and e~(Cd)= eK(xy). SOagain we get 1 as desired.

+

+

+

Proof: Let H = CG(zl). Certainly H acts on B- = [B, zl] and B+ = CB(zl) = BslBT (cf. 27.9). Then H acts on B+ = B f l B+ and as C is absolutely irreducible on BT and B ~with , each space the orthogonal complement in B+ of the other, we may assume H is absolutely irreducible on B+. In particular, up to a scalar multiple, the restriction .j.+ of y to B+ is the unique quadratic form on B+ preserved by H. - - Next by 9.9.4, H acts on End(B-) via conjugation; that is, for g E H and p E End(B-), g : p H p9, where 4(b) = p(bg-l)g for b E B-. Further for a E B+, it follows from 9.9.5 that A, E End(B-), where A, : b I+ b * a, and the map a H Xa is in Hom(B+, End(B-)). Also by 9.9.4, Ag, = A,,.

168

Chapter 10 The Griess Algebra and the Monster Define T E L ~ ( B + )by T(a, c) = TT(X,A,). Then

91. The a u t o m o ~ h i s mgmup of the Griess algebra

169

that CH(Bs) acts on B- via (x @ 6)g = x @ 6g for some representation g : 6 H 69 of CH(Bs) on B;. Similarly by 28.6, for r E R:

so H preserves T. Hence by the uniqueness of ?+, T = pS.+ for some p E R. Let xi, i = 1,2, be distinct elements of X and p = A,,,. Then by 28.10, p(y @ 6) = x3-i 8 6 if y = xi and 0 otherwise. Hence

so as 7(x1x2, xlx2) = 2, we conclude p = 212. On the other hand let T E R and p = Xy(,). Then by 28.6:

Further by 28.3, (X,<(T))= a , / d and (<(r),t(r)) = C x a $ / 8 = 4, so the second term in (*) has projection 3/2 az/64 on x @ 6. Hence ~ ) (*) over our standard basis summing the contributions to T T ( ~ from for B- we get

.

But for g E CH(Bs), we have observed that y(r)g = G(g)y(r) for some ~ ( g E) R, so Ay(T)(x@6)g= ((x@~)*Y(~))s = (X@~)S*Y(T)S = (x@6g)* c,.(g)y(r) = +(g)Xy(,)(x €4 69). Pick x with v = x - 2(x, t(r))t(T) # 0. Then by (*), (u@Erg)/8 = (v@Sr)g/8 = @6)g= ~ r ( g ) A ~ ( ~ ) ( x @ 6g) = q.(y)(v @ 6gr)/8. As u # 0 we conclude [T,g] induces a scalar on B? and then conclude the image P(CH(Bs)) of CH(Bs) in PGL(B?) commutes with the image P ( r ) of r. So as Q1 is generated by R , P(CH(Bs)) 5 c ~ ~ ~ ( ~ ~ ) ( P=(p(Q1) Q ~ ) since ) Q1 is irreducible on ~ 1Therefore . CH(Bs) I Q ~ D ,where D is the subgroup inducing scalars on B-. Finally as D preserves y, D = (zl)CH(B-). Thus we have shown CH(Bs) Q I C H ( ~ - B,). Then as CH (Bs) acts on Ry(r) for each T E R,[ c ~ ( B ~ ) , Q ~CH(B+), ] so CH(BS) = gl. In particular Q1 9 H. Further as Cal(r) = CQ, (y(r)), and H permutes {RY(T) : r E R), H permutes R by conjugation. So as Col is the subgroup of Out(Q1) permuting R , our proof is complete.

<

+ <

Lemma 31.4: G is finite.

= 212 x ( l + 312 . az)/64 = 26(24 + 48) = 9 - 2'. x

So as y(x 8 6, x 8 6) = 1, p = 9 2', a contradiction. L e m m a 31.3: C = CG(zl).

-

Proof: Let H = CG(z~).For (j,i) E {(T,-), (s,-), (s,r)), and b E Bjl let A; = Ab.€ End(Bi) be defined by Ab : a b* a as in 9.9.5, and recall Ai : b o A; is in End(Bj, End(Bi)). Further by 9.9, H acts on X i ( ~ j ) via conjugation by ( ~ k ) g= A& for g E H. Now for x, y E X, T E R , 28.9 and 28.3 say

-

&Y(Y(T))= (XY,E(T)@ t ( r ) ) ~ ( = ~ )a z a y y ( ~ ) / 42'=(~). In particular we conclude that {Ry(r) : T E R ) is the set of weight spaces for Xr(Bs), and hence as H acts on XT(Bs), H permutes these weight spaces. Moreover we see that CH(Bs) fixes each weight space. Next we saw during the proof of the previous lemma that for distinct x, y E X , X i V interchanges x @ 6 and y @ 6 and annihilates u @ 6 for all u E X-{x, y). So as CH(Bs) commutes with all these maps, we conclude

Proof: Consider the stabilizer of the forms P and 7 on B = B @R C. It suffices to show is finite. The proof of the previous lemma applies equally well to to show C = CG(zl). As the stabilizer of the forms /3 and 7, is an algebraic group. As C is irreducible on B, G is reductive. Thus if the connected component GO # 1 then by a standard result from the theory of reductive algebraic groups, CG(zl) is infinite, a contradiction. Thus Go = 1; that is, G is finite.

Lemma 31.5: G is simple.

Proof: By 31.3 and 27.6, Q1 = F*(C) is extraspecial. Further zl is fused to 22 in N and into C - Q1 under (C, N), so the result follows from Exercise 2.4. Remarks. We define the simple group G constructed in this section to be the Monster. By construction it satisfies Hypothesis H(Col,l2), and indeed is even of Monster type in the sense of the next chapter. Griess constructed a 196,883-dimensional algebra and the Monster in 1980; see [Grl]. Then in [Gr2], Griess adjoined an identity to obtain the 196,884dimensional Griess algebra. Later Conway [Co3] and Tits [T2] supplied simplifications of some of Griess' arguments, which we

170

Exercises

Chapter 10 The Griess Algebra and the Monster have incorporated in our treatment. In the remainder of this subsection we briefly summarize the approaches of Griess, Conway, and Tits, and compare them with the approach adopted here. Early in the study of the Monster it was determined that the smallest possible dimension for a nontrivial irreducible C-module for the Monster is 196,883. Moreover a calculation by Norton showed that such a module would admit an algebra structure invariant under the Monster. To obtain an identity for this algebra, we add an extra dimension. Let C be the centralizer of an involution z in a potential Monster. Griess determined the 196,884dimensional RC-module B which has a chance of extending to the Monster, and the algebra structures on B preserved by C. From these structures he picked one which can extend to the Monster and guessed the action of an extra automorphism u of the algebra not contained in C. This treatment is a tour de force, but very long and complicated. Let G = (C, u ) . To identify G as the Monster, Griess observes that G preserves a Z[1/6] lattice in B closed under the algebra multiplication. He then reduces modulo p for each prime p > 3 to obtain a GF(p)algebra B(p) and a representation ~ ( p :) G -+ G(p) 5 O(B(p)). He proves C C?r(p) is the centralizer of z?r(p) in G(p), so G(p) satisfies Hypothesis 31(12,Col) of the introduction. Then Griess appeals to a theorem of S. Smith [Sm] which determines the order of a group satisfying H(12, Col), to conclude g = IG(p)l is independent of p. Hence IGl = g, so G is finite and satisfies H(12, Col). Conway had the tremendous advantage of knowing the facts about B that Griess generated. This knowledge led Conway to a wonderful, inspired construction. He begins with the Parker loop L and uses L to construct the group N and its factor group N discussed here in Sections 14 and 27. He then extends N~ = N n C to C and uses lVl to construct an RC-module B and an algebra structure on B which is visibly preserved by C. Next he restricts this algebra to Nl, chooses an Nl-monomial basis XI, for B, and writes down bases X2 and X3 of B and bijections x i I+ 2 2 I+ 23 among the three bases. Conway refers to thiis set up as a dictionary. Then he sketches a proof that the bijections define isometries of B induced in N. Therefore N also preserves B. Finally Conway exhibits a vector t E B whose orbit under O(B) is finite. Hence O(B) is finite and Conway concludes by proving C = c~(~)(z)* Tits begins with C and the CC-module C @R B of Griess, which we also denote by B. He writes down algebra structures on B preserved by

171

C and proves that for certain of these algebras, CO(B)(~)= C. He then proves O ( B ) is reductive as an algebraic group, and hence as CO(B)(~) is finite, O(B) is finite. He does not prove C is proper in O(B). Our argument follows that of Conway initially, in that we use L to construct N, we use Nl to construct C and the action of C on B, and we take Conway's definition of the algebra structure T on B. At thiis point (in Section 28) we transfer emphasis to the trilinear form p determined by T and the bilinear form 7, and calculate the monomials of /3 on our standard basis. Then in Section 29 we calculate the action of Nl on our basis and extend the Nl-module B to an N-module in such a way that N preserves 7. Further in Section 30 we check that N preserves 0 and hence also T . Finally in Section 31 we use Tits's approach to prove C = CG(zl) and G = O(B,P, 7)is finite. Our theory of large extraspecial subgroups then immediately says G is simple.

Exercises 1. Let GJ E L with 4(cg) = c E C, r = r(cg,@) E 7Z2, u = f, E A, w = fX - v, and u E A.: Prove . (1) ii G E A2 if and only if u E Az(v, *2). :ii C E A:), with representatives (2) Ni, has two orbits on {ii E u = -XI, x $ C, and u = -Ax fd+Xl x E c n dl d E C2(c). (3) w E A4 with coordinate frame fx+, - 2fd, d E V , d f7 c = 0, and +(fx+c 7 2Xd1 x E c. (4) ii 271 E A2 if and only if (u,w) = 0 or f32, and (u, wo) = -32 for some wo in the coordinate frame of w. (5) N* is transitive on {ii E A$ : 2ij ii E A)! with representative u = --Ax, x E c. ~ (6) CN+(T) has two orbits on pairs (rZ,rg) E 7Z3 x 7Z3 such that T T = rg, with representatives ( ~ ( x1), , T(x,Q)), x $ c, and (r(x,d), ~(YlGJd))14(d)E C2(c)1 {x, Y) = c n 4@). (7) CN+(T) is transitive on pairs (r2,r3) such that ~ 2 =~rZ2, 3 with I), r(x, cgs)), x E C. representatives (~(x,

+

+

+

+

+

173

32. Subgroups of groups of Monster type 32. Subgroups of groups of Monster type

Chapter 11 Subgroups of Groups of Monster Type

In Chapter 10 we constructed a finite simple group G possessing an involution z such that F*(CG(z)) = Q is extraspecial of order 21+24, CG(z)/Q Col, Q/(z) is isomorphic to the Leech lattice modulo 2 as a CG(z)/Q-module, and z is not weakly closed in Q with respect to G. We say that a group G satisfying these hypotheses is of Monster type. In this short chapter we investigate groups of Monster type. In particular we see that such a group contains simple subgroups of type Fp, for p = 2, 3, 5, 7, and 24. See Section 32 for the definition of groups of type Fp. Now from Chapter 5, there are twenty-six sporadic groups. In C h a p ter 6 we constructed the five Mathieu groups. In Chapters 8 and 9 we constructed the three Conway groups plus Suz, J2, HS, and Mc. Hence each of these twelve sporadic groups is a section of the Monster. Similarly the sporadic groups Fl, Fz, F3, and F5 are of type Fp, and hence sections of the Monster. Held's group H e is of type F7 and the largest Fischer group Fz4 = M(24)' is of type F24, so these groups are sections of the Monster. Finally the Fischer groups M(22) = Fz2and M(23) = F23 are sections of F24, and hence also of the Monster. This is best seen by viewing Aut(Fz4) = M(24) as a 3-transposition group. Thus we have existence proofs for twenty of the twenty-six sporadic groups.

Define a finite group G to be of Monster type if G possesses an involution z such that Q = F*(CG(z)) E 21+24, CG(z)/Q Col, Q/(z) is isomorphic to the Leech lattice modulo 2 as a CG(z)/Q-module, and z is not weakly closed in Q with respect to G. Similarly define a finite group G to be of type Fp if G possesses an involution z such that F*(G) = Q is extraspecial, z is not weakly closed in Q with respect to G, and (a) p = 2, CG(z)/Q S Coal Q 21+22, and Q/(z) is isomorphic to the subspace vL/(v) of A/(v) as a CG(z)/Q-module, where vE and CG(z)/Q Coz is the stabilizer in Col of v . (b) p = 3, CG(z)/Q Z As, and Q 2 21+8. (c) p = 5, CG(z)/Q S A5wrZ2, and Q 2 2'+'. (d) p = 7, CG(z)/Q E L3(2), Q Z21+6, and IzGnQ1 = 29. (e) p = 24, CG(z)/Q Z2/U4(3)/Z3, and & 2'+12. Throughout this section we assume G is of Monster type. We let H = CG(z), H = H / ( z ) , Q = F*(H), and H* = HI&. By hypothesis we may identify Q with as a GF(2)H*-module. We adopt the notation and terminology of Chapter 8 in discussing this module. Let B(z) = zG n Q - (2). By hypothesis there is zg E 8(z). Let s = zg and U = Q n Q g .

x2

Lemma 32.1: (1) H is transitive on B(z) and g(z) = x 4 . (2) CH(s)* is the split extension of E211 by M24, dim(0) = 12, (Qgn H)*= 02(CH*(5)) induces the group of transvections on 0 with center (S), Q Z LIi2, and o/(S) is isomorphic to the 11-dimensional Todd module for CH(s)*/O2(CH(s)*). Proof: By 23.2 and 8.3.4, H has two orbits on involutions in Q-(z) with representatives ti, i = 2,4, where fi E &. Now the stabilizer in H* of ii is isomorphic to Co2, M 2 4 / E 2 ~for~ i = 2,4, respectively, by 23.1 and 23.2. By 25.8, Co2 is simple, so 02(CH. (i2)) = 1. Thus 8.15.8 and 8.15.7 imply that (1) holds, dim@) = 12, Oz(CH(s))* = (Qg n H)* induces The the full group of transvections on 0 with center (g), and Q Z final remark in (2) follows from 23.10.5.

0i2.

Lemma 32.2: (1) H is transitive on involutions in Q - zG. (2) Let t E Q - zG be an involution. Then CG(t)/(t) is of type 8'2. Proof: We saw during the proof of the previous lemma that (1) holds and that CH(t) 2 Co2/(Z2 x 21+22). Further as the Monster is of Monster type with (using 27.2 and its notation) &(s) in the role of s and E+ in the role of U, and as ~l(+l(s)~l(cu,,)) = 2 L Y E i2with

174

Chapter 11 Subgroups of Groups of Monster Q p e

PART I11

~ ) ~ ( s ) @ ~ ( Ea , E+, ~ ) we may choose t E U. Hence 8.14 completes the proof of (2).

Lemma 32.3: Let B

E

Syl5(CH(s)) Then CG(B)/B is of type F5.

Proof:We appeal to 8.13 and 26.5. Lemma 32.4: Let Z3 Y < CH(s). Then (I) If Y is not 3-central in CH(s) then CG(Y)/Y is of type F3. (2) I f Y is 3-centralinCH(s) thenCG(Y)/Y is of typeF24. Proof: Use 8.13, 26.4, and 26.6. Lemma 32.5: Let B

E

Sy17(CH(s)). Then

(1) CG(B)/B is of type F7. (2) CG(B)/B E He. (3) AutG(B) E+ Z6 and some element of order 2 inverting B induces

Chapter 12 Coverings of Graphs and Simplicia1 Complexes

an outer automorphism on E(CG(B)).

Proof: Use 8.13 and 26.2 to prove (1). Further by 26.2, AutH(B) r Z6 2 Aut(B), so AutG(B) E Z6. Also by 26.2, an involution t E H inverting B induces an outer automorphism on CH(B)/B02(CH(B)) and hence also on CG(B). Next by Exercise 14.5 and by 44.4, CG(B)/B % L5(2), M24, or He. As t induces an outer automorphism on CG(B)/B and Out(M24) = 1 (cf. Exercise 7.5), the second case is out. So to prove that (3) holds, we may assume CG(B)/B G L5(2). Next there is an element y E H of order 3 faithful on B centralizing CH(B)/B. Then as L5(2) contains no such element of order 3, y centralizes E(CG(B)), so 31 divides ICG(y)I.However, CQ(9) = CQ(B)*[CQ(y) ,B] with CQ(B) 2 21+6 and [CQ(y),B] 2 21+6k as [CQ(y),B] is extraspecial (or. (2)) by Exercise 2.2, and as each faithful irreducible GF(2)B-module is of dimension 3. Therefore by 32.4, CG(y)/(y) is of type F24, and hence has order prime to 31. Remarks. The general structure of the local subgroups of the Monster considered in this chapter was first studied in 1973 by the group at Cambridge, particularly Conway, Harada, Norton, and Thompson, and independently by Griess; see the discussion in Chapter 5. While I don't know the specific approach used by the Cambridge mathematicians or Griess (as most of that work remains unpublished), it must have been local group theoretic as no other techniques existed which could make a dent in the problem. Of course our analysis depends upon the theory of large extraspecial subgroups. That theory had not been systematically developed in 1973.

A covering of a graph or simplicial complex K is a surjective morphism d : k + K that is a local isomorphism. A precise definition appears in Section 35. The graph or complex is simply connected if it admits no proper connected coverings. In this chapter we establiish basic results on coverings including criteria for deciding when a graph is simply connected. Then in Chapter 13 we generate machinery for reducing the question of the uniqueness of a group G, subject to suitable hypotheses, to the question of whether certain graphs associated t o the group are simply connected. This machinery is the basis for our uniqueness treatment of the sporadic groups. In Section 33 we consider certain equivalence relations on the set P of all paths in a graph A. We find in Section 35 that these invariant relations correspond to certain fiberings of the graph A; that is, surjective locally bijective morphisms onto A. Among these fiberings are the coverings of A and of the simplicial complexes K with A as the graph or 1-skeleton of K. We also find in Section 35 that each covering corresponds to a local system on the graph or complex. This discussion is broader than is strictly necessary for our purposes, but it has the advantage of putting the concepts in a larger context which hopefully makes them easier to understand. Further the expanded discussion is useful in the study of various simplicial complexes associated to finite groups, such as the Brown and Quillen complexes.

33. The fundamental gmupoid

176 Chapter 12 Coverings of Graphs and Simplicia1 Complexes Finally in Section 34, we investigate criteria for proving that a graph or complex is simply connected. For example, in Section 35 we see that the graph is simply connected if and only if each cycle is a product in the fundamental groupoid of conjugates of triangles. We term a graph with this property to be triangulable. In Section 35 we record various results useful for proving that a graph is triangulable.

33. The fundamental groupoid In this section A is a graph. We direct the reader to Section 3 for our notational conventions and terminology for graphs. In addition let P = P(A) be the set of paths in A. For each p = xo..-xr E P we write org(p), end(p) for the origin xo and end xr of p, respectively. Write p - q or simply pq for the concatenation of paths p and q such that end(p) = org(q). Thus if q = yo y, with yo = x, then pq is the path pq = zo . - - z , + ~of length r s such that zi = xi for 0 i 5 r and Zi = yi-,. for T 5 i 5 T 3. Write for the path x, . .xo. A path p = x0 - .- x, is a cycle or circuit if xr = 20. Define an equivalence relation N on P to be P-invariant if the following four conditions are satisfied:

+

<

-

(PI1) If p N q then org(p) = org(q) and end(p) = end(q). (PI2) rr-l org(r) for all r E P. (PI3) Whenever p N pt and q qt with end(p) = org(q), then also Pq ~'9'. (PI4) x xx for all x E A. Define the kernel of an equivalence relation N on P to be the set ker(-) of all cycles s such that s org(s). Given a set S of cycles of A, define a relation on P by p -s q if p and q have the same origin and end and pq-l E S.

(Cl) (C2) (C3) (C4) (C5)

rr-l E S for all r E P. If p E S then p-l E S. If p, q E S and org(p) = org(q), then pq E S. If p E S then r-Ipr E S for each r E P with org(r) = end(p). If p is a cycle and r E P with org(p) = end(r) and r-lrp E S,

then p E S. (C6) xx E S for all x E A. Lemma 33.2: A set S of cycles of A is closed if and only if NS is a P-invariant equivalence relation on P. Proof:Write for NS. Observe that N is reflexive if and only if (Cl) holds and is symmetric if and only if (C2) holds. Assume S is closed. We must show N is transitive and satisfies (PI1)(PI4). Notice (PI1) is satisfied by definition of while (Cl) implies (PI2) and (C6) implies (PI4). Also N

N

N

(i) If a, b E P with ab E S then ba E S. For as ab E S , a-laba E S by (C4), and then by (C5), ba E S. Next (ii) I f a , b , c ~ P w i t h a b E S a n d c ~ S t h e n a c b € S . Namely by (i), ba E S, so by (C3), bac E S, and then acb f S by another application of (i). Similarly

N

N

N~

Lemma 33.1: Let Nker(w).

N

be a P-invariant equivalence relation. Then

-

=

(iii) If a, b,r E P with arr-lb E S then ab E S.

-

For by (i), rr-lba E S , so by (C5), ba E S, and then ab E S by (i). q q l l E S, and then if end(p) = Now if p pl and q q1 then ppll, -1 -1 org(q), (pq)(plqi)-l = pqq1 pl E S by (ii), so pq plql. That is, (PI3) holds. Finally if p q T then pq-l,qr-l E S, so by (C3), pq-lqr-l € S and hence pr-l E S by (iii). That is, p N r , so is transitive and we have completed the proof that if S is closed then is an invariant equivalence relation. Conversely assume is an invariant equivalence relation. We have seen that (Cl) and (C2) hold. Let p, q E P with end@) = org(q) = X. Then if p, q E S,we have p x q, so by (PI3), pq x.x = x, and hence q-l.x-q x, pq E S. That is, (C3) holds. Also wheng E S then q-lpq so (C4) holds. Notice (PI4) implies (C6), so it remains to establish (C5). But if q-lqp E S then x q-lqp N p by (PI2) and (PI3), so p E S, completing the proof. N

N

N

N

Proof:Let p, q E P , org(p) = x, and S = ker(-). If p q then x = org(q) and end(p) = end(q), so pq-l is defined and pq-I qq-l N x. Thus pq-l E S and hence p NS q. Conversely suppose p -S q. Then pq-l E S, x = org(q), and end(p) = end(q). Now p = p end(p) pq-lq x . q = q. N

N

N

-

Define a subset S of P to be closed if S is a set of cycles satisfying the following six properties:

N

N

N

-

-

178 Chapter 12 Coverings of Graphs and Simplicia1 Complexes The closure ( T ) of a set T of cycles is the intersection of all closed subsets containing T. It is easy to check that the intersection of closed sets is closed using the axioms (C1)-(C6). Thus

Lemma 33.3: The closure ( T ) of any set T of cycles is a closed set. Define the basic relation to be the relation - B ~ ~where , B a s is the . = is the P-invariant smallest closed subset of P. Write = for - B ~ ~Thus equivalence relation generated by the paths of length 0. Notice

Lemma 33.4: = is characterized by the property that if anda-b thena-b.

-

is P-invariant

Given a set S of cycles of A and p = xo .- - xr E P , write Is(p) for the paths obtained by replacing xi with s or s-I for some i and s E S with origin xi. Write DS(p) for the paths obtained by replacing a subpath t of p such that t or t-' is in S by org(t). Let Basl denote the set of all paths x x and x y x with x E A and y E A ( x ) . For p = 2 0 . X , let I ( p ) = I B ~@)~ and , D(p) = DBasl ( p ) . The processes I and D are called insertion and deletion, respectively. Notice D ( p ) is empty if and only if p has no subpaths in Basl. For 0 5 li, ki E Z , define ~ ' l lkn ~ ' n ( precursively ) in the obvious manner. Regard P as a graph by decreeing that p is adjacent to q if q E I ( p ) U D(p). Write b]for the connected component of p in P. The following observation gives a characterization of the basic relation.

-..

Lemma 33.5: p

-

= q if and only i f ; f ] = [q].

-

Proof: Let be the relation p ,., q if and only if M = [q]. Then is an equivalence relation satisfying ( P I l ) , (PI2), and (PI4). If p q there exists a path p = po,.. ,pn = q in P. Then for a, b E P with end(a) = org(p) and org(b) = end(p), we see that ap = a p ~. ,.. ,apn = aq and pb = pob,. .. ,pnb = qb are paths in P , so is P-invariant. To complete the proof we prove that if p q then p zz q, and appeal to 33.4. Proceeding by induction on the distance of p from q in PIit suffices to show that if q E I ( p ) u D(p) then p = q. So as r is P-invariant it suffices to show x v x r x for all x , v E A with v E A(x), which follo-nrs from (C4).

.

N

-

Write @(A)for the set of equivalence classes [p] of the basic relation F. Given a set S of cycles of A, defineI S ( ; f ] )to consist of the classes [a]such that a E Is(b) and b E [p]. Define Ds([p]) analogously. Let Ps be the

179

33. T h e fundamental groupoid

graph on @(A)with [p] adjacent to [q]if and only if [q]E IS([p])uDS(b]). Write bIS for the connected component of b] in PS. For p E (S),define the S-degree degs(p) to be the distance from b]to [erg@)] in PS. Arguing as in the proof of 33.5 we obtain:

Lemma 33.6: Let S be a set of cycles of A. Then p if Ms = [qls.

-(s) q if and only

Lemma 33.7: Let S be a set of cycles and p E ( S ) . Then

.

(1) If degS(p) = 0 then p r org@) (2) If degs(p) = 1 then p E r-ltr for some t E S U S-l and T E P. (3) If degS(p) > 1 then p = cd for some c, d E (S) with degS(c) <

degs(p) > degs(d). Proof:Part ( 1 ) is trivial, so assume n = degs(p) > 0.Then there exists q E (S)of degree n-1 such that either q = ab and p = asb or q = asb and p zz ab for some s E SUS-I. In the first casep r asb abb-Isb = qb-lsb and in the second p qb-ls-lb. But degS(b-lsb) = 1 and if n = 1then q = org(q), so the lemma holds.

Define an inve~sionon a category C to be a contravariant functor I n v : C + C such that I n v ( x ) = x for each object x in C and ~ n =vid. ~ Thus for each morphism f : x + y, I n v ( f ) : y -+ x and ~ n v ~ (=f f ). Define a pregroupoid to be a category C together with an inversion I n v on C . The composition map on C will be termed the pregroupoid product. A morphism of pregroupoids a! : ( C ,I n v ) -+ (C', Inv') is a covariant functor a! : C -+ C' preserving inversion; that is, a ! ( I n v ( f ) )= I n v f ( a ( f ) ) for each morphism f in C. The kernel of the morphism a! consists of those morphiims f : x -+x such that a!(f) = idz for some object x . A groupoid is a category in which each morphism is an isomorphism. Notice a groupoid is a pregroupoid with the natural inversion map Inv : f H f-l. A morphisrn of g~oupoidsis just a covariant functor a : C -+ C',since each such functor preserves the natural inversion. We can make P into a category as follows. The objects of the category P are the vertices of A, the morphisms from a vertex x to a vertex y are the paths from x to y, and the composition of suitable paths p and q is the product p .q. Further we have an inversion I n v : p 4 p-l on P, so P is a pregroupoid. Similarly given any invariant relation on P, write 13 for the equivalence class of p E P under and write for the set of equivalence classes of -. Given 3 E define m g @ ) = org(p) and end@) = end@). This is N

N

180 Chapter 12 Coverings of Graphs and Simplicial Complexes well defined by (PI1). Further if p', 4 E with end(@)= org(6) define fi-4 to be the equivalence class of p .q. This is well defined by (PI3). Define ) to be the equivalence class of p-l. If p q then pq-l E k e r ( ~ so ) hence q-l p-l. Thus is by (C4) and (C5), q-lp E k e r ( ~ and well defined. Finally observe p is a groupoid. Namely the objects of are again the vertices of A, the morphisms from x to x are the equivalence classes fi, for p a path from x to y, and composition is the product in p. Each morphism in p is an isomorphism by (P12). Thus we have essentially shown:

- -

Lemma 33.8: Let be an invariant equivalence relation on A and P = P(A). Then (1) P is a pregmupoid. is a groupoid; in particular @(A) is a groupoid called the fun(2) damental gmupoid. is a surjective morphism of pre(3) The map p ++fi of P onto groupoids which is the identity on the set A of objects and with kernel ker (w) . (4) The map [p] -t @ of @(A) onto @ is a surjective morphism of groupoids which is the identity on the set A of objects. (5) If C is a groupoid with object set A and a : P -r C is a morphism of pregroupoids with kernel S which is the identity on A, then S is a closed subset of P and p NS q if and only if a @ ) = a(q). N

Given an invariant relation N on A and x E A, denote by ji(A,x) the subgroupoid of consisting of all fi such that p is a cycle with origin x. Observe that ii(A,x) is a group under the groupoid product. Write rl(A, x) for iil(A,x) when is the basic relation =. Then rl(A, x) is the fundamental group of the graph A at x. The last three lemmas in this section are not used elsewhere in Spomdic Groups, so they can be skipped if the reader chooses. However, they give a solution to the word problem in the fundamental groupoid, and hence seem worth including.

Lemma 33.9: If p

EP

with D(p)

# 0 then D(I(p)) = I(D(p)).

Proof: Let p = xo ...xr and for s E Basl with xi = org(s), write D(p, i, s) for the path obtained by replacing the subpath s of p beginning at X i by xi, and write I(p, i, s) for the path obtained by replacing xi by s. Observe that if s, s' E Basl with org(s) = xi and org(sl) = xt, then

+

(a) D(I(p, i, s), t, sl) = I(D(p, t, s'), i - l(sl), s) for t l(sl) (b) D(I(p, i, v ) , t l(s), s') = I(D@, t, s'), i, s) for t 2 i.

+

181

33. The fundamental groupoid F'urther if d = D(p, t, s') and q = I(d, j,s), then either j 5 t and by (b), q = D(I(p, j, s), t E(s), s') E D(I(p)), or j > t, in which case the jth entry d j of d is xi, where i = j l(8') 2 t l(sl), and thus by (a), q = D(l(p,i, s), t ,s') E D(I(p)). So in any event, I(D(p)) C_ D(I(p)), so to complete the proof we must show D(I(p)) E I(D(p)). Let q = I(p,i,s) and suppose d = D(q, j,sl) E D(q). If j 5 i 1(s1) then by (a), d = I(D(p,j, s'), i - l(sl), s) E I(D(p)). Similarly if j 2 i Z(s) then qj = xi, where t = j - l(s) 2 i, so by (b), d = I(D(p,t,sl),i,s) E I(D(p)). Thus we may assume i-l(sl) < j < if l(s). Observe first that if j 5 i and i l(s) 5 j 1(s1) then s is a subpath of s'. So as s and s' are of the form xx or xyx, we conclude s = st and then that d = p E I(D(p)). So we may assume this is not the case. Next s, s' have length 1 or 2. Suppose first l(s) = 1. Then s = xx and asj
+

+

+

+

+

+

+

< +

+

+

Define the basic degree bas(p) of p E ker(=) to be the distance of p from org(p) in the graph P.

Lemma 33.10: For p E Bas, p

E

I~~S(P)(~~~(~)).

Proof: Let x = org(p). The proof is by induction on n = bas(p). If n = 0, then p = x and the lemma is clear, so take n > 0. Then the set & of paths q E Bas with bas(q) = n - 1 and p E D(q) U I(q) is nonempty. Let q E Q. By induction on n, q E Pel(%). Thus if p E I(q), then p E P ( q ) , so we may assume

(*) If q E Q then P 4 I(q) SO P E D(q). Then D(q) # 0,so q # x and n > 1. If n = 2 then q E Bas1 so as p E D(q), p = x, contradicting n = 2. Hence n > 2. Now q E I(r) for some r E I ~ - ~ ( xand ) as n > 2, D(r) # 0.Then p E D(q) C D(I(r)) = I(D(r)) by 33.9. Thus p E I(s) for some s E D(r) and bas(s) bas(r) 1 = n - 1. That is, s E Q, contradicting (*).

<

+

Lemma 33.11: For p = xo . - xr E Bas,

< i. Proof: This follows from 33.10 by induction on bas(p).

35. Coverings of gmphs and simplicia1 complexes

184 Chapter 12 Coverings of Graphs and Simplicial Complexes an %-l-~ath X I = YO,- - .,Yk = X 3 in A(xo,xz).As Yi+l E Rm-l(yi), Pi = X O ~ ~ X ~ E Y S~ =+C3(A) ~ X Oby induction on m. Thus as p is in the closure of the paths pi, 0 < i < k (cf. 34.3), also p E S.

.

Lemma 34.8: Let p = xo -. x5 be a pentagon in A such that xi: nx i n xk # 0. Then p E C4 ( A ) .

Proof: Let x E x& n x f n xk. Then p is in the closure of the triangle ~ ~ 2 and x the 3 ~squares X O X ~ X ~ Xand X ~X O X X ~ X ~ X O . Lemma 34.9: Let p = xo - - xs be a hexagon in A with x i nxk n x i # 0. Thenp E Cq(A).

Proof:Let x E x& n x i n s f . Then p is in the closure of the squares XXiXi+lXi+2X, i = 0,2,4.

-

+

Lemma 34.10: Let = --s, where S = C4(A).Let 3 denote the set of paths p of A of length 4 such that p q for some path q of length 3. Let p = xo . x, E P be such that p q for any q E P of length less than n. Then

<

~ )each i n - 2. (1) xi+2 E A ~ ( xfor (2) If n 2 4 then d(A(xo,xz), A(x2,x4)) > 1.

-

(3) p contains no subpath in F. (4) If x E A(xo,2 2 ) then p ~ 0 2 x 2..xn.

Proof: If (1) fails then p q, where q is the path obtained by deleting xi+l from p, contrary to the choice of p. Thus ( 1 ) is established. If x1 # x E A(xo,x2)then x o x ~ x ~ x xEoS, so X O X X ~ ~0x122and hence (4) holds. If p contains a subpath in 3 then without loss of generality it is q = X O - . - X ~ .Thus t h e r e i s r = yo...y3 w q . N o w p = q.x4...xn r xq .- .xn which is of length less than n , contrary to the choice of p. So (3) holds. Finally assume n 1 4, x E A(xo,xz), y E A(x2,x4), and y E xL. By (4) we may take x = X I and y = 23. But then (1) supplies a contradiction. N

185

simplices of K such that each subset of a simplex is a simplex. The morphisms of simplicial complexes are the simplicial maps; that is, a simplicial map f : K 4 Kt is a map f : X --r X' of vertices such that f ( s )is a simplex of Kt for each simplex s of K . Example The clique complex K ( A ) of our graph A is the simplicial complex whose vertices are the vertices of A and whose simplices are the finite cliques of A. Recall a clique of A is a set Y of vertices such that y E x L for each x, y E Y . Conversely if K is a simplicial complex then the graph of K is the graph A ( K ) whose vertices are the vertices of K and with x * y if ( x ,y) is a simplex of K. Observe that K is a \ subcomplex of K ( A ( K ) ) . Given a simplicial complex K and a simplex s of K , define the star of s to be the subcomplex stK(s)consisting of the simplices t of K such that s U t is a simplex of K . Define the link LinkK(s)to be the subcomplex of stK(s) consisting of the simplices t of stK(s)such that t n s = 0. The dimension of a simplex s is Is1 - 1 and the dimension dim(K) of K is the maximum of the dimensions of the simplices of K if those dimensions are bounded, with dim(K) = oo otherwise. An n-simplex is an n-dimensional simplex. Define a morphism d : A + A of graphs to be a local bijection if for all a E ,'I

N

N

35. Coverings of graphs and simplicial complexes We continue the hypotheses and notation of Section 33, In particular A is a graph and P = P(A) is the set of paths of A. Recall that a simplicial complex K consists of a set X of vertices together with a distinguished set of nonempty subsets of X called the

is a bijection. Define d to be a fibering if d is a surjective local bijection. The fibering is connected if its domain f is connected. The fibering is a covering if d , : a L -+ d ( a ) l is an isomorphism for dl a E I?. We say A is simply connected if A is connected and A possesses no proper connected coverings. Similarly define a simplicial map q5 : L --+ D of simplicial complexes to be a covering if q5 is surjective on vertices and a local isomorphism; that is, for each vertex x of L, the map dz : stL(x) --r stD(d(x))is an isomorphism. Recall that the n-skeleton Ln of a simplicial complex L is the subcomplex consisting of all simplices of dimension at most n. Notice that the 1-skeleton LI of L and its graph A ( L ) are essentially the same, modulo the identification of the edges of A ( L ) with the 1-simplices of L.

Lemma 35.1: (1) If 4 : L -, D is a simpEcial map then the induced map : A ( L ) -t A(D) of vertices is a morphism of graphs. #J

186

Chapter 12 Coverings of Graphs and Simplicia1 Complexes

35. Coverings of graphs and simplicial complexes

(2) If d : A -t A is a morphism of graphs then the induced map d : K ( A ) -+ K ( A ) of vertices is a morphism of simplicial complexes.

(4) Let D be a simplicial complex with graph A. Then end : D

Lemma 35.2: If q5 : L

-+

K is a covering of simplicial complexes then

(1) q5 is surjective on simplices. (2) For each simplex s of L, q5 : stL(s) -+ stD(q5(s))is an isomorphism.

Lemma 35.3: Let d : A -+ A be a morphisrn of graphs. Then (1) d is a fibering if and only i f d : K ( A ) ~, K ( A ) ~ is a covering

of simplicial complexes. (2) d is a covering of graphs if and only i f d : K(A) -+ K ( A ) is a covering of simplicial complexes. Given a P-invariant equivalence relation N, recall from Section 33 that P/N = P is the set of equivalence classes $ of N . Each member of 15 has the same origin and end, so we set end($) = end@). Thus we have a map end : P -,A. Make P into a graph by decreeing that @ is adjacent to 4 if p q-(yx), where y = end(q) and x = end(p) E A(y). Notice that if p N q (xy) then q p . (yx),SO our graph is undirected. Denote by C ( A , x ) the set of paths with origin x. Write %(A,x)= C(A,X ) / N for the set of classes 3 with p E C ( A ,x). Recall that % ( Ax, ) denotes the group of 13 E 2 ( ~ x ) ,with p a cycle and n l ( A ,x ) = % ( Ax, ) for as the basic relation r;n l ( A , x ) is the fundamental group of the graph A. If D is a simplicial complex with graph A and an invariant relation, define D to be the subcomplex of K(P) with simplices s such that end(s) is a simplex of D. Let %(Dlx ) be the corresponding subcomplex of %(A,x ) and %(Dlx ) the correspondinggroup. Let nl (D,x ) = f (D,x ), where N = N S for S the closure of the set of all 2-simplices of D. Then nl(D,x ) is the findamental group of D. N

N

Lemma 35.4: Assume A is connected and lence- relation on P. Then

N

is a P-invariant equiva-

(1) The connected components off' are the sets ~ ( A , x )x ,E A. (2) The map end : $ H end(p) is a fibering of A and induces a connected fibering of g ( ~x ), onto A. (3) The fibering end is a covering if and only if k e r ( ~contains ) all triangles of A.

187 -t

D is a covering of simplicial complexes if and only if ker(-) contains all &simplices of D. (5) The fibering end : g ( ~x), -, A is an isomorphism i f and only i f ker(-) is the set of all cycles in A. (6) a! : f l ( A , x ) -t A U ~ ( ~ ( A , X ) ) is- a faithful wpresentation of f l ( A , x ) such that iil(A,x ) acts regularly on each fiber of the fibering of end : ~ ( A , X ) -, A , where for 4 E %l(A,x),a(J): $ H 8-1 .$. Proof: The proofs of parts (I), (2), and (6) are straightforward. Our fibering end is a covering if and only if whenever { x ,y, z ) is a triangle in A and p E P with end(p) = x, then the equivalence classes of p, p . (xy),and p. ( x z )form a triangle in p. i3ut the latter holds if and only if p . ( x z ) p (xy) (yz) if and only if p (xzyx) p-l E S = k e r ( ~ ) if and only if each triangle xyzx is in S. That is, (3) holds. Similarly by definition, end : fi -, D is simplicial, while the map is a covering if and only if for each simplex s = { x ,y, z ) of D, the unique preimage of s under end containing j3 is a simplex of D if and only if xyzx is in S, so (4) holds. ) A is a morphiim of graphs surjective on vertices As end : ~ ( A , x-+ and edges, end is an isomorphism if and only if end is an injection. Now if end is injective and p is a cycle with origin x then end@) = end(Z) = x, so p x E S, and hence S contains all cycles. Conversely if S contains all cycles and end@) = end(@ then pq-l is a cycle so pq-l E S and hence p q. Thus (5) holds. N

.

.

.

.

N

N

Define a local system on the graph A to be an assignment F : x I+ F(x), ( x ,y ) H Fxa,of each x E A to a set F ( x ) and each edge ( x ,y) to a bijection Fzy : F(x) -,F(y),such that

(LS1) FVz o Fzy = idF(x) for each edge ( x ,y) of A. Example 35.5 If d : A -, A is a fibering we obtain a local system F~ on A defined by Fd ( x )= d-I ( 2 )and F&, ( a )= d ~ ' ( ~ ) .

Recall from 33.8 that P is a pregroupoid; that is, P is a category possessing the inversion Inv :p H p-l. Given a local system F on A, we can extend F to a pregroupoid homomorphiim from the pregroupoid P to the groupoid of sets and bijections. Namely if p = xo - . xn P ( A ) is a path, define Fp : F(org(p))-,F(end(p))recursively by F,, = idF(,,) and Fp = Fx,-lx, o Fq, where q = xo %,,-I. We say paths p and q are F-homotopic if Fp = Fq and write p N F q.

.

. ..

188 Chapter 12 Coverings of Graphs and Simplicia1 Complexes L e m m a 35.6: (1) If p,q are paths with end(p) = org(q) then Fpq = Fq o F,. (2) FP O Fp-' = idF(org(p)) (3) F-homotopy is a P(A)-invariant equivalence relation. (4) F is a pregroupoid homomorphism from the path pregmupoid P ( A ) of A to the groupoid of sets and bijections. Proof: Part (1) is immediate from the definitions while (2) follows from (LS1). Then (1) and (2) imply (4),while (4) and 33.8.5 imply (3).

Let D be a simplicial complex with graph A. A local system on D is a local system on A such that:

(LS2) FxZ = FyZ0 Fxy for each 2-simplex ( x ,y, z ) of D. Example 35.7 If d : L + D is a covering of simplicial complexes then by 35.3.1, d induces a fibering d : A 4 A of the graphs of L , D and hence a local system F~ on A by Example 35.5. Then Fd is a local system on D. We can view D as a category whose objects are the simplices of D and morphisms are inclusions of simplices. Lemma 35.8: If F is a local system on D then F induces a functor F from D to sets via F ( s ) = F(x,) for some xs E s and Fs,t = Fp for some path p from xs to xt in t. Notice the definition in 35.8 is independent of p as t is simply connected. Lemmas 35.6.4 and 35.8 show that a local system on A or D is a local system in the sense of Section 7 of [Q]. Given a local system F on A, define AF to be the disjoint union of the sets F ( x ) , x E A, and define dF : AF -+ A to be the map with d i l ( x ) = F ( x ) for each x E A. Make AF into a graph by decreeing that u is adjacent to v if there is an edge ( x ,y) of A with u E F ( x ) and Fxar(u)= v. Notice (LS1) says this relation is symmetric, so AF is an undirected graph. Similarly if F is a local system on D let DF be the subcomplex of K ( A F ) with simplices s such that d F ( s )is a simplex of D. Lemma 35.9: Let F be a local system on D. Then (1) dF : DF -+ D is a covering of D. (2) F = F ~ Fis the local system of dp. (3) If q5 : L -+ D is a covering then DF* = L and q5 = dF+.

189

35. Coverings of graphs and simplicial complexes Proof: Let d = d F , x E A, and u E F ( x ) . Then u1 = {Fxy(u): y E x L ) and the map Fxy(u)I+ y is the restriction d, of d to uL. In particular d, : u1 4 x L is a bijection so d is a fibering. Further if t = { x ,y, z ) is a Zsimplex in D, then FXz(u)= Fyz(Fxy( u ) ) = Fxyz(u),so Fxe(u) is adjacent t o F,,,(u) in AF and hence d;l(xyzx) is a triangle in AF. So d;l(t) is a 2-simplex of DF and (1) holds. Parts (2) and (3) follow from the definitions.

Lemma 35.10: The map F H dF is a bijection of local systems on D with coverings of D. The inverse of this bijection is d I+ F ~ . Proof: This follows from 35.9. Indeed by Exercise 12.4, this bijection induces an equivalence of categories.

L e m m a 35.11: Let d : L -,D be a connected covering of complexes, A the graph of D, F = F ~ and , N the relation on P = P ( A ) defined by p q i f Fp = Fq. Then N

is a P-invariant relation such that u uvwu for each 2simplex {u,v , w) of D. (2) For a a vertex of L and x = d ( a ) , the map

(1)

N

defined by +(@)= Fp(a) is a covering with d o .II, = end. (3) For p E P with org(p) = x , there exists a unique path q in L with org(q) = a and d(q) = p. Further Fp(a)= end(q). For p a cycle of A, p E ker(--) if and only i f Fp = id. (4) Proof: The first statement of (1) is just 35.6.3, while (LS2) implies the second. By definition of --,$ is well defined with d o $ = end, and as L is connected, Il, is a surjection. By (1) and 35.4.4, end : % ( D , x )+ D is a covering. Thus d and end are local isomorphisms of simplicia1 complexes so locally = d-l o end is an isomorphism, and hence II,is a covering. Thus (2) holds. Part (3) follows by induction on the length of p. In ( 4 ) ,Fp = Fz if and only if p x if and only if p E k e r ( ~by ) definition of N. SO as Fx = id, (4) holds.

+

N

Lemma 35.12: Let S be the closure of the tsimplzces of the connected simplicial complex D, = --s, and x a vertex of S. Then end : E(D, x ) -+ D is the universal covering of D. Proof: By 35.4.4, e = end : %(D,x ) -+ D is a connected covering. be Conversely suppose d : L --, D is a connected covering, and let

--

190 Chapter 12 Coverings of Graphs and Simplicia1 Complexes the relation induced by d as in 35.11, 11, : C(D,x)/= -t L the map of 35.11, and f = end : C ( D , x ) -+ D. Then by 35.7, Fd is a local system, so (LS2) implies F: = id for each triangle p determined by a Zsimplex of D. Thus S k e r ( z ) by 35.11.4. Hence g : ~ ( D , x--+) C ( D , x ) / z is a covering, where g(p) is the =-equivalence class of p, and f o g = e. Then by 35.11, h = $ o g is acovering with d o h = e.

(1) If p is a cycle G-conjugate to a path of Ai then F: = id. (2) If each triangle of A is G-conjugate to a triangle of A. then d is a covering. (3) If I' is connected and the closure of the set of G-conjugates of cycles of Ai, i E I , is the set of cycles of A then d is an isomorphism.

Lemma 35.13: For a connected simplicia1 complex D, the following are equivalent:

Proof: Let F = Fd. Given an edge ( x ,y) of A, a E d-l(x), and P = Fx,y(a),we have d(ag) = d ( a ( g X ) )= d ( a ) ( g Y )= xg and d(Pg) = ~ g , so F,,,,(ag) = pg. That is, FXglyg(ag)= F,,,(a)g. Proceeding by induction on the length of a path p with origin x and using 35.6.1, we conclude Fpg(ag)= Fp(a)g. Next if Fp(a) = a and @ E d P 1 ( x )then by transitivity of k e r ( Y ) on dA1(x),there is g E ker(Y) with P = ag. Then Fp(P) = Fpg(ag) = Fp(a)g = ag = f l , so Fp = id. Let p be a closed path with origin x which is G-conjugate t o a path of Ai for some i E I. We claim Fp = id, so by the previous paragraph it remains to show Fp(a) = a for some a E d-l(x). By hypothesis pg is a path in Ai for some g E G. Then if Fpg(6)= 6 for some 6 E d-'(xg) then 6 = Fpg(6)= ~ ~ ( 6 ~SO -Fp(a) ~ ) =~a for , a = bg-l. Hence we may take p to be a path in Ai. But for each path q = xg xr in Ail since the restriction e of d to ri is an isomorphism of ri with Ail e-l(q) = e-l(xo). .e-l(x,) is a path in ri with Fp(e-l(zO))= e-l(xr) so in particular if q is a cycle then so is e-l(q), and therefore Fq = id. Therefore Fp = id, completing the proof of the claim. Therefore (1) is established. Next by 35.11 the relation p q if and only if Fp = Fq is an invariant ) all cycles G-conjugate t o a member relation and by ( I ) , k e r ( ~contains of hi. Thus under the hypothesis of (3), k e r ( ~ contains ) all cycles of A and hence d is an isomorphism by 35.15.2. Similarly (1) and 35.15.1 imply (2).

c

(1) D possesses no nontrivial connected coverings, (2) P ( A ) is generated by the &-simplicesof D. (3) Fp = id for each local system F on D and each cycb p of A. (4) ?rl(D,x)= 1 f o r x a vertex of D.

Proof: Parts (1) and (2) are equivalent by 35.12. Finally (3) and (4) are equivalent by 35.11.4. Lemma 35.14: A is simply connected i f and only if A is triangulable. ProoE This is a restatement of the equivalence of parts (1) and (2) of 35.13 when D = K ( A ) . Lemma 35.15: Let d : I? -t A be a fibering of graphs with A connected and the invariant relation induced by d as in 35.1 1. Then N

(1) d is a covering if and only if k e r ( ~ contains ) all triangles of A. (2) d is an isomorphism if and only i f ker(.v) contains all cycles of A.

Proof: By 35.11 we have a fibering II,: ~ ( A , X ) --+ L with d o 11, = end. Thus for each a E A, d z l = II,o endzl. Therefore if end is a covering then endzl is a morphism so d z 1 is the composition of morphiims and hence a morphiim. That is, d is a covering. But by 35.4.3, if k e r ( ~ ) contains all triangles then end is a covering, so d is a covering. Conversely if d is a covering then F$ = id for each triangle p of A, so k e r ( ~contains ) all triangles by 35.11.4. Thus (1) holds. The proof of (2) is similar, using 35.4.5 in place of 35.4.3. -

-

191

35. Coverings of graphs and simplicia1 complexes

Lemma 35.16: Assume d : I' -+ A is a jibering of gmphs such that A (ri: i E I ) and (Ai :i E I ) are families of gmphs such that for all i E I , ri C I?, Ai A, d ( r i ) = A i , the inclusions I'i --+F , Ai -t A are morphisms, and d : ri -+ Ai is an isomorphism. Assume &&her that X : G -+ Aut(I') and Y : G -+ A u t ( A ) are representations of a group G such that for all a E I' and g E G, d(a(gX)j = d(&)(gY) and ker(Y) is transitive on the fibers of d. Then

-is connected and

- ..

Remarks. The fundamental groupoid of a simplicial complex K is a standard construction in combinatorial topology; for example, the edge path groupoid of Chapter 3, Section 6 of [Sp] is essentially the fund* mental groupoid. Similarly the fundamental group of K is essentially the edge path group of K as defined in [Sp]. Of course the fundamental group of K is isomorphic t o the fundamental group of the topological space of K; compare Theorem 3.6.16 in [Sp]. Our notion of a covering of a graph or simplicia1 complex is suggested by the standard notion of a covering of topological spaces or by Tits's

192 Chapter 12 Coverings of Gmphs and Simplicia1 Complexes definition of a covering of geometries in [TI]. The reader is cautioned, however, that in the combinatorial group theoretic literature, the term "graph covering" is sometimes used as we use the term "fibering." The concept of a covering of a graph or simplicial complex was introduced in [ASl] and [AS3], and our treatment of these topics comes from those references.

Exercises i

I

! !

i

Exercises 1. Let d : L --i K be a universal covering of connected simplicial complexes, let x E L be a vertex, and let Aut(L, d) be the subgroup of Aut(L) permuting the fibers of d. Prove (1) The map $ :Aut(L, d) -+ Aut(K) defined by +(g) : d(x) +-+ d(xg) for x e L and g E Aut(L, d) is a surjective group homomorphism with kernel sl(L, x) . (2) If s is a simplex of L then the map : Aut(L, d)t -, A U ~ ( K ) ~ ( ~ ) is an isomorphism for each t C s. 2. Let p = xo . . .sf be an r-gon in a graph A. Prove (I) If r is even and A ( ~ - ~ ) / ~ ( X fl O A(xfI2) ) is connected then p E CT-l(A). # 0 then (2) If r is odd and A(X(,-~)p,x(,+~) n A('-~)/~(XO) P E Cr-l(A). 3. Define a path p in a graph A to be reduced if D(p) = 0. Prove (1) The equivalence class Ip] of p under the basic relation contains a unique reduced path rp. (2) rp E Dn(p) for some nonnegative integer n. 4. Let D be a simplicial complex. Define a morphism f : F -t E of local systems on D to be a family of maps f, : F(x) + E(x), x E A(D) = A, such that for each x and y E A(x), fy o FXy = Exy of,. Define a morphism 4 : (L, d) -+ (E, d) of coverings of D to be a simplicial map 4 : L - t L with 204 = d. Prove (I) If : (L, d) -+ (5,d) is a morphism of coverings then f d : Fd -+ F~is a morphism of local systems, where f$ = q51Fq(.. (2) Iff : F -+ E is a morphism of local systems then q5f : (D*, dF) -+ ( ~ ~ , isda morphism ~ ) of coverings, where 4&,) = j,. $J

(3)

-

.

.

4J' ( ~ ~ , dis~an) isomorphism (F f E) rt ( ( D ~dF) , ) of the category of local systems on D with the category of coverings of D. The inverse of this isomorphism is d I-+ Fd, 4 I-+ f4.

5. Given a local system F on D let limXEA(F(X))be the disjoint union of the F(x) modulo the equivalence relation ct generated by the identifications FZy: F ( x ) -+ F(y), (x, y) an edge of A. Prove (1) For u, v E U x E A ( F ( ~ )u) ,N v in limXeA(F(x))if and only if there exists a path p from x = dF(u) to y = dF(v) with Fp(u) = v. (2) If D is simply connected then the map u I-+ ii is an isomorphism of F(x) with limXEA(F(x)).

193

36. Amalgams

Chapter 13 The Geometry of Amalgams

In this chapter we put in place the machinery we will use to establish the uniqueness of most of the sporadic groups. The general approach is as follows. Let 7.1 be some group theoretic hypothesis and G a group satisfying 7.1. We associate t o each such group a coset graph A on a coset space G/G1 defined by some self-paired orbital of G on GIG1, as in 3.2. We pick some family 3 = (Gi : i E I ) of subgroups of G and show that the amalgam A ( 3 ) of this family is determined up to isomorphism by 7.1 independently of G. The family 3determines a coset geometry I' = r(G, 3) and a geometric complex C(G,3)as in Section 4, and indeed any completion p : A ( 3 ) 3 G of A(3) determines a corresponding geometry and complex. The family 3is chosen so that A is isomorphic to a collinearity graph of I' and so that the closure of the cycles in the collinearity graphs of proper residues of I? is the set of all cycles of A. Further if L :A ( 3 ) -+ G is the universal completion of A(3) then we obtain a fibering d : A 3 A from the collinearity graph o f f onto A, and as the cycles in residues of I'lift to cycles in residues in f , the fibering d is even an isomorphism. Hence G G E G for any pair of groups G, (? satisfying 7.1, giving us our uniqueness. This is a broad outline of the approach. In Section 36 we develop the theory of amalgams and geometries necessary for this procedure. Then in Section 37 we specialize to a particular class of rank 3 amalgams and geometries which will be sufficient to establish the uniqueness of most of the sporadic groups. Our starting point is a &tuple U = (G, H, A, A H ) consisting of our group G and graph A together with a suitable subgroup H of G and a graph AH, which will be

195

a residue in our geometry. We term such 4tuples satisfying a few more axioms uniqueness systems and we study these objects in Section 37. We are usually able to choose our uniqueness system so that each triangle of A is fused into AH under G. Thus it remains to show that A is triangulable in the sense of Chapter 12 in order to implement the procedure described above. So in the process of proving the uniqueness of a sporadic, we usually show some graph A associated t o the sporadic is simply connected via the results in Section 34. Finally in Section 38 we see how to obtain well-behaved uniqueness systems from a flag transitive group of automorphiims of a suitable rank 3 string geometry. In Chapter 14 we apply this point of view t o the Zlocal geometry of M24and a truncation of the projective geometry of L5(2) to obtain uniqueness proofs for those groups. Similarly in Chapter 16 we use the approach to prove the uniqueness of J2,Sua, and Col.

36. Amalgams Let I = (1,. .. ,n ) be a set of finite order n. An amalgam of mnk n is a family A = ( a , , : P j - t PK:J C K C I ) of group homomorphisms such that for all J C K C L, cYj,KaK,L = aJ,L.

Example 36.1 Let 3 = (Gi : i E I ) be a family of subgroups of a group G. For J c I let J' = I - J be the complement to J in I and a s in Section 4, define G = Gj. Define PJ = GJr. Thus PJ n PK = Gj t n GK, = GJ#UKt = G(jnK)# = PJnK. Also for J C K c I, define aJK : Pj -+ PKto be inclusion. Then

njE

is an amalgam.

A morphism q5 : A 3 A of rank n amalgams is a family q5 = (q5j : Pj 3 Fj : J

c I)

of group homomorphisms such that for all J C K C I the obvious diagram commutes:

Pj

4.1 Pj

aJ K ---",PK

1.4

&J,K

PK

196

Chapter 13 T h e Geometry of Amalgams

A completion P : A -,G for A is a family P = ( P j : Pj -+ Gj t i :roup homomorphims such that G = ( P j P j : J c I ) and for all J C K c I

36. Amalgams

! I

the obvious diagram commutes:

Proof: Suppose /3 : A --+ G is a faithful completion. Then injective, so Lj is injective.

197

P j = L j$ is

Lemma 36.6: Isomorphic amalgams have isomorphic universal completions.

4 :A + A is an isomorphism of amalgams and L : A --P G(A) and : A --+ G(A) are universal completions, then $b : A -+ G(A) is a completion, so there exists $J : G(A) -+ G(A) with Lj$ = +i;j for each J. Similarly we have 4 : G(A) -+ G(A) with = 4-I/, j. Then = $-I.

Proof: If

The completion

: A --+

G is said to be faithfil if each PJ is an injection.

Example 36.2 Let 3 = (Gi : i f I ) be a family of subgroups of a group G with G = (3). Form the amalgam A(3) of Example 36.1. Then the identity maps idJ : Pj -+ Pj form a faithful completion id = (idJ :J c I ) with id : A(3) -+ G.

I

4

The free product F(A) of the groups Pj, J C I, in an amalgam A is the free group on the disjoint union of the sets Pj modulo defining relations for the groups Pj. We have the following universal property:

Given a completion P : A 4 G of A let F(P) = (Gi : i E I ) , where Gi = P , I / ~ ~I?(@ , , = r(G, 3(P)), and C(P) = C(G,3(P)) be the geometry and geometric complex of P, as defined in Examples 4 and 9 of Section 4. Further for i E I , define the collinearity graph A(P, i ) of C(P) at i to be the graph on the set GIGi of objects of r ( P ) of type i with x adjacent to y if there exist chambers C, of C(P) for u = x, y with u E Cu and C, n C, a flag of type it.

Lemma 36.3: If (cpJ : Pj --+ G) is a family of group homomorphisms then there exists a unique group homomorphism cp : F(A) -+ G with gcp = gcpJ for all g f P j and all J c I.

Lemma 36.7: Let /3 : A -+ G be a faithful wmpletion of an amalgam A and L : A -+ G(A) the universal completion of A. Let cp : G(A) --+ G be the surjection of 36.4. Then

Define the free amalgamated product G(A) of the amalgam A to be the free product F(A) of the groups Pj, J c I , modulo the relations g - l ( g ~ j , K )= 1 for 'J c K c I and g E Pj. Write i j for the image of g f F(A) in G(A) and let L = ( ~: Pj j -+ G(A)) be defined by g~J = i j for g E Pj. Then

Lemma 36.4: L : A G(A) is a universal completion for A. That is, if p : A --+ G is a completion of A then there exists a unique group = PJ for all J C I . homomorphism $ : G(A) + G such that ~ j $ --+

Proof: By construction

L :A

4

G(A) is a completion of A. Suppose

p : A -t G is a completion. By the universal property of the free product recorded in 36.3, there exists a group homomorphism cp : F(A) + G defined by gcp = gPJ for g E Pj. As 0 is a completion of A, ga jKPK = gPJ for all J C K C I , so g - l ( g ~ j K ) E ker(cp). for each g E Pj. Thus cp induces a group homomorphism $J : G(A) --+ G defined by fi$ = g v = g P j for g E Pj. This is the map of 36.4.

Lemma 36.5: If A possesses a faithfil completion then the universal completion L : A -+ G(A) is faithful.

(1) cp : G(A)i

--+ Gi is an isomorphism with cp(G(A)j ) = G j for each i E J E I . (2) cp induces a morphism cp : C(L)+ C(P) of geometric complexes which is a covering of simplicia1 complexes, via cp : G(A)ig I--+ Gi~(g). (3) Assume for some &ed i E I that: (*) Gi, = GrtGr and Gi n Gi = (Gij n qj : j E it)for some t E Gil - GI. Then the covering cp of (2) restricts to a fibering cp : A(L,i ) --+ A(P, i ) of collinearity graphs. (4) Assume (*) and (**) The closure of the set of cycles of A(P,i) conjugate under G to a cycle of the wllinearity graph at i of Linkqp) ( G j ) ,as j ranges over i', is the set of all cycles of A(P,i). Then cp : G(A) + G is an isomorphism.

Proof: Let H = G(A). By 36.5, L is faithful. Thus cp : Hi Gi is the composition cp = L;'/~~Iof isomorphisms, so (1) holds. As cp(H,) = Gi, the map cp : Hig I--+ Giq(g) of (2) is well defined and as cp : H -+ G is surjective, cp : C(L)--, C(P) is surjective on vertices. -+

198

37. Uniqueness systems

Chapter 13 The Geometry of Amalgams

From the definition of C ( L ) in Example 9 of Section 4, the chambers of C(L) are of the form SI,, = {His : i E I ) , x E H , and P ( S ~ ,=~Sz,rp(x) ) is a chamber of C ( P ) , so cp : C ( L ) -+ C ( P ) is a morphism of geometric com~ ) Link(Gi) C(Gil3 ( P ) i ) 1 plexes. By 4.5, Link(Hi) C(HilJ ( L )and while by ( I ) , cp : C ( H i 1 3 ( ~ ) * ) C(Gi13(P)i)is an isomorphism, SO cp : Link(Hi) -+ Link(Gi) is an isomorphism and hence cp : C ( L ) C ( P ) is a covering of simplicia1 complexes. That is, (2) is established. ) A = A(P,i).By (2), cp restricts to a surjective Let A = A ( L , ~and morphim of graphs cp : A -+ A. Further as G is transitive on chambers of C ( P ) , each chamber through Gi is conjugate under Gi to 3 ( P ) . Also Gi, is the stabilizer of the wall W = 3 ( P ) - {Gi) of type i' and GI is the stabilizer of F(P). Assume (*). Then Gr = GztGz,so Git is 2-transitive on the chambers through W and GI is transitive on Link(W)- {G;). Hence Gi is transitive on A(Gi).Then by (1)and (2),HI is transitive on Link(U)- {Hi), where U = 3 ( ~ - {Hi), ) so Hi is transitive on A(Hi). Next Git E A(Gi) and by (*), Gi n G: = (Gij n : j E i f )is the stabilizer in Gi of Git. Let s E Hi, with cp(s) = t. By ( I ) , cp : Hi -+ Gi is an isomorphism with v(Hij n Hfj) = Gij n GIj SO (p(Hijn Hfj) : j E it)= Gi nGi. Of course q(Hi nH,S) Gi nGf,SO p(Hi n Hf ) = Gi nGi. Thus by ( I ) , cp :Hi/(Hi n H f ) -+ Gi/(Gi nGI) is a bijection and hence cp : A(Hi) -+ A(Gi) is a bijection and (3) is established. anjisomorphism ) of Finally by (2),cp : LinkC(')(Hj)-+ L ~ n k ~ ( ~ ) is( G simplicia1 complexes and hence induces an isomorphism of the collinearity graphs at i of these links. Hence by 35.16.3, under the hypotheses of (4),cp : A -+ A is an isomorphism. That is, the map cp : Hi + Gi is an isomorphism and the map cp : H/Hi -+ G/Gi is a bijection. Hence cp : H -+ G is an isomorphism and (4) is established. -+

-+

qj

<

37. Uniqueness systems Define a uniqueness system to be a Ctuple U = (G, H, A , A H ) such that A is a graph, G is an edge and vertex transitive group of automorphisms of A , H G, AH is a graph with vertex set xH and edge set ( x ,y)H for-some x E A and y E A(%)nxH, and:

<

.

(U) G = (H1Gx),Gx = (Gx,y,Hx),and H = (H({x,Y)),HX). In this section we assume U = (G,H, A , A H ) is a uniqueness system and (3,y) an edge in AH. Notice by definition y E A ( x ) , so x # y. Let G1 = Gx, G2 = G ( { x , ~ ) GQ ) , = H , and J = J ( U ) = (Gi : i E I ) , where I = {1,2,3). The amalgam of U is the rank 3 amalgam

199

A(U) = A ( 3 ( U ) )as defined in Example 36.1. Notice P12 = H , P23 = Gx, Pi3 = G ( { x , Y ) ) ,Per = HztZI1Pi = H ( { X ~ Y )J'2) , = HZ, and P3 = G q . We say that a rank 3 amalgam A is residually connected if Pij = (Pi,Pj) for all distinct i,j E I . Now: Lemma 37.1: Assume A is a residually connected rank 3 amalgam and @ : A 4 M is a completion of A. Then

(1) I'(/3) is a residually connected geometry. (2) If P13 = PIP3 then r(P)has a string diagram and M is flag transitive on I?@). Proof:Part (1)is 4.5.3. If P13 = Pl P3 then by 4.2, the diagram of C(P) is a string (in the sense of Section 4) so (2) follows from 4.11.1. Lemma 37.2: (1) There is t E H with cycle ( x ,y). (2) The amalgam A(U) is residually connected with P13 = PIP3.

Proof: As x # y, ( U ) says H({x,y)) # Hz, so (1) holds. Now G({x,y)) = Gx,y(t).Thus Pi3 = PIP3. This observation together with condition (U) in the definition of uniqueness system shows the amalgam is residually connected. Observe that the inclusion map P : A(U) -+ G is a faithful completion of the amalgam A(U) by Example 36.2. Form the geometry I' = r(P) and its collinearity graph A(@,1) as in Section 36. We call the objects of I' of type 1 points and the objects of type 2 lines. By 37.1 ar~d37.2, I' has a string diagram, so by Exercise 1.5, two points a, b of I? are incident in A@, 1) if and only if a and b are incident in I? with a common line of r. Hence the term collinearity graph. Moreover as in 3.2:

Lemma 37.3: The map Glg H xg is an isomorphism of A(P, 1) with A. Further the line G2g of J? is identified with the edge {xg,yg) of A via this isomorphism. Thus we identify A with A(@,1) via the isomorphism of 37.3 and write A for both. to be a pair of isoDefine a similarity of uniqueness systems morphisms a : Gx -+ G~ and 5 : H -+ B such that a = 5 on Hz, HxC = H ~ Gx,ya , = GI,s, and H({x,y))C = ~ ( { ,$)) f for some edges ( x ,y), ( I , # ) of A H ,AB, respectively. We say the similarity is with respect to ( x ,y), ( I ,g ) if we wish to emphasize the role of those edges. The similarity is an equivalence if there exists t E H with cycle (x,y) such C all b E Gxs. that (bt)a= ( b c ~ ) ~for

~,a

Chapter 13 The Geometry of Amalgams

37. Uniqueness systems

Define a m o r p h i s ~of uniqueness systems U,U to be a group homomorphism d : G -+ G such that the restrictions d : H -+ H and d : Gx -+ Gz are isomorphisms defining a similarity of U with U.

If each triangle of A is G-conjugate to a triangle of AH then d : A -+ A is a covering. (4) If AH is a base for A then U E U and & Z G.

Lemma 37.4: (1) The geometry of points and edges of AH is isomorphic to the residue of H in I?. (2) If d : G -+ G is a morphism of uniqueness systems U,U, then A(U) 2 A(U) and d induces a covering d : f -t r of geometries and a fibering d : A -+ A of graphs defined by d : Gig I-+ Gi(gd) and (5g)d = x(gd), respectively. The restriction d : Afi AH is an isomorphism and ker(d) is transitive on each fiber of d on A.

Proof: Let d : G be the homomorphism supplied by the universal property of By construction G = ( a , & ) . By 36.5, the universal completion is faithful, and by 37.1, G is flag transitive on f'. Then arguing as in the proof of 37.3 using Exercise 1.5, we have a natural e-equivariant bijection between the lines of f and the edges of A, so as is flag transitive on f', G is also edge transitive on L\. Next as p is faithful, d : H -+ H and d : Gz -,Gx are isomorphisms. If K is the kernel of the action of G on A then K 5 G~ and K a C. Therefore Kd < Gx is normal in G so as G is faithful on A, Kd = 1. Then as d : G%-+ Gx is an isomorphism, K = 1. That is, G is faithful on A. The other conditions in the definition of uniqueness system are properties of the amalgam A(U) and hence are shared by U . Thus 0 is U is a morphism a uniqueness system. Remarks above show d : fi of uniqueness systems. By construction, d defines an equivalence of 6 with U. We have established (1) and (2). Notice that if d : L\ -+ A is an isomorphism then d : G -+ G defines an equivalence of the actions of G on A and G on A, so d : G -+ G is an isomorphism and hence d : -,U is an isomorphism. Thus to prove (4), it suffices to assume AH is a base for A and prove d : A 4 A is an isomorphism. But 37.4.2 supplies the hypotheses of 35.16.3, so that lemma completes the proof of (4). Similarly 35.16.2 implies (3).

200

-+

Proof: Part (1) is a consequence of 37.1, 37.2, and 4.11.2. Assume the hypotheses of (2) and pick t as in 37.2.1. As G({x, y)) = (t)Gx,y, as d : GZ,li -+ Gx,y is an isomorphism, and as Ed = t for some E E with cycle (2,8), d : (?((2,5)) -+ G({x, 9)) is an isomorphism. Thus as d is a similarity, d induces an isomorphism A(0) % A(U) of amalgams. As c i d = Gi and cijd = Gij for each i,j, d is a well-defined surjective morphism of geometries and graphs. Now Gi is incident with Gjh in I? when h E Gi. But as d : Gi Gi is an isomorphism, d : Gi/Gij -+ Gi/Gij is a bijection and hence d : i=' -+ I" is a local bijection. Also A(x) = yGz and A(Z) = gG5 with G q j d = Gx,yrso d : Zg +-+ x(gd) is a bijection of A(x) with A(%) and hence d : A -+ A is a fibering of collinearity graphs. By 37.1 and 37.2, G and G are flag transitive on their respective geometries, so the same argument shows d is a local isomorphism of geometries. ) and only if g E k e r ( d ) ~ so ~ , ker(d) is tranNotice Gig E d-I ( G ~ if ) . d is a covering of geometries, (1) says sitive on the fiber d - l ( ~ ~ As d : AR -+ AH is an isomorphism. -+

Let L : A(U) 4 G(A(U)) be the universal completion of A(U). Write G(A(U)), H for HL, Gz for G X ~etc. , Let L\ = A(', 1) be the collinearity graph of G. Then G? is indeed the stabilizer of some f E A. Let A$ be the collinearity graph of the residue of fi in f' and u = (G, H, A, Ah). We say AH is a base for U if the closure of the G-conjugates of all cycles of AH is the set of all cycles of A.

6 for

Lemma 37.5: (1) u is a uniqueness system equivalent to U. (2) There exists a morphism d : U -+ U of uniqueness systems.

201

(3)

e

e.

-+

-+

Lemma 37.6: If U and A(U) r A@).

a

are equivalent uniqueness systems then

<

Proof: Assume a : Gx G5 and : H -+ H define an equivalence of our systems and let t be the element of H supplied in the definition of equivalence. Define /I : G({z, 9 ) ) -+ G({z, Q)) via p : btn H (bc~)(tC)~ for b E Gx,y and n E Z. The map is well defined as a and C agree on Hz. The map is a homomorphism as (bt)a = ( b ~ ) for ~ c all b E GXlv.As a is injective on Gx the map is an isomorphism. Now by construction : H -+ H, (Y : Gx -t Gz, and /I : G({x, y)) -+ G({z, g)) define our isomorphism of A(U) with A(u). -)

<

We are now in a position to state one of the principal results of this section:

Theorem 37.7: Assume U,U are equivalent uniqueness systems such

Chapter 13 The Geometr~lof Amalgams

202

that A H ,AB are bases for A , A , respectively. Then U is isomorphic to 0. ProoE By 37.6, A(U) G A@). Then by 36.6, G is also the universal completion of A(u). Then by 37.5.4, U 2 u Z U. Corollary 37.8: Assume U and U are equivalent uniqueness systems, A is triangulable, each triangle of A is G-conjugate to a triangle of AH, and U also satisfies these hypotheses. Then U U. In order to apply Theorem 37.7 and its corollary, we need effective means for verifying the equivalence of uniqueness systems. Theorems 37.9 through 37.12 supply such means.

Theorem 37.9: Assume U and are similar uniqueness systems and for some edge (2,y) of A H , Aut(GX,*)nC(Hx,y)= 1. Then U is equivalent to U . Theorem 37.10: Assume a : Gx -t G3 and i : H -+ H define a similarity of uniqueness systems U and a with respect to edges ( x ,y ) , (1,f j ) , and there exist K 5 Gx and t , h E N H ( K ) such that: (1) CA,~(K)(K n H ) = A ~ ~ z (( H K )). (2) t has cycle (x,y), th E Gx, and he E N*(Ka). (3) G x , = (K,,Hx,y).

Then a ,

!

37. Uniqueness systems

203

where for h E H , Z(xh) is defined by Z(xh) = ~ ( x )Further ~ . the same conditions hold for 0. (2) Z ( x ) a = Z(x)( = Z(1) and Z(y)C = Z($). (3) A u t ( H ~ )N ( Z ( x ) )= Aut~ut(~)n~(Z(x))(H~)Aut~ut(~,)(H~). (4) N ~ u t ( ~ ) n ~ ( z ( x 5 ) )N ( H x H ( { Y))Hx)C(HX). ~, Then U and u are similar. We next prove Theorems 37.9 and 37.10. We are supplied with a similarity a,6 :U 4 21 with respect to ( x ,y), (2,g). In Theorem 37.10 we also have a subgroup K of G,; in Theorem 37.9 let K = Gx,y.By 37.2 there ) Aut(K) be exists t E H with cycle ( x ,y). Let K = K a and 4 : N ~ ( K-t ) a* : 0 I+ Oa. the conjugation map, and define a* : Aut(K) -t A u ~ ( Kby For g E NG(K)write ga* for g$a*, where 11, : NG(K)-+ Aut(K) is the conjugation map. Thus (ka)ga*= (kg)a for each k E K. Therefore for g E NG, ( K ) ,ga* = ga4 as (kg)a = (ka)gff= (ka)gU'#'. Now for k E K n H and h E NH ( K ) , ( ~ o ) ~=Q(kh)a * = (kh)c = (kc)" = jka)hc, so ha* = hcq5 on I? n B,and hence hi$ h-'a E C A U t ( ~ ) (nHK). But in Theorem 37.9, CA,t(K)(Kn H ) = 1, so ha* = hi4. In particular this holds for h = t , so (kt)a = (ka)ta' = ( k a ) g for all k E K = GX,,, completing the proof of Theorem 37.9. Similarly in Theorem 37.10, hypothesis (1) of Theorem 37.10 says

.

c, define an equivalence of U and U.

Hypothesis V: The uniqueness system U = (G,H, A , A H ) satisfies the following four conditions for some edge ( x ,y) of A H :

Theorem 37.11: Assume U,U are uniqueness systems satisfying Hypothesis V with respect to edges ( x ,y ) , (3,f j ) and a : Gx -+ GZ and C : H -t H are isomorphisms such that GXlya= G*,B, HxC = Hz = Hxa, and H ( ( x , y))( = a ( { %g)). , Then U and U are similar.

- Theorem 37.12: Assume U and U are uniqueness systems, ( x ,y),(%, g) are edges in A H , A k , and a : Gx -+ GZ and : H -+ H are isomorphisms such that Hxa = HZ and: (1) Them exists Z ( X )char G, with Z ( x ) 5 Hy, and either (a) Gx = N G ( ~ ( z ) 07), ) Gx,, = NG, ( Z ( y ) ) , (b) Hx = NH ( Z ( X ) and

Then picking h as in hypothesis (2) of Theorem 37.10, th E Hz by hypothesis, so (th)a*= (th)a4by an earlier remark, and hence ( t c ~ * = )~~* (th)a*= (th)ac$= ( t h ) ~= 4 (t@)hC'#', so ta* = (t<$b)hS'+h-'a*= tC4. We have shown (kt)a = ( k c ~ ) for ~ c all k E K. Also for b E (bt)a= (bt)c= ( b ~ )=~ (ba)Y, c so by hypothesis (3) of Theorem 37.10, for all a E G,,y. This establishes Theorem 37.10. (at)a = Next the proof of Theorem 37.11. So assume U and U are uniqueness and z : H -+ systems satisfying Hypothesis V and that a : Gx 4 0% H are isomorphisms satisfying the hypotheses of Theorem 37.11. Then az-I E Aut(H,), so by ( V l ) ,there exist 6 E NAut(H)(Hx)and C E IH= NAut(G,)( H z ) with az-I €6 on Hz. Let a = €-la and i = Sz; then a = ( on li,.Further by (V2), we may pick c E N ( G ~ )Therefore . G~,,C-' = G&, for some g E Hx and hence GX,,a = ( ~ $ , , ) a= Geg= G ~ , ~with ( ~g(ga) ~ )E A@). Similarly by 37.2 there is t E H with cycle ( x ,y) and by (V3) we may pick 6 E N(HxtH:,). Then st6 = yh for some h E Hz, so t6 has cycle (x,yh) and HXtu6= (Hz n H$)6 = Hz r l Ht,6 = Thus

-

~2,~.

38. The uniqueness system of a string geometry

Chapter 13 The Geometry of Amalgams

204

HXd<= Hz,ytSz = H:,~Z = H;J = Hz,ghzrwhile

See Section 4 for definitions and terminology. We recall that by 4.2 and 4.5, group theoretically these hypotheses are equivalent to the assertions that G = (Gx, G1, Gn), Gx = (GX,l,Gx,,), GT = (Gs,zlGx,l), and

= R({z, jihz}) - ri,,vh,,

GZ= G I , ~ G ~but , , Gx # Gx,~Gx,nand G,

= Hx,Ya I so H((x,y))< = H({l,ji(hz))). Finally H3,ghz- = G3,g(ga), so ji(hz) and ji(ga) are fixed points of Hz,g(hz)on AB(%). But as Hji.is transitive on AB(f), NHz(Hz,?jhz)is transitive on the fixed points of HZ,ghzon An(%), and then by (V4). G3,gh, = G Z , for ~ each such fixed point f . In particular G'3,gga = G2,ghz,SO a and define a similarity of our systems. This completes the proof of Theorem 37.11. Finally we prove Theorem 37.12. Assume the hypotheses of that theorem and let t E H have cycle (x, y). Observe that hypothesis (a) of Theorem 37.12 implies hypothesis (b), so we may assume (b) holds. Notice that Hz< = NH(Z(x))C = NR(Z(5)) = H,. Then arguing as in the proof of Theorem 37.11, using hypotheses (3) and (4) of Theorem 37.12 in place of (Vl) and (V3), respectively, we may adjust a and so that a = on Hz while still retaining hypotheses (1) and (2). For example, for 6 E Aut(H) n N(Hx) n N(Z(x)), st6 = yh for some ~ Z(xtS) = Z(yh) = z ( ~ ) So ~. h E Hz, so Z(y)S = Z(x)% = Z ( X ) = ~ z with for = Sz with Z(y)z = Z(jj), we have Z(y)C = ~ ( ~= )Z(jjhz) jiha E AR(3). Next by hypothesis (I), Z(x) Hy, so Z(y) = Z(xt) = z ( x ) ~ (H,)~ = Hyt = Hz. Thus as a = C on Hz, Z(y)a = Z(y)c = Z(jj). Further Z(z(tC)) = z(z)~C = (Z(X)C)~C= (Z(X)~)[= Z(y)c = Z(ji). Also Z(ft6) = Z(g) = Z(ZQ for some F E H, so tC E Nfi(Z(f))F = HeF and hence 3 5 = 5f = ji. That is, tc has cycle (2,ji). Next Gx,ya = (NcE(Z(y)))a = Ncf(Z(jj)) = G'zlg. Finally HXly= NH(Z(X)) n N H ( ~ ( Y )so ) = NR(Z(f)) nNp(Z(%)) = Hji.,~. Therefore H({x, y})C = (Hx,y(t))C = H2,g(tC) = H({%,g)). Therefore a and C define a similarity of our systems, and the proof of Theorem 37.12 is complete.

<

<

<

<

<

<

38. The uniqueness system of a string geometry In this section we assume: (r0) G is a flag transitive group of automorphisms of a residually connected rank 3 string geometry I' and (x, I, a ) is a flag in I' such that the residues of x and a are not generalized digons.

# G,,XG,,z.

(*)

The three classes of objects of type 1, 2, and 3 will as usual be called points, lines, and planes; respectively. Given an object a E F, write r ( a ) for the residue of a, ri(a) for the objects of type i in I'(a), and &(a) for the kernel of the action of Ga on I'(a). Let p be a prime and P(a) = Op(Q(a)). Define the collinearity graph of I' to be the graph whose vertices are the points of I' and with points adjacent if and only if they are incident with a common line of I?. Notice that by 38.1, this agrees with the notion of "collinearity graph" in Section 36. Let A be the collinearity graph of J?.

Lemma 38.1: rl(1) C r l ( a ) and r3(1)

r3(~)+

Proof: This is Exercise 1.5. Lemma 38.2: A is connected.

Proof: This follows from 4.6. Consider the following hypotheses: ( r l ) Each pair of distinct collinear points x, y is on a unique line x y. (I'2) If x, y E Fl(a) are collinear then x y E r2(n). (r3) Each triangle of A is incident with a plane. (r4) G,,l is 2-transitive on rl(l). (I'5) Gx,l = (Gx,y,llGx,l,,) for x # Y E rl(l).

+

+

Theorem 38.3: Assume (G, ,'I x, 1, n) satisfies hypotheses (Pi) for i = 0,4,5. Define A to be the collinearity graph of I', H = G,, and AH Me collinearity graph of the residue of a. Then (1) U = (G, H, A, AH) i s a uniqueness system. (2) If hypotheses (I'i), 0 2 i 5, hold then each triangle of A is G-conjugate to a triangle of AH.

<

Proof: Let y E rl(l) - {x}. By flag transitivity and (r4),G is edge transitive on A. By (r4) there exist t E Hl with cycle (x, y) and H1 = These observations together t). By (r5), Gx,l = (Gx,y,l, with (PO) show that 2.4 satisfies condition (U) for uniqueness systems. Therefore (1) is established.

206

Chapter 13 The Geometry of Amalgams

< <

!

Next assume hypotheses (ri), 0 i 5. If abc is a triangle in A then by (I'3), abc is G-conjugate to a subset of F1(x) = AH, SO without loss of generality abc c AH. By ( r l ) and (r2), the lines a b,a c, b+ c are incident with n, so abc is a triangle in AH and (2) is established.

Lemma 38.6: Assume

Each triangle of A is contained in a line of r. If y E a 2 ( x ) then there exists a unique plane n incident with x and y. Further A(%)r l E I'l(n). (3) Ench square and pentagon of A is incident with a plane. (4) Assume a is a plane with d(x,rl(a)) = 1 and the wllinearity graph of r ( a ) is of diameter 2. Then A(x) n rl(u) = {u} for some point u, and A ( X )r)~I'l~(a) = u1 f rll(u). (1)

(2)

-

<

Write d(x, y) for the distance between vertices x and y in A and for E T). The following two lemmas can be used to study M24 and its 2-local geometry; see Exercise 13.2.

S,T E A let d(S, T ) = min{d(s, t) : s E S, t

Proof: First if axb is a triangle in A then a and b b e distinct points at distance 1 from x on a b, so by (a),x E a b. Thus (1) holds. Let y E h2(x). Then there is u E A(x, y) and by (b), there is n E r3(u x , u y). Hence by (c), n = n(x, y) = n(u x, u y) is the unique plane incident with x and y and u x and u y. Similarly if v E A(x, y) then n(v x,v y) = n(x, y) ='n, SO v E rl(a). Finally if w E A(x) n A ~ ( then ~ ) x, w E A ~ ( n~(x) w), SO by (a), there is v E A(y) n (x w). Then v E A(x, y) c n, so w E x v G n. Therefore (2) is established. Notice (2) implies (3). Suppose a is a plane with d(x, a ) = 1 and the collinearity graph of r ( a ) is of diameter 2. Then there is u E A(%)n a. If u # v E A(x) n a then by (I), d(u, v) = 2, so by (2), x E A(u,v) S a, contradicting d(x, u) = 1. So u is unique. Clearly uL n a C_ A ~ ~ (no. X) Conversely suppose y is a point on u at distance 2 from x, but y !$ A(u). Then d(u, y) = 2 so x E A ~ (n~A(u) ) a by (2), a contradiction.

+

+

+

+

+

+

I

<

(rl) and

For each line k of I' there exists a unique y E r l ( k j with d(x, y) = 4 % k) . (b) Each pair of lines of r(x) is incident with at least one plane. (c) If d(x,z) = 2 then Ir3(x,z)l 5 1. Then

Example 38.4 Let V be a 5-dimensional vector space over the field F of order 2. From Example 1in Section 4, the projective geometry PG(V) of V is a rank 4 geometry whose objects are the points, lines, planes, and hyperplanes of V with incidence equal to inclusion. Let I' = P G ( v ) ~ be the truncation of PG(V) obtained by suppressing the hyperplanes of V. Thus r is a rank 3 geometry over I = {1,2,3} whose objects are the points, lines, and planes of V. Pick a flag (x, 1,n) from r ; that is, x C 1 C n is a chain of subspaces of V of dimension 1, 2, 3, respectively. Observe that I' is a rank 3 string geometry, since by Example 6 in Section 4, the residue r ( l ) r PG(~)@PG(v/~)~. Similarly r(x) S PG(V/X)~ and r(n) S PG(n). Let G = GL(V) L5(2). Then G is flag transitive on I?, so Hypothesis (r0) is satisfied. Indeed visibly Hypotheses ( r l ) through (r5) are satisfied, so Theorem 38.3 supplies us with a uniqueness system U = (G, H,A, AH). The vertices of A are the points of V and as each pair of points of V are collinear, A is a complete graph. Hence A is of diameter 1,so by 34.5, A is triangulable. Therefore by 35.14, A is simply connected while by Theorem 38.3 and 37.5.4, G is the free amalgamated product of A(U). Finally by Exercise 13.1, if is a uniqueness system equivalent to U then G r G.

Proof: Form the semidirect product S of M with (a), and let Z = Z(R) and E = Cs(R). Then E = Z x (a). Observe that if A (a)with A 9 S then [M,A] M n A = 1, so as (a)is faithful on M, A = 1. It follows that a is a pelement and as @(Z)= 1, cr is of order p. Now as a centralizes a Sylow psubgroup of M , E splits over Z as an S-module, (cf. 12.8 in [FGT]) so S = MCs(M) and hence a E Inn(M).

207

(a)

+ +

L e m m a 38.5: Assume M is a finite group with F*(M) = Op(M) = R and @(Z(R)) = 1, and a E Aut(M) centralizes a Sylow p-subgroup of M. Then cr E Inn(M).

38. The uniqueness system of a string geometq

+

+

+

+

+

+

+

Lemma 38.7: Assume the hypotheses of 38.6 and in addition assume (d) The collinearity gmph of r ( n ) is of diameter 2 and &generated. (e) A has diameter 3. (f) For each z E ~ ~ ( the 2 gmph ) on r2(x) defined by l* k if and only if d(I'l(x(l, k)),z) = 1 is connected, where ~ ( 1k), is the unique plane through 1 and k. Then A is 4-generated. Proof: Let S = C4(A) be the closure of ths set of all cycles of A of length at most 4; we must show S contains all cycles of A. By hypothesis (e) and 34.5 we must show S contains each T-gon for r 5 7. By definition, S

208

209

Chapter 13 The Geometry of Amalgams

38. The uniqueness system of a string geometry

contains all squares and triangles. By 38.6.3, each pentagon is incident with a plane, so by hypothesis (d), S contains all pentagons. Let p = x0 -xT be an r-gon. Assume first that r = 6 and let a = T ( x ~ , x ~Now ) . if d(xo,a) = 1, then by 38.13.4, A(xo)n n = { x ) with x E A(x2,x4).But then 34.9 says p E S. In general by (f) there is a sequence of lines x3 yi, 0 5 i 5 n , with yo = xq, yn = 2 2 , and 7r(x3 yi,xg yi+1) of distance 1 from xo. Let u, E A(xo, yi) with uo = x5 and % = X I . Proceeding by inX O by the previous paragraph, duction on i, X O X ~ X ~ X ~ ~ E~ US,~ while X O U ~ Y ~ X ~ Y E~S. +Thus ~ U by34.3, ~+~X the~product X O X ~ X ~ X ~ ~ ~ + ~ of these two hexagons is in S , so p E S by induction on i. Therefore all hexagons are in S , so it remains to consider the case r = 7. But z3,xq E a3(x0)n (x3 x,), so by hypothesis (a), there is x E A2(xo)n ( 2 3 xd), and hence p is in the closure of two hexagons and a triangle.

Assume also that distinct points of r ( a ) are incident with at most one line in r ( a ) . Then (1) ( r l ) and ( r 3 ) hold. (2) If1,k E I ' ~ ( X then ) tr3(l,k)l = 1.

--

+

+

+

+

r3(1,

r(n),

I C ~ + ~

+

The remaining lemmas in this section give criteria for establishing the hypotheses (N), 1 5 i < 5. With the exception of 38.8, these lemmas will not be used elsewhere in Sporadic Groups, so the reader can skip them. They will be used to treat the uniqueness of other sporadic groups not considered here.

Lemma 38.8: Assume ( r 6 ) Gx is primitive on planes and lines through x. ( r 7 ) Q(x)Q(x)is transitive on rl(l)- { x ) . Then (1) Q ( x ) is transitive on rl(1)- {x) and GX,! = Gx,y,lQ(x)for Y r l ( q - {XI. (2) ( r l ) , (F4), and ( r 5 ) hold. Proof: By 38.1, &(a)fixes r l ( l ) pointwise. Further by (r7),Q(x)Q(a) is transitive on r l ( l )- { x ) , so Q ( x ) is too. Thus (1) holds. Notice that as Q(x) 5 GV,x,l,(1) implies (r5). Next if y is on a line k E r 2 ( x )distinct from 1, then by (r6), Gx = (Gx,~, G x ,= ~ (Gx,y,l,Gz,y,k,Q ( x ) )= Gx,vQ(x).Then A(x)= YQ(x)= 1 - { x ) . Thus by flag transitivity and 38.2, 1 is incident with all points of r, so by flag transitivity and 38.1, G, = G,,xGT,l, contrary to (*). Thus ( r l ) holds. As Q(x) 5 GV,r,( 1 ) and flag transitivity imply (r4).

Lemma 38.9: Assume the lines and planes in r ( x )form a linear space; that is, each pair of lines in r ( x ) is incident with a unique plane of r ( x ) .

Proof: Let I, k E r 2 ( x ) .As r ( x )is a linear space there is a unique plane a E r 3 ( x ,1, k). Therefore by 38.1, a is the unique member of k); that is, (2) holds. Further if y E r l ( l ,k ) is distinct from x then by 38.1, x, y E r l ( a ) are distinct points incident with the lines I , k of so by 1 = k. That is, (I'l) holds. If xyzx is a triangle then by (2) hypothesis, X O there is n E r 3 ( x y,x z ) so xyz is incident with a by 38.1. Hence ( r 3 ) holds. Notice that we are not claiming, however, that xyzx is a triangle in the collinearity graph AH of r3(a);that is, y and z may not

+

+

be collinear in r(n).

Lemma 38.10: Assume (r1)and

(r8) I f k is a line with Q(n) 5 Gk then k Then (r2) holds.

E rz(n).

Proof: If x, y E rl ( a ) are collinear then &(a)5 Gx,y Thus x y E r z ( a )by (r8).

+

< Gz+y by (rl).

Lemma 38.11: Assume ( r l ) and P(7r) is weakly closed in NG(P(r))= NG(G,) = G,. Then ( r 2 ) holds. Proof: By 38.10 it suffices to prove (r8).Let &(a)5 Gkand a E rs(k). Then by the weak closure of P(a) and Sylow's Theorem, there is g E Gk with P(a)g = P(u). Then as NG(P(a))= NG(GV)= G*, ag = C , SO k E r2(a).

Lemma 38.12: Assume ( r l ) and (r4), and assume for collinearpoints x,y and1 = x + y that (a) P(1) = ( P ( l )n P(x))(P(l)n P(Y)),and (b) ( P ( x ): P ( X )<~ IP(l)Q(x) ~ : P(l)kQ(x)lfor all k E r2(x) such that r3(l,k) is empty. Then ( r 3 ) holds.

+

Proof: Suppose xyzx is a triangle in A and let k = x z. By ( r 4 ) and flag transitivity, IP(y) : P(y),l = IP(x) : P ( X ) ~ ~ . Now IP(l)Q(x): P(l)kQ(x)lis the length of the orbit kP(1)of k under P(1) and by (a), kP(1) = k(P(1)n P(y). So

210

Chapter 13 The Geometry of Amalgams (aP(l) n P(y1.z 5 n P(y))k) 5 IP(Y) : P(Y).z~= IP(x) : P(x)yl. Hence by (b) there is n E r3(lrk). Then by 38.1, xyz is incident with ?r.

Lemma 38.13: If G # G,Gx and GX,y is transitive on the planes of r(x) not through 1 for y E rl(l), then (F2) holds. Proof: If x, y E r l ( u ) for some plane a not incident with I then as Gx,y is transitive on planes in F(x) not incident with 1, r3(x) = r3(y). Then by 38.2, r3(x) = r3(v) for all points v, contradicting G # G,G,. Thus (r2) holds.

Remarks, The idea of establishing the uniqueness of a group G with respect to some hypothesis 31 by first proving (a) a certain amalgam A associated to G is determined up to isomorphism by ? independently i of G, and then proving (b) the collinearity graph of the completion of A via G is simply connected, was introduced in [AS11 by Yoav Segev and the author. Most of the material in this chapter comes from that reference.

Exercises

--- - --

1. Let U and U be equivalent uniqueness systems such that Aa is a base for U . Assume either (a) IGI = 1 ~ 1 or 1 (b) G is simple. Then G 2 G is the free amalgamated product of A(U). 2. Let G = M24, 11 the 2-local geometry for G, and A its collinearity graph. Prove (1) G, I'satisfy hypotheses (f0)-(F5) of Section 38, and hence determined a uniqueness system U via Theorem 38.3. (2) A is Cgenerated. (3) Each triangle and square of A is incident with a plane of I?. (4) G is the free amalgamated product of A(U). (5) If u is a uniqueness system equivalent to U then G r G. - (Hi: Use Exercise 7.1 togetherwith 38.6 and 38.7.) 3. Assume hypotheses (I'O), ( r l ) , (r2), and (r4) of Section 38 with GI finite. Prove (1) G is transitive on triples x, y, ?r with x, y E rl(?r) and x, y collinear. (2) If for each triple in (I), (\r3(1)I1p)= 1, F*(Gx,y) = Op(Gz,y), @(Z(Op(G~,v))) = 1, and Z(G,,y,,) = 1, then the uniqueness

Exercises system U of Theorem 38.3 is equivalent to any uniqueness system similar to U. (3) Let K = Gr,(q and assume Iri(l) vert is odd for i = 1,3, GI = KG1,T,F*(K) = 02(K), cP(Z(02(K)))= 1, and Z(K,) = Z(G,). Let U be the uniqueness system defined by Theorem 38.3, suppose U is a uniqueness system defined via Theorem 38.3 by a pair G, f satisfying hypothesis (N), i = 0,1,4,5, and a,4 is a similarity of U with U with respect to (x,-y), (2,g) such that GngX+yC = Then U is equivalent to U. (Hint: Use Theorem 37.9 to prove (2) and Theorem 37.10 to prove (3).)

eii,f+ii.

211

39. Some &local subgroups i n L5(2), M24, and He

213

39. Some 2-local subgroups in L5(2), MZ4, and He

Chapter 14 The Uniqueness of Groups of Type M2,, He, and L,(2)

In this chapter we consider groups G possessing an involution z such that H = CG(z) is isomorphic to the centralizer of a transvection in L5(2). We saw in 8.9 that F*(H) = Q Z D: is an extraspecial 2-group. Thus we can use the theory of large extraspecial subgroups developed in Section 8 to study G. To eliminate the trivial case G = H , we assume in addition that z is not weakly closed in Q with respect to G. Recall that by 8.10, M24 satisfies these hypotheses, and of course so does L5(2). From Chapter 5, there is one more simple group satisfying the hypotheses: the sporadic group He discovered by Held. We begin our study of G in Sections 39 and 40 by generating various facts about the 2-local subgroups of G which will define a coset geometry r for G. This culminates in lemma 40.5, which singles out the three possibilities for these subgroups and the geometry I?. We then begin to use the machinery of Chapters 12 and 13 to establish that each case gives rise to at most one group. In particular in Section 41 we associate a uniqueness system to G when G is of type M24 or L5(2) and use the results in Section 38 to prove G is unique. Then in the remaining sections we analyze the final case leading to He, eventually using Lemma 36.7 to prove uniqueness here. In addition to our uniqueness result we also derive the order of He and the structure of various local subgroups of We.

In this section z is an involution in a finite group G such that H = CG(z) is isomorphic to the centralizer of a transvection in L5(2). Let Q = 0 2 ( H ) and Z = (z). Let T E Sy12(H). Assume t is not weakly closed in Q with respect to G. For X 2 G write A(X) for the set of elementary abelian 2-subgroups of X of maximal order and set J(X) = (A(X)). We regard H as the stabilizer in GL(V) of an incident-point-hyperplane pair Vl, V4 of a 5-dimensional GF(2)-space V and let

be a T-invariant chain of subspaces with dim(&) = i . Let U1 be the subgroup of H of all transvections with center Vl, U2 the subgroup of transvections with axis V4,A1 the subgroup centralizing V2 and V/V2, and AZ the subgroup centralizing V3 and V/V3. Lemma 39.1: (1) Q E D: and H is the split extension of Q by L3(2). (2) The subgroups Ul and Uz are normal subgroups of H isomorphic to El6. Moreover Q = UlU2, Ul n U2 = Z, CH(Ui) = Ui, and H splits over Ui. (3) A(T) = {Al,Az) with Ai a T, Ai 2 E64) and A1 n A2 2 E16. Further A1 and A2 are the max-imal elementary abelian subgroups of J(T) = A1A2, T splits over Ai) Ai n Ui is of n;mk 3, and Ai n U3-i is of rank 2. (4) The transpose-inverse automorphism of L5(2) centralizing z and acting on T interchanges Ul and Uz and A1 and A2. (5) H has four orbits on involutions of Q: { z ) , Ul - 2,U2 - 2,and the set 'l;j of involutions in Q - (Ul U U2). (6) If t E TQ then ICH(t)l = 29 and Z = ~ ~ ( t ) is( the ~ ) second member of the derived series for CH(t). (7) NH(A1) is of index 7 in H and has seven orbits Ail 1 i 7 , on A? of lengths 1, 6, 2, 6, 12, 12, 24, respectively. Indeed A1 = { z ) , Az = U l n A l - 2 , A3 = U 2 n A l - Z , a n d A 4 = T Q n A 1 . F o r a € A 7 and h E CH(a) -A1, CAI( h ) E8 contains z, two members of A2, and four members of A7. (8) CH(Al fl Ul) = AIUl with A1 and Ul the maximal elementary abelian subgroups of A1Ul.

< <

214

Chapter 14

39. S o m e 2-local subgroups in L5(2), M2& and He

T h e Uniqueness of MZ4,H e , and Ls(2)

~z#

<

(9) N H ( J ( T ) ) = T has six orbits ri, 0 i 5 5, on z(J(T))# of lengths 1, 2, 2, 2, 4, 4, respectively. Further ro = { z ) , ri = (Ui n Z ( J ( T ) ) )- Z for i = 1, 2, and r3= TQn Z ( J ( ( T ) ) ) .

Proof: These facts can be checked inside of G L ( V )

Proof: Adopt the notation of 39.1.7, and assume is not contained in zG. Then z is weakly closed in U2 by 39.1.5. Thus As contains no member of zG. By 39.4, A4 contains no conjugate of z. Let n = I Z ~ 1. ~

L5(2). The check

is left as an exercise, but see 8.8 for (1).

Lemma 39.2: T E Sy12(G). Proof: As Z = Z(T), N G ( T ) 5 H.

Lemma 39.3: Either NG(Ui) = H or NG(Ui) is the split extension of Ui by L4(2) acting faithfully on Ui.

AS z

E

Lemma 39.4: Let 7 be the set of involutions G-conjugate to an element of TQ.Then z 4 7. Indeed if t E TQ then C H ( t )E Sy12(CG(t)). Proof: By 39.1.6, S = C H ( t )is of order Z9 and Z = ~ N G ( Z ) = H , SO S E Sy12(CG(t)).

Lemma 39.5: Either

/

~ 1 c# zG or Uz#

( ~ Thus 1 . NG(S) <

zG.

Proof: This follows from 39.1.5, 39.4, and the hypothesis that z is not weakly closed in Q with respect to G.

Lemma 39.6: (1) Al and A2 are not conjugate i n G . (2) Ai is weakly closed in G for i = 1 and 2. (3) NG(Ai) controls fusion in Ai (4) NG(Ui n Ai) = NG(Ai) n NG(Ui).

.

ProoE By 39.1.3, A(T) = { A 1 , A 2 } is of order 2 and by 39.1.7, A1 and A2 are normal in T. So (1) and (2) follow from 7.7.2, and (3) follows

from ( 2 ) and 7.7.1. Evidently NG(Ai)n N G (U,) 5 NG(AinU,). The opposite containment follows from 39.1.8.

Lemma 39.7: If U? C zG then either

<

( I ) NG(Ul)/Ul 2 L4(2), N G ( A ~ ) Nc(U1), and zG n -41 G Ux, or (2) uU2#C zG.

Then n = CiEldi, where I = {i : A, C zG} and di is the order of Ai. We have 1,2 E I but 3,4 4: I . Thus n = 7,19,31,43, or 55 by 39.1.7. But by 39.6.3, NG(A1) is transitive on zG nA1, so n divides the order of L6(2). It follows that n = 7 and zG n A1 = Al n u;#.Thus Nc(A1) 5 NG(A1n Ul). So NG(A1) 5 NG(Ul) by 39.6.4. In particular as NG(Al) is transitive on Al n NC(Ul) $ H. Now 39.3 completes the proof.

u?,

Lemma 39.8: Assume NG(Ul)/Ul

<

Uil CG(Ui) = CH(Ui) = Ui. Therefore NG(Ui)/Ui GL(Ui) = L4(2). But H/Ui is the stabilizer of z in GL(Ui), and hence maximal in GL(Ui). Thus NG(Ui) = H or NG(Ui)/Ui L4(2). In the latter case as H splits over Ui, so does T , and hence NG(Ui) splits over Ui by Gaschutz's Theorem (cf. 10.4 in [FGT]).

Proof:

215

S L4(2).

Then either

(1) U? = zG fl N G ( u ~ )O ,r

(2) U?

c zG and NG(Al)/A1 E L3(2) x S3.

Proof: Let M = NG(Ul). Then N M ( A 1 ) / A 1 L3(2) x Z2 with A1 =

I

!

;

B1 $B2 the sum of two equivalent natural modules for N M (A1)OO/A12 L3(2), and with B1 = Ul n A1 and B; # B2 for u E Ul - B1. Thus M1 = N M ( A l ) has three orbits I'i, 1 5 i 5 3, on A? of order 7 , 14, and 42, respectively, where rl = B?, f 2 = Bz# U B,u#, and f 3 C 7. Thus by 39.4, zG n A1 is f l or f l U f 2 of order 7 or 21. Let I' = zG n A ~L, = NG(Al), K = k e r (L), ~ ~and assume I' is of order 21. Then IL : MII = 3, so L / K 5 S3. In particular M p 5 K, so M,00 = K m q L. If M I 1 L , then B1 = Z ( 0 2 ( M 1 ) )q L, contradicting 39.6.3 and our hypothesis on fusion of z in A1. So L / K S3. Then as [u,M I ] 5 All L = C L ( M r / A 1 ) M I JSO L / A 1 !Z L3(2) x S3. Also n Al E I?, so u,# c zG by 39.1.5. That is, (2) holds. So assume f G Ul. Then L acts on (r)= Ul n All so L = M I by 39.6.4. Now if v is an involution in M - Ul then as M/Ul = GL(Ul), either v induces a transvection on Ul and v is fused into A1 under M , or m([Ul,v ] )= 2 and v is fused into A2 under M . Thus if v E zG, v is of the second type and we may take v E A2. Then [v,U1]= Cv,(v) = Ul nA2, so each involution in vU1 is in v(Ul n A2) C A2, and is conjugate to v under Ul. Notice we may assume such a v exists, or else (1) holds. Let M* = M/Ul. Then A; contains six conjugates of v* and nine transvections. We have seen that if r* is a conjugate of v* then all four involutions in rUl are in zG, while if r* is a transvection then no member of rU1 is in zG. Thus zG n A2 is of order 3 6 . 4 = 27. Further if r* is a transvection with r E U:! n A2 then r(A2n U l ) contains two members of 7 and two members of U2 - 2.So there are

~z#

+

A

~

I

Chapter 14

216

The Uniqueness of M24, H e , and Ls(2)

Lemma 39.9: If zG n NG(Ul) = U? then Ul a G. Proof: Let M = NG(Ul). By our hypothesis and 7.3, M is the unique point of G / M fixed by z. Let x E A1 - UI. Then D = Cu, ( x ) = UI n Al is a hyperplane of Ul but N*((x, U l ) ) is not transitive on D#;instead it fixes the center [Ul,x] of the transvection x. Hence G = M by Exercise 2.7.

Lemma 40.2: Either (I) (2)

c

Ai zG for i 5 3, while A4 E 7 and by 40.1 we may choose notation so that A6 C_ zG. So A = zG n Al is of order 21, 33,45, or 57 = 3 19. But by 39.6.3, NG(Ai) is transitive on A, so the order of A divides that of L6(2). Hence IAj = 21 or 45. Suppose 1 A1 = 45. Then N H ( A l ) has two orbits A4 and As on A? - A of order 6 and 12, with A4 C 7. Thus either 7 n A1 = A? - A is of order 18, or 7 n A 1 = A4 and As are orbits of NG(A1).We may assume the latter. But then (A4)= Q n A1 5 N G ( A 1 ) ,contradicting that A is of order 45.

z G n ~ = U ~ U U or~ , G = NG(Ui) is the split extension of Ui by L4(2) for i = 1 or 2.

Proof: By 39.5 and the symmetry supplied by 39.1.4, we may assume

u?,

Uf zG. Then by 39.1.5 and 39.4, either (1) holds or rG n Q = and we may assume the latter. Now 39.3, 39.7, 39.8, and 39.9 complete the proof.

1

In this section we continue the hypotheses and notation of the previous section. In addition we assume zG n Q = u,#U u,#.This hypothesis is justified in view of lemma 39.10. Recall the definition of J(T) in&, tion 39 and observe that by 39.1.3, J ( T ) = AlA2 is of order 2* and index 4 in T, with Z ( J ( T ) )= @ ( J ( T ) )= Al n A2 2 El6.

Lemma 40.1: (1) N G ( J ( T ) ) / J ( T )2 S3 x S3 has two orbits zG n Z ( J ( T ) ) and 7n Z ( J ( T ) ) on z(J(T))# of order 9 and 6, respectively. (2) zG n A, contains a NH(Ai)-orbit of length 12 for i = 1 and 2.

zG for i 5 2, while

r3 C 7,

-

Lemma 40.3: If zG n A1 is of order 21 then Nc(Ul)/Ul 2 L4(2), NG(A1)/A1 L3(2) x S3, and A? = (zG n -41)U (I.n A i l . Proof: Let M = N G ( A l ) and M* = MIA1. Then

I

Let I = N G ( J ( T ) ) and K = k e r I ( M ) . By 40.1, IM : I1 = 7 , so M* / K * % M / K S7. Now I* r S3 x Sq. Further if K* = 1 then M* S7. But S7 has a unique subgroup isomorphic to S3 x S4 and that subgroup is maximal but not of index 7. So K* # 1. Thus either O3(I*)5 M* or J ( T ) 2 K , since 03(1*) and J(T)* = 02(1*) are the minimal normal subgroups of I*. However if J ( T ) 5 K then I = M by a Frattini argument.

<

I

Proof: Let B = Z ( J ( T ) ) . By 39.1.9, T = N H ( J ( T ) )has orbits ri, 0 I i 5 5, on B#. Notice J ( T ) is weakly closed in T, so by 7.7, N G ( J ( T ) )

controls fusion in B. Also by 39.1.9,

/ z G n A I J=21, or =45 and 1 7 n A l l =18.

Proof: Adopt the notation of 39.1.7. By the hypotheses of this section,

Lemma 39.10: Either

40. Groups of t y p e L5(2), M24, and He

217

so from the orders of ri listed in 39.1.9 we conclude I' = zG n B is of order 5, 9, or 13. The last case is out as 13 does not divide the order of L4(2). In the first case AutG(B) is of order 20 = IrllAutH(B)I with Sylow 2-group A u t H ( B ) 2 Ed. But then a Sylow 5-group P of AutG(B) is normal in AutG(B) and centralizes an involution in AutG(B). This is impossible as P is irreducible on B. So Irl = 9 and I7nBI = 6. Further AutG(B) is of order 36 with Sylow %group E4. We conclude AutG(B) G S3 x S 3 and for b E r - Q , lbTl = 4. Next the NH(Ai)-orbits on A# are listed in 39.1.7. As IT : CT(b)l = 4, the length of the orbit Aj of b under NH(Ai) is not divisible by 8. But by 39.1.7, orbits of N H ( A i )on Ai - Q have length 12 or 24. Hence d = 12.

eighteen elements of A2 fused into U2- Z and eighteen in 7.As elements in U2nAl - Z are not fused to those in TQn Al in M I = NG(A1),these elements are not fused in G by 39.6.3. Hence these eighteen element sets are the orbits of NG(A2) on ~f distinct from zG n A2 by 39.6. By 39.1.4 and 39.1.7, there is an element a E zG n A2 with C H ( a )= A2(h) = R of order 27 and CA, ( R )= B contains five elements from zG and two elements b and c from U2 - Z . Then as ING(A2) : NG(A2) n CG(a)l = 27, R is of index 2 in some subgroup S of NG(A2) fl CG(a) and S acts on (b,c ) and hence fixes z = bc. But then S _< C H ( a ) = R, a contradiction.

(1) (2)

40. Groups of type L5(2), M24, and He

f

i

<

The Uniqueness of MZ4,He, and L5(2)

Chapter 14

218

So 03(1*) a M*. Let X be of order 3 in I with X* = 03(1*) and let Bo = CM(X). From the action of X on J(T), CA, (X) = 1, so NM(X) is a complement to Al in M and (NM(X) : Bol = 2. By Gaschutz's Theorem (cf. 10.4 in [FGT])there is a complement B to X in Bo. Claim ( B : H n BI = 7. As IBo : Bo n HI = 21, it suffices to show B contains a Sylow 3-group P of Bo n H. This follows as P is inverted by some b~ B n H a n d B n H a B O n H . So 7 = IB : H r l BI = /zBI and hence by 39.1.7, zB is composed of two orbits under H n B of length 1 and 6, and (zB) = Ul n A1. Thus B NG(Ul nA1) NG(Ul) by 39.6.4. Thus NG(Ul) $ H , so 39.3 and 39.8 complete the proof.

/

Lemma 40.4: Assume zG n A1 is of order 45 and let M = NG(A1), a E T n Z(J(T)), and I = CM(a). Then (1) MIA1 S S6/Z3. (2) I/A1 2 S5 and CA,(Ioo) = B E Eq with NM(B) = X I for some X of order 3 with XAl g M.

<

Proof: By 40.2, n, = 1zG n Ail = 21 or 45. firther if n, = 21, then by 40.3, NG(Ui)/Ui 2 L4(2) and NG(Ai)/Ai S L3(2) x S3. On the other hand if ni = 45 then NG(Ai)/Ai E S6/Z3 by 40.4. Thus in this case H = NG(Ui) by 39.3 and 39.8. So the lemma holds.

1

Proof: First by 40.2 and 40.3, zG n Ai and 7 n Ai are the orbits of NG(Ai) on A#. Thus it remains to show that each involution in G is fused into A1 or A2. Now A1 and A2 are the maximal elementary abelian subgroups of J(T), so each involution in J(T) is in Al or A2. Next if NG(Ui)/Ui is isomorphic to L4(2), then each involution in NG(Ui) is conjugate under NG(Ui) t o an element of J(T) (cf. the proof of 39.8). So we may assume G is of type He. Let Li = NG(Ai)". Then each involution in Li is Li-conjugate to a member of J(T) and there are two Li-classes of involutions in NG(Ai) - Li, each of which has a representative in L 3 4 . So the proof is complete.

I

I

219

for i = 1 and 2.

Lemma 40.6: G has two conjugacy classes of involutions: zG and 7.

i

-Lemma 40.5: Up to a permutation of 1 and 2, one of the following holds: (1) NG(Ui)/Ui % L4(2) and NG(Ai)/Ai E L3(2) x S3 for i = 1 and 2. (2) NG(Ul)/U1 L4(2), NG(UZ)= Hl N G ( A ~ ) I A2~L3(2) x S31 and NG(A2)/A2 S6 123.

E S6/Z3

,

I

<

NG(Ui) = H and NG(&)/Ai

We will say that G is of type L5(2) in case 40.5.1 holds, G is of type MZ4if 40.5.2 holds, and G is of type H e if 40.5.3 holds. We will consider each type in succeeding sections.

<

Now by 40.1, NI(J(T))* S S4 is of index 5 in I * and as in the proof of the previous lemma, J(T)* is not normal in I*,so I * < S5. Thus I * S5. Let B = C ~ ( ~ ( ~ ) ) ( O ~ ( N ~ ( J ( TThen ) ) ) )B . E E4 and ( ( ~ / ( a ) ) ~ ) Al/(a) is an image of the Sdimensional permutation module for I*. Hence as an element of order 3 in I is fixed point free on A1/B, B a I , Further by 40.1, o~(NG(J(T)))* % Z3 x A4 is transitive on B#. There= NM(B) is of index 6 in M , so M*/K* < S6, fore L = o'(NG(J(T)))I where K = kerL(M). As F*(I*) is simple and o ~ ( N ~ ( J ( T ) ) ) *S Z3 x A4, it follows that IK*l 5 3. As IM*I = 31S611it follows that K* Z% Z3 and M*/K* E Ss, as desired.

(3)

and Mz4

1

Proof: Let M* = MIAl. Then

=

4 1. ~ r o u p ofs type L5(2)

,

41. Groups of type L5(2) and

M24

We continue the hypotheses, notation, and terminology of Sections 39 and 40. In addition we assume G is of type L5(2) or M24 as defined after 40.5. Let G1 = NG(Ul), Gz = NG(A1), and GQ = NG(A2). Let 3 = (GI, G2, G B )and form the coset geometry r = r(G, 3)as in Example 4 of Section 4. We adopt the notation and terminology of Sections 4 and 38 in discussing r.

Lemma 41.1: I' is a residually connected string geometry and G is flag transitive on I?. Proof: Let L = (3)and suppose for the moment that L = G. From 4.5 and 4.11 it suffices to prove GI, = (Gls,2s,Gls,3s) for each s f S3 and G2 = GI2Gz3. But T 5 G123 and from 40.5, for each choice of distinct i, j there is Ki T: Ki G, with Gii/& maximal in Gi/Ki, SO indeed Gi = (Gij, Gik). For example, K1 = Ul, G1/Ul = GL(Ul) L4(2), Gl2/Ul is the maximal parabolic of G1/Ul stabilizing a 3-dimensional

<

220

Chapter 14

The Uniqueness of M241He, and L5(2)

41. Groups of type Ls(2) and

subspace of Ul, and Gl3/Ul is the maximal parabolic stabilizing a 2dimensional subspace of Ul. Similarly K2 E E64, IG2 : G121 = 3, and IG2 : G231 = 7, SO G2 = G12G23. Thus it remains to show L = G. Notice that L satisfies the hypothesis of G so all results in Sections 39 and 40 can be applied to both G and L. In particular by 40.6, G and L have two classes of involutions with representatives z and t E 7 and zG n L = zL and tG n L = tL. Further by 39.1.2, CG(z)= H GI 5 L, so by 7.3, L is the unique fixed point of z on GIL. Then by 7.5 it suffices to show t fixes a unique point of GIL, and hence by 7.3 it suffices to show CG(t)I L. By 39.1.9 we may pick t E E = Z ( J ( T ) ) .Let M = CG(t)and I = M n L. We may assume M # I. For v E zG n I , L is the unique fixed point of v on G / L , so I is the unique fixed point of v on M / I . Proceeding by induction on the order of G, we may assume L Z L5(2) or M24. Suppose first that L r L5(2). We view L as GL(V) (where V is the 5-dimensional space over the field of order 2 of Section 39) and observe V2 = [V, t ] and V3 = CV(t),so I G23 = NL ( J ( T ) ) .By 40.1.1, IzG n El = 9 and I / J ( T ) E S3 has orbits of length 3,6 on zG n J ( T ) . We pick z in the orbit of length 3 and observe that by 7.5, there is v E zG n J ( T ) with K = CM(zv)$ I. Let B = (t,zv), D = I fl K , and K* = K / B . Observe that as CG((zv,t ) ) $ L, zG n (zv,t) = 0 and hence zv E cE(02(1)) % E4. So B = cE(02(1)) and D = 02(1). Next D* is the unique point of K*/D* fixed by x* for each x E zG n D. But Al q D while D = 02(1) has two orbits on involutions of AT of length 3,12, and each orbit has a representative x* with x E zG. Therefore 7.5 supplies a contradiction and completes the proof. Therefore L = M24. Thus it suffices to prove:

<

<

Lemma 41.2: Lett E I

n Z ( J ( T ) )and assume G is of type M24.

(2) CG(t)has three orbits on zG fl c G ( t ) with representatives z, 22 E E A1

(ab)$= abt. Therefore I has two orbits on zG n A2 with representatives z and z2 = ab, respectively. Next each involution in Ioo-A2 is fused under I into A1 -A2. By 40.2, 1Pn A ~ I = 21 and by 40.1, (zG~ z ( J ( T ) = ) (9, so 1zGnAl -A21 = 12. But 12 = INI(J(T)): CI(zl)lfor zl E zG n Al - A2, so I is transitive on zG n I* - ~ 2 . Now CT(t)is transitive on the involutions in CT(t)- C T ( B )by Exercise 2.8, and hence each is fused to s. Further ICI(s)l is divisible by 3. But all involutions in H centralizing an element of H of order 3 or 7 are in Q so by 39.1.5 and 39.1.6, if t E 7 ,and r E z G n C G ( t ) ,then 21 is prime to ICG((t,r ) ) ( .Therefore zG n (s,t ) = @, completing the proof of (2),modulo (1). So it remains to prove (1). Let M = CG(t).Suppose g E M and u,ug E zG n ~ M n . We claim g E I. By (2),u,ug E J ( T ) , so u,ug E Al u A2 by 39.1.3. If us E A2 then conjugating in I we may take us E Z ( J ( T ) ) = Al n A2. Now u E Ai for i = 1 or 2, and we saw in the previous paragraph that C M ( u )is a 2-group, so Ai is weakly closed in CM(u).Similarly Ai is weakly closed in CM(ug),so g E NM(Ai). So as NM(Ai) <_ I , our claim holds in this case. Thus we may assume u, ug E Al - A2. But then again g E NM(A1)5 I. So the claim is established. Then the claim and 7.3 show I is the unique point of M / I fixed by u E zc n I . Let K = C I ( B )and CG(B)*= CG(B)/B.Then K* is the unique point of CG(B)*/K*fixed by u*. However, each involution in K* is fused to z* or z f , so by 7.5, K = CG(B). Finally let M* = M / ( t ) . Then I* is the unique point of M*/I* fixed by b* = B*, z*, and zl, and each involution in A4 is fused to one of these involutions, so again 7.5 shows M = I and completes the proof.

Then

(1) CG(t)I G3.

Z ( J ( T ) )and zl

221

- A2.

Proof: Let I = CG,(t). By 40.4.2, I/A2 r S5, B = B ( t ) = CA,(Iw) 2 Eq, and there is X of order 3 in G3 with NG,(B) = X I and G3 = A2NG3( X ) .Then X I acts faithfully on A2/B as GL2(4)extended by a field automorphism, so in particular each involution in A2 - B is fused into zB under C I ( X ) .From 40.1.1, z B n 7 = {a). Let s E I n T - I* be an involution and b E B - ( t ) .As z E Z ( T ) , s centralizes z and zB n T = {a), and hence also za. Therefore as bS = bt, za = t , and then

Let ( x ,I , T ) be the flag of I' whose member of type i is stabilized by Gi and more generally adopt the notation and terminology of Section 38. Thus, for example, A is the collinearity graph of I?. Recall this is the graph whose vertices are the points of I' and with points adjacent in A if they are incident with a common line of I'. By 41.1, Hypothesis (I'O) of Section 39 is satisfied by the pair G, I?. We observed during the proof of 41.1 that hypothesis (I'6)of 38.8 holds. By 41.1 and 4.11, I'(xi) zi I'(Gi,3i)for xi of type i. But we essentially saw the structure of Gi and Fi during the proof of 41.1. Namely we saw that G1/Ul r L4(2) with GL2/Ul and Gl3/Ul the maximal parabolics stabilizing a point and line in some natural module for G1/Ul. In

222

Chapter 14 The Uniqueness of

M24,

He, and L5(2)

particular Ul = Q(x). Similarly 1 has (G2 : = 3 points and U1 is transitive on the two points of 1 distinct from x, so hypothesis (r7) of 38.8 holds. Hence by 38.8, G and I? satisfy hypotheses ( r l ) , (r4), and (l'5), so we can appeal to Theorem 38.3 to obtain a uniqueness system U for G. Further as hypothesis ( r l ) of Section 38 is satisfied, each pair of points is incident with at most one line. In particular if y # x is a point incident with 1 then Gx,IV GI.

<

Lemma 41.3: Let I = G13. Then I = Aut(1). Proof: Let A = Aut(I) and I * = IIAn. As Z(I) = 1 we can identify I with Inn(I) A. Notice I*= I f x I; with I f E S4 and I; E S3. Also J(T) is not normal in I, so by 39.1.3, A1 = J(Oz(I)) char I. Let X be of order 3 in I with X* = 03(I*). Then If = O~'(C~.(X*))and I; = CI. (If) are A-invariant, so as Sk= Aut(Sk) for k = 2,3, A = ICA(I*). Then by a Frattini argument, A = IB, where B = CA(I*)nNA(X). As NI(X) is a complement to A2 in I, [B,NI(X)] = 1. Next Ul r l A2 = CA2(02(I))is A-invariant and I 2 is absolutely irreducible on Ul n A2, so B 5 C(U1 n A2). Further there is Y of order 3 in NI, (X) with CA, (Y) = Ul nA2. Then B acts on [Y,A2] and as NI (Y) nNI (X) is absolutely irreducible on [Y,A2], B centralizes [Y,A2]. But now B centralizes I = NI(X)A2, so B = 1 and A = I. Lemma 41.4: If G is of type M24 or L5(2) and the same type as G with uniqueness system u then U is equivalent to U. Proof: Let I = G13. By 41.3, Aut(I) = I. Further if x # y E r l ( l ) then Gx,v is the pointwise stabilizer of rl(l), so from 40.5, Gx,y = NGl (A1)w is the split extension of A1 by L3(2). We first use Theorem 37.11 to show U is similar to Li. By 39.3, we have an isomorphism a : GI -, and by 39.1, Ala = All so from the previous paragraph, Gz,va = NG1(A1)OOa= N ~ ~ ( A ~=) = Also is determined up to conjugation in G1 by its isomorphism type, so we may choose a with G1,3a = 6 1 , ~ . Next either G is of type M24 and G3 is the split extension of A2 by S6/Z3 or G is of type &(2) and the split extension of A2 by L3(2) xS3. In particular in either case, Gg is determined up to isomorphism, so we have an isomorphism C : G3 -+ (23 and arguing as in the previous paragraph, we may pick 5 with G1,3C = and G3({z, y))< = c3((3, 8)). Thus to show U is similar to G, it suffices to verify Hypothesis V of Section 37. As I = Aut(I), (V1)-(V3) are satisfied. Further G?r,x,a,

223

42. Groups of type He and Gx,y are the subgroups of G3 and G1 fixing rl(l) pointwise, so N ~ ~ , ~ ( G n , x=, yG1,2,3 ) 5 G1,2 = NG,(Gxly), so (V4) also holds. Finally to complete the proof we appeal to Theorem 37.9. Namely G x , is the split extension of A1 by L3(2) and G?r,x,a,contains a Sylow 2-subgroup of GXla,,SO 38.5 says Aut(Gxly) n C(G?F,Z,y)= 1. Lemma 41.5: If G is of type L5(2) then G Z L5(2), while if G is of type M24 then G E M24. Proof: First if G is of type L5(2) then by 41.4, U is equivalent to the uniqueness system U of L5(2), so the lemma follows from the discussion in Example 38.4. So assume G is of type M24; here we appeal to Exercise 13.2. Namely let G = M24 and U the uniqueness system for G. By 41.4, U is equivalent to U, so by Exercise 13.2, G r

c.

Notice that 41.5 shows: Theorem 41.6: Up to isomorphism L5(2), M24 are the unique groups of type L5(2), M24, respectively.

42. Groups of t y p e He

In this section we continue the hypotheses and notation of Sections 39 and 40. In addition we assume G is of type He. Let t E Z(J(T)) fl 7 and define B = B(t) = c ~(02(cG(t) ( n NG(Z(J(T))))). ~ ~ Let~ Y 6 SYIQ(NG(Z(J(T)))),Li = NG(A~)n CG(B), and L = IL11 L2). BY 40.4.2, B E Eq, Li/Ai E L2(4), and = Cy(Li/Ai) 2 Z3 with C L ~ ( ~ ) a complement to Ai in Li and Y acting on B. Let li E CL~('&)be an involution acting on Y. As Yl # Y2 but li inverts Yo = Cy(B), we have: Lemma 42.1: (11, 12) induces S3 on Y, fies Yo, and is transitive on the remaining three ~ubgroupsYl, Y2, Y3 of Y of o d e r 3. Lemma 42.2: CL,(q) = CG(qB) for i = 1 and 2. Proof: Let Ti= T n CL,(K), so that Ti E Sy12(CL,(x)). There is a conjugate v, of z in &B _< A 3 4 (cf. the discussion in paragraph NG(A3+), we have one of the proof of 41.2) and as CG((K,vi)) C G ( 5B (vi)) = Ti. Now the lemma follows from Exercise 16.6 in [FGT].

<

Lemma 42.3: (11, 12)

S3, so l$ = .1:

Proof:As l1 and l2 are involutions, W = (ill 12) % D2* for some n. As W induces S3on Y, 3 divides a. Let 1 = ll12. It remains to show l3 = 1.

~

~

224

Chapter 14

The Uniqueness of M24, He, and

Ls(2)

But by 42.1, l3 E CG(YB)5 C L i ( X )nCG(Y)= Yo by 42.2. So we may assume = (13). Next by 40.1 there is an involution v E NT(B) inverting Y . By 42.2, CG(YB)= Yo, so [v,I] E CG( YB ) = Yo. Thus there is an element k of order 3 in 1Yo nC(v).Then k centralizes CB(v)= ( t ) ,and hence also B. But (1) is a Sylow 3-group of CG(B)f l N ( Y ) ,so E9 !S ( k ,Yo)= ( I ) S Zg , a contradiction.

42. Groups of type He

225

Lemma 42.7: (1) CG(t)= L(u),where u is an involution in NG(J(T))CG(B)(2) CG(t)is transitive on involutions in uL. (3) u E 7. (4) NG(B)= Y L ( u ) and N G ( B ) / BEi Pr3(4).

Proof: Let M = CG(t),M* = M / ( t ) , K = L(u),and B = (t,b). Then K = N M ( B )by 42.6, so K* = CM.(b*).Next K* has five classes of Lemma 42.4: L = L1 U Lll2Ll = L2 U L211L2. involutions with representatives b*, z* , u*, b*u*, and b*z*. Notice bu is of order 4 while b, z, u, and bz are involutions. By 40.1, (t,z) and Proof: It suffices to show L: E L1 U L112L1.Let K = NLl ( J ( T ) ) Then . (t,bz) contain conjugates of z but B contains no such conjugate. Also K = BOB1B2Y0, where B 5 Bi, Bo = Z ( J ( T ) ) ,At = BOBi, and Bo/B tu E u B , so once we show u E I ,(u,t ) contains no conjugate of z. Use and Bi/B are the Y-invariant Csubgroups of Ai/B. Thus li interchanges the argument in paragraph three of the proof of 41.2 to see u $! P. Bo and Bi via conjugation. We have shown z* and b*z* are not conjugate to b*, u*, or b*u* in Next 12K1112 = 12A2Y0B11112 = 12A2YOllB012= l2A2&l1l2B2 = M*. Also C M ( z )and CM( b t ) are contained in K. Finally z* is not fused = A ~ Y O Z ~Ll12Ll. B ~ Therefore ( ~ 1 1 ~ 2=) 12Kll12Bo h C to b*z* in M* by 7.7 as z* and b*a* are in the center of the Sylow 2Lll2Ll. subgroup CT(t)* of M*, but z* is'not fused to b*z* in G.(CT(t)*). Finally Li = KUKliB3-,. So L: = ( K U K E ~ B=~ K ) '~~~ U ( K Z ~ B ~ ) ' ~ . Thus by 7.3, K* is the unique point of M*/K* fixed by z* and b*z*, We have seen E L112L1.Also K'Z C L2 = K U K12B1 2 and then M* = K* by 7.5. L1 U L112L1. So the proof is complete. So (1) and (3) are established. Part (2)follows as T n L is transitive on the involutions in u(T n L ) by Exercise 2.8. As Y is transitive on Lemma 42.5: LIB 2 L3(4) with B 5 L = Lm. B#, (NG(B): CG(t)(= 3 and NG(B)= YCG(t).Also NG(B) acts on Proof: First B 5 Li = L r , so as L = ( L l ,L2), B 5 L = Lm. Let the projective plane I' constructed during the proof of 42.5, with B the 3 = {L1,L2) and I? = I'(L,F) the coset geometry of Example 4 in kernel of this action. So 5.2 completes the proof of (4). Section 4. The largest subgroup of Ll nL2 normal in Ll and L2 is B , so It follows from 42.7 that B is a TI-set in G. Let A = B~ and form the B is the kernel of the action of L on I?. We show I' is the projective plane A with vertex set A and A adjacent to C if A # C and [A,C ] = 1. graph P over GF(4),and ILIBI = 1L3(4)1.But by 5.2, A U ~ ( P2)L3(4). ~ So We refer to the members of A as root 4-subgroups of G. L = Aut(P)* 2 L3(4). By 42.4, L is 2-transitive on the points and lines of I'. Thus each pair of Lemma 42.8: (1) A ( B ) is an orbit of L of length 105. points is incident with a line and as L2 = (A1,A!) for g E L2 - NL(L1) (2) For A E A ( B ) , (A,B ) = Z ( J ( S ) )for some S E Sy12(G),ABIB while A1 5 L! for each h E L1, L2 is the unique line incident with L1 is the root group of a transvection of LIB, and { A ,B ) = AB n A. and Llg. Similarly each pair of lines is incident with a unique point. Proof: Let A E A(B). Then A 5 CG(B)= L, so conjugating in L we So is a projective plane of order IL2 : Ll n L2( 1 = 4. Then as may take A 5 J ( T ) E Sy12(L).Then as A1 and A2 are the maximal L i I B 5 Aut(r), = P by 18.6. Further there are 21 points of I' as elementary abelian subgroups of J(T), A i. Ai for i = 1 or 2. Then 'I = P, so / L I B /= 21)L1/BI = IL3(4)1. as NG(Ai) controls fusion in Ai and is Ztransitive on B ~ G ( * ' )L, is Lemma 42.6: L = CG(B). transitive on A ( B ) and we may take AB = Z ( J ( T ) ) .Now by 40.1, { A ,B ) = AB n A and JA(B)I= IL : NL(A)l = 105. As Z ( J ( T ) ) / Bis a Proof: Let M = CG(B)and M* = MIB. Assume L # M. As L* is root group in LIB, the proof is complete. transitive on its involutions and CMl(a*)= CM(a)* L*, L* is strongly embedded in M* by 7.3. But L* has 9 . 5 . 7 involutions but no subgroup Lemma 42.9: (1) H is transitive on H n A of order 42. of odd order divisible by 5 9 .7, contradicting 7.6. (2) T = J(T)Q.

c

-

<

226

Chapter 14

The Uniqueness of M24, He, and L5(2)

(3) (t) = B n Q. (4) H has two orbits on 7

n H with representatives t

I

and b E B - Q.

Proof: By 42.7, zG nCG(t) E L = CG(B). Also NG(B) is transitive on zG nL, while Cc(t) has two orbits on zG nL. Therefore H is transitive on H nA and has two orbits on 7 n H with representatives t, b E B. By 39.1.5, tH = TQ = 7 n Q, so b E B - Q and (3) holds. Also lA n HI = lH : NH(B)I = 21H : NH(Z(J(T)))I = 42. Next J(T) = CH(b) is of index 4 in T. But as b $! Q, IQ : CQ(b)l 2 4, so (2) holds.

227

If y = b then as b E Z(L), xb is an involution, so t = xb. Then x = tb E B, contradicting x E zG. Suppose y = s. If A = Ax then ( s x ) ~E A, so as t E (sx), s x = t and x = st = z. There are lsLl = 9 5 - 7 choices for s and hence 9 . 5 7 pairs (x, y) arise is this subcase. If A # Ax but (A, Ax) is abelian then sx is of order 4 but the involution in (sx) is ssx # t. If (A, Ax) E J ( T ) ~then (x, A) is a 2-group and (A, Ax) E Sy12(L), so x E (A, Ax). But then (A, Ax) is abelian. By 42.10, when y = s the only subcase remaining satisfies K = (A,Ax) S L2(4). Then x E NL(K) = BK, and as x E zG, x = cr for some c E B# and some involution r E K. Notice ( x s ) ~= ( r ~ E) K~, so as t E (xs), lrsl is odd and c = t. Now there are 16 21 choices for K by 42.10. Further there are fifteen choices for s in K and as ITS[ is odd, there are twelve choices for r given s, Thus there are 16 21 . 180 pairs (x, y) in this subcase. Finally we have the case y = u. Here x = cr, c E B#, r E A E A(B). If [x, u] = 1 then x = tu. But tu E 7 by 42.7.2. If A = A" then as [x, u] # 1,xu is of order 4 and t = [x, u] = [c,u][T,u]. Thus [r,u] = 1 and c = b or bt. As u induces a field automorphism, CL.(u*) L3(2) has 21 involutions, so u acts on 21 root groups. That is, there are 21 choices for r and two choices b and bt for c, so there are . luLl = 16 15, so there are 32 9 5 7 pairs (x, y) 42 pairs ( x , ~ )Also in this subcase. If A # A" but (A, A") is abelian then xu is of order 4 but t # [x, u]. Suppose (A, A") = J(T). Then A Ai and as Ai <1 T (u, J(T)), (A, A") Ail a contradiction. Finally suppose (A, A") = K E &. Then rru is of order 3 or 5 and the involution t in (xu) is mu. Thus c = b or bt and as A # Au there are six choices for r in K. So there are 12N pairs (x, y) in this subcase, where N is the number of pairs (v, M ) with M E K~ and v E uL n NG(M). ThenN=201LnEl=20-16.21. We have shown that

-

Let E be the set of subgroups K of G with K Z L2(4) and Syl2(K) E A.

-

-

Lemma 42.10: Let A, C be distinct members of A(B). Then L is transitive on s n L of order z4 - 3 7 and one of the following holds:

-

(1) C E A(A) and ABC is L-conjugate to Al or A*. (2) (A, C) E J ( T ) ~(3) (A, C) E E is conjugate to a complement to Ai in NL(Ai).

Proof: Let L* = LIB. By 42.8.2, A(B)* is the set of root groups of L*. Hence by Exercise 14.1, (A*, C*) E for i = 1 or 2, or (A*, C*) E J ( T ) * ~ ,or (A*, C*) % L2(4) is conjugate to a complement to Af in NL. (Af). Further there are 24.3-7subgroups of L* of the last type, and L is transitive on such subgroups. As {A, B) = A B n A and J ( T ) = (A, C) for A E (A n Al) - A2 and C E (A n A2) - All the lemma follows.

-

We now calculate the order of G using the Thompson Order Formula. For u = z or x, let a(u) be the number of pairs (x, y) such that x E zG, y E 7, and u E (xy). By the Thompson Order Formula 7.2:

<

Thus we need to calculate a(z) and a(t). This is tedious but not difficult given the right approach; the details appear in the next two lemmas.

Lemma 42.11: a(t) = 5 . 7 - 9.481. Proof: Let L(u)* = L(u)/B. Thus L(u)* is L3(4) extended by the field iutomorphism u. Let x E zG and y E 7 with t E (xy). By 7.1, (x, y) is dihedral and t E Z((x, y)), so (x, y) 5 CG(t) = L(u). We calculated during the proof of 42.7 that L has four orbits on I n L with represent* tives t, b E B - (t), s E A E A(B), and u, and L has two representatives on zG n L with representatives z = ts and zl = bs. Observe that as t E (xy), t # Y-

42. Groups of type He

I

<

>

Lemma 42.12: ajz) = 4 . 9 . 7 19. Proof: By 42.9, we may take y E B. If [x,y] = 1 then x = yz. But if y E Q then yz E yQ, so y 4 Q. Thus y = b or bt and x = yz. As there are 42 choices for B by 42.9.1, there are 84 pairs (x, y) in this subcase.

228

Chapter 14

He, and L5(2)

The Uniqueness of

Suppose x E N H ( B )but [x,y] f 1. Then (x,y) Z D8 but the involution in (xy) is in B and hence not z. Suppose x E T . Recall Z ( J ( T ) )= B x B1 for some B1 E A. By the previous paragraph, Bx = B1. Thus z = yyx, which forces y E Q and hence y = t. Now there are 64 involutions in T - N H ( B )and 16 of these involutions are in 7,so there are 48 choices for x. As there are 42 choices for B, we have 42.48 pairs ( x ,y ) in this subcase. Suppose y = t but x 4 T. Then as N H ( ( t , z ) )= T and z is in (xt), (t,t x ) Z Ds. This is impossible as [v,vx] 1 for each involution v E Q and each involution x 4 Q. So y @ Q and x @ T . Let H* = H / Q . Then x*y* is of order 3 or 4. Suppose Ix*y*l = 4. Then yx E T and z G ( y , yx). As y # t , we have a contradiction as in paragraphs two and three. So Ix*y*l = 3. Thus there is h of order 3 in (xy) and y inverts h. Now CH(h)= (h) x CQ(h)with CQ(h)2 D8 and [y,CQ(h)]= 1. Further each involution in yCQ(h) is in zG, so the three elements of y(h) are the three choices for elements of H n 7 inverting h, and x = zyh is determined by y and h. Finally there are 7. 27 choices for h, so there are 2' .3 .7 pairs ( x ,y ) in this subcase. We have shown that I=

I

/

i

1 1

I 1

i

/

. .

Lemma 42.13: (GI = 2''. 33 52 73 -17 Proof: We apply the Thompson Order Formula 7.2:

lGI = lcG(~)la(t) + IcG(t)la(z). We have ICG(z)l= 2'

-

-

. 3 7 and ICG(t)l= 29 32 5 .7. So

lG( = (2". 3.7)(5.7.9.481)

.

+ (2'

32 5 7)(4.9.7 19) +

+

= 2''. 33. 5 . 72 (481 6 19) = 210 .33 - 5 .72 .595,

I

I I

/

/ 1I

1

as claimed.

Proof: First; CG((t,X ) ) = B X , so by Exercise 16.6 in [FGT],either C G ( X ) 5 NG(B) or C G ( X ) / X 2 L2(4). But by 42.13, X i: P E Sy15(G) with /PI = 25, so the latter holds. Then C G ( X ) = X x I with B 5 I 2 L2(4).Thus I E 3. Next X 5 K E L n 3. Then there is an involution k E K inverting X. Notice k centralizes BR, where R is of order 3 in NG(B)nCG(K).Then k acts on I = CG(X)OO,so as Aut(I) E S5 and [k,BR] = 1, [k,I]= 1. Then I 5 C G ( ~5 )Cc(B(k)), ~ so I centralizes (X,B(k)) = K. Now by symmetry, K = C G ( X I )for XI E Sy15(I).So as N G ( X )= I N G ( X I ) by a Frattini argument, NG(X)5 NG(K).Hence as K is transitiv~on Sy15(K),K is the unique member of X containing X . Next by 42.8, Z ( J ( T ) )n A = { A ,B ) and we may take A E Sy12(K). So there is g E NG(J(T))with Bg = A and g2 G NG(B).Now Ig E L n 3 and as NG(B)n NG(Bg)is transitive on { D E L n 3 : Bg 5 D),we may take I9 = K. Then as K = CG(I),Kg = CG(Ig)= CG(K)= I , so g interchanges I and K . We conclude (3) holds as ING(BX)I = 240.

Lemma 42.15: (1) Up to conjugation in L there is a unique subgroup S of L with S n l nonempty and S r S3. (2) CG(S)E Sg with CG(S)* E E. Proof: By Exercise 14.2, up to conjugation in L* = L I B there is a unique subgroup So* S3.Further CL.(Sz) = 1. Then So = B x S with S % S3 and if s is an involution in S then sB contains a unique member of I which we take to be s. Hence (1) is established. Let K E L n 5. Then K contains an S3-subgroup and the involutions in K are in 7 ,so we may take S 5 K . By 42.14, CG(K) = I E Let M = CG(S),so that I 5 M. Now CG(S(t))= B(u) 2 D8, with u E 7 inducing a field automorphism on L*. Thus I(u) E &. Then CI(u)Z S3, so as CG(S(t))is a 2-group, u 4 t M .Hence by a transfer argument (cf. 37.4 in [FGT]) u 6 o ~ ( M ) Then . B = Co2(M)(t),so O ~ ( M=) I by Exercise 16.6 in [FGT].Thus (2) holds.

-

.;

Lemma 42.16: Let X be of order 3 in L. Then (1) N G ( X ) / X E S7 with C G ( X )= CG(X)". (2) In the representation ofNG(X)on (1,. ,7), transpositions and products of two distinct commuting transpositions are in 7, while products of three distinct commuting transpositions are in zG.

..

Lemma 42.14: Let X be of order 5 in L. Then (1) X is contained in a unique member K of &. (2) C G ( X )= CG(K)E E. (3) NG(KCG(K))interchanges K and CG(K)and is of index 2 in S5wrZ2.

229

42. Groups of type H e

I

I 1

Proof: Let M = N G ( X )and M* = MIX. Then CM.(t*) 2 D8 x S3. Let s E 7n L with X ( s ) r S3. By 42.15, CM*(s*)2 2 2 x $5. Next we may take XD1 9 NG(D1)for some conjugate Dl of A1 with

Chapter 14 The Uniqueness of

230

M2q1 He,

and Ls(2)

43. The mot 4-group graph for He

<

and A is regular on A n (A34 - Z(J(T))). SOG is transitive on the set S of pairs (A, C) from A x A with Ag < A, and (A, C)g = J(T), with l6lG : NG(T)I = IS1 = JA~N;, where N? = ~A:(B)I. Thus

(s, B) 5 NM(Dl) = MI. Then Ml 2 s6/23 with X CMl(X)03 by 40.4. Let v = st and R E S Y ~ ~ ( C ~ ( ( S ,Then ~ ) ) )v. E zG and CMl (v)* 2 2 x Sq with 02(CM1(v)) 5 02(CG(V)). Hence CM, (v) is maximal among subgroups K of CG(v) with R E Sy12(K). But @(R) = (t), so as R E Sy12(CM(t)), we conclude R E Syl2(M). Hence CM,( 4 = CM (v). Therefore t*, s*,v* are representatives for the conjugacy classes of involutions of M*. Further we are in a position to apply the Thompson Order Formula 7.2. Namely the parameters in the order formula for M* are the same as those for S7 since the involution centralizers and fusion pattern in M* are the same as in S7. Therefore by the Order Formula, IM*J = /S71.Then as IM* : M;( = 7 and M: 2 86, we conclude M* E 5'7. Further in the representation of M* on {I,. . . ,7), t* is the product of two transpositions as CM=(t*) 2 S3x Dg ,s*is a transposition as CM. (s*) % 2 2 x S5, and v* is the product of three transpositions as CM.(v*) 2 z2 X 6'4. SO (2) holds.

43. The root 4-group graph for He In this section we continue the hypotheses and notation of Section 42. In particular A is the commuting graph on the set of root 4-subgroups of G. We will show that A .is 4generated in the sense of Section 34. Throughout this section let M = NG(B).

Lemma 43.1: M has seven orbits on A:

(4

{B). A(B) of order 105. A ~ ( B )= {A E A : (A,B) = J(S), S E Sy12(G), B D 5 S, D E A?), i = 1,2, of order 8 . 105. firther A(A, B) = A n Z(J(S)) is of order 2. (4) A ~ ( B ) = {A E A : (A, B) E 8) of order 26 21. Further A(A, B) = Sy12(I) is of order 5, where I = CG((A,B)) E Z. (5) A ~ ( B )= {A E A : (A, B) 2 Sq) of order 24 . 32 .5. Further NM(A) = NA(B) x NB(A) x L3(2). (6) A ~ ( B )consisting of those A E A such that (A, B) = CG(X) -. for some X of order 3, and of order 27 -35. Further NM(A) = NM(R) E ~ 4 1 3 for ' ~ some ~ R E Sy13(G).

(2) (3)

-

-

-

<

Proof; By 42.8, A(B) is an orbit of M of length 105. Next by 42.8, A n (Ai - Z(J(T))) is of order 4 and by 42.10, if A E A n (Ai - Z(J(T))) and C E A n (A3-i - Z(J(T))) then (A, C) = J(T)

231

and M is transitive on A:(B). As CG(J(T)) = Z(J(T)), A(A,C) = A n Z(J(T)) is of order 2 by 42.8. Next if A E A$(B) then by 42.14, CG((A,B)) = K E E f l L and (A, B) = CG(K). So as L is transitive on Z n L by 42.10, and as B is regular on (A, B) n A - {B), L is transitive on A ~ ( B ) Further . I A ~ ( B= )I 413 n LI = 26.21 by 42.10. By 42.7 there is u = t g E CG(t) - L. Then B acts on I = (u,t) and t E CG(u) 5 NG(Bg), so [Bg, t] = (u) and hence Bg acts on I. Thus (B, Bg) acts on I and from that action we conclude (B, Bg) 2 Sq. That is, Bg E A:(B). Conversely if A E A ~ ( B )then NA(B) = (v) is of order 2 and CB(v) = ( c ) for some c E B#. Conjugating in M we may take c = t. Then as [v, B] # 1, v 4 L, so by 42.7.2, v E uL. Hence s orbit under M. Further A = B(v) is L-conjugate to Bg, so A j ( ~ ) . i an ~ A ~ ( B= ) IIuM1 = 2 4 - 3 2 - 5and NM(Bg) =CM(u) = (u) x (t) x L3(2) as u induces a field automorphism on L I B (cf. 42.7.4). By Exercise 14.2, L is transitive on its subgroups X of order 3 and by 42.16, NG(X)/X E S7. Represent I = NG(X) on C = {I,... ,7) and let I * = I I X . Then B* is the 4-group in I*moving some set C(B) of order 4. Further CG(X) = (A, B) for some A E A n I if and only ~ B if C(A) U C(B) = C. Then NM(A) = NlnM(A) S ~ 4 / 3 l +and ) I. Hence (A;(B)~ = 4 1 ~ =~271.35 and M is is regular on A ~ ( B n transitive on A ~ ( B ) . To complete the proof, observe that the sum of the orders of the seven orbits listed in 43.1 is 8,330 = 2 . 5 72 17 = IG : MI = 1A1, so these are all the orbits of M on A.

Lemma 43.2: All triangles in A are fused under G into A1 or A2. Proof: Let ABC be a triangle. We may take AB = Z(J(T)) by 42.8. Then C < CG(AB) = J(T), so as A1 and A2 are the maximal elementary abelian subgroups of J(T), ABC _< Ai for i = 1 or 2. Lemma 43.3: A ~ ( B )=

u:=~A:(B).

Proof: Let A E A. If A E A:(B) then A(A, B) # 0 by 43.1 and hence A ~ ( B )C A 2 ( ~ )Conversely . assume d(A,B) = 2 and let C E A(A, B). Then C B I B is a root group of LIB. But by 43.1, CLIB(A) contains a root group of L I B only if A E A:(B) for 1 < i < 3.

232

Chapter 14

The Uniqueness of Mz4, He, and Ls(2)

Lemma 43.4: Each square in A is fused under G into K I , where K, I are a pair of commuting members of 8. Proof: If p = Bo Bq is a square then B2 E A 2 ( ~ oand ) B1, B3 E A(Bo,Bz), with B1 E A 2 ( ~ 3 )so, the remark follows from 43.1 and 43.3. For example, if C2 E A:(&) for i = 1 or 2, then by 43.1.3, A(Bo,B2)= { A ,B ) with A E A(B) rather than A E A ~ ( B ) .

Lemma 43.5: If A E A:(B) for i = 1 or Z1 then B E A$-~(A). Proof: By 43.2, (A,B ) = J ( S ) for some S E Sy12(G)and B with D E. A?. But then A E D' S with D' E

AE~.

<


Remark 43.6. We sometimes view M/B as acting as PrL3(4)on its natural %dimensional projective module W over GF(4).Observe that in this representation, each X E E n L is a characteristic subgroup of the stabilizer of some decomposition W = c ( X )@ a ( X )of W as the sum of a point c ( X )and a line a(X). Similarly each A E A ( B ) is the root group of transvections determined F'urther we by an incident-point-line pair (el(A),e2(A))= (c(A),a(A)). may choose notation so that A1 is the group of transvections of V with a common center and A2 is the group of transvections with a common axis. Then C E A(A) if and only if c(C)= c(A) or a(C)= a(A) and in while in the second ABC E A:. the first case ABC E Notice A 5 X if and only if c(A)E a ( X ) and a(A)= c(X) c(A). Finally C E A ~ ( Aif)and only if ei(A)E e3-i(C)but ei(C)$ e3-i(A).

AF,

Lemma 43.7: (1) A ~ ( B=) AQ(B) (2) A is of diameter 3.

+

U A$(B).

Proof: Observe that 43.1,43.3, and (1)imply (2), so it remains to prove (1).Further d(B,B') 1 3 for B' E A:(B), i = 1,2, by 43.3, and by 43.1, A:(B) is an orbit of NG(B)on A, so it suffices to exhibit A E A?(B) with d(A,B ) 5 3. Now by 43.8, NL(A1) is the split extension of Al by I E E. Next by 40.5,

o 2 ( ~ G ( ~ 1=) O ) *~ ( N G ( A ~ ) ) / ~ ~ , ~Gi (A6NC(~~)) Then 0 2 ( ~ ~ ( A 1is)represented )* as Ag on a set St of six points and if C1 E A n I and Cz E such that (Cl,C2)fixes no point of a, then (Cl,Cz)* Z Sq. Hence by 43.1, C2 E A!(c~). Therefore there exist . ABlB2B is a B-invariant D E A? and A E NG(D)n A ~ ( B )Further path in A, where B1 E A n CD(A)and B2 E A n CD(B).This shows d(A,B) 1 3 for A E A;(B).

43. The root 4-group graph for He

233

Let A E A ~ ( Band ) adopt the convention of Remark 43.6. Thus BNM(A)is the stabilizer of a nonincident-point-line pair ( p , I ) o f W . Then one orbit of NM(A)on A ( B ) consists of those C with (c(C), a(C)) = (q,k), where q # p, k # 1, q I , and p 4 k. Then Y = NM( A )1NM 1 ( C )2 S3 with O3( Y )= X having eigenspaces Wl and W2 on W of dimension 1 and 2, respectively. Then X $ L,so X is faithful on B and q < Wz and Wl 5 k, so X centralizes C. We claim C E Ai(A); then as d(A,C) 3, the lemma is established. ) X is faithful on B, X is also faithful on A. As As A E A ~ ( Band C $ NM(A),C 4 A(A). As [X,C]= 1 and A = [A,X],C 4 Ai(A). Finally we check using the representation of NG(C)on A(C) supplied , there exists no B E A(C) such by Remark 43.6 that if A E A ~ ( c )then that Y = NM(A)n NG(C)E S3 with 0 3 ( Y )5 CG(C).

<

In the remainder of this section let w = N C , ( ~ ) be the invariant equivalence relation on the paths of A generated by C4(A)(cf. Chapter 12).

Lemma 43.8: Let p = Bo .. .Bq be a path with Bg E A ~ ( B ~Assume ). there exists no path q of length 3 with p q. Then (1) Bq E A ; ( B ~ for ) j = 1 or 2. c(C) $ a ( X ) and (2) For C E A(B2,B4) and X = CG((Bo,B2)), 4x1 4 a ( c ) . (3) There exist three P E A ; ( B ~n) X with BoPB2B3B4 p. N

Proof: Without loss of generality B = B2;we adopt the notation of REmark 43.6. As p is equivalent to no path of length 3,34.10.1 says Bi+2 € A ~ ( Bfor ~ )0 < i < 3. Similarly by 34.10.2, d(A(Bo,B), A(B4,B ) ) > 1. For i = 0,4, let Xi = CL(B)i,and if Bi E A:(B) let cj = c(Xi) and a; = a(Xi). For example, by hypothesis, Bo E A ~ ( B )Assume . B4 E A$(B) and let c be a point on a0 n a4 and Ci E Xi n A with c(Ci) = c. Then by Remark 43.6, C4 E c;, contradicting d(A(B,Bo), A(B,B4)) > 1. So Bq E A ~ ( Bfor ) j = 1,2, and without loss of generality j = 1 and A(B, B4) = {C,D) with BCD = A1. Notice B3 = C or D by 43.1. As d(C,A(B, Bo))> 1, c(C) 4 a0 and QJ 4 a(C). Thus (1) and (2) are established. Next let P E A n Xo. If c(B3) 4 a(P) and a(P) 4 c(B3) then P E A g ( ~ 3and ) p BoPBB3B4. But by ( 2 ) and as a(P)= QJ 4- c(P)with c(P) € ao, there exists a unique P E A n X Owith c(B3)E a(P);namely that P with c(P) the projection of c(B3)on ao. Similarly the unique P' E A n Xo with c(P) E a(B3) satisfies c(P) = a(Bo)n ag. So (3) holds.

1 234

Chapter 14

The Uniqueness of Mz4,H e , and Ls(2)

In the remainder of this section assume A is not Cgenerated, in the sense of Section 34. Thus there exists a nontrivial cycle p = Bo - Bn. Choose p with n minimal subject to this constraint. Then by 34.4, p is an n-gon.

.-

Lemma 43.9: Bi+z

E A!(~)(B~)

I

43. The root 4-group graph for He

!

the structure of NG(Al), Y = CG(E) nNG(A1) is the split extension of Al by (BoyB3) S4 and E# C_ ZG.

N

Lemma 43.10: (1) j(i) = 3 - j(i (2) n is even. (3) n > 7.

<

I

Proof: Without loss of generality B2 = B. By 43.9, we can assume B f A;(Bo); that is, j(0) = 2. Then by 43.5, Bo E A ~ ( B ) .Then (B, Bo) = J(S) for some S E Sy12(G) and Z(J(S)) = BIB; with {B1,Bi} = A(B, Bo) and BBIB; E A?. Similarly (B,B4) = J(R) and Z(J(R)) = B3Bi with BB3Bi E A? for i = 1 or 2. Finally (B1, B3) = J(T). We claim J ( T ) = B1BiBB3Bi. Assume not; then without loss of generality B i $ T. Then as BBIBi is the member of A? containing BB1, B1 5 A2, so B3 E A % ( B ~ )Similarly . B3 E A~(B:). That is, c(B3) E a(B1) and c(B1) $ a(B3). But p BOBiB2...Bn, SO also (B',, B3) = J(T1) and as B3 E A~(B;), B1 $ TI, so c(B3) E a(B;). Therefore c(B3) E a(Bl)na(Bi) = c(B1), so c(B3) = c(B1), contradicting [Bl, B31# 1. Hence the claim is established. In particular Z((B, Bo))B = A1 = Z((Bil B3))Bi and A(Bo,B) E Z((B, Bo)) E J(T). By symmetry A(BTB4) C J(T), so (B1l 83) = J(T) = (A(BolBz), A(%, B4)). - - Now as Z((B1,B3))Bl = All j(1) = 1. That is, (1) holds. Also by (I), j(2r) = j(0) = 3 - j(1) = 3 - j(2s + 1) for all integers r, s, so (2) holds. in S and Y = (Ah, Az). Next let A$ = BoBIBi be the member of Then A1 A Y and A1 n A2 and A1 n A; are of index 4 in All so E = Az flA1 n A', is of order at least 4 and contained in Z(Y). Then from

-

I

. ..

N

<

(a) NG(J(S)) r l NG(B) is transitive on those D E ~ fwith : B3 D NG(Az) and (A1,D)/A2 S4(b) There exists a hexagon q = Co. C6 with 'Ci+2 E A;(il (Ci), C,+3 E A;(C,) for 0 i 5 6 and (q) = H.

<

<

I

--

Now if q satisfies (b) then conjugating in H we may take Ci = Bi for 0 5 i 3, and then the hypotheses of (a) are satisfied with D, = Z(J(T))C4. Similarly our typical nontrivial hexagon p satisfies the hypotheses of (a) with Dp = Z(J(T))Bq, so by (a), (p) = (Dp, J(S)) E (D,, J ( s ) ) ~ = = H ~Therefore . we have shown that if p is nontrivial then E n E # 1, to complete the proof. So it remains to establish (a) and (b). For (b), let H* = H/Q and q* = C t C,* a hexagon in the commuting graph of involutions of H*. Let q be any lift of q* in H to A with Co = C6. Let G; = Gz/02,3(Gz) and observe that G; acts on A? as Spq(2) on the fifteen points of its symplectic space with (A;, D*) g Sq, A5, for D orthogonal, not orthogonal to Al in this space, respectively. Further NG,(B3)* E 55 has two orbits of length 5, 10 on A? with the set 0 of members of A? containing B3 of order 5. So as B3 $ A1 and NG2(A1) = NG2(J(T)),a subgroup X of CG, (B3)nNG(J(T)) of order 3 is transitive on the three members D of 0 with (D*,A;) G Sq. But as [B3, X] = 11 [X, NG(Al)"] A11 so X NG(J(S)).

<

(dG

N

AF

< <

AB

-

+ 1) for all 0 < i 5 n.

As n 2 5 is even, t o prove (3) and complete the proof of the lemma, i 3). Also K = we may assume n = 6. Notice Y = (Bi : 0 CG((BolB3)) g L3(2) by 43.1.5 and E is a Csubgroup of K. As E# C zG,we may assume z E E. Then as 3 divides the order of Y 5 CG(E), E Q. Let H* = H/Q.By 42.9, B t r Z2 for all i, 0 i 3, so Y* GZ S4. Then from the structure of H , (YI = ICH(E)I, so Y = CH(E) = CG(E). As A2 5 Y and A2 is weakly closed in NG(A2), there . P = (B3, Bq, B5,Bo) satisfies is no Y-invariant member of ~ fBut the hypotheses of Y except that there is a member of normal in P, so E = Z(P) must live in the second class of Cgroups of K. In particular if r = B3C4C5B0 is a path such that El = Z((r)) E BK,then P - BOB5B4B3r 1. Thus to complete the proof it suffices t o show we can choose r subject to this constraint such that Bo B3 . r 1. Indeed we will show that this holds when Z((r)) n E = 1. Toward that end we let Gz = NG(A2) and show:

< <

with j(i) = 1 or 2 for each i.

Proof: Assume not. Then without loss of generality B:! E A~(Bo).Now by minimality of n and 34.10.3, Bo -..B4 is not equivalent to any path of length 3, so by 43.8, Bq E A?(BZ) for j = 1 or 2, and ~A:(B~)n A(BolB2)1 2 3. By symmetry, Bn-z E A;(BO) for k = 1 or 2, and I A $ ( B ~ - ~n ) A(Bo,Bz)l 2 3. Hence as lA(Bo, Bz)l = 5, there exists P E A ~ ( B ~ B3) - ~ ,n A(Bo, B2). NOWby 34.10.4, p BoPBl. Bn, so without loss of generality P = B1. But then as P E A ~ ( B ~ B3), -~, 43.8.1 supplies a contradiction.

235

---

I

<

<

236

Chapter 14

44.

The Uniqueness of M241He, and L5(2)

Let I = NG(J(T)) and E = Z(J(T));notice I = G12. Now I / E = Il/E x 12/E,where Ii/E r S4.So as S4 = Aut(S4), the subgroup D of Aut(1) normalizing I1factors as D = IA, where A = CAut(I)(I/E). Next for X E Syls(I), X = X1 x X2 with Xi E Sy13(Ii) and E = [E,Xi]. Then E = I n A and A = ECA(X)with CA(X) = CA(NI(X)) 5 CA(E) as Nr(X) is absolutely irreducible on E. Similarly [CJ(T)(Xi), CA(X)] <_ CE(Xi) = 1 as NI(X) is absolutely irreducible on CJ(T)(Xi). Hence as J(T) = E C . J ( ~ ) ( X ~ ) C J ( ~ ) ( [CA(X), X ~ ) , J(T)] = 1. But then CA(X) centralizes I = N I ( X )J(T),SO CA(X) = 1. Thus D = ICA(X) = I. That is, JAut(I) : I)= 2 and I is the stabilizer in Aut(I) of I1 and I 2 and hence also the stabilizer of A1 and A2. In particular as Ajai = Aj for j = 1,2 and i = 1,2,3, ala;l induces an inner automorphism on I = G12, so by Exercise 14.3 we may pick a1 and a 2 so as to agree on I. Similarly GI23 is a well-determined subgroup of index 2 of I and the argument above shows I is the stabilizer in Aut(G123) of Al and A2, and hence by Exercise 14.3 we may choose 0 3 so that a 3 agrees with a1 and a 2 on G123. Finally Gs4 = NG4(B) E Z2/(A4 x A5) and Gi4 = G24 = G124 = NG4(E) is a well-determined subgroup of index 2 in S4wrZ2.Then it is an exercise to show Aut(GQ4)= NAut(G4)(G34)S S5 X S4, Aut(G124) = N ~ ~ t ( ~ + ) ( G f22 S4w~Z3, 4) and C A ~ ~ ((G1234) G ~ ) = 1, so by Exercise 14.3 we can choose a 4 so that a = (ai : i E I ) defines an isomorphism of the amalgams A(3) and d(F).

Lemma 43.11: A is 4-genemted. Proof: By 43.7.2 and 34.5, n 5 7. Then 43.10.3 supplies a contradiction.

44. The uniqueness of groups of type H e In this section we continue the hypotheses and notation of Section 42. Our aim is to show that, up to isomorphism, there is at most one group d = M = NG(B). Let B 5 K i E Z of type He. Let Gi = NG(Ai) ~ n G3 and K2 = CG(Kl). Choose K1 so that Z(J(T)) E Syl~(KlK2).Let G4 = NG(KIK2). Finally let I = (1,. . ,4) and 3 = (Gi : i E I ) and I? = I?(G,F).

.

Lemma 44.1: Up to isomorphism there is at most one quasisimple group L with Z(L) zz E4 and L/Z(L) ?2L3(4). Proof: Let be the universal covering group of L3(4) and 2 = z(E) (cf. Section 33 in (FGT]). Let % = (s2: s E 2 ) and = i/&. Then is the Iargest perfect central extension of L3(4) whose center 2 is an elementary abelian 2-group, so there is a surjection of onto L. Thus it suffices to show 121< 4. Let 112 be a maximal parabolic of 2/8. Then I contains a Sylow so by 33.11 in [FGT], I is a perfect central extension 2-subgroup of of I/Z. Let E = 0 2 ( I ) . Then E = E/Z is the natural module for I*= I/E 2 L2(4). As E is the natural module for I*, I is transitive on E#, so each element in E# is an involution, and hence E is elementary abelian. Hence there are involutions in I - E, so as I is perfect, E = [E, I]. ( ~ ,by 17.12 in [FGT]. But as a GF(4)-module, Thus 121 <_ I H ~ E)I 1 = dirn(fl1(f, E)) by Exercise 14.6, so indeed 121<_ l ~ l ( i , E ) = I 4.

e

e,

Lemma 44.3: Up to isomorphism there is at most one gmup of type He.

See Example 36.1 for the definition of the amalgam A(F) of the family F of subgroups of G.

Lemma 44.2: If G is a group of type He then the amalgams A(F) and A($) are isomorphic. Proof: By 44.1, L is determined up to isomorphism, so as M is the split extension of L by S3 with the involution inducing a field automorphism on LIB, there is an isomorphism as : M + M . Similarly for i = 1,2, Gi is the split extension of Ai by S6/Z3 and in particular is determined up to isomorphism. So there exist isomorphiims ai : Gi --, Gifor i = 1,2. Finally by 42.14, G4 is a determined subgroup of index 2 of the wreath product L2(4)wrZ2, so there is an isomorphism a 4 : G4 -+

c4.

The uniqueness of groups of type He

I

I

Proof: Let A = A(3). Applying the construction of Example 36.2, we obtain a faithful completion P : A 4 G and by 36.4, a surjection rp : G(A) -+ G of the universal completion G(A) onto G. Observe that A = A(P, 1) is the collinearity graph of I' defined in Section 36. We now appeal to 36.7 applied to the completion P : A -+ G and the collinearity graph A. Observe that hypotheses (*) and (**) of 36.7 are satisfied. For example, 43.2, 43.4, and 43.11 say (**) holds. So by 36.7, G(A) Z G. Next by 44.2, if G is a group of type He then A(3) r d ( F ) . So also G(A) S G. Theorem 44.4: Assume G is a finite group containing an involution z such that CG(z) is isomorphic to the centralizer of a tmnsvection in

238

Chapter 14

L5(2) and z is not weakly closed i n 0 2 ( C G ( z ) )with respect to G. Then one of the following holds: (1) G is isomorphic to the maximal parabolic of L5(2) stabilizing a point of the natural module for L5(2). (2) G L5(2). (3) G r M z 4 . (4) G E H e .

I

,

ProoE By 39.10 either (1) holds or zG r l Q = U1# U Uf, and we may assume the latter. Then by 40.5, G is of type L5(2), M24, or He. In the first two cases (2) or (3) holds by 41.5. In the third, G is determined up to isomorphism by 44.3, and hence as the group He of 32.5 is of type He, (4) holds. Remarks. Held [He] was the first to consider groups G possessing an involution z such that C G ( z ) is isomorphic to the centralizer of a transvection in L5(2). He proves that if G is simple then G is isomorphic to L5(2) or MZ4,or G has order 21°. 33. 5' -73 - 17 and determines much of the local structure of G. G. Higrnan and J. MacKay then proved the existence and uniqueness of H e using the machine. Our treatment of the problem bears little resemblance to Held's. For one thing, more local group theoretic techniques are available today than in 1969, making possible a more conceptual, less computational proof. For another, Held felt free to quote extensively from the literature, whereas the treatment here is self-contained. And of course we actually establish the uniqueness of He.

I

,

(6) { B E R : center(A) 4 a x i s ( B ) and center(B) $! a x i s ( A ) ) of order 64. Further ( A ,B ) S L2(4). 2. Let L = L3(4). Prove ( 1 ) L is transitive on it subgroups of order 3. ( 2 ) L is transitive on its subgroups isomorphic t o S3, and if So is such a subgroup then CL(So) = 1. 3. Let A , B be subgroups of a group G I 2 , B be subgroups of a group G, a : A + A, ,f3 : B 3 B be group isomorphisms, I = A n B , I = An B, l a = I = I b , and 7 = a@-l E A u t ( I ) be the automorphism induced Prove on I by (1) If 7 E A u t B ( I ) then for some c E B, Po = ec/3 : B -t B with a = Po on I, where tc: b +-+ bc is the automorphiim of B induced by c. ( 2 ) Let ai : Ai -+ Ail 1 5 i 5 n, be isomorphisms for subgroups Ai 5 G , Ai < G, Ii = A i n B , I. = Ainii, J = n , % n B , J = AinB, I.a.- I . - I ,P, . and Jai = = Jp. Assume a, = aj on J for all i, j, AutB(J,) = Aut(Ji) for each i, A u t s ( J ) = A u t ( J ) , and C B ( J )= 1. Prove there exists c E B such that tcP = Po = ai on Ii for each i. 4. Let V be a &dimensional orthogonal space over the field of order 2 and L3(2) 2 L IG = O ( V ) . Prove (1) V has sign +1 and L stabilizes a maximal totally singular subspace U of V . ( 2 ) P = NG(U) is transitive on the set W of eight totally singular complements W to U in V. Further N p ( W ) = L1 % L. (3) If L E then L has orbits Oi, i = 1,2,3, of length 7,7,21 on singular points of V. (4) If L $! ~f then L is transitive on W, L n L1 is transitive on Oi for each i, and L has two orbits of length 7,28 on singular points of V . 5. Let G be a finite group, z an involution in G , and H = C G ( z ) ,and assume Q = F*(H) is extraspecial of order 128 with H / Q r L3(2) and 1zG n QI = 29. Let Go = L5(2), zo a transvection in Go, and Ho = CG, (zO).Prove ( 1 ) H / ( z ) E Ho/(zo), so in particular 39.1.2 and 39.1.5 hold. (2) H is transitive on the set W of pairs ( W l , W2)such that W i is a hyperplane of Ui with z 4 Wi. (3) Let (Wl,W z ) E W and L = N H ( W l ,Wz).Then L % SL2(7) or L3(2).

,

-,

LP

Exercises 1. Let L = L3(4), R be the set of root groups of transvections in L , and A E R. Prove N L ( A ) has six orbits on R: (1) { A ) . ( 2 ) { A # B E 52 : center(A) = center(B)) of order 4. Further (A,B)=AxB. (3) { A # B E $2 : axis(A) = a x i s ( B ) ) of order 4. Further ( A ,B ) = A x B. (4) { B E 52 : center(B) # center(A) E a x i s ( B ) # a x i s ( A ) ) of order 16. Further ( A ,B ) E Sy12(L). ( 5 ) { B E R : center(A) # center(B) E axis(A) # a x i s ( B ) ) of order 16. Further ( A ,B ) E Sy12(L).

239

Exercises

T h e Uniqueness of MZ4,He, and L s ( 2 )

1

ni

240

Chapter 14

The Uniqueness of M24t He, and L5(2)

(4) If L E SL2(7), s is an involution in Q - (z),and S E Sy12(CH(s)), then a2(s) = (z). (5) H 2 Ho. (Hint: Use Exercise 14.4 to prove ( I ) . ) 6. Let L r L2(4) and V the natural GF(4)L-module. Prove

!

I

I

dim(lT1 (L,V)) = 1.

Chapter 15

The Group U4(3)

*

;

,

;

The group U4(3) is the image PSU4(3) of the special unitary group SU4(3) in PSL4(9). Thus this unitary group is a classical group (cf. Chapter 7 in [FGT] for a discussion of the classical groups and unitary groups in particular), but it exhibits sporadic behavior and is crucial in the study of various sporadic groups. For example, we saw in Exercise 9.4 that U4(3)is the stabilizer of a point in the rank 3 representation of the McLaughlin group. Further we find in Chapter 16 that U4(3)is a section in a blocal of Suz. In Section 45 we prove a uniqueness theorem for U4(3)characterizing the group in terms of the centralizer of an involution. In the end we construct a geometric complex for the group which we prove to be isomorphic to the complex of singular points and lines in the Pdimensional unitary space over the field of order 9. This last part of the proof can also be used in Exercise 9.4 to identify the point stabilizer of Mc in terms of the information in Lemma 24.7. In Chapter 16 we use the characterization of Section 45 t o pin down the structure of the centralizer of a certain subgroup of order 3 in groups of type Suz. This result is used in turn in the proof of the uniqueness of Suz and in 26.6 to determine the structure of the centralizer of the corresponding subgroup of order 3 in Col, while 26.6 is used in 32.4 to establish the existence of a subgroup of the Monster of type F24.

Chapter 15 The Group U4(3)

242

In this section we assume the following hypothesis: Hypothesis 45.1: G is a finite group, z is an involution in G , H = CG(z), 0 2 ( H ) = Hi H2 with Hi r SL2(3),Hi 9 0 2 ( ~and ) ,H I n Hz = ( z ) , Q = 0 2 ( 0 2 (=~F*(H), )) and z is not weakly closed in Q with respect to G. Let H = H / ( z ) and H* = H/Q. Lemma 45.2: (1) [ H I ,H2] = 1 and H permutes { H I ,HZ). (2) Q 2 Q;. (3) Q is a 4-dimensional orthogonal space of sign -1 over GF(2) and E' 0, ( 2 ) . H* 5 (4) 0 2 ( H )is transitive on the eighteen involutions i n Q - ( z ) .

0(v)

Proof: As Hi 9 0 2 ( H ) ,[ H I Hz] , I Hl n H2 = ( z ) < Z(Hi), so as Hi = o ~ ( H ~[)H, I ,Hz] = 1. Then { H I ,H2) is the set of normal subgroups of o ~ ( H isomorphic ) to SL2(3) and hence is permuted by H. That is, (1) holds. F'urther ( 1 ) implies (2) while (2), 8.3, and 8.4 imply (3). By (3) there are 18 involutions in Q - ( z ) and if t is such an involution then l o 2 ( H ) : C0'2(H)(t)l= 18, so (4) holdsBy Hypotheses 45.1 and 45.2, there exists u = zg E Q (u,Z ) = E Q T E SyZ2(H).

- ( z ) with

Lemma 45.3: Assume Q n Qg = E and let N = (Q,Qg) and R = CQ(u)CQg( 2 ) . Then (1) N is the split extension of R by Y E S3 with C R ( Y ) = 1.

(2) R is isomo~phicto a Sylow 2-subgroup of L3(4) and both members of A(R) are normal in N . (3) R*o~(H*) is the subgroup of O q ( 2 ) of index 2 whose Sylow 2subgroup R* is a 4-group containing no transvections. (4) All involutions in R O ~ ( H are ) i n zG. (5) For X E Syl3(H), NR02(H)(X) = X T X With TX r D8 and ( z ) = C T (XI. ~ Proof: By 8.15, N J R E S3 and R is special with center E and R / E = R 1 / E @ R 2 / E the sum of natural modules for N / R . Let s E Q - R be an involution. By the Baer-Suzuki Theorem (cf. 39.6 in [FGT]),s inverts some element y of order 3 in N , so Y = js,y ) e!S3 is a complement to R in N . Further [ R I E ,s] = ( Q n R ) / E and Q n R = CQ( E ) 2 Z2 X D8.

I

Next R* E R / ( R n Q ) 2' E4 and [R,Q n R] 5 E , so [EL,R*] 2 E. Now if r* E R* induces a transvection on Q then [Q,r*] is a nonsingular point of Q. However, is singular and [EL,r*] < E , so either E = [ ~ l , r *5] [&,r*]or ( ~ ~ , r=* 0] so EL is the axis of r* and hence E = ( E ~= )[Q,T*]. ~ Therefore R* contains no transvection and (3) holds. Next there is exactly one more irreducible R 3 / E for Y on R / E and R, = ( r y ) ~where , 1 and ri, 1 5 i 5 3, are coset representatives for E in Q n R. Thus we may pick rl and ~2 to be involutions and r3 # of order 4. Then as Y is transitive on (R,/E)#,all elements in Ri are involutions for i = 1,2, while Stl(R3) = E. In particular R1 r R2 r El6. If x E C R I ( r 2 )- E then x* # 1 and [x,Q n R] = 1, so x* induces a transvection on Q, contrary to (3). Hence CR,(r2) = E and then as Y is transitive on ( R ~ / E ) #CR, , ( r ) = E for all r E R2 - E. It is now an easy exercise to verify that R is isomorphic to a Sylow 2-subgroup of L3(4) with {R1,R 2 ) = A(R). By construction, R, 9 N . So (2) is established. By (2), if 1 # r* E R* then [Q,rk]= C*(rk), so by Exercise 2.8, if ? is an involution then all involutions in FQ are conjugate to f . Thus all involutions in rQ are contained in ~ ( z )Therefore . each involution in R O ~ ( H) Q is conjugate to si, 1 i 5 3, where 1 and si are coset representatives for E in Qg n R. Thus 45.2.4 implies (4).Recall also that two of the si are involutions and the third is of order 4. Finally for X E Sy13(H), R O ~ ( H = ) Q N R O ~ ( H l ( Xby ) a Frattini argument with NQ(X)= ( z ) , so N R O ~ ( H ) ( X=) X T X 7 where Tx E s ~ , ! ~ ( N ~ ~ a ( with ~ ) ( Tx X) /) ( z )Z E4. We have seen that if 1 and t i , 1 5 i 3, are coset representatives for ( z ) in Tx then as iiis an involution, two of the ti are involutions and the third is of order 4. Thus T X D8, completing the proof of (5). Lemma 45.4: Either (1) Q n Q g = E , or (2) Q n ~g = v r ~ 8 there , exists zk E v - E, and i f we set N = ( Q ,Qg, Q ~ )N,/ V r L3(2) is faithful on V .

<

<

Proof: As Q has width 2, 8.15.7 says V = Q n Qg is of rank at most 3. So as E 5 V , either V = E or V Si 8 8 , and we may assume the latter. By 45.2.4, V# C zG, so there is zk E V - E . Now is a totally singular line in Q, so as each such line admits the action of a subgroup of 0 2 ( H )of order 3, N H ( V ) is transitive on Thus V 5 Q~ for zk E V - E, so 8.16 completes the proof, since Q ~ H ~ ~ H ~ = C ~ ( V ) = V .

v

v#.

244

Chapter 15 The Group U4(3)

Define G to be of type U4(3) if Hypothesis 45.1 is satisfied with IQ fl Qg1 = 4 for z9 E Q - ( z ) and IHI = 27 . 32. In the remainder of this section assume G is of type U4(3).We will show G % U4(3). Define R, &, i = 1,2, and N as in the proof of 45.3. Recall T E Sy12(H)with E g T .

Lemma 45.5: (1) H = R O ~ ( H ) . (2) T E Syl2(G), T E Sylz(N), and IT1 = g7. (3) R = J ( T ) , A ( T ) = { R l ,Rz), and & is weakly closed in T with respect to G. (4) G has one conjugacy class of involutions. Proof: As JHI = 27. 32 = I R O ~ ( H ) ~ , (1) holds. As ( I ) = Z ( T ) and T E Sylz(H), T E 312(G). As E T and IN12 = IT[, T E Sy12(N). Thus (2) is established. If s is an involution in T R then IR/E : CRIE(s)l = 4 = ICRIE(s)I and ( s ) E 2 Dg, so by Exercise 2.8, sR is the set of involutions in sR. But we saw during the proof of 45.3 that if we choose s E Q then CR(s) = CQnR(s) 2 Dg, SO m(CT(s)) = 3. Thus as m(&) = 4 and A ( R ) = { R l , R2} with R, 9 T, (3) holds. Finally 45.3.4 implies (4).

-

Lemma 45.6: NG(R,) is the split extension of Sp4(2)' on R,.

4 by A6 acting as

Proof: Let Mi = NG(&). First CG(&) = CH(&) = &, SO Mi/& is faithful on 4.Next as R, is weakly closed in T it follows from 7.7 and 45.5.4 that Mi is transitive on R?. In particular IMiI = 15 IH fl Mil = 16 IA61, so (Mi/&( = IA6(. Finally the map

-

-

is a bijection of ~ f with i the hyperplanes of 4, so by Exercise 15.1, Mi/& _< Spq(2) '15'6. Hence Mi/& is the unique subgroup A6 of S6 of index 2. The extension splits by Gaxhutz7sTheorem (cf. 10.4 in [FGT]) as T splits over &.

-

Lemma 45.7: Let X E Sy13(H) and Tx E Sy12(NH(X)).Then

(1) If Z3 % U 5 x is TX-inuariant and ICH (U)12> 2 then C H ( U ) is the split extension of X by CT,(U) S Eq. (2) N G ( X ) = V T x , whew V = 0 3 ( N G ( x ) ) E81 and TX E Sy12(NH( X ) )is isomorphic to DR. (3) If Z3 ^. U 5 X with-CTX(U) ^. E4 then V _o NG(U) and N G ( u ) / v s4.

'

Proof: By 45.3.5, N H ( X )= X T x , where T x P Dg and ( z }= CTx(X). Thus (2) E Sy12(CG(X)),so C G ( X ) = V ( z ) ,where V = O(Ca(Z)), and N G ( X ) = V T x . So to prove (2) it remains to show V 2 Egl. Next by 45.6, Ui = Nx(&) % Z3 and CR(Ui) G E4, so as (2) E s y l 2 ( C G ( X ) )X, = Ul x U2. Then (1) is established. U = Ul cenAs z inverts V / X , V / X is abelian. By ( I ) , Z3 tralizes an involution t E TX - (z). By Exercise 8.1 in [FGT],V = Cv(z)Cv(t)Cv(tr) = XCv(t)Cv(tz). But as t E .zG and (VI is odd, ICv(t)l divides 9. Further as t z is conjugate to t in T x l ICy(t)l = ICv(tz)(. Thus as U = C v ( ( t , z ) )Z Z3, V = X or IV1 = 81. Suppose V = X . Then X E Sy13(G). So as ING(Rl)13 = 9 , a Sylow 3-subgroup X I of MI = NG(R1) is Sylow in G. This is impossible as an element of order 4 in NM,(Xl) is faithful on X I , whereas T x E Sy12(NG(X))and T x contains no such element. En. So lVl = 81 and V / X = G / X @ I/'t,/X, where V, = CV(s)X I HenceifVisnot Esl thena = [v,w]# 1 for v E G - X ar~dw E &,-X inverted by z. Notice (a) = [V,V ] .Further there is an involution s E T x with tS = t z . Then we may choose us = w, so as = [w,v]= a-l. AS (a) = [V,V],by symmetry between t and s, s inverts a also. Hence st E CTx(a) is of order 4, contradicting (1).This establishes (2). Let K = NG(U). Then NMl(U)= T x W , where Es r W E Syk(M1) and CM,( U ) S Z3 x A*. As ( z ) = Z ( T x ) and TX E Sy12(NH(u)), T x E Sy12(K). Also S = CTx ( U ) I E4 and as W is transitive on S# with C H ( U )= S X , for all s E s#,CG((s,U ) )= SCV(s). So W acts on ( C K ( ( s 1 U ):) s E s#)= S V , and I = T X W V = ( C K ( s ) , N K ( S ): s E s#).Similarly if T E Tx - s is an involution then ( z , ~E) sy12(cK(r)) and CV(r) E SgZ3(CG(~)), so as CG(r) is a {2,3)-group, CK(r) = (r,e)Cv(r) 5 I. Thus if I # K then I is strongly embedded in K in the sense of Section 7. This contradicts 7.6, as T 81 zK since r inverts U while z centralizes v. Therefore I = K and (3) is established. Lemma 45.8: Let W = X n HI E Sy13(H1). Then (1) F*(NG(W))r 31+4 has a complement H2 in N G ( W ) , where IH2 : H21 = 2 and H2 has quaternion Sylow 2-subgroups of order 16. (2) P = V F * ( N G ( W ) )E Syba(G), /PI =36, a n d V = A ( P ) . Proof: Let M = NG(W). Then NH(W) = Hz@), where t E T x is of order 4 inverting X.In particular C Q ( W ) ( ( )= S E Sy12(NH( W ) ) , where CQ(W)= Q r l Hz 2 Q8. Thus S is quaternion of order 16 and N H ( W ) = H2W, where & = (()Hz. In particular as ( z ) = Z ( S ) ,

Chapter 15 The Group U4(3)

246

S E Sy12(M).Then as S is quaternion, by a result of Brauer and Suzuki (cf. Theorem 12.1.1 in [Go]), M = D ( H n M ) ,where D = O ( M ) .Notice H~ is a complement to D in M , so to prove (1) it remains only to show D s 31+4. Next as W = D n H, z inverts D I W , so DIW is abelian. Further W 5 V and V is abelian, so V < M , and then V n D = [V,z]W is of index 3 in V . Pick U = Ul and U2 as in the proof of 45.7. Then from 45.7.3, there is a conjugate F of U2under NG(U) with F = [F,z]. Hence F 2 V n D and by 45.7.3, NG(F) _< NG(V) with ING(F)/ = 23 - 35. In particular as D is of odd order, ICD(F) : V n D ) = 1 or 3. But as DIW is abelian, [D,F] 5 W , so ID : CD(F)I = 1 or 3. Thus ID : D n V l divides 9. Now if D 5 V then Hz = [V n H2,Hz] 5 C ( D ) , contradicting D $ C ( z ) .So ID1 = 3a, a = 4,5. Then as z inverts D / W and GL2(3)contains no quaternion subgroup of order 16, I f 2 is irreducible on D / W and a is ) even, so ID1 = 35 and @ ( D )5 W . Indeed D = ( ( v n~ ) ~ is2 generated by elements of order 3 and D is of class at most 2, so D is of exponent 3. Finally as F $ Z ( D ) and & is irreducible on D I W , W = Z ( D ) , so D r 31+4 and (1) is established. Let P = V D , so that P E Sg13(M). As W = Z ( P ) , P E Sy13(G). Next CDIw(U)= (V n D ) / W , so for d E D - V , Cv(d) 5 D. Then as D E 31f4, IV : CV(DO)(> IDo/(V n D)I for all Do < V with V n D proper in Do. This shows {V) = A ( P ) , and completes the proof of (2). Lemma 45.9: NG(V) is the split extension of V by As acting faithfully as Rq(3) on V , with wGnvthe set of singular points of V and uiGflv, i = 1,2, the two classes of nonsingular points of V . Proof: Let M = NG(V) and M* = M I V . Then from 45.8, V = C G ( V ) n N G ( W ) = CG(V),so M* is faithful on V . Again by 45.8, N M ( W ) * = NM(P)* is the split extension of P* E Eg by Zq and P E Syls(M). By 45.8.2, V is weakly closed in M with respect to G , so by 7.7, PG n V = for each <: V . We claim P* is a TI-set in M*. For if d E P n Pm - V for some m E M - NG(P),then as dim(Cv(d)) = 2, C v ( d ) = W W m is invariant under (P,Pm) ( 2 ) . Then (P,Pm) ( z ) induces GLs(3) on CvCd), so N M ( P )contains a Cgroup, a contradiction. Next D8 Tx E Syl2(H n M ) , so a s ( z ) = Z(Tx), Tx E Sgl2(M). Then as M* GL4(3), IM*l = 23 .32 5" 1 3 ~with , a, b E {O,l). In . as P* is a TI-set in M*, particular IM* : NM=(P*)I= 2 . 5". 1 3 ~But (M* : NM*(P*)I= 1 mod 9, so we conclude IM*l = 23 . 32 - 5 = 1 4.

vM

1 I /

247

In particular IM* : NM*(P*)I= 10 = IP*l+ 1, SO as P* is a TI-set in M*, M* is 2-transitive on Sy13(M*) and hence on wMas N M ( W ) = NM(~)Similarly from 45.7.3, = 15, so as V contains 40 points, these 40 points are wMU U? U Now by Exercises 15.2 and 15.3, M* acts as RT(3) on V . The extension splits by Gaschutz's Theorem as we can find a complement to V in P contained in F*(NG(W)).

(UP/ u?.

, I

wG

Let I' be the rank 2 geometry with point set and line set vG, where incidence is defined by inclusion. Let A be the colliiearity graph of I?. It turns out I? is the geometry of singular points and l i e s in the &dimensional orthogonal space for PSI; (3) (E U4(3))over GF(3).This is essentially proved in the next two lemmas. See also 24.7, the Remark preceding 24.7, and Exercise 9.5.

Lemma 45.10: (1) Each element of onler 3 in P is contained in V or D = F*(NG(W)). (2) W is weakly closed in D with respect to G . (3) A is the commuting graph on wG. (4) Each line has ten points and each point is on four lines. (5) lA(W)l = 36. (6) D is regular on A ~ ( wof) order 35. (7) For W' E (w),(W,W') E so (W,W ' ) S SL2(3). (8) I? is a generalized quadrangle. (9) A is of diameter 2 (10) JGI = 27. 36 5.7.

HP,

I

I

v

<

45. u4(3)

f

- -

-

Proof: Let x E P - V be of order 3. Then from 45.9, x has Jordon blocks of size 1 and 3 on V with [V,x,x] = W . Then all elements of order 3 in xV are contained in xCV(x)[V,x] = x wL = x(V f l D), so (1) is established. Let K = NG(W). By 45.9 there are ten points on the line V , while there are IK : K n M)I = 4 lines through W . So (4) holds. Observe also from 45.9 that wMn V n D = { W ) . Also the sets ((vk n D)/w)#, k E K , contain 32 vectors of (D/w)#, with the remaining 48 vectors forming an orbit under K . We claim that if w E W and y E G , then wYW is not such a vector and hence & f vk for some k E K. Then as w M n V n D = { W ) , ( 2 ) holds. Namely ID : CD(wY)I = 3 with W < @(CD(W9))< Dv, and Dy n D is abelian so ID n Dvl 27. Hence C D " ( W )$ Dl establishing the claim. By (1) and (2), each member of wGn K - { W ) is contained in a

<

I

248

Chapter 15 The Group U4(3)

Exercises

unique member of v K , so (3) holds and (3) and (4) imply (5). Further we see all triangles of A are contained in lines. For Wl E AnV, Wl is transitive on the three lines through W distinct from V and V n D is transitive on the nine points on any line vk # V distinct from W, so G is transitive on paths WWlW2 with d(W, W2) = 2. In particular K is transitive on A ~ ( w ) . Next let W # W2 E Sy13(Hl). Then W2 E A and Wl = X n Hz E A(K W2), so W2 E A2(w) with H 2 = NK(W2) a complement to D in = A ~ ( w )of order 35. Further for each K. Thus D is regular on k E K, and Wg E A n vk,vkn If2 E A(W2, WO),so d(W2, WO) I 2. This proves (8) and (9). Then lAl = 1 + lA(W)l+ la2(w)l = 112, so [GI = 112. lKl, and (10) holds.

in Aut(A) of wI,so Wp+ = W' and hence G T = ( W * U ~ ( ~ )=) ~GI, * completing the proof.

WF

Lemma 45.11: Up to isomorphism, U4(3) is the unique group of type u4 (3). Proof: Let GI be a second group of type U4(3). The group of 45.8 is determined up to conjugacy in Out (D) r GSp4(3), so the split extension K = NG(W) of D by H2 is determined up to isomorphism, so there exists an isomorphism a : K -+ K' with Hza = Hi and V a = V'. Let I = Hz n V and W # J E A n HI. Set I' = la and let W' # J' E Hi n A'. Define p :A + A' by W p = W', ( I ~ ) P= I ' for ~k~ E K, and (Jd)p = J ' for ~ d~ E D. As D is regular on A2(w) and K is transitive on A(W) with NK(I)a = (VNR2(V))a = V'Nq (V') = NKI(IJ),the map fi is a well-defined bijection of A with A'. We claim p is an isomorphism. By construction A(W)P = A(W1). Next by 45.10.8, I is adjacent to if and only if is on the unique line incident with W and I ; that is, when I V. So I is adjacent to if and only if k E NK(V) if and only if k a E NKt (V') if and only if I' is adjacent to (I'~)P. Similarly I is adjacent to J~if and only if d E D n V if and only if d a E D' n V' if and only if I' is adjacent to (Jd)p. Thus A(I)@= A(I1), Thus to complete the proof of the claim it suffices to show J is adjacent to J~if and only if J' is adjacent to (Jld)p. But J is adjacent to Jd if and only if the line J Jdthrough J and Jdcontains a unique point of A(W), which we may take to be I. Then [V, z] = DnDV, where I = WY, and thus [V,z] is regular on J I - {I), so J is adjacent to Jdif and only if d E [V, z] if and only if d a E [V', za] if and only if J' is adjacent to (Jd)p. So the claim is established. Now the isomorphism P : A -+ A' induces an isomorphism P* : Aut(A) -,Ad(At) (cf. Section 1). Finally W is the pointwise stabilizer

+

+

249

Remarks. The group U4(3) was first characterized via the centralizer of an involution by Phan in [Ph].

Exercises 1. Assume V is an n-dimensional vector space over the field F of order 2 and h : v I-+ h(v) is a bijection of V# with the set of hyperplanes of V such that (1) v E h(v) for all v E v#. (2) If u,v E V# with u E h(v) then v E h(u). Then the map f : (u, v) w f (u, v) is a symplectic form on V, where f (u, v) = 0 if u E f (v) and f (u, v) = 1 otherwise. 2. Assume G is a finite group, S4 Z H 5 G with IG : HI = 15, P E Sy13(G) with E9 g P a TI-set in G, and Nc(P)/P 2 Z4 is faithful on P. Prove: (1)H = NG(O2(H)) with 0 2 ( H ) weakly closed in H with respect to G. (2) G is rank 3 on GIH with parameters k = 6, 1 = 8, X = 1, and p = 3. (3) The rank 3 graph of G on G/H is isomorphic to the graph A of all 2-subsets of a 6-set, with s adjacent to t in A if s n t = 0. (4) G E As. 3. Let G = As, F the field of order 3, P E Sy13(G), H = NG(P), Vo the 1-dimensional FH-module with H/CH(Vo) Z Z2, and V = voGthe induced FG-module. Let I be the 4-dimensional orthogonal space of sign -1 over F regarded as an FG-module, recalling that G r L2(9) SZ, (3). Prove (1) A nontrivial FG-module U is a homomorphic image of V if and only if there is a 1-dimensional FH-submodule Uo of U with Uo FH-isomorphic to Vo and U = (UoG). (2) V is FG-isomorphic to the dual space V*. (3) CV(P) is FH-isomorphic to the sum of two copies of Vo and dim(C[v,P](PI) = 1. (4) Soc(V) = (CIV,pl(P)G) is irreducible. (5) V is indecomposable with Soc(V) Z V/J(V) !+ I. (6) If U is an irreducible FG-module and Uo an FH-submodule of U which is FH-isomorphic to Vo, then U Z I.

1I

46. Groups of type Col, Suz, J2, and J3

251

Lemma 46.1: (1) Q is an orthogonal space of dimension 2w and sign e over G F ( 2 ) and H* Z Rfiw(2) acts as the commutator group R(Q) of the orthogonal group O(Q) of Q. (2) H is transitive on the involutions in Q - ( z ) , and all such involutions are in zG. Proof: By definition of Hypothesis X ( w , RZw(2)),H* Z R5,(2) and Q is extraspecial of order 21+2w.Then by 8.3, Q is an orthogonal space over GF(2) of dimension 2w and H* preserves this structure. In particular H* < o(Q). As H* R5,(2), this forces Q to have sign c and H* = R(Q). That is, (1) holds. As R(Q) is transitive on the singular points of Q , which by 8.3.4 correspond to the involutions in Q distinct from z, (2) holds.

Chapter 16

Groups of Conway, Suzuki, and Hall-Janko Type

Lemma 46.2: (1) E = Q n Qg. (2) (Qg17 H)* = 02(cH* ( g ) )% E22("-1) &i!h C H I( E ) / ( Qf ~l H)* fig-,( 2 ) and (Qg n H)* and (Q n H g ) / E the natuml module for In this chapter we prove the uniqueness of the Conway group Coil the Suzuki group Suz, and the Hall-Janko group J2. More precisely we consider groups of type X ( w , Rzw(2)) and define G to be of type Col if ( w ,e) = (4,+I), G to be of type Suz if ( w ,e) = (3, - I ) , and finally G to be of type J2 if ( w ,e) = (2, -1) and G has more than one class of involutions. Then we prove Col is the unique group of type Col, S u z is the unique group of type Suz, and J2 is the unique group of type J2. In the process we generate much information about the three groups. For example, for each group we obtain the group order and determine most of the conjugacy classes and normalizers of subgroups of prime order. This determination i s completed in Chapter 17.

46. Groups of t y p e C o l , Suz, J 2 , and J3

In this section we assume G satisfies Hypothesis Z ( w , R5,(2)) with ( w ,e) = (2, - I ) , (3, - I ) , or (4,+I). (See the Preface for the definition of groups of type X ( w , L).) In particular z is an involution, H = CG(z), and Q = F*(H). Let H* = H I & , H = H / ( z ) , zg E Q - ( z ) , E = (z,zg), and R = (Q n Hg)(Qgn H ) . Let AH be the set of subgroups X of H of order 3 such that [Q,XI S

Proof: As H* = R(Q) and E is a singular point in the orthogonal space Q, o ~ ( c ~ *S(! E2aw-z E ) ) and C Q ( E ) / Eare the natural modules for

In particular

cH(E) is irreducible on O 2 ( C H * ( ~ )But ) . by 8.15, 1 #

n9HI*A cH.(E), so (

~ gHnI * = O ~ ( C ~ . ( E and , ) then (1) holds by 8.15.8. Finally (1) and earlier remarks imply (2). (

~

Lemma 46.3: (1) R is special with center E . (2) N G ( R ) / R r Rfiw-2(2) x S3 with R I E the tensor product of the natural module for its factors. (3) N G ( R ) splits over R .

'

Proof: Parts (1) and (2) follow from 46.2 and 8.15. Let X be of order 3 in N G ( R ) with X R a NG(R). Then C R ( X ) = 1 so N G ( X ) n NG(R) is a complement to R in NG(R), establishing (3). Lemma 46.4: H ~ ( R (2), ~ , Q ) = 0.

'

(28. I

Proof: If w = 3 or 4 let X be of order 3 in L = R5,(2) with C a ( X ) = 0. Then the commuting graph on AXL is connected as L = ( N L ( X ) ,N L ( X ~ ) ) for X # Xg 5 C L ( X ) .Hence the lemma follows from an observation of Alperin and Gorenstein which appears as Exercise 6.4 in [FGT].

So take w = 2. Here if V is a 5-dimensional GF(2)L-module with Y L satisfies [V,L ] = [ V , Y ] ,so if C V ( Y ) # C V ( L ) then V is an image of the permutation module U for L on L / Y . As the 4dimensional composition factor of U is not isomorphic to Q , this is a contradiction. Thus V splits over [V, L], so 17.11 in [FGT] completes the proof.

[V, L] r Q then Dlo

<

Lemma 46.5: Let w = 3 and Y E AH. Then C G ( Y ) is quasisimple with C G ( Y ) / Y U4(3)and ING(Y) : CG(Y)I = 2. Proof: Let NG(Y)* = N G ( Y ) / Y , K = C G ( Y ) , and P = C Q ( Y ) . Then F * ( C H ( Y ) * )= P* with 0 2 ( c H ( y ) * )2 SL2(3) * SL2(3) and 4 = ICH(Y) : 0 2 ( C H ( Y ) ) I By . 8.13 and 46.1.2, all involutions in P are P and g E K ; then by 46.2.1, in z K . In particular we may take E E = PnPg. Thus K* is of type U4(3),as defined in Section 45. Therefore by 45.11, K* g U4(3). Next as I N H ( Y ) : CH(Y)I = 2, ING(Y) : KI = 2. So it remains to ' show Y < Km. But there is h E H with yh 5 HI a 02(cH(y)), H1 E SL2(3), and Y = [Yh , s] for some 2-element s E CH( Y ) .Then yh = [yh, s] 5 K w and if Y 31, K w then by 45.8, yh < @(So)for some So E s ~ z ~ ( N ~ (Thus Y ~ Y) )<. @ ( S )for S E Sy13(K), contradicting K=YxKm.

<

Lemma 46.6: Let w = 4, X C Q ( Y ) E Q:. Then

€

AH, and X 5 Y


with Y

% Eg

and

(1) C G ( X ) is quasisimple with C G ( X ) / X S Suz. (2) C G ( Y ) is quasisimple with C G ( Y ) / Y 2 U4(3).

Proof: To prove C G ( X ) / X E S u z , argue as in 25.9. Actually this argument only shows C G ( X ) / X is of type S u z but proceeding by induction on the order G and appealing to 48.17 in an inductive context, up to isomorphism there is a unique group of type Suz. Notice (2) now completes the proof of (I), so it remains to prove (2). Let P = F*(CG(z)). As in 25.9, C Q ( X ) E Q:. Then there is h E N H ( Y ) with Y = X X ~ and xh inverted in C H ( X ) .Thus by 46.5, xh 5 CG(Y)OO. Finally as h E N G ( Y ) , X 5 CL(Y)OO,establishing (2). Lemma 46.7: Let w = 3 and I' the set of involutions i E H such that i* is of type cp. Then ( I ) H is transitive on J?. (i)l= 2'. (2) POTi E r, IcH

253

46. Groups of type C o l , Suz, J2, and J3

252 Chapter 16 Groups of Conway, Suzulci, and Hall-Janlco Type

Proof: We recall from Exercise 2.11 that i* is of type c2 if [Q, i]is 2dimensional and not totally singular and that if i E I' then i z E i Q , each involution in i Q is in i&> and iQ= i Q 6 , where : Q = C g ( i ) and Q; = [Q,i]. Pick Y E AH;by 46.5, C G ( Y ) / Y 2 U4(3), so there is an element t E C H ( Y )with t2 = z and t* E c2. Now t centralizes Y [ Q , Y ]G SL2(3) with Q t = C Q ( Y )n Q?, so by the previous paragraph each involution i E t Q is conjugate under Y Q to t u for some fixed u E [Q, Y ]of order 4. In particular there are 24 involutions in t Q , so as H* is transitive on c2 of order 270, H is transitive on r of order 24 270. Thus C H ( i ) = IH1/(24 270) = 2'.

-

Lemma 46.8: H is determined up to isomorphism independently of G. Proof: Let Ho '.

9

the universal covering group of H and

HI ). Then , by 46.4 and 8.17, P % Q and Let Z = Z ( H )and P = [ o ~ ( H H = H / U for some complement U to Z ( P ) in Z and H / P 2 L , where L = Lo/O(Lo)ia(02(Z(Lo)))and Lo is the universal covering group of H*. Let (n)= Z ( P ) . Suppose first w = 2. Then L Z SL2(5) (cf. 33.15 in [FGT]).Thus Z 2 Ed. Also H* has one class of involutions and m([Q, t ] )= 2 for each such involution t, so I? is transitive on involutions in I? - Q by Exercise 2.8. Thus if z Z is an involution in H / Z - P Z / Z then x2 = a is the unique element of Z such that t2 = a for some t E H - P Z with t2 E Z. Noticeas Z E E4, Z = ( a , ~and ) IUI = 2 . We claim U = (a).For if not U = ( a n ) , so for each t E H Q with t2 E ( z ) , t2 = z. That is, Q contains all involutions in H. This contradicts 46.3.3 which says H contains an S3 subgroup. Suppose next that w = 3. The argument is similar. This time by Exercise 16.6, Z Y Eq and each involution in H* of type c2 l i i s to an element of H I P of order 4. Now by 46.7 and its proof, H has two orbits t f , i = 1,2, on the set f' of t E 81 with t2 E Z and t P Z of type q in H I P 2 r H* with t 2 = t l x for some element x of order 4 in C p ( t 2 ) , Itrl = 8, and It;[ = 24. Then as involutions of H* of type c2 lift to = a x 2 = an. In this elements of order 4 in H I P , cr = # 1; then case we claim U = ( a n ) . If not U = ( a ) and t = tlU E I' (where I' is defined in 46.7), so 46.7 supplies a contradiction as ltQl = Itr( = 8. Finally take w = 4. We lift X E A H to a Sylow 3-group in its preimage in H, and hence regard X also as a subgroup of H. Now X centralizes

-

ti

ti

254 Chapter 16 Groups of Conway, Suzulci, and Hall-Janko Type an involution i E H with i* of type c2, and by the previous paragraph i lifts to t E H of order 4 with t2 E U. By Exercise 2.11, H* has two classes of involutions of type a4 with representatives jl and j2, where [Q,!,] = Ai are the two classes of maximal totally singular subspaces of Q. Further if a is a transvection in O(Q)then cu induces an outer automorphism on H* with A? = A2 and jr = j2. By Exercise 16.7, m ( Z ) = 3 and ji lifts to an element ti of order 4 in H I P with Z = (al,ag,n),where t; = ai. By 8.17.4, a l i i s to an automorphism of H, which we also denote by a. Then tq E t3-,Z, so a interchanges a1 and 0 2 . Hence up to conjugation under a , U = (al,a2), ( a l ,nu2),or (nal,na2). Next Q induces the full group of transvections on Ai with center ( z ) , so cH(Ai) = QCG(Ai)and CG(Ai)/Ai E cH*(Ai) g Efj4. Further the are of type c2 or jl, so as t2 E U , @(CG(A,)) =1 involutions in cH*(Ai) if and only if ui E U. Now if G = Col then (cf. the proof of 46.12) @(CG(Ai))= 1 for exactly one of i = 1,2, say i = 1. Thus in that case a1 E U but a2 $ 4. Therefore U = (ol,n q ) . Further t2 E U is fixed by a , so t2 = n a l w . Now in the general case we know t 2 E U, but of the three possibilities for U, only the second contains t2 = nsla2, so indeed H is determined up to isomorphism and the proof is complete. Notice in the process of proving 46.8 we have also proved several other facts. First Lemma 46.9: Let L 3 0 = Aut(A) be the automorphism group of the Leech lattice, K < L be quasisimple with K / Z ( K ) E Suz, and Z ( L ) 5 J 5 K with J/Z(L) a mot J2-subgroup of L/Z(L). Then the root 4involutions of K / Z ( K ) % Suz, L/Z(L) %' Col, and J/Z(L) r J2 lift to elements of K , L, and J of order 4, and J is quasisimple.

See 49.1.1 for the definition of a root J2-subgroup of Col. Now Z ( K ) %' and Z ( L ) G Z2, so from the proof of 46.8, CK(z)/O3(Z(K))= I;TK is HO,K/O(HO,K), where HO,Kis the universal covering group of C K ( z ) / Z ( K )Similarly . fiL = & , L / ( o l ) ~ ( ~ Oin, lthe ) language of the - -prooE of 46.8. Our analysis in Section 48 will show that each root 4 involution i of K / Z ( K ) contained in H is of type c2 in H* and hence a lift ;in riICor HL is of order 4 from the proof of 46.8. Also Z2 generates Z ( L ) ,so as we can pick ;E J , J is quasisimple. Zg

Lemma 46.10: If w = 4 and jl and jz are representatives for the two

1

255

46. Groups of type Col, Suz, J2, and J3 classes of involutions of H* of type a4 then there exist involutions i E H with i* = jl, but no involution j E H with j* = j2.

I

For if ti E H with tiPZ = ji then we showed that t: E U but t i

4 U.

L e m m a 46.11: If w = 2 then H is transitive on the involutions in H - Q. Namely we saw that H is transitive on involutions in H - Q, whiie since an involution t E H -Q is of type c2 on Q, t z E tQ by Exercise 2.11, completing the proof.

Lemma 46.12: Assume w = 4 and let T E Sy12(H).Then (1) J ( T ) = A r E 2 i i . (2) NG(A) is the split extension of A by M24 with A the Golay code module for NG(A)/A. NG(A) has two orbits on A#: the octad involutions with repre(3) sentative z and the dodecad involutions. (4) The isomorphism type of the amalgam H

6

N H ( A )+ NG(A)

is determined independently of the group G of type Col. Proof: By 46.8, there is an isomorphism a : H -, H of H with the centralizer H of a 2-central involution f in G = Col. By 22.5, G contains a subgroup M which is the split extension of A Z E211 by M24 with A the 11-dimensionalGolay code module for M24. By 23.10, CA(A)is a point and N = N ~ ( A ;= )~ ~ ( 2 and then by 23.1 and 23.2, that point is By Exercise 7.4, A = J ( M ) . Let A = Aa-l. The isomorphism a : A -+ A induces an isomorphism a* : GL(A)A -t GL(A)Aof the semidirect product of A by GL(A)with the semidirect product of A by GL(A).Now M = N ~ ( A ) satisfies conclusions ( 2 ) and (3) of the lemma. In particular we may regard M as a subgroup of GL(A)A;let M = ~ a - li: GL(A)A. Then N H ( A ) = ~ ~ ( A ) a - lM is isomorphic to N ~ ( Avia ) a , so as A = J ( M ) , also A = J ( T ) and hence (1) holds. Similarly NH(A)is the split extension of A by the split extension Ho of El6 by L4(2). Further NH(A)is the stabilizer of the octad involution z in the Golay code module for a complement Mo to A in M containing Ho. Let h E 02(Ho); then by 8.10, CHo(h)= CM,(h) and hence by 8.8 is isomorphic to the centralizer of a transvection in L5(2). Let N = NG(A) and N* = NIA. We claim C N . ( ~ *= ) CEi,(h)*Z CHo(h).First CG(A) = CH(A) = A, SO N* is faithful on A and

<

) .

256 Chapter 16 Groups of Conway, Suzuki, and Hall-Janko Q p e Ho z Hz = CN.(z). Now h E Q - A, so h induces a transvection on Q n A Z E32 with center z and axis B = CQnA(h). Further V = CHo(B) r E16 with HO/V the stabilizer of z in GL(B). So if we set Hl = CN(h) n N(B), CH;(B) = CHi(B) n C(z) = V* and Hi/V* is maximal in GL(B), so either CHo(B)* = H: or H:/V* = GL(B). The latter is impossible as 2; fixes h* and Aut(V*) n C(h*) contains no L4(2)-section, so Hf 5 C(V*), whereas V* = CH; (V*). Thus Hf = CHo(h)*. We next show B = [A,h], so CHo(h)* = CN(h)*. Indeed as A is the Golay code module for Mo, [02(Ho),A] 5 Q n A, so [A, h] 5 CQnA(h) = B. On the other hand M contains no transvection, so m([A, h]) > 1 and then as CHo(h)is irreducible on B/(z), B = [A,h]. So CN(h)* S CHo(h) is the centralizer of a transvection in L5(2). Further O~(HO)#= hHo C 02(CHo(h))= Qo, so by 44.4, either N = NH(A) or N* G L5(2), M24, or He. The first case is impossible as A = J(T), so A is weakly closed in T with respect to G, and hence by 7.7, N controls fusion in A. Thus z is fused to zg in N. As Ho 2 Hz is a 2-local subgroup of N*, 40.5 says N* is not He. Suppose N* 2 L5(2). Then IN : NH(A)I = 31 = IQ n A#I, so as Q n A# zG n A = zN, Q n A# = zG n N. This is a contradiction as NG(R) n NG(A) does not act on Q n A from 46.3. So N* E M24. As Ho is a complement to A in NH(A) and NH(A) contains the Sylow 2-subgroup T of M, N splits over A by Gaschutz's Theorem (cf. 10.4 in [FGT]). Next A = (zN) with HG = CN. (2) and m(Q n A) = 5 with Hz the stabilizer in GL(Q nA) of z. Thus by Exercise 7.2, A is the Golay code module as an N*-module. Thus (2) holds. By 19.8, (3) holds. We have shown there is an isomorphism : N -, IV with ( H n N)(' = H n = ( H n N)a. To prove (4) it remains to show we can pick a and so that a = C on H nN = I. For this we observe I/A = Aut(I/A). For example, I/Oz(I) = GL(O2(I)/A) and H~(GL(o~(I)/A),02(I)/A) = 0 by the Alperin-Gorenstein argument (cf. Exercise 6.4 in [FGT]) applied to the class of subgroups of order 3 that are fixed point free on 02(I)/A. Then the claim is easy. Thus by Exercise 14.3.1 we may assume a = C on IIA. Finally A is generated by the orbits of I on zN of length 30 and 240 (cf. 19.2) and as a = on I/A, a = on these orbits, so a = 6 on I.

c

c

<

<

When w = 4, 46.12 says there are two G-classes of involutions in A corresponding to the octads and dodecads via the identification of A with the Golay code module. Further z is an octad involution and we refer to

4 6.

Groups of type Col, Suz,

257

J2,and J3

the second class as the dodecad involutiow of G. We will see eventually that G has one more class of involutions: the root 4-involutions.

Lemma 46.13: Let w = 4, T E Sy12(H), A = J(T), and d dodecad involution. Then (I)

(2)

E

A a

d is contained in a unique G-conjugate of A; in particular CG(d) 5 NG(A). CG(d) is the split extension of A by Aut(M12) and is determined up to isomorphism.

Proof:By 46.12, N = NG(A) is the split extension of A by M24 with A the Golay code module, so as d is a dodecad involution, CG(d) n N satisfies (2); for example, see 19.8.3. Further by 46.12.1 and 7.7, N controls fusion in A. Therefore if CG(d) < N then (I) holds by 7.3. Thus we may assume CG(d) $ N and it remains to derive a contradiction. By 26.3 the result holds if G = Col so by 46.12.4, CH(d) 5 N and hence CG((u, d)) < N for u E zN. SO as CN(d) controls CG(d)-fusion in A, 7.3 says each u E zG n A fixes a unique point of CG(d)/CN(d). This puts us in a position to appeal to Exercise 2.10. Let d E B 5 A with B maximal subject to CG(B) & NG(A). Let K = CG(B), M = N n K, and K* = K/B. Then by Exercise 2.10, M* is transitive on A*#. Thus 211-m - 1 divides IM1, where m = m(B). However, we see in a moment that m 5 3, whereas 211-m - 1 does not divide IM241 for 1 5 m I: 3. So it remains to show m 5 3. But by Exercise 2.10, z 4 B, so each member of B# is a dodecad involution. Thus m 5 3 by Exercise 7.3. Lemma 46.14: Let w = 2 and R

< T E Sy12(H). Then

(1) R is isomorphic to a Sylow 2-subgroup of L3(4) and NG(R) is the split extension of R by Z3 x S3. (2) R = J(T) and A(R) = {A1,A2) with Ai E El6 and R = NT(A~). (3) NG(R) n NG(Ai) is transitive on Ai - E and E#. (4) NG(Ai) controls fusion in Ai. (5) Either (a) NG(Ai) is the split extension of Ai by GL2(4) acting naturally on Ai and G has one conjugacy class of involutions, or (b) NG(Ai) NG(R) and G has two classes of involutions with representatives z and a E Ai - E. Moreover Q is the weak closure of z in H.

<

258 Chapter 16 Groups of Conway, Suzuki, and Hall-Janko Type Proof: By 46.3, IR1 = 64 with NG(R) the split extension of R by X = Xo x X3 with Xo S3 and X3 Z Z3,and R / E is the tensor product of the natural modules for Xo and X3. Further Xo is faithful on El so Cx(E) = X3. Now X has two more conjugate subgroups X1 and X2 of order 3 and R = CR(X1)CR(X2)E with CR(Xi) 2 E4 for i = 1,2. Therefore Ai = ECR(Xi) % Els with R = AlA2 and A1 n A2 = E. Moreover X2 induces a GF(4)-structure on A1 preserved by CR(X2)X1X2, so CR(X2)X1X2 acts as a Bore1 group of GL2(4) on All establishing (1)-(3)If b, bh E Al then b E Al n A:-', and then by (2), A?-' = At for some c E CG(b), so ch E NG(A1) with bch = bh. Hence (4) holds. Next by 46.11, H is transitive on involutions in H - Q, while by 46.12 each involution in Q is fused to z in G, so either G has one class of involutions or two classes with representatives z and a E A1 - E and Q is the weak closure d z in H . In the first case by (4), N G ( A ~is) transitive on ~ fso ,ING(Al)I = 15 lNH(A1)l = 2*. IGL2(4)1. Let K = NG(A1). Then IK : NK(R)I = 5 so the permutation representation n of K on K/NK(R) maps K into S5.We conclude AlX2 = ker(r) and K s r As. Thus (a) holds in this case. In the second case K acts on zG n A1 = E#, so K 5 NG(R) and (b) holds.

-

If w = 2 and G has one class of involutions we say G is of type J3.If w = 2 and G has two classes of involutions we say G is of type J2. In previous sections we have already defined G to be of type Col if w = 4 and of type Suz if w = 3. Thus the hypotheses of this section can be restated to say that G is of type J 2 , J3, Suz, or Col. In the remaining sections in this chapter we prove that up to isomorphism there is a unique group of type J2, Suz, and Col, respectively. There is much parallelism in the proofs, so we could prove a few more lemmas simultaneously for each type, but soon the proofs would have to diverge. We choose to begin the separation after the next lemma, and even that lemma makes appeals to facts established later in this chapter.

Lemma 46.15: G is simple. Proof: If G is not of type J2we appeal to Exercise 2.4. To apply that result we need only show Q is not the weak closure of z in H. This is true if G is of type J3since G has one class of involutions and we have or Col this seen there are involutions in H - Q. If G is of type SUZ follows from 48.4, 49.4, and their proofs.

I I

I f

i 1

I I

I

4 7.

259

Groups of type J 2

So assume G is of type J2.Then by 46.13, Q is the weak closure of z in . by 8.11, M = F*(G) H , so we need to work harder. Let M = ( z ) ~Then is simple. Then Q = (zG n H ) 2 M and as Q is weakly closed in H , G = MNG(Q) = M H by a Fkattini argument. Thus it remains to show H 5 M, so assume not. As Q 5 M and H/Q is simple, Q = H n M . Let X E AH.By 47.3.1, K = CG(X) = (zGnK), so K 5 M. This is a contradiction as X is not contained in Q = H f l M . 47. Groups of type J2 In this section we assume G is of type J 2 as defined at the end of the previous section. Moreover we continue the notation of that section. In addition let R 5 T E Sy12(H); by 46.14, A(T) = {A1,A2) with Ai G Elel R = A1A2, and Al n A2 = E. L e m m a 47.1: ( I ) For t E A1 - El CG(t) 2 E 4 x A5 and U(t) = O2(CG(t)) is a TI-set in G with NG(U(t)) A4 x A5, z E E(CG(t)), and X3 a Sylow 3-subg~oupof CG(E(C~(t)))in AH. (2.) I G = ~ 27 33 52 .7.

. .

Proof: By 46.14.5, G has two classes of involutions with representatives z and t. This allows us to use the Thompson Order Formula 7.2 to calculate the order of G. For x = z or t define a(x) = IP(x)l, where

P(x) = {(u,v) : u E zG, v E tG, x

E

(2121)).

Then by the Thompson Order Formula, IG/ = a(z)lC~(t)l$- a(t)ICG(z)l. Thus we must determine a(%)and ICG(x)).We recall that if (u,v) 6 P(x) then as (u, v) is dihedral and x E (uv), x E Z((U,v)) so (u, U) 5 CG(z). Observe that ICG(z)l = /HI = 27 e3.5. We claim a(z) = 26 - 3 .5. For if (u,v) E P(z) then by 46.14.5, u E Q, while by 46.1, v E H - Q. Then uv 4 Q,SOUV # zand hence [u,v] # 1. But ifil E CQ(v) then [u,u] = 1, so [il,v] # 1 and (u, v) 2 Dl6. Conversely for each involution u E Q and each involution v E H - Q with [il,v] # 1, z E (u,v), SO ( u , ~ E) P(z). Thus P(z) consists of such pairs. Now tG n H = tH is of order z3 3.5, while C- (t) contains a unique singular point, so there are eight choices for u wit v = t. Hence a(z) = 26 - 3 . 5 , as claimed. Suppose that CG(t) NG(R). In the notation of the proof of 46.14, we may take t E CR(X1), so CG(t) = XlAl. Now (u,u) E P(t) if and

B

<

260 Chapter 16 Groups of Conway, Suzuki, and Hall-Janko Type only if u f E# and v = ut, so a(t) = 3. Also ICG(t)l = 3 . 24, so

IGl = a ( z ) l C ~ ( t )+l a(t)lHI =26.3.5.24.3+3.27.3.5 =27.32-5-(1+8)=27-34.5.

I

4 7. Groups of

261

type J2

I

Lemma 47.2: Denote by 5A15B the conjugacy class of subgroups of G of order 5 centralizing a root 4-involution, 2-central involution of G , respectively, and let P E Syls(G). Then (1) 5~ and 58 are the conjugacy classes of subgroups of order 5 in G. (2) For X E 5,4, N G ( X ) Dl0 X A5, E(CG(X))= (A n N G ( X ) ) , X is inverted by a conjugate zY of z, and E ( C G ( X ) ) is a wmplement to QY in Hy . N G ( Y ) = Dlo x Dl0(3) ForY E (4) P = CG(P) and NG(P) is the split extension of P by Dl2.

Now Sylow's Theorem supplies a contradiction. For let P E Sy15(H). Then N H ( P ) E D20 so ( z ) E S y 1 2 ( c ~ ( P )and ) ING(P)~ =~4. But sB, lG : PI E -2 mod 5, so as JG: NG(P)I = 1 mod 5, ING(P) : PI -2 mod 5. Thus as ING(P)( 2 = 4, v = 0 3 ( N G ( P ) )E Sy13( N G ( P ) )is of order 33. As N H ( P ) has order prime to 3, z inverts V . Now we may Proof: Let N = NG(P). Then by 47.1 and Sylow's Theorem, IP( = 52 take t to invert P and V = CV(t) x C v ( t z ) with ICv(t)J = 32 and and I N : PI = J G :PI -= 2 mod 5. ICv(tz)l = 3, contradicting ICG(t)I3 = 3. Let t be a root 4-involution, X E Sy15(CG(t)),M = C G ( X ) ,and So K = CG(t) NG(R). Let K* = K / ( t ) . NOWA E Syl2(K) and M* = M I X . By 47.1, U ( t ) E Sy12(M) and U(t)* = CM*(t*).Also E is strongly closed in A, so by Thompson Transfer (or use the more taking X 5 P , P 5 M , so M $ CG(t). Thus by Exercise 16.6 in general 37.4 in [FGT])applied to A*, K* has a subgroup Kg of index [FGT],M* G As. Further we may take z E E(CG(t))to invert X , so 2 with E* E Sy12(K$). Then as E* = CK;(z*), Exercise 16.6 in [FGT] z centralizes CG(E(CG(t))) n NG(U(t))r A4 and hence [z,E ( M ) ]= 1. says K i E A5. As above we may take t E CR(X1) = U ( t ) . As U ( t ) Finally N G ( X ) = E ( M ) C M ( t )= E ( M ) x X ( z ) % A5 x D10, so (2) centralizes X I E , [U(t),K:] = 1, so K = U ( t )x E ( K ) with E ( K ) E A5. holds. Notice E = [A1,X1]E Sy12(E(K)).Further X3 acts on CG(U(t))= K Next let Y = P n E ( M ) , so that P = X x Y 2 E25. Without loss of and centralizes X l E , so [ E ( K ) X3] , = 1 and NG(U(t)) = X3U(t) x generality t inverts Y . Now ( z ) E Sy12(CH(Y))with N H ( Y ) = Y ( t ,z ) , E ( K ) Z A4 x A5. Now z E E 5 E ( K ) so X3 5 CG(z) = H and hence so C G ( Y ) = O ( C G ( Y ) ) ( z )with z inverting O ( C G ( Y ) ) / Y .Therefore X3 E Sy13(H) = AH.That is, (1) is established. O ( C G ( Y ) ) = Y x [O(CG(Y)),z]is abelian, so C G ( Y ) CG(P) = Now we use the Thompson Order Formula to complete the proof of (2). C M ( P )= P and thus N G ( Y ) = P ( t ,z ) 2 Dl0 x Dlol so (3) holds. We have shown ICG(t)l = 24.3.5, ICG(z)I = 27.3-5, and a ( z ) = 26.3-5, As ( t , z ) E S y l P ( N G ( Y ) with ) X , Y the fixed points of ( t , z ) on P so it remains to calculate a(t). If ( u ,v ) E P ( t ) then u E zG nK E E ( K ) and X 4 yG,(t,z ) E Sy12(N)and then by the Burnside pcomplement a n d v ~ t ~ n ~ ~ _ ~ - ~ ( ~ ) , s o v = w x , w ~ ~ ( t ) ~ a n d x ~ z ~ ~ { l ) . Theorem (cf. 39.1 in [FGT]),N = O ( N ) ( t , z ) .As ( N : PI E 2 mod 5, Then t E (uv) = (wux) and ( W U X ) ~= ( u x ) E~ E ( K ) , so lux1 is odd and IO(N) : PI = 3 mod 5. Then as N I P 5 GL2(5),N I P E D12. Therefore w = t. Thus Iuxl = 1, 3, or 5 and there are 15, 60, 120 pairs ( u ,v) of the (4) holds and N has two orbits on the six subgroups of P of order 5, so respective type. Therefore a(t) = 195 = 3.5.13. So by the Thompson (1) holds. Order Formula, Lemma 47.3: Denote by 3.4,3~3the conjugacy class of elements of G of order 3 centralizing a 2-central, root 4-involution of G, respectively, and let P E Sy13(G). Then

<

establishing (2). We call the conjugates of U ( t ) the root 4-subgroups of G and the conjugates o f t the root 4-involutions of G. Denote by A the set of Gconjugates of members of AH;that is, A is the set of subgroups of G of order 3 centralizing a conjugate of z.

(1) For x E 3 ~ (x)G , = A, c G ( x ) is quasisimple with C G ( x ) / ( x ) A6, and i V G ( ( x ) ) / ( ~ )PGL2(9). (2) For E 3 8 , N G ( ( Y ) )'2 S3 x 444(3) P Z 31+2, NG(P) is the split extension of P by Z8 faithful on P / @ ( P ) ,and Z ( P ) E A is weakly closed in P with respect to G. (4) 34 and 38 are the conjugacy classes of elements ofG of order 3.

262 Chapter 16 Groups of Conway, Suzuki, and Hall-Janko Qpe 6 Proof: First let D E Sy17(G). Then ING(D) : Dl r IG : Dl mod 7. Further by 47.1 and 47.2, no element of order 2 or 5 centralizes an element of order 7, so JNG(D): Dl = 2a 3b with a = 0,1, 0 b 3. Thus ING(D): Dl = 6 or 27. Next let K = E(CG(t)) and X E Sy13(CG(K)).Thus X E AH and X = (2)for some x E 3 ~Now . NH(X) = SX, where S is semidihedral of order 16 and Cs(X) = CQ(X) r Dg. Let X 5 I E SyZ3(XK) and I P. Choose I to be z-invariant. Then I = X x Y, where Y = [I,z] is generated by y G 3 ~ Indeed . setting M = CG(Y) and M* = MIY, we have U(t)* = CM.(t*), so by Exercise 16.6 in [FGT], either U(t) a M, so (2) holds, or M S Y x A5. In either case P $ CG(Y),so P is nonabelian and as X and Y are the only subgroups of I fixed by z, X = Z(P). Now let L = CG(X). Then K 5 L and Kg E Sy15(K) C Syl5(L) as a Sylow 5-subgroup of G is self-centralizing. Also P E Sy13(L) and CQ(X) E Sy12(L),so ILI = 23. 33. 5.7=,c = 0,l. If c = 1we may take D L and INL(D) : Dl = IL : Dl 2 mod 7, so DX < NL(D). But by a Frattini argument in NG(X), I NG(X)r l NG(D) :NL(D)I = 2, so as ING(D): Dl = 6 or 27, ING(D) : Dl = 6, contradicting NL(D) # DX. Hence c = 0 and IL : KXI = 6. So as K GS A5 it follows that L/X r A6. Then as P is nonabelian, L is quasisimple and P E 31f 2. Also as S is semidihedral, NG(X)/X S PGL2(9), so (1) is established. Further we have seen that if (2) fails then CG(Y)2 Z3 x A5. But by 47.2, only B of type 5~ centralizes an element of order 3 and a Sylow 3-group of CG(B) is of order 3. Thus 3~ = 3f3, whereas such elements have nonisomorphic centralizers. Hence (2) holds. As X = Z(P), NG(P) 5 NG(X),so by (I), P is the split extension of P by Zs transitive on (PIX)#. Thus NG(P) is transitive on elements of P of order 3 in P - X, so all are in 3 B . Thus X is weakly closed in P, and (3) and (4) are established.

-

<

Notice in the process we have also shown:

Lemma 47.4: The nonnalizer of a Sylow 7-subgroup of G is a Fbbenius group of order 21. -For we showed during the proof of the previous lemma that if D E Sy17(G)then ING(D): Dl = 6 or 27. But by 47.1 and 47.3, elements of order 2 and 3 do not centralize elements of order 7, so the lemma holds. Notice we have now determined the order of each group G of type J2, its conjugacy classes and normalizers of subgroups of prime order, and the isomorphism type of the normalizers of Sylow groups of G.

4 7.

?

I i

< <

<

. -

i

I

Groups of type

J2

263

The remainder of this section is devoted to a proof that, up to isomorphism, there is a unique group of type J2. In the process we generate a great deal of information about the set A of subgroups of order 3 centralized by a 2-central involution. Thus in the remainder of this section let X E A and M = NG(X).

Lemma 47.5: (1) M has four orbits on A: {X), A(X), A?(X), and A;(X). (2) A(X) = {Y E A : (X,Y) E A*), IA(X)I = 36, and NM(Y) E A5 for Y E A(X). (3) A:(x) = {Y E A : (X,Y) 2 SL2(3)), ]A!(X)~ = 135, and NM(Y) E Syl2(M) for Y E A:(x). (4) A$(x) = {Y E A : (X,Y) As), IA;(X)~= 108, and NM(Y) 2 Z2 x Dl0 for Y E A ~ ( x ) . Proof: By 47.2, X 5 E(CG(F)) = L for some F E 5.4, L E A5, and CG(L) = D 2 Dlo. Then there exist Y, X2 E xLwith (XIY) S Aq and (X,X2) = L. In particular NM(X2) = (M n L) x D s 2 2 x DlO, so I X ~= IIM : NM(X2)1= 108 and X2 E A ~ ( x ) . Next 02((X,Y)) = U is a root 4subgroup of G, so by 47.1,

with K S A5. Then K = NM(Y), lyMl= IM : KI = 36, and Y E A(X). Finally we may take z E M. Then CM(z) = XT, T E Syl2(M), and = (xQ) = (X,X1) SL2(3) for suitable X1 E AH. Then NG((X,X~)) NH ((XIXi) = T(X,Xi), SO NM(X~) = T, Ix?~ = IM : TI = 135, and X1 E A:(X). Now so A = {X) u Y

U X? U

XY, and the lemma holds.

We now regard A as a graph with A(X) the set of vertices adjacent to X in A. In order to prove the uniqueness of groups of type J2, we prove A is simpjy connected. By 35.14 it suffices to show each cycle of A is in the closure C3(A) of the set of triangles of A. This is accomplished in a series of lemmas. .

Lemma 47.6: Let Y E A(X) and K = NM(Y). Then (1) A(X) n A:(Y) = {Ye : e E E(M) - KX) is an orbit of KY of order 15. (2) {Ym : m E M - E(M)) is an orbit of KY of order 18 such that for each M E M - E(M), ymXcontains one member

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47. Groups of type J 2

264 Chapter 16 Groups of Conway, Suzukz, and Hall-Janko a p e of A(X, Y ) and two members of A(X) fl A ~ ( Y ) .In particular IA(x) n A;(Y)~ = 12. (3) lA(X,Y)l = 18. (4) For each U E A, zG n CG((X,Y, U)) # 0. (5) Each triangle of A is conjugate to (X, Y, Yl) or (X, Y, Y2), where f i E A n (X, Y) - (X, Y) and Q E A(X, Y) - ~ ~ ( ~ 1 .

Proof: Let L = E ( M ) . Then L acts 4transitively as As on LIKX, so I' = {Ye : e E L - K X ) is an orbit under K X of order 15 with K acting as A5 on r / X of order 5. Further for each Ye E I', K n K e % A4 and we may take E = 0 2 ( K nKe). Then (X, Y, Ye) < CG(E) = RX. Therefore (X, Y) = XIA1, XI, (X, Ye) = X[A2, XI, and (Y, Ye") 2 SL2(3) for each x E X . Thus I? C A:(Y). Similarly for t E M - L, K acts as L2(5) on L / x K t of order 6, so 19 = ytXKis of order 18 and K acts as L2(5) on 8/X. Further for ymE 6, K f l K m Dl0 and (X, Y, y m ) CG(K n Km) 2i A5 by 47.2. acts on a set I? of order 5 with Fixr(X) of In particular CG(K n P) order 2 and Fix(X) n Fix(Y) and Fix(X) n Fix(Ym) distinct points of 'I as Y,Ym E A(X) but Ym $ yX.Thus ymXcontains a unique Yl E A(Y) (i.e., the Y1 with Fixr((Y,YI)) # 0 ) and two members of A$ (Y). We have established (I), (2), (3), and (5). Further we showed (4) holds in the case U E A(X). If U E A:(x) then CL(U) = S E Sy12(L), so each point of L/XK is fixed by some involution of S. In particular if s E S fixes X K then s E K n S, so (4) holds in this case. Finally if U E A ~ ( x ) then CL(U) 2 Dlo and again some involution in CL(U) fixes the point K X of LIKX, so (4) holds here too.

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Lemma 47.7: (1) A is of diameter 2 with A ~ ( x )= A$(x) U A;(x). (2) lA(X,Y)l = 4 for Y E A2(x). then A(X, Y) G A n (X, Y) and A(X, Y) is con(3) If Y E nected.

Proof: Count the set Ri of pairs (U,Y) with Y E A:(x) and U E A(X, Y) in two ways. Let A i = IA(X,Y)J and A J = ~ JA(U) n A:(X)[ for U E A(X). Then ~A:(x)~A~ = [nil= lA(X)lai, while 47.5 and 47.6 say IA(X)( = 36 and ( A ; ( x ) ( / ~ ~= 9. Therefore (2) holds and of course (2) implies (1). Further if Y E A ~ ( x )then K = (X,Y) E A5 and by inspection A(X, Y)n K is of order 4 and connected, so (3) holds. Namely, as in the proof of the previous lemma, if we view K as acting on a set I' of order 5 then U E A n K is in A(X) if Fixr(X) n Fixr(U) # 0 and U E A ~ ( x ) otherwise, which makes the calculation easy.

265

We now set up the machinery to analyze AH. First 0 has five singular points which we label S = {sl, . ,s5); then si = 2i = 24%is the image of two involutions of Q. We choose X _< H. Then X is contained in four complements to Q in H conjugate under CQ(X), and if A is such a complement then A* 2 A acts as A5 on S. Notice each a E A5 is determined by its action on S. For example, we may choose notation so that X = (2) with x = (sl, s2, s3), which we usually write as x = (1,2,3). Then CQ(X) = (24, z5) E Ds and [Q, x] = ( ~ 1 ~zlza) 2 , 2 Qs. Notice U E A n A is in A(X) if U and X have a common fixed point on A and U E A;(x) otherwise.

..

Lemma 47.8: A(X, Y) is connected for Y E A ~ ( x ) .

Proof: Without loss of generality (z) = Z({X, Y)), so X* = Y*. Then Y = Xa for some a E Q and without loss of generality a = zl. Let A be a complement to Q in H containing X as above and let Uj = (ui) < A with ui = (i,2,3), for i = 4,5. Then Ui E A(X) and a E CQ(Ui) so Y = Xa E A(Ui). Also U4 E A(U5). NOWZi E CQ(U9-i) - CQ(Ui) for i = 4,5, and CQ((X,Y)) = {z4,z5), so since IA(X,Y)I = 4 by 47.7.2, we conclude A(X, Y) = {U4, U z ,U5, UF ). Finally A4 Z (U4, U5) 2 (U4, U5)zi = U (,: US-i) 2 (U4,U5)Z4Z5= ( U p , Up), SO A(X, Y) is connected.

Lemma 47.9: A is simply connected. Proof:By 47.7, A is of diameter 2, so by 34.5, it suffices to show each rgon is in C3(A) for r < 5. By definition of C3(A) (in Section 34) triangles are in C3(A). Further if Y E A 2 ( x ) then by 47.7 and 47.8, A(X,Y) is connected, so squares are in C3(A) by 34.6. Thus it remains to show that if p = Xo. - X 5 is a pentagon, then p E C3(A). By 47.6.4 we may take (Xo, X2, X3) < H. Let Xi = (xi). Suppose (Xo,X2} = A r A5 and X3 E A?(xO). Without loss of generality x0 = (1,2,3) and x2 = (3,4,5). Now I = (X2,X3) E A4 2 (Xo,X3}*, so we may take xg = (1,3,4). As (xO,x3) SL2(3), xg 4 = B for some b E CQ(X2) = (zl,z2). A. ~ u ast I E A4, I < Further CQ(I) = (z2), so I < B n BZ2 and B = z1 or ziz2. Let v = (2,3,4) E A; then V = (v) G A(Xo, X2). If b = zl then b E CQ(u), so V = vb B and then V E A(X3). Similarly if b = 2122 then V = VZ1 < At' = BZ1z2Zl= BZ2, SO again V E A(X3). Therefore in any case V E X$ n X$ n X: , and then 34.8 says p E C3(A). Suppose X2,X3 E A ~ ( x ) . Without loss of generality, zt; = (1,2,3), xa = (3,4,5), and xz = (1,4,5). Then there is U = (u) E A(X2, X3)

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266 Chapter 16 Groups of Conway, Suzuki, and Hall-Janko Tgpe

48. Groups of type Suz

with u* = (1,3,4) and by the previous paragraph, if V E A(U,XO)then XoXlX2UVXo, XoVUX3X4Xo, and UX2X3U are in C3(A), s o p is too by 34.3. This leaves the case X2, X3 E A$(xo). Now if U E A(X2, X3) with U E A$(x~), then as in the previous paragraph, p E C3(A), so assume otherwise. Then Xg* = X.j or X,*, say the former. But then up to conjugation in H, (Xo,X2,X3) CH(E) = RXO, with z € (XO1X2)and Z((Xo, X3)) = e = z9 or zzg. Then replacing z by ze, we reduce to a previous case.

K({X, Y)) = (t), where t E CG(GXY) with xt = Y. Therefore P = YC = (xt)[ = ZtC.Hence the pair a,C forms a similarity of U with U , in the sense of Section 37. Indeed as t E CG(GXY) and tC E CG(GXy), (bt)a = bar = (ba)tc for each b E GXY, so the pair forms an equivalence of 2.4 and U.By 47.6.5, each triangle of A is G-conjugate to a triangle of AK. By 47.9, A is simply connected. Thus we have achieved the hypotheses of Corollary 37.8, and that result completes the proof.

Lemma 47.10: Up to isomoqhism there is a unique group of type J2.

48. Groups of type Suz

Proof: By 25.10, a root J2-subgroup J of Col is of type J 2 or J3. (Cf. 49.1.1 for the definition of a root J2-subgroup of Col.) Further J contains root Cinvolutions and 2-central involutions of Col, so 3 has more than one class of involutions, and hence 3 is of type J2. That is, there exists a group of type J2. Thus it remains to show that if G is a group of type Jz then G E G. For thiis we use Corollary 37.8. Thus we must first construct a uniqueness system for G. Let Y E A(X); by 47.5, Gxy g As. Let Dl0 2 D I GXy and K = CG(D); by 47.2, K 2 As. We let AK = A n K regarded as a subgraph of A. Finally let U = (G, K, A, AK) and observe that U is a uniqueness system for G in the sense of Section 37. For example, (M, K ) is of type J2, so by 47.1.2, G = (M, K). Also E(M) is the unique maximal subgroup of M containing Gxy, so as Kx $ E(M), M = (Gxy1Kx). Next E(M) is the unique covering of A6 over Z3 (cf. 33.15 in [FGT]) and M is the split extension of E(M) such that M I X 2 PGL2(9), so there exists an isomorphism a : M + M, where of course M = NE(X) for some X E A. Further as M is transitive on its As-subgroups . (cf. Exercise 5.1 in [FGT]), GXya = GRP for some P E ~ ( x ) Let K = CG(Da) and form the uniqueness system a = (C?, ElA, AR) just as U was constructed. Then K g As g K and Kx S S3 = Aut(KX), so by Exercise 14.1.1, a : KX + Kx extends to an isomorphism C : K -,

In thiis section we assume G is a group of type Suz as defined at the end of Section 46. Further we continue the notation of Section 46. As usual let A be the set of G-conjugates of members of AH and regard A as a graph with vertices adjacent if they commute.

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K. -

Now CG(GXy ) 2 A4 is a Borel group of K , so CG(Gx y)C is one of the two Borel groups of K containing X. These two groups are interchanged by T E CA,t(k)(Kx), so replacing c by CT if necessary, we may assume CG(GXY)C = CE(GRT)_. Next for some Z E X , YCQ = P, where cz is conjugation by 3. Thus replacing a, C by acz,C q , we may assume YC = Y. In particular

Lemma 48.1: Let X E Sy15(H). Then (1) X is contained in a unique subgroup L of H with L 2 A5 and CQ(L) = CQ(X). (2) CG(X) = XCG(L) with CG(L) 2 i41jand CG(CG(L)) = L. A Sylow 3-subgroup Y of L is in A. (3) NG(L) = (L x CG(L))(a), where a is of order 4 with (a)CG(L) = CG(NL(Y)) Mlo and (a)~ / ( a r~ S)g

.

Proof: First CQ(X) = D E D8 and we may take E 5 D.Now by 46.3, X is contained in a complement Lo t o R in NG(E) and Lo = L x L l with L 2 A5 and L1R = (Q,Qg). Then X < L 5 H and Nc(XE) = NL(X) x L I E with L I E E! S4 and D ILIE. Now CH(D) = CQ(D)L and as NH(X) = NL(X) x D 5 NH(L), L is the unique complement to CQ(D) in CH(D) containing X , so (1) holds. Let I = CG(X). Then L I E II and D E Sy12(I) as D E Sylz(In H). Let E' be the second 4group in D. By the uniqueness property in (I), L also centralizes Sq g LiE' 5 NG(E1), so K = (LIE, LiE') < CI(L). Now Y E Sy13(L) AH. Let S = NL(Y) and M = CG(Y). Observe that D8 E D = CQ(S) as CQ(DY) 2 Ds and for s E S an involution, (s)CQ(DY) is semidihedral. Further as s inverts Y, s induces an outer automorphism on M/Y 2 U4(3). Therefore by Exercise 16.1.5, CM(S) 2 Mlo. Therefore as K 5 CM(S), K % A6. Further NI(Do) 5 K X for each 1 # Do 5 Dl so K X is strongly embedded in I. However, K X has no subgroup of odd order transitive on 1zKl, so by 7.6, I = K X . Therefore K = CG(L). Now NG(K) = KNH(D) = (K x L)(a), where

Ea = E' and (Y induces an outer automorphism on L. We may choose (Y E CG(S), SO as C G ( S )2 Mlo, (3) holds. Lemma 48.2: Let Y E AH. Then (I) C G ( Y ) has one class of involutions zCc( Y). (2) C G ( Y ) is transitive on involutions s inverting

Y and each such s E H is of type c2 in H*. Further C G ( ( s ) Y )Z Mlo, N G ( Y ) = ( s ) C G ( Y ) ,and N G ( Y ) / Y S Z2/U4(3) is determined up to isomorphism.

Proof:Part (1) follows from 46.5 and the fact that U4(3)has one class of involutions. By ( I ) , zG n C G ( Y )= zCc(*), so ( 3 ) holds. We saw during the proof of 48.1 that there exists an involution s E H inverting Y with C G ( ( s ) Y )2 Mlo; indeed by Exercise 16.1.5, ( s ) C G ( Y ) / Yis a uniquely determined subgroup of Aut(U4(3))transitive on the involutions not in U4(3), so N G ( Y ) / Y is determined up to isomorphism and C G ( Y ) is transitive on involutions inverting Y . Finally each involution in H* is of type a2 or c2 (cf. Exercise 2.11) and if i E H inverts Y then [Q, Y,ij is nonsingular, so i* is not of type a2. Thus (2) is established.

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Lemma 48.3: Let X E Sy15(H), A4 g I < C G ( E ( C G ( X ) ) ) and , U= O2(1). Then

u#,

typeczinH*. . (3) N G ( I ) = I C G ( I )( s ) , where s is an involution inducing a graphfield automorphism on C G ( I ) and I ( s ) Z S4. (4) Each involution i n N G ( I )-CG(I) is fused into U under G, while zCc(I) is the set of involutions in C G ( I ) . Proof: Let Lo = CG(CG(X))be the the As-subgroup of H containing X supplied by 48.1; thus I I Lo. From 48.1, a Sylow 3-group Y of I is in AH. Also Lo has one class of involutions, so u E U is conjugate t o an element inverting Y and hence u* is of type c2 in H* by 48.2. From the action of Lo on Q there is a unique singular point in [Q, U ] , which we choose t o be E. Thus U R = CH(E'/E) n C G ( E ) and Y centralizes E . Now R = [ R , Y ] C R ( Y )and using the structure of N G ( R ) described in 46.3, the argument of 46.14 shows the product is central with factors isomorphic to Sylow 2-subgroups of L3(4) and with CR(u) = CR(U) = U x S , where S = C R ( Y ) . In particular d ( S ) = { A l , A 2 ) with S = A1A2

and A l n A 2 = E . Notice NG(E)nNG(U)= I S ( Y l , s), where (Yl,s ) g S3 is faithful on E, Z3 Z Y1 centralizes I , and I ( s ) Z 5'3. We may pick s to centralize u. Then IUS(s)l = 2', so by 46.7.2, U S ( s ) = CH(u). Let M = C G ( I ) ; then S E Sy12(M) and S = M n H . Let KO = Cc(Loj; then KO 5 M and by 48.1, KO r A6. Without loss of generality z E S n KO E Sy12(Ko),so A1 n A2 n KO = ( I ) and NK,(Ai n K O )2 S4 for i = 1,2. Then arguing as in 46.14, we conclude N M ( A i ) is the split extension of Ai by L2(4) acting naturally on Ai. Let L = C G ( Y ) , so that L / Y E U4(3). Let S So E Sy12(L). By 45.5 and 45.6, S = J(So) and NL(Ai)/AiY 2 A6 acts as Spd(2)' on Ai. In particular there exists a unique subgroup Ni of NL(Ai) with Ni/fii g L2(4) and Ai the natural module for Ni/Ai. Therefore K = ( N l ,N2) is a uniquely determined subgroup of U4(3).We claim K Z L3(4). For from 24.4.0, L3(4) 2 G5 G7, in the notation of Section 24, while by Exercise 9.4, G7 2 U4(3).Further as S = J(So) is isomorphic to a Sylow 2-subgroup of L3(4) we may take S 5 G5, and then by uniqueness of Ni, K = ( N 1 ,N 2 ) = (75 2 L3(4). So K r L3 (4). In particular K has one class of involutions and S = C M ( z ) 5 K , so K is strongly embedded in M . Hence as K has no subgroup of odd order transitive on its involutions, M = K by 7.6. So (1) is established. Also as I ( s ) E S4 2 Aut(I), I M ( s ) = N G ( I ) . By 48.1 there is t E N G ( I ) acting on KoLo with Ko(t) 2 Mlo and t inducing an outer automorphism on I . It follows that s induces a graph-field automorphim on K. Thus ( 3 ) is established. Each involution in N G ( I )-I C G ( I ) inverts a conjugate of Y and hence is fused into U under H by 48.2. Each involution in U E - E is fused into U under [R,Y ] ,and zK is the set of involutions in K , so (4) holds. Thus it remains to show CG(u) IN G ( I ) , so assume not. Let D = CG(u) and B = N D ( I ) . Then S = C D ( z )<_ B and zG n B = zK = z B , so by 7.3, z fixes a unique point of D / B . Then by Exercise 2.10.1, D has a normal subgroup Do with D = BDo and S E Sylz(D0). But now K = B n Do is strongly embedded in Do, while K has no subgroup of odd order transitive on its involutions, so 7.6 supplies a contradiction.

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(3) y G n ~ = y H = h H .

(1) C G ( I ) r L3(4). (2) For u E CG(U)5 N G ( I ) , U is a TI-set i n G, and u* is of

269

48. Groups of type Suz

268 Chapter 16 Groups of Conway, Suzuki, and Hall- Janko Q p e

I

I

I

<

We term the conjugates of the subgroup U of 48.3 mot 4-subgroups of G and involutions fused into U under G as root 4-involutions. We call conjugates of the groups I and L of 48.3 and 48.1, root A4-subgroups and ~ o o A5-subg~oups, t respectively. ,

i

Lemma 48.4: (1) G has two conjugacy classes of involutions: the class

270 Chapter 16 Groups of Conway, Suzuki, and Hall-Janko Type

48. Groups of type Sux

zG of 2-central involutions and the mot 4-involutions. (2) Let t l = z and t2 be a mot 4-involution. Let G be a group of type Suz with corresponding involutions 6 , i = 1,2. Then there e$st isomorphisms (Yi : CG(ti) -+ CC(fi) S U C ~that (t? n CG(ti))ai = f y t l CG(fi) for all i,j E {1,2). (3) 1GI = 1G1. (4) Each involution inverting a member of A is a root 4-involution.

I

NM(A) transitive on the ten singular points and 30 nonsingular points of A* (cf. 45.9). Without loss of generality z E NM(A). Then CA(z) = B 2 E27 with B* a nondegenerate subspace of A* containing two singular points B; and B$.Now from the structure of H, B is abelian and by 48.2.3, B n A = {Y,Yl, Y2) with = Bt. Thus NM(A) is 3-transitive on A n A (Y) of order 10 and A Z E3a. Now for 1 # Po* 5 P*, IA : CA(Po)l > IP{l, so {A) = A(P), establishing (1).Further as lA n A1 is prime to 3, P E Sy13(G), so (3) holds. By 7.7, N = NG(A) controls fusion in A, so IN : NM(A)I = IAnAl = 11 and NIA is a Ctransitive extension of Mlo. Therefore by Exercise 6.6, N/A % Mll. As NM(A) splits over A, N splits over A by Gaschutz's Theorem (cf. 10.4 in [FGT]). So to show NG(M) is determined up to isomorphism it remains to show the representation of L = Mll on A is determined. But A = V/W for some L-submodule W of the induced ; where : K -+ GF(3) is the sign representation of module V = ,< K r Mio. In particular V is monomial with basis X and L preserves the quadratic form on V making X orthonormal. As NM (A) is irreducible on A/Y but A does not split over Y, A is not self-dual as an NIA-module, and hence W is not a nondegenerate subspace of V. Thus W has a totally singular L-subspace U-isomorphic to the dual of A and V is uniserial as a GF(3)L-module. In particular A is the unique 5-dimensional image of V and hence is determined as a GF(3)L-module. So (2) is established. As N is 2-transitive on yGn A of order 11 there are )'2I( = 55 2dimensional subspaces Yl Y2 with E A nA and each such subspace contains two conjugate points not in A n A. Hence N is transitive on the remaining 110 points of A.

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Proof: Part (3) follows from (1)and (2) and the Thompson Order Formula 7.2. Thus it remains to prove (I), (2), and (4). Let u,fi be root Cinvolutions of G, G, respectively; take u E H. By 48.3 there is an isomorphism 0 = a 2 : CG(u) -, CG(fi) and setting e = zp, we have zG n CG(u) = zCc("), so (zG nCG(u))/3 = zC n ce(a) is the set of 2-central involutions of G centralizing a. Next by 46.8 there is an isomorphism a = a1 : H -+ H = Ce(Z). By 46.1 the involutions in Q, Q are 2-central in G, G, respectively. Thus it remains to consider involutions in H - Q. Now H* E 52;(2) has two classes of involutions: those of type a2 and c2 (cf. Exercise 2.11). By 48.3.2 and 46.7, each involution of H of type c2 is a root Cinvolution, so it remains to deal with the involutions of type ag. Also this observation together with 48.2.2 implies (4). Let Y E AH. Then CH(Y) contains an involution i of type a2, and in : = Q; * [Q, Y]. But [Q, Y] E Qs, so z the notation of Exercise 2.11, Q is its unique involution and hence by Exercise 2.11, the set of involutions I(iQ) in iQ satisfies I(iQ) c Q i C CG(Y). Therefore each involution of type a2 is 2-central in G by 48.2.1, completing the proof.

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+

Lemma 48.5: Let Y E A and P E Sy13(NG(Y)). Then

Lemma 48.6: Let I be a root A4-subgroup of G and B E Sy17(CG(I)). Then NG(B) I NG(I), so CG(B) = B I and NG(B) is the split extension of B I by a of order 6 with a2E CG(I) and (a3, I) 8 4 .

(1) J ( P ) = A 2 E3s and CG(A) = A. (2) NG(A) is the split extension of A by Ml1 and is determined up to isomorphism. (3) P E S Y ~ ~ ( G ) (4) NG(A) is 4-transitive on A n A of order 11. (5) NG(A) is transitive on the remaining 110 points of A.

Proof: Let M = NG(Y) and M* = M/Y. Thus by 48.2.2, M* is E(M)* % U4(3) extended by an involution u* inverting Y, and CM.(u*) g Mlo. In particular if A is the preimage of J(P*) then Esl E A* with NM.(A*) the split extension of A* by Mlo and A* has the structure of a Cdimensional orthogonal space over GF(3) with

271

I

.

'

Proof: Let U = 02(I); by 48.3, CG((u)B) = BU for each u E u#,so by Exercise 16.6 in [FGT], CG(B) = B I or B x K with K G As. Now in the latter case Y 5 S 5 K with S 2 S3. But by 48.2.2, G is transitive on such subgroups and indeed CG(S)G Mlo, contradicting B 5 CG(S).

Lemma 48.7: IGI = 213 3

.5

-7.11.13.

Proof: Let M = Col act on the Leech lattice modulo 2. By 25.9 there is a quasisimple subgroup G of M with Z(G) = D of order 3 and G/D of type Suz. So by 48.4.3 we may take G = G/D.

We observe first that CA(D) = 0 by Exercise 9.6. Therefore D centralizes no element m of M of order 23. For Cii(m) is a 2-dimensional nondegenerate subspace of the orthogonal space A containing two singular points and hence admits no automorphism of order 3. On the other hand [GI divides jMI, so /GI = 2a - 3b . 5C 7d lle 13f. As z is 2-central and lH12 = 213, a = 13. By 48.5, b = 7 and e > 0. So as lMlll = 11, e = 1. By 48.6, d = 1. We interject a lemma:

(2) lA(Y)I = 23 5 7 = 280 and NM(Y1) 2 Z2/(Z4 * ~ ~ 2 ( 3 ) ) / 3 ~ + ~ for Y1 E A(Y). (3) A ~ ( Y )= {fi E A : (Y,Y~)r SL2(3)} is of order 35 . 5 - 7 = 8,505 and NM(Y2) is a complement to CM(z), when z is the involution in (Y, Yz) . (4) A3(y) = {Y3 E A : (yY3) 2 A5} is of order 23 35 7 = 13,608 and NM(Y3) r Z2 x Mlo. (5) a 4 ( y ) = {Y4 E A : (Y,Y4) 2 A4) is of order 2 . 35 = 486 and NM (Y4) E L3(4) extended by a graph-field automorphism.

I

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Lemma 48.8: (1) G has two classes 5,4 and 5 8 of subgroups of order 5. (2) G has a subgroup K = (LA x LB)(a) with LA A5 a root ASsubgroup of G, LB As, ( a ) ~ * / ( a ~%)S5, and (a)Lg Mlo, XC E Sy15(LC) with NG(XC) I K t and Xc E 5 c for C = A, B. (3) For P E Sy15(G), NG(P) is the split extension of P by Z4 x S3

=

=

=

, ;

while also so A ~ ( Y )= qMand the lemma holds. Lemma 48.10: (1) A ~ ( Y )= {X E- A : dA(Y,X ) = i ) . (2) G is transitive on triples (Yo, Yz, Y3) with d(Yo, Y,) = 2 for i = 2,3 I

In the remainder of this section let Y E A and M = NG(Y). As usual A(Y) denotes the vertices in A adjacent to Y but distinct from Y; that is, those X E A with (X, Y) % E9.

< i < 4,

and f i E A(Y2). Moreover Y& n Y; n Y* # 0 for each such triple. (3) For Y3 E A3 (Y), M is transitive on a2(Y) n A(Y3) of order 90.

Proof: By construction (1) holds for i = 0,l. Further if we set M* = M/Y, A(Y)* is the set of root groups of transvections in E(M)* U4(3) with the map Yl H Y; a bijection of A(Y) with the set of such root groups, so for each A, B E A(Y), (A, B)* E E g or SL2(3). Hence A2(y) is the set of members of A a t distance 2 from Y by 48.9. In particular ) A ~ ( Y are ) at distance at least 3 from Y. the members of A ~ ( YU E h 2 ( y ) ; without loss of generality z E (Y,Yo) = S. Then Let by 48.9, NM(Yo) is a complement to Y in H n M. So by Exercise 16.2, NM(Yo)has three orbits Fi,1 _< i 5 3, on A(Y) with = A(Y) nH = A(Y, Yo) of order 8, r2of order 128 and consisting of those Y2 E A(Y) with Y$ E A(Y2), and I'3 of order 144 consisting of those Y3 E A(Y)

Notice by 48.8 that c = 2 and we also showed f = 1 during the proof of 48.8, so the proof of 48.7 is complete.

Lemma 48.9: (1) M has five orbits Ai(y), 0 A'(Y) = {Y) and A'(Y) = A(Y).

Proof: The argument is much the same as 47.5. Let M* = MIY, so that M* is U4(3) extended by an involution s with CMl( s * ) % Mlo. We find members X of A ~ ( Y )such that NM(K), and hence also = J M: NM(x)I, is as claimed. For example, by 48.2 and 48.5, the map Yl I-+ Y{ is a bijection of A(Y) with the set of root subgroups of transvections of U4(3), so (2) holds (cf. 45.8 and 45.10). We choose Y E AH and let Y2 E YQ.We pick Y3 and Yq so that (Y, X) is a root A5-, root Aq-subgroup of G, for i = 3,4, respectively. Finally we observe that

lxMl

faithful on P.

Proof: Ip the notation of 48.1, let X = XA, L = LA, LB = CG(L), XB E Syz5(LB), and P = XAXB. Then by 48.1, K = NG(L) has the structure described in (2) and NG(X) < K. Further L contains a root Csubgroup U and CG(XB) n CG(u) = UXB for u E U# by 48.3, so by Exercise 16.6 in [FGT], XBL = CG(XB) and then NG(XB) < K. Therefore P = CG(P) n CG(X) = CG(P) and NK(P) is the split extension of P by Z2 x Zq. In particular if S E Sy12(NK(P)) then S E Sy12(NG(P)) as XA and Xc are the eigenspaces of S on P and XA 4 xg. SimilarIy if P 4 Sy15(G) then P is of index 5 in PI < G and as a Sylow 2-subgroup S of NG(P) is abelian, we conclude from the structure of GL2(5) that Pl q NG(P). Thus S acts on Z(Pl), so Z(P1) = XC for C = A or B, contradicting NG(XC) 5 K. So P E Sy15(G). Thus either the lemma holds or NG(P) 5 K. In the latter event 1 r IG : NG(P)I r 213 - 37 . 7 . 1 1 . 1 3 f / 2 ~ E 13f E 3f mod 5, so f = 0. But by 48.6, if B f Sy17(G) then ING(B) : BI = 72, so if f = 0 then 1 = IG : NG(B)I = 213 . 37 . 52 11/72 = -1 mod 7, a contradiction. Thus f = 1 and the lemma holds.

273

48. Groups of type Suz

272 Chapter 16 Groups of Conway, Suxuki, and Hall-Janko Q p e

on A, with i

48. Groups of type Suz

274 Chapter 16 Groups of Conway, Suzuki, and Hall-Janko a p e with Yg E A2(y3). As NG(&) is transitive on vertices at distance 2 from Yo and there exist vertices at distance 3, d(Yo,Y,) = 3 for i = 2 or 3. Let P E Sy13(M) and A = J(P). Then from 48.5, A r E3s and NM. (A*) and NM. (Z(P*)) are the two maximal parabolics of M* with Z(P*) = X * for some X E A(Y). Then picking m E CM(Z(P)) NM(A) and letting K = (A,Am), we have from 48.9.2 that K = 0 2 ( c G ( x Y ) ) 2 ~ ~ ~ ( 3 ) / Choose 3 ~ + ~Xo. E A n A - {X, Y) and set X2 = X c . Then X2 f A2(xo) and we choose notation so that z E (Xo, X2). Indeed for X3 E Am f l A - {Y, X, X2), X3 # X i 5 Am, so X3 E A2(xO) and X i E A(X3). Thus (Xo,X2,X3) is a triple with d(Xo,X2) = 2, X3 E A(X2), and X i E A(X2,X3). But we showed in the previous paragraph that G is transitive on such triples, so (Yo, Y, fi) E (XO,X2, ~ 3 )Then ~ . as d(Xo, X3) = 2, also d(Yo, Y2) = 2. Therefore by the previous paragraph, d(Yo,Y3) = 3 and M is transitive ,: on the members of A at distance 3 from Y. Finally Y E X+ n X: nx so (2) is establiihed. To complete the proof of (3) we show Y3 E A3(yO).Then we count the set R of pairs (Y, f i ) with Y E A2(yO)and Q E A(Y) n A3(yo) in two ~ IA3(Yo)[@ ~ = 23 35 7 @, ways: (35. 5 7). (Z4 32) = I A ~ ( Y= ~IRI) = where a = IA(Y) n A 3 ( ~ ) = l lr21= 144 and P = lA(Y3)nh2(yo)l. We conclude @ = 90, so that (3) holds. It remains to show Y3 E h3(yO). We consider the action of H* R i (2) on the orthogonal space Q. Now the stabilizer L* of a nonsingular point ii of Q is isomorphic to Ss. Represent L* on {I,.. ,6) and let Xi = (xi) E L n A for i = 0,2,3, with x3 = (1,2,3), xo = (3,4,5), and x2 = (4,5,6). Then (X;, X i ) G A5, so by 48.9, X3 E A3(xO).Similarly X3 E A(X2). Finally (X;,Xz) Z A4, so replacing X2 by a suitable conjugate under Q, we may take X2 E A2(xO).Thus (Xo, X2,X3) is conjugate to (Yo,Y, Y3), completing the proof.

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.

Lemma 48.11: Let Y3 E A3(y). Then (1) NM(Y3) is transitive on A(Y) n A"Y~) of order 90, 180, 10, f o ~ i = 2,3,4, respectively.

(2) A(Y) n A2(y3) is connected. (3) F o r X E A(Y)nA3(&), A(Y,X)nA2(y3) #a.

Proof: We represent M* = M/Y on the set t of singular lines of a 6dimensional orthogonal space V over GF(3), with A(Y) corresponding to C via the equivariant map X H [V,X*]. Notice X1 E A(X) if and only if [V,XI] n [V,V2] # 0.

275

Let N = NM(Y3). By 48.9.4, Z2 x Mll E N* = CM.(@) for some involution @ E M* E(M*). Then by Exercise 16.1.8, N* has three orbits tjl1 < j 5 3, on t of order 10, 180, 90, which therefore correspond to the orbits of N on A(Y). By 48.10.3, the orbit of length 90 is A(Y) n A2(y3). Then (2) holds by Exercise 16.1.9. Further if X E A(Y) with [V,X*] E t2then by Exercise 16.1.8, X E A(X1) for some X1 E A(Y) n A2(y3), so da(X, f i ) = 3. On the other hand by 48.10, A4(y) # 0, so as M is transitive on A3(y), A4(y3) n A(Y) # 0.We conclude (1) holds. Moreover X I E A(Y, X ) fl A2(y3), so (3) holds.

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Lemma 48.12: Let & E A4(y). Then NM(&) is transitive on A(&) and A(&) c A ~ ( Y ) . Proof: Let X = IA(&) n A3(y)I and a = IA4(y) n A3(y)I for Y3 E A3(y). Then counting pairs Y4) with & E Ai(y) and Q E A(G) in two ways, we get

(z,

from 48.9 and 48.11. We conclude X = 280 = IA(Y4)J, so A(Y4) C a 3 ( y ) . F'urther by 48.11, G is transitive on triples (yY3,Y4) with & E A ~ ( Y )and Yq E A(Y3), so NM(Y4) is transitive on A(&) n A3(y) = A(&).

Lemma 48.13: A is simply connected. Proof: By 48.10, A has diameter 4, so by 35.14 and 34.5 it suffices to show each r-gon p = Xo .X, is in C3(A) for 4 < r < 9. If r = 4 this holds by 34.6, once we show A(Y, 5)is connected for fi E a 2 ( y ) . But taking a E (Y, Y2), (Y,Y2) = Y[Q, YJ = S and A(Y, Y2) = A1 U A2, where Ai = A n Si and 0 2 ( c H ( s ) ) = S1* S2with Si 2 SL2(3). So as A(Y,Y2) is connected. Ai c A(X) for X E If T = 5 then p E C3(A) by 34.8 and 48.10.2. If r = 6 then p E C(A) by 48.11.2 and Exercise 12.2.1, while if r = 7 we appeal to 48.11.3 and Exercise 12.2.2. Finally by 48.12 there are no 9-gons in A and 48.12 plus Exercise 12.2.1 handle the case r = 8.

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Lemma 48.14: Let L !2! As, V a 4-dimensional orthogonal space of sign -1 over GF(3) regarded as a faithful GF(3)L-module of dimension 4, A = N G ~ ( ~ ) ( Land ) , U = U(L, V) the largest GF(3)L-module such L] < V. Then that V < U, CV(L) = 0, and [U,

(1) L is absolutely imducible on V, and AIL Z D8. (2) d i r n ( ~ l (v)) ~ , = 2.

276 Chapter 16 Groups of Conway, Suzuki, and Hall-Janko Type A has two orbits on hyperplanes of U containing V with representatives Ul and U2 such that NA(Ul) g Zz x S6and NA(U2) S z2 x M10. Proof:The first statement in part (1) follows as a Sylow %subgroup of L fixes a unique point of V. For the second observe that from Section 45, the semidirect product S = LV is a local subgroup of X = U4(3) and if B = Aut(X) then B / X % Ds, B = XNB(S), and V is self-centralizing in l&(S) = D, Thus D/V 5 A with DILV E D8, Further as L is absolutely irreducible on V, ICA(L)I = 2, while IOut(L)I = 4, so indeed D j V = A. Next L has two conjugacy classes of subgroups K~ and K~~ with K 2 As, for some a E A (cf. Exercise 5.1 in FGT]). By Exercise 15.3, V r [VI, L]/CV,(L), where VI is the permutation module for L over F on the cosets of I = K or Ka. Notice, however, that UK = VK/CVK(L) is not isomorphic to UKa = VI(./C~~.(L) as K fixes a point in the ~ , 2 2. former but not the latter. Thus d i r n ( ~ l (V)) ~ , = 1. This is because W/CW(K) = Observe next that d i r n ( ~ l (V)) U(K, V), where W is the 6-dimensional permutation module for K , since a Dlo-subgroup of K fixes a point in Vf - V for any extension Vf of V by the trivial K-module. Thus if Z is an GF(3)L-module with CZ(L) = 0 and [Z, L] = V, and dirn(Z/V) > 2 then there exist points X , Y of Z fixed by K, Ka, respectively, with V X = V Y. Thus V X 2 UK g UKo, a contradiction. So (2) is established. Finally by the universal property of U (cf. 17.11 in [FGT]), the action of A = NGL(")(L) extends to U. Let e E A induce scalar action on V. Then (e) = CA(L) and as CV(L) = 0, Cu(e) = 0, so AIL S D8 is faithful on U/V and hence has two orbits on the points of U/V with representatives Ui/V, i = 1,2. We may pick Ul r UK, so NA(Ul) = LNA(K) 2 Z2 x S6. As U2 is the restriction to L of the 5-dimensional irreducible for Mll discussed in the next lemma, NA(U2) % Z2 x MIO. (3)

+

+

+

Lemma 48.15: Let L S Ml1, V a faithful 5-dimensional GF(3)Lmodule induced from the sign representation of Idlo, and K = LV the semidirect product. Then (1) V is determined up to isomorphism as a GF(3)L-module, so K is determined up to isomorphism. (2) L has two orbits on the points of V of length 11 and 110. (3) If X is a point in the orbit of length 11 then

48. Groups of type Suz

277

Proof: Parts (1) and (2) were established during the proof of 48.5. Moreover Lx % Mlo and E = E(Lx) 2 A6 and from the proof of 48.5, V is isomorphic to the dual of the module U2 of 48.14.3 as an Emodule. Thus by 48.14.3, NGqV)(E)= LX(t), where t is an involutory automorphism of K centralizing L and inverting V. So Aut(Kx) = Lx (t)U, where U = U(Lx, V) in the language of Section 17 of [FGT]. But by 17.11 in [FGT] there is an injection U/X --+ U(LX, V/X) and U ( L ~V/X)/(V/X) , r (U(E, V/X)/(V/x)) n c ( L ~ )z z3 by 48.14.3. Thus (U : VI < 3. Now take S to be the quasisimple subgroup of Col with Z(S) r Z3 and S* = S/Z(S) E S z , supplied by 46.6.1. Then there is Kg 2 S* with Kg E K , and as KO contains a Sylow 3-subgroup of S, KO does not split over Z(S). Thus H'(L, V) # 0, so V < U ( L ,V). But as LX contains a Sylow %subgroup of L, U(L, V) U, so as (U : VI 5 3, we conclude U = U(L, V). Thus Aut(Kx) = Lx(t)U = NAut(K)(KX), completing the proof of (3).

<

Lemma 48.16: Let P E Sy13(G) and A = J ( P ) . Then (1) J = ( A n P ) . (2) G is transitive on triangles of A and each such triangle is fused into A n A.

Proof: By 48.9 if X, Y E A with (X, Y) a 3-group then [X,Y] = 1, so B = ( A n P ) is abelian. Then as CG(A) = A 5 B, (1) holds. Then (2) follows from (1) and 48.5.

We now establish the main theorem of this section:

Lemma 48.17: Up to isomorphism there is a unique group of type SUZ. Proof:By 25.9 there exists a group of type SUZ. Thus it remains to prove then G % c. As usual we construct a that if G is a group of type SUZ uniqueness system U for G and appeal to Corollary 37.8. Let P E Sy13(NG(Y)), A = J ( P ) , and K = NG(A). Let AK = A n A; by 48.5, K is Ctransitive on AK of order 11 and AK is a complete subgraph of A Thus AK contains a second member X in Z(P). Let U = (G, K, A, A,) and observe that U is a uniqueness system in the sense of Section 37. Pick P E A; by 48.2.2 and Exercise 16.3 there exists an isomorphism a : G y -+ C?.);. Let P = P a , A= J(P),K = N&), dR= AnA, and 2 = Xa the second member of in Z(P).Then U = (G, K, A, ELR) is also a uniqueness system.

.

278 Chapter 16 Groups of Conway, Suzuki, and Hall-Janko Q p e By 48.5.2 there is an isomorphism C : K -+ K with XC = X and YC = F. Thus KyC = KCYC= KT = Kya. Now we appeal to Theorem 37.12 with Z(Y) = Y. By construction, hypotheses (1) and (2) of ~ ) ver( K ~ ) . Theorem 37.12 hold. By 48.15.3, Aut(Ky) = A U ~ ~ ~ ~ ( Thii ifies hypothesis (3) of Theorem 37.12. As K is 2-transitive on K/KX, hypothesis (4) of Theorem 37.12 holds. Thus Theorem 37.12 says our uniqueness systems are similar. Next by 48.9.2, F*(GXY) = 03(GXY) and Z(KXy) = 1. So as P 5 Kxy, 38.10 says CAut(Gxv)(KXY)= 1. Therefore Theorem 37.9 says our uniqueness systems are equivalent. Now by 48.16, each triangle in A is fused into AK under G. Also by 48.13, A is simply connected. So Corollary 37.8 completes the proof of the main theorem of this section. We close thii section with several results which establish the existence of certain subgroups of G and various properties of the graph A. These results will be used in the next section in our proof of the uniqueness of groups of type Col.

Lemma 48.18: Let U be a root 4-subgroup of G and V E Sy13(CG(U)). Then (1) CG(V) = V x E(CG(V)) with E(CG(V)) Z Ag and NG(V) =

TOCG(V), where Qs E To E Sy12(E(CG(U)) n NG(V)) and ToE(CG(V))/CTO( E ( c ~ ( V ) ) 9 )!S6. (2) NG(Vo) NG(V) for each 1 # Vo 5 V. Proof: Let L = E(CG(U)) and I = CG(L), so that L r L3(4), I is a root A4-subgroup of G, and NL(V) = VTo with Q 8 E To faithful on V. Further NG(V) n Nc(I) = I(t) x VTo, where t = uh induces a graph-field automorphism on L and u E u#. Without loss of generality (z) = Z(To). Then as zG n NG(I) C L, V(z) 5 (zG n CG(t)) = L ~ so , (I, rh) = K 5 CG(V(z)). Next as U is weakly closed in T E Sy12(I(t)), T E Sy12(CG(V)). By 46.1, all involutions in Q are in zG, so as no involution in T is in zG, T (7 Q = 1. So K n Q = 1 and K E K* H*. Let KO be a root As-subgroup of G containing I, as described in 48.1. By 48.1, NG(Ko) contains a conjugate of VTo, which we take to be VTo. Notice that by 48.1, CT,(Ko) = (to) is of order 4. Then to E CH(Ko) < Q, so [to,K ] < Q n K = 1, and therefore K* 5 CH. (h)9!S6, so K E A6. Now K V is strongly embedded in CG(Vo)for each 1 # Vo 5 V and no subgroup of K V of odd order is transitive on the involutions of KO,so by 7.6, K V = CG(Vo). Thus the lemma is established. Notice ToK/(to) S Sfjas TOKO/(tO)g S5.

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49. Groups of type C o l We refer to conjugates of the subgroup E(CG(V)) of 48.18 as mot As-subgroups of G .

Lemma 48.19: (1) G is transitive on triples (Xo, Xi, X2) from A with d(Xo, XI) = 4 and X2 E A(X1). (2) For each such triple, (Xo, XI, X2) is a root As-subgroup of G, Xz E A3(x0), and Cc((Xo,Xi,Xz)) % Z ~ / E S .

Proof:Part (1) is 48.12. Then (1) and 48.18 imply K = (Xo,X2,X3) is a root As-subgroup of G and Z4/E9 E V(tO) < CG(K), with the notation chosen as in the proof of 48.18. As (to) = CH(K), CG(K) = O(CG(K))(tO)with z inverting O(CG(K)). Therefore O(CG(K)) is abelian so O(CG(K)) = O(CG(K)) n NG(V) = V. Lemma 48.20: For each X E A: (1) A(X) n H # 0,and (2) Xz E A ~ ~ ( x ) .

Proof:If X E A ~ ( Y for ) i > 2 then zG nNG((X,Y)) E: CG((X,Y)) by ) it remains to establish (1). 48.3.4 and 48.1. Thus X Z E A ~ ~ ( xand If z E NG(X) then by 48.4.4, X < H, so (1) holds. Suppose X Z E A(X). Then by 48.9.2, O ~ ( N G ( X X ~2 ) )~ ~ ~ ( 3 ) / so 3 z~ centralizes + ~ , and ) )lies in CG(zl) n NG(XXZ)E some conjugate zl in 0 2 ( ~ ~ ( x X Z D8/(& x SL2(3)), where we check that z centralizes some member of A(X, XZ) n C(z1)Finally suppose X z E A ~ ( x ) and let (22) = Z((X,XZ)). Then z E Qo = 02(C~(.t2))= QI * Q2 * Q3, where QI = [Qo,X] and 9 2 and Q3 are the two quaternion subgroups of CQ,,(X). Then z = xy, where x E Q1 - (3)and y E CQo(X). Then x is of order 4 so as z = xy is an involution, y is also of order 4. Therefore y E Qi for i = 2 or 3, and hence z centralizes Xi E A(X) with [Xi, Qo] = Q5+, 49. Groups of type Col In this section we assume G is of type Col as defined as the end of Section 46, and we continue the notation of Section 46. Again A denotes the set of Fconjugates of members of AH, regarded as a graph whose edges are pairs of commuting members of A. Recall that for X E A, CG(X) is quasisimple with CG(X)/X 2 Suz by 46.6. Defme the root Cinvolutions, root 4-subgroups, root Aqsubgroups, etc. of G to be the Gconjugates of the corresponding elements or subgroups of CG(X).

279

280 Chapter- 16 Groups of Conway, Suzuki, and Hall-Janko Q p e Lemma 49.1: Let B be of order 5 in H with CQ(B) 2 QSD8. Then

Let X E A, U a root 4-subgroup of CG(X), u G U# a root 4 involution, and K = E(CG(UX)). Then K % SL3(4) by 49.2 and 48.3. Now if u = z j for some j E G then setting J = ~j and J*= J / Q ~ , we have either U 5 CQj(K) or K * Cj. (U*) 5 P* for some maximal parabolic P * of J * Z Rsf(2). In the former case K* cj.(O) = P*, a maximal parabolic. So in any event K* r SL3(4) is contained in a maximal parabolic of 52$(2), which is not the case as a Levi factor of such a parabolic is solvable or A8. So octad involutions are not root 4-involutions. Similarly by 46.12, the centralizer of a dodecad involution contains no SL3(4)-section, so dodecad involutions are not root 4involutions. Therefore (1) holds. Then (1) and 48.4 imply zCc(X) = rGfl CG(X) and therefore (2) holds.

(1) E(CG(B)) = L 2 J2. (2) CG(L) r As is a root A5-subgroup of G. (3) NG(L) = NG(CG(L)) = (CG(L) x L)(P), where P is an involution inducing an outer automorphism on L and CG(L).

Proof: As in 25.10, CG(B)/B is of type J2, so by 47.10, c G ( B ) / B J2. We claim CG(B) splits over B, so L = E(CG(B)) % J2.For by 47.2 there is Lo CG(B) with Lo/B = L l I B x L2/B Dl0 x A5. Now both L1 and L2 split over B; for example, L2 splits over B as L2IB is perfect of 5-rank 1 (cf. 33.14 in [FGT]) while L1 splits as O(L1) = Co(tl)(d) x [O(Ll),d] for d an involution in L1. Thus as Lo contains a Sylow 5-subgroup of L, L splits over B by Gaschutz's Theorem. Next L contains some X E A; let M = CG(X), so that M I X 2 Suz. Then by 48.1, CM(B)/X S Zg x As and CM(E(CM(B))) = K x X with K 2 As a root As-subgroup. Now CL(X) contains E(CM(B)) and hence an As-section, so by 47.3, X 5 E(CM(B)). Then also X 5 E(CM(B)) IE(CM(I)) for a root Ad-subgroup I of K , so we have shown:

<

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Lemma 49.2: If X E A and I is a mot A4-subgroup of CG(X) then x ~ ~ ( 1 xr1~ ~ ~ ( 4 1 . Returning to the proof of 49.1, we have [Q, B] = [Q n M, B] is Kinvariant so K acts on CQ( [Q,B]) = CQ(B) and then K = [K,B] centralizes CQ(B). Therefore K centralizes (CQ(B),E(CM(B))) = L. Then by a Frattini argument and 48.1, NG(L) = LNM(L) = LK(a), where a is of order 4 in M and induces an outer automorphism on K , and (a)E(CM(B))/X r Mlo. Now by 47.3, NL(X) = E(CM(B))(y) with NL(X)/X 2 PGL2(9) so there is an involution P E yaE(CM(B)) inducing an outer automorphism on K and L.

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49. Groups of type Col

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/ i

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Lemma 49.4: (1) G has three classes ty, 1 < j 5 3, of invodutions: the octad involutions, the dodecad involutions, and the mot 4-involutions. (2) If G = Col and ai : Hi -+ Efi is the isomorphism of amalgams supplied by 46.12.4, then ai(t7 n Hi)= fy n Hi for i = 1 , 2 and j = 1,2,3. (3) X E A is inverted by dodecad involutions and root 4-involutions, but not by octad involutions. (4) H is transitive on its root 4-involutions and if u is a root 4involution in H then ICH(u)I = 215 . 3 . 5 . Proof: First by Exercise 2.11, H* has five classes of involutions of type a2, c2, cq, ad, and a:. We choose T E Sy12(H), let A = J(T), and choose notation so that (AnQ)/(z) = [a, Q]for some a of type a4. (Thii is possible from the proof of 46.12.) Then the members of A* are of type a2 and aq. By 46.10, if j* E a&then j is not an involution. Thus

(a) If i is an involution in H then i E Q or i* E a2, c2, aq, or cq. Next by Exercise 2.11: (b) H has (at most) two orbits on involutions of type a4 with r e p resentatives a and az, while H is transitive on involutions of type cq. (c) Each involution of type a2 and a4 is fused into A under H.

We call the conjugates of the group L of 49.1.1 root J2-subgroups of G. Recall the definition of octad involutions and dodecad involutions from Section 46; in particular zG is the set of octad involutions of G .

Lemma 49.3: (1) The classes of octad involutions, dodecad involutions, and root 4-involutions are distinct. x ~ X E A ~ . (2) H ~ A = A ~ =~ O

For ad-involutions this follows from (b). Similarly there is an involution of type a2 in A such that each involution in iQS (we use the so (c) follows for notation of Exercise 2.11) is fused into A under cH(;), a2-involutions from Exercise 2.11.

Proof: With notation as in 46.12, as A is weakly closed in T , NG(A) controls fusion in A by 7.7, so octad and dodecad involutions form different classes.

(d) H is transitive on its involutions v of type c2, each such involution is a root CinvoIution, and \CH(v)l = 215 - 3 5.

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I

282 Chapter 16 Groups of Conway, Suzuki, and Hall-Janko a p e Namely we may choose B as in 49.1, F the root As-subgroup centralizing E(CG(B)),and u E HnF a root Cinvolution inverting B. Then u* E c2 and u centralizes K = C H ( B )with ~ Q$ = CQ(u)= 0 2 ( K ) * QG. Next C H ( B ( u )contains ) X E A and each involution in uQ$ is fused to u or u j under KQ, where j E C Q ( B X )- (z).Finally from 46.7, u and u j are fused in C H ( X ) Thus . each c2-involution of H is conjugate to the root Cinvolution u. Further luHl = lc21X,where X = 48 is the number of involutions in uQ. So ICH(u)l= ICH.(u*)llQ1/48= 215. 3.5. We now adopt the notation of (2); then H = HI and Hz = NG(A) M. Let a = a2, ( = a l l and M* = MIA. Now a(.zM)= a ( ~ ) = ~ f (M ~ ) is the orbit of octad involutions in A under M and similarly if d E A is a dodecad involution then a ( d M )= dL is the orbit of dodecad involutions in under M. Therefore by (c), all involutions of type a2 and a4 are octad or dodecad involutions. Indeed H fl M has two orbits on those i E A-Q of type ak for each k , one orbit octad and the other dodecad, so (e) H has two orbits on involutions of type ak for k = 2,4, and in each case one orbit consists of octad involutions and the other of dodecad involutions. Further as the pair a , C is an isomorphism of amalgams, a of type a2 or a4 is octad in G if and only if ((a) is octad in G. Similarly as a , ( is an isomorphism of amalgams and each involution of M is fused into Hn M under M , once we show [ ( t ynH ) = iy n H for each j, the same holds for M. Thus it remains to prove all involutions of H of type c4 are dodecad, since by (d) the q involutions are root Cinvolutions. Next each involution i inverting X centralizes a member of zG n C G ( X ) ,so by 49.3, we may take i E H. But only involutions of type c2 and c4 invert X* in H*, so to prove (3) it also suffices to show c4involutions are dodecad. Finally observe that CG(d)= CM(d)has more than two orbits on cG(d)nzG,so H has at least three orbits on d G n ~ forcing , involutions of type c4 to lie in dG. Namely HnM has two orbits on dodecad involutions in A, so CG(d)has two orbits on octad involutions in A. Further there are octad involutions in CG(d)- A fused into Q - A under M. 49.5: Let I be a root 4-subgroup of H and U = 02(1) a root 4-subgroup of G. Then

. .. .Lemma

(1) N G ( ~ O 5)NG(I)for each 1 # Uo 5 U. (2) NG(I) = ( I x E ( N G ( I ) ) ) ( ~ where ) , E(NG(I)) G2(4), 7 in-

duces a field automo~phismon E(NG(I)),and I ( y ) r S4. (3) The octad involutions in NG(I) are the long ~ootinvolutions

283

49. Groups of type Col of E(NG(I)),the dodecad involutions are of the form uu for u E U# and u a short root involution of E(CG(I)),and all other involutions are root 4-involutions of G.

Proof: The argument is much like that of 48.3. First from the proof of 48.3 we may choose X E AH to centralize I and with U CR(X).Let Y E Syls(I),so also Y E AH. Arguing as in the proof of 48.3, we have R = CR(Y)* [R,Y ]with [Y,R] a Sylow 2-subgroup of L3(4) and CR(Y) the "R" for CG(Y).Moreover U = CR(Yl)for some Yl of order 3 with NG(E)n CG(U) = U x ( C R ( Y ) ( Yx~ C G ( Y Y ~ E )and )) CG(YYIE) L2(4).Further for u E u#, CR(U)CG(YYIE)(~) ICH(U),where T E NG(E)with E ( T )3 Dg and I ( T )2 S4.Then by 49.4.4, this containment is an equality. First CG(I)I: CG(Y).Let Kl = CG(I)fl NG(X).Then by 48.3 and 49.2, Kl = E ( K l ) ( a ) ,where E ( K l ) r SL3(4)and a induces an outer automorphiim on E ( K l ) inverting X . Now pick B and L as in 49.1 with X L and I contained in the root As-subgroup CG(L). Then by 47.3, as we saw during the proof of 49.1, N L ( X ) = E(NL(X))(ao)with N L ( X ) / X 2 PGL2(9). Notice C E ( N L ( X()a)) = D 2 Dl0 and by 47.2, K2 = CL(D)r As. Let KO = (K1,K2),so that KO I CG(I).We will show CG(I)3 G2(4). As a induces an outer automorphism on E(K1)inverting X , cu induces a graph or field automorphism. As a centralizes an element of order 5 in D 5 E(K1),cu induces a graph automorphism. Now K1 5 C G ( X Y ) and by 46.6.2, C G ( X Y ) is quasisimple with C G ( X Y ) / X Yr U4(3).We saw during the proof of 48.3 that E ( K l ) ,and hence also K l , is determined up to conjugation in C G ( X Y )and hence also in CG(Y).Further by 48.8, K2 = E(CG(YD)),so KO = (K1,K2) is determined up to conjugacy in CG(Y).Thus as C G ( Y )is determined up to isomorphism independent of GI without loss of generality G = Col. Let G act on A, the Leech lattice modulo 2, and set V = CA(U).Now by Exercise 9.6, Cii(Y) = 0 and each u E U# inverts a conjugate of Y , so dim(CA(u))= 12. Thus 0 # dim(V) 5 12. Further letting F = GF(4), we have that as C V ( Y )= 0, Y induces an F-space structure on V preserved by KO. Similarly Ci(X) = 0, so Kl is faithful on V . Then as dimF(V) 6 and a induces a graph automorphism on E ( K l )r SL3(4),V = Vl @V2, where Vl is the natural FE(K1)-moduband V2 = Vla is dual to Vl. In particular dimF(V)= 6 and V = CA(u)for each u E u#. Moreover Kl is determined up to conjugacy in GL(V).

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284 Chapter 16 Groups of Conway, Suzulci, and Hall-Janko n p e

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Lemma 49.7: Let X E A, Px

Lemma 49.6: IG( = 221 - 3 . 5 a7 .11.13.23. Proof:Let G = Col . By 46.12.4 there is an isomorphism cri : Hi + H~~ i = 1,2, of amalgams, where H1 = H , Hz = M = NG(A), and A = J ( T ) for T E Sylz(H). By 46.13, CG(d) < M for d E A a dodecad involution.

E Sy13(NG(X)), and

J = J(Px). Then

(1) J g E3a and J = C G ( J ) . (2) NG(J) is the split extension of a group K with Z(K) of o d e r 2 inverting J and K/Z(K) E M12. (3) K is 5-transitive on J n A of order 12.

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Then by 49.4, G has three classes of involutions 1 j < 3, and u l and a 2 induce isomorphisms ai : CG(ti) --+ Ce(ti), i = 1,2, such that ai(tF n CG(ti)) =:i n Cc(fi) for each i , j. Finally by 49.5, there is an isomorphism a3 : CG(t3) + CC(f3) with ~ x ~ ( t y n ~ = ~ f( ?t n~C)E)( ~ ) . Therefore by the Thompson Order Formula 7.2, IGI = Then 22.12 completes the proof.

Next V = [V,Dl 63 Cv(D) with dirnF(Cv(D)) = 2. Thus

where L, E GLz(4) with [V,Dl L+] = 0 and [CV(D),L-] = 0. Now L2(4) E K2 L+L-, so as (a)X 5 K2 with CV(X) = 0, K2 is a full diagonal subgroup of L+L-. Then as (a)X is contained in a unique such subgroup, K2 is uniquely determined. Thus KO = (K1, Kz) is determined up to conjugation in GL(V). Finally K, = G2(4) < GL(V) is generated by subgroups K1 and K2 as above (cf. [A3]) so KOE K*, completing the proof that KO G2(4). Next by 48.3 there is an involution y fused into U in CG(X) with (y)I S4 and y inverting Y and inducing a graph-field automorphism on E(K1). We pick y E NH(E) and let (u) = Cu(y). By the first paragraph of this proof, (y)UCKo(z)= CH(u). In particular y acts on (Kl,CH(U)) = U x KO and then on KO, and setting M = CG(u) and K = UKo(r), we have CM(z) K. As 7 induces a graph-field automorphism on Kl, y induces a field automorphism on KO G2(4). Then as G2(4) is transitive on its field automorphiims, K is transitive on involutions in K - UKo, and each is a root Cinvolution. Next KO2 G2(4) has two classes of involutions: the long root elements in zK0 and the short root elements v fused into K2 in KO, which are therefore root Cinvolutions of G. Further, from 48.3 each member of Uz is a root Cinvolution of CG(X). Thus all involutions in K are root 4involutions except those in zK and possibly involutions fused into VU#. But CK(z) = CH(u) contains dodecad involutions, so these involutions are dodecad. It remains to show K = CG(u). Assume not. We have shown tGnK = zK and CM(t) K , so K is the unique point of M / K fixed by z by 7.3. Thus if M # K we can apply Exercise 2.10 to obtain a contradiction. Namely by Exercise 2.10, M has a normal subgroup Mo with M = MOK and Mo n K = KO. Then if V is maximal in the set U = U(Mo) of Exercise 2.10, then as zK n V = 0, V is either (v) or the root group of v in KOfor some short root element v of KO.But then CKo(V)/V does not have one class of involutions, contradicting Exercise 2.10.2.

285

49. Groups of type Col

Proof: The proof is much like that of 48.5. Let M = NG(X) and M* = MIX. By 48.5, J(P5) 2 E35 and if Jo is the preimage of J ( P 2 ) then (CG(X) n NG(Jo))/Jo 2 M11 with CG(X) n Nc(Jo) 4transitive on Y * n~J$ of order 11, where Y E AH centralizes X. Now by a Frattini argument INM(Jo) : Cc(X) n NG(Jo)I = 2, SO as Mll = Aut(Mll), NM(Jo)/Jo Z Z2 x Mi1. Let t be an involution in NM (Jo) with It, NM(Jo)] J o Then t inverts X and as NM (Jo) is irreducible on J;, t either inverts or centralizes J$. As no involu~ subgroup ~ in/ its centralizer, 3 ~ it is the former. Then tion has an ~ NM (Jo) n CG(t) = KOis a complement to Join NM(Jo) and as t inverts

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Jo,J~~E3s.~rtherforeach1#Po~Px,~Jo:CJo(Po)~>~Po~,~~ J = Jo.As CG(J) < CM(J) = J , J = CG(J), so (1) is established. Also K = NG(J) n CG(t) is a complement to J in NG(J). Next we may take z E KO and CJ(z) E Esl with AH n Cj(z) = ~ ~ n nU {X) ~ ( of~order ) 4 and AH n X Y = {X, Y) for some Y E AH n CJ(z). SO by 49.3, A n CJ(z) = YNMn~(j)U {X). But from 48.5, each subgroup of J of order p is fused into Cj(z) under KO, so A n J = yK0U {X) and y k H Y * is~ a bijection of yK0with Y * ~ o . Therefore a s I Y * ~ O = I11, lA n JI = 12. Now KO is 4-transitive on A n J - (X) and J is weakly closed in NM(J), so NG(J) is transitive on J n A. Hence K is 5-transitive on A n J. From 48.5, (t) is the kernel of the action of KO on A n J and hence also of the action of K. Then by Exercise 6.6.7, K/(t) M12. In the remainder of this section let X E A and M = NG(X).

Lemma 49.8: M has five orbits Ai(X), 0 5 i

< 4,

on A, where

49. Groups of type Col

286 Chapter 16 Groups of Conway, Suzukz, and Hall-Janko Q p e

287

Ao(X) = {X) and ) (1) Al(X) = A(X) is of order 25.5.11.13 = 22,880 with N M ( X ~= E(NM(XX1)) edended by E4 and E ( N M ( X X ~ ) ) / X X % ~ U4(3) (2) A2(X) = {X2 E A : (X, X2) S SL2(3)) is of order 34. 5.7.11 13 = 405,405 with NM(X2) a complement to X in H n M,when z E (XI X2). (3) A3(X) = (X3 E A : (X,X3) E As) is of order 25 35 11-13 = 1,111,968 with NM(x3) z2 X J2. (4) A4(X) = {X4 E A : (X,X4) E Ad) is of order 2.35 -11 = 5,346 with NM(X4) g A~t(G2(4)).

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.

Proof: The proof is entirely analogous to that of 47.5 and 48.9, and is left as an exercise.

Lemma 49.9: Define J as in 49.7 and let J 5 P E Sy13(G). Then (1) J = ( P n A ) . (2) G is transitive on triangles in A. Proof: As NG(J) is 5-transitive on J n A, (1) implies (2). By 49.8, if Y E A with (X, Y) a 3-group then [X, Y] = 1,so B = (Pfl A) is abelian. Then as CG(J) = J < B, (1) holds. Lemma 49.10: A is of diarneteF2. Proof: This follows from 49.8, which says A(X, Y) # 0 for each y

, --

--

E A.

Lemma 49.11: (1) G is transitive on triples (Xo,X2, X3) from A with Xz E A4(Xo) and X3 E A(X2) n A~(xo). (2) (Xo, X2, X3) = K =" As is a root A6-subgmup of G, X3 E A3(Xo), and NG(K) = ( K x CG(K))(T),where CG(K) 2 U3(3), K(T) 2 S6,and CG(K) (7) G2 (2). (3) x$-nx+nx,I #a. Proof: Let X4 E A4(X), so that I = (X4,X) g Aq is a root A4subgroup and L = CG(I) r G2(4). Then A(X, X4) consists of the tenters Y of Sylow 3-groups of L and from the proof of 49.5, NL(Y) S Zp x SL3(4), so L is transitive on A(X, X4) of order IL : NL(Y)I = 25. 5.13. Thus I' = A(X) - A(X,X4) is of order 25 . 5 . 1 0 13. Pick X3 E I' such that there is Xp E A(X, X4,X3). By 48.19, K = (X, X3, X4) is a root As-subgroup of G, X3 E A3(X4), and CG(KX2) z4/31t2. Let p be of order 4 in NK(XX3). Then P is faithful on XX3 so P induces an automorphism on CG(XX3)/XX3 = D* U4(3) with I 03(CD. (X2+)nCG(P)).Hence by Exercise 16.1.10, CG((P)XX3)E U3(3)

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=

for suitable choice of p, as CAut(K)(I) fl NAut(K)(X3) = (7),where T is a transposition in S g Aut(K), since T acts on some conjugate of /3 under XX3. Then as lGz(4) : G2(2)1 = [I'llwe conclude that L is transitive on I? and (2) holds. Transitivity of CG(I) on I' gives (1) and (3).

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Lemma 49.12: Let I be a root A4-8ubgroup of G and V of order 7 in CG(I). Then CG(V) =" Z7 x A7 and NG(V)/V S Z3 x S7. (2) NG(E(CG(V))) is of index 2 i n PGLp(7) x S7. (3) Let W be of order 3 in CG(E(CG(V))). Then W is fused into (Q, Q9) and NG(W) r S3 x Ag. (1)

Proof: By 49.1 we have a subgroup L1 x L2 of G with L1 a root A5-subgroup of G containing I and J2 E L2. Further by 49.11, L1 is contained in a root As-subgroup K l and U3(3) CG(Kl) = K2 5 CL2(Kl)-Next by Exercise 16.5, there exists h E Lp with Kz flK$ = K 2 L3(2). Then F = (K1, K!), and indeed CG(F) = C G ( K ~fl ) CG(K~= ) ~K2 fl ~ , =h K . Now S4 S < K with the involutions in S in zG, HJ without loss of generaIity z E 02(S). Then a Sylow 2-subgroup of S is contained in Q, so without loss of generality E = 02(S), and then by 46.3, S (Q,Qg) with S n R = E. F'urther for W E Sy13(S), by 46.3, S3 S So = (Q, 99) n N(W) is a complement to R in ( 9 , ~ ~ ) and NG(W) n NG(R) = So x KO,with KO = CG(WE) & As.iThen So = Ns(W) and CG(S) = CG(WE) 2 As. Next CG(I) g G2(4) by 49.5, and lG2(4)I7 = 7, so we may take V 5 K. Then K = (V,S) so F 5 CG(K) = CG(V) n CG(S) and we conclude F = A7 or As. But taking U = 02(1), NG(U) fl CG(V) = ( I x X)(T) x V, where (T)I 2 S4 and X(T) =" S3.SO F E A7 and F V is strongly embedded in CG(V). Then by 7.6, F V = CG(V). That is, (1) and (2) hold. Now a 2-central involution d in KOis diagonal in the product U x Ul of root 4-subgroups in KO,so by 49.5, d is a dodecad involution. Then W 5 CG(d) = NG(A) n CG(d), where A = J ( T ) for some T E Sy12(G) by 46.13, so if G = Col then W is the subgroup of order 3 discussed in 26.4, and by 26.4, NG(W) E S3 x Ag. In particular by 46.12.4, CG((d,W)) is determined up to isomorphism as is the fusion of involutions in CG((d,W)). Similarly i b ( ( u , W)) 2 Z3 x (Z2/(E4 x A5)) with fusion of involutions determined, so by the Thompson Order Formula

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288 Chapter 16 Groups of Conway, Suzuki, and Hall-Janko Q p e 7.2, ICG(W)I is determined and thus is 3 . IAgl. So ICG(W) : WKol = 9 and then as KOr As, CG(W) r Z3 x Ag. We term the U3(3)-subgroup of 49.11 a mot U3(3)-subgroup of G. We term the An-subgroups of G, n = 7,9, appearing in 49.12 as root An-subgroups of G. Let C be the closure of the set of all triangles of A and all squares XoX1X2X3Xo with Xi+2 E A4(Xi) for each i. Lemma 49.13: C = C4(A). Proof: Let p = XoXlX2X3Xo be a square in A. We first observe that the graph on A(Xo, X2) obtained by joining X to Y if X E Aa(Y) is connected. This is because for K = (A(Xo, X2)), K = (NK(X), NK(Y)). Thus the result follows from 34.7. Lemma 49.14: A is &generated. Proof: By 49.10 and 34.5 it suffices to show each 5-gon p = Xo Xg is in C4(A). If X2 E Ai(XO) for i = 2 or 4, we do this by showing X; fl X$~X; # 0 and appealing to 34.8. Namely if X2 E A4(XO)this follows from 49.11.3. If X2 E A2(Xo) we will see it follows from 48.20. For by 49.8, NG(Xo)flNG(X2) is a complement to X2 in NG(X2) fl CG(z)when z is the involution in (XO,X2). By 48.20 there is X E A(X2, X3) fl H. NOW

x ~x;nx,lnx,i.

Thus we may assume Xi+2 E A3(Xi) for each i. In particular K = CG((Xo, Xz)) r J 2 and there is X E A4(Xo)nA4(X2)n (XOlX2). Then M = CG((X,X2)) S G2(4) and M is transitive on A(XZ)- A(X) with CG((X, W,X2)) = U(W) 2 U3(2) for W E A(X2) - A(X) by 49.11. We produce Y E A such that for j = 0,2,3, Y # A3(Xj). Then for 0 5 i 4, we pick a geodesic pi from Xi to Y in A. Now p is in the closure of the cycles qi = pi - p$l Xi+1Xi for each i, so it suffices t o show qi E C4(A). But if d(Xj,Y) = 1 for j = i or i 1 then qi is of length 4 and hence in C4(A), while if d(Xj,Y) = 2 then a t least one k = j or j 1 is 0, 2, or 3, so the cycle qi of length 5 contains Xk, Y with Y 4 A3(Xk), and hence qi E C4(A) by earlier reductions. Thus it remains t o produce Y. If X3 E A(X, X2) let Y = X. Thus we may assume X3 $4 A(X, 1 2 ) . In this case we let V = U(X3) and observe that K n V contains an element of order 2 or 7. [ M : KI = z5 . 13 and IVI = 25 33. 7, so lK n VI 1 IVI/IM : KI > 14. Therefore as maximal subgroups of V of odd order are of order 21 and 27, either K n V contains an element of order 2 or 7, or some Yo E A. In the latter case let Y = Yo. If k E K n V is of order 7 then L = (X3,X2,Xo) 5 CG(k) and by

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49. Groups of type Col

49.12, L is a root AT-subgroup. So L~ = L/02(L) E AT. Similarly if k is an involution in K fl V then k E zG so we may take k = z and I& : CQ(Xi)l = 4, so ICQ(L)I 2 8. If CQ(L) Z Q8 then L centralizes Yo E AH with CQ(L) = [Yo,Q], and we let Y = Yo. Otherwise we may take L 5 CG(E), so LRIR is a subgroup of CG(E)/R 2 As, so again L* r A7. Thus in anv case L* r A7. Then we may pick Y E A n L with (Xi*,Y*) not isomorphic to As, and hence Y 4 A3(Xi). Namely representing L* on (1,. . . ,7)we may take Xi* = (xi*),where x$ = (1,2,3), x; =. (4,5,6), and xf, = (1,4,7). Then take Y* = ((4,5,7)). Lemma 49.15: G Col. Proof: As usual we construct a uniqueness system for G; then we appeal to Theorem 37.7. Let W be the subgroup of order 3 in (Q,Qg) discussed in 49.12.3. Then by 49.12.3, K = NG(W) 2 S3 x Ag. Let G = Col and w a corresponding subgroup of G. Then there is an isomorphism :K 4 I?. E A ~ Then . = XC, so that Pick X E AK = K fl A and let U = (G, K, A, A,) and 0 = (G, I?, A, AR) are uniqueness systems. By Exercise 16.4, there exists an isomorphism a! : Gx 4 Gz. Now K x E S3x (z2/(z3 x A6)). Let A(KX) be the subgroup of Aut(Kx) permuting the root Csubgroups of KX. Then A(KX) = AutK(Kx) S S3 x S3 x S6. Thus by Exercise 14.3.1, we can choose C so that C = a on Kx. Pick Y E A(X) n K and let = YC. Then P = Ya, GXYa = and K({X, Y))C = K((R, P)), so a , C define a similarity of U with U. Next Gxy is of index 2 in Aut(E(Gxy)) = NG(XY) = L, where E(Gxy) 2 U4(3)/Eg is quasisimple. Now CL(Kxy) = CG(KXy) Z(Kxy) = 1, so

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C A ~ ~ ((KxY) G ~ ~= )1Therefore by 37.9, a,6 define an equivalence of U with U. We now appeal to Theorem 37.7 t o complete the proof. By 37.7, it remains to show AK is a base for A. Thus by 49.13 and 49.14, it suffices to show that each triangle and each square p = Xo . -Xq with Xi+2 E A4(Xi) for all i in A is fused into AK under G. But by 49.9, G is transitive on triangles of A and by 49.5 and 49.8, G is transitive on squares satisfying the hypotheses of p. Thus as AK contains triangles and such squares, our proof is complete. Remarks. The general structure of the the groups J2and J3was determined by Janko in [J2].The uniqueness of Jzas a rank 3 permutation

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Exercises

290 Chapter 16 Groups of Conway, Suzukz, and Hall-Janko Q p e group on the cosets of U3(3) was proved by M. Hall and Wales in [HW]. Suzuki was the first to investigate Suz; see, for example, [Su]. Soon after, Suz was discovered to be a section of Col and much of the structure of the two groups was determined by Conway and Thompson. The general structure of Suz and Col was investigated in a systematic way by N. Patterson in his thesis [PI, where Patterson also produced uniqueness proof%for Suz and Col. To identify the groups, Patterson appealed to a theorem of B. Stellmacher [St] on groups generated by a class A of subgroups of order 3 such that for each distinct X , Y f A, (X,Y) is isomorphic to Eg, A4, A5, or SL2(3). S. K. Wong also worked on Suz and Col independently and he and Patterson published their work jointly in [PWl] and [PW2].

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=

=

Exercises 1. Let (V,f ) be a &dimensional orthogonal space of sign -1 over F = GF(3), G = A(V, f ) the group of similarities of the space, and G = PA(V, f). Let L = E(G). Prove (1) L 2 ~ n ; ( 3 ) r u4(3), G = AU~(L), and GlL DB. (2) Let z be an involution in G with [V, z] of dimension 4 and sign +l. Then F*(CE(2)) = Z Q; and is of type U4(3). (3) Let K = GF(9), K# = (C), C2 = i, and (vK, fK) a 3-dimensional orthogonal space over K with basis X = {xl, x2, x3) with f K ( ~ i , ~ j ) = O f ~ r a l jl, ifK(xi)= # fK(xilxi) = 1 f o r i = 1 , 2 , and f K(x3) = [. Let T = be the trace from K to F. Take V to be vK regarded a s an F-space. Then (V,To f K, is 6-dimensional of sign -1, so we may take f = To f K . (4) Let /3 = i1 be scalar multiplication on vKvia i and regard P E G. Then CL(/3) = LK(r, - I ) , where LK = E ( A ( v ~fK)) , r Q3(9) r L2(9) and T is the semilinear map T : ajxi H agxi with X ~ T= -C(xl x2), X ~ = T <(x2 - xl), and x37 = x3. (5) Let A be the set of G-conjugates of elements a E CC(z) such that a2 E (Z) and Ca(a) 2 D8. Then G is transitive on A and for each a E A, a is an involution, l is transitive on involutions in a l l and Cz(a) S Mlo. (6) E A. (7) (P,CL(P)) = H has two orbits S1 and S2 on singular points of V, where S1 = {Fv : v is K-singular) is of order 40 and S2 = {Fv : fK(v) = zti) is of order 72. Further L~ has two orbits Sg, e = &I, of order 36 on S2, where S$= {Fv : fK(v) = ~ i ) .

(8) H has three orbits Lj, 1 < j < 3, on the singular lines of V, where Ll = (1 : 1 is a K-singular point of vK)is of order 10, t2={1:11nS11=2, 11nS;I=l, e = f l ) i s o f o r d e r 1 8 0 , a n d L3 = (1 : Jln S$I = 2, c = f 1) is of order 90. Representatives are K(xl +ixz), F ( x l + FiCx2, and F < x l + FCx2, respectively. That is, x l ixz is K-singular, fK(<xj) = i, fK(i[x2) = -i, f K ( ~ x l 4x2) = -i, and f K (<XI eiCx2) = 0. (9) Let L be the graph on the singular lines of V with 1 adjacent to k if 1n k # 0. Then L3 is a connected subgraph of L. (10) Up to conjugation there exists a unique automorphism y of E of order 4 with 72 $4 i such that 03(Ci(X)) r l C(yi contains 31+2 , where X is the center of a Sylow %subgroup of L. Further Cz(7) U3(3) and C ~ ( 7 ) 1 ( 7 ) G2(2). (Hint: See Section 2 in [A51 for some help.) 2. Let V be a Qdimensional unitary space over GF(9), L = SU(V) 2 SU4(3), L* = L/Z(L), and H the stabilizer in V of {U, u ~ }where , U is a nondegenerate line of V. Prove (1) H = CL(z*),where z is the involution in L with U = [V,z]. (2) H has three orbits ril1 5 i 5 3, on the set r of singular points of V. (3) rl = (I' r l U) U (I' n uI) is of order 8. (4) r2= {(u v) : (u) E I? n U, (v) E J?n u ~ is}of order 128 and W + Wz is a singular line for W E r2. (5) I3' consists of the points (u v), where (u), (v) are nonsingular points of U,u ', respectively, with (u, u) = -(v, v). M h e r lrsl = 144 and W W z is a nondegenerate line for W E r3. 3. Let G = U4(3), Go the covering group of G, and G = G ~ / o ~ ( z ( G ~ ) ) . Prove (1) Let M A ~ l E s lbe a maximal parabolic of G, M the preimage of M in G, and z = z(M). Then 2 = Z(G) and M ~ / O ~ ( Z ( M ~where ) ) , Mo is the covering group of M. (2) A Sylow 3-subgroup of the Schur multiplier of G is isomorphic to Eg. (3) 03(M) = W is the dual of the MIW-module U(L, V) of 48.14. (4) B = AU~(G) Aut(G) and N ~ ( M )has two orbits on points of 2 with representatives 21 and where N ~ ( z ~ ) / wS Z2 x S6 and ~ ~ ( 2?Z2z2 ) x Mlo. (5) ~ / 2 2is the unique quasisimple group L such that Z(L) n Z3, L/Z(L) 2 G, and L admits an involutory automorphism a! such that M(cu)/03(M) 2 Mlo.

+

I

i

1 1

3

+

, , i

xi

I

-t,

i

!

j i

291

+

+

292 Chapter 16 Groups of Conway, Suxuki, and Hall-Janko n p e (Hink Use 48.14 and its proof; for example, note the role played by M in the first paragraph of the proof of 48.14.) 4. Prove a Sylow 3-subgroup of the Schur multiplier of Suz is of order 3. (Hint: Use Exercise 16.3.2 and 46.5.) 5. Let G = J2. Prove (1) G has a subgroup K 2 U3(3). (2) G is rank 3 on R = G / K with parameters n = 100, k = 36,1= 63, X = 14, and p = 12. (3) For a E R with K = Ga, /3 E A(@),and 7 E I'(a),Kp E L3(2) and Ky = C K ( t )for t an involution in K. (Hint: Use 49.11.2 for (1).Then observe that if t is an involution in K then 02(CG(t))T)NG(CK(t))is not contained in K. Conclude C K ( t ) = K-, for some 7 E R. Similarly observe that for P E Sy17(K), NG(P)is not contained in K and I r L3(2) is the unique proper subgroup of K of index at most 36 containing P.) 6. (1) If X is a group with Z ( X ) 2 Z2 and X / Z ( X ) 2 Qg, prove X z2 x Qg. (2) Let S = Sp4(3). Prove (a) S has a subgroup SL2(3)wrZ2. (b) S / Z ( S ) Z RZ(2) and involutions of type a2 in RZ(2) lift t o I involutions of S , while involutions of type c2 lift t o elements of order 4. (3) Let G = R6(2), G9 the covering group of G , and G = Go/ 0 2 ( Z ( ~ o ) )Prove . G % Sp4(3),so a Sylow 2-subgroup of the Schur multiplier of G is of order 2. 7. Let G = ~ 8 + ( 2Go ) , the covering group of G , and G = G ~ / O ~ ( Z ( G ~ ) ) . Let M be the stabilizer of a point in the natural module for G , Mo the covering group of M, M = M ~ / O ~ ( Z ( Mand ~ ) )& , f the preimage ; of M in G. Prove (1) M is the split extension M = LA, where L Z As and A 2 EG4is , % Z2. the natural module for L. Further H ~ ( LA) (2) M is the split extension M = LA, where , ! 2 As/Z2 is the 1 covering group of As and A = [ o ~ ( M M ]) , Z2 x D:. (3) Z(G)= Z ( M ) 2 Eq and M = M / ( Z ~ ~ where ~ ) , z E fr, y E A, and X Z ( M ) ,y ~ ( are~ involutions ) of type c2 in G. (4) z(&) = ( 1 , a2,b2,c2), where ~ z ( G and) ~ z ( G are)involutions of I type a4 in G and CZ(G)is of type c2 in G. i (Hint: See Exercise 2.11 for the discussion of involutions in G. In (2), imitate the proof of 8.17.)

Chapter

Subgroups of Prime Order in Five Sporadic Groups

=

I

I

We have considered five of the sporadic groups in detail: M24, He, J2, Suz, and Col. In particular we have characterized each of these five groups in terms of a hypothesis H(w,L ) for suitable w and L, sometimes with some extra conditions. That is, we have proved that there exists a group G satisfying hypothesis H(w, L ) and that G is unique up t o isomorphism. We have also determined the order of G , and we have generated much information about its subgroup structure. In particular we have determined most of the subgroups of G of prime order and their normalizers in G. This information is summarized in five tables at the end of this c h a p ter. In the case of M24 and J2 all the necessary facts have already been established. Namely Section 21 describes all subgroups of M24 of prime order and their normalizers. Similarly Section 47 contains the corresponding information for J2. However, some work remains to be done for Suz, Col, and He. We complete the discussion of the subgroups of prime order for these three groups in this chapter.

50. Subgroups of SUXof prime order In this section we assume G is Suz. Further we continue the hypotheses and notation of Sections 46 and 48. Fkom 48.7, G has order 213 .37. 52 . 7 . 1 1 13.

-

294

By 48.4.1, G has two classes of involutions. Their centralizers are described in 46.1 and 48.3. The subgroups of G of order 5 and their normalizers are described in 48.8. The normalizer of a subgroup of order 7 is described in 48.6. According to Table Suz, G should have three classes 3 ~3B, , and 3 c of subgroups of order 3. The class 3 4 is A and the normalizer of a member of A is described in 46.6. The normalizer of a group in 3 c is described in 48.18. Let X = ( 2 ) € A and Y = ( y ) E A ( X ) ; then ( x y ) E 3B. To verify that N G ( ( X Y ) is ) as described in Table Suz, using 48.9.2, we only need to show NG((xy))5 N G ( { X , Y ) ) . If not, xy = xlyl for some X 1 = { x l ) E A and some YI = ( y l ) E A ( X 1 ) with { X , Y ) # ( X 1 , Y l ) . Let M = C G ( X ) and M* = M / X , so that M* r U4(3) and y* = (xy)* is a long root element of M*, Then x y E C M ( X 1 ) .But by 48.9, C M ( X 1 ) * contains no long root elements unIess X 1 E A ( X ) or b2(x). Suppose X 1 E A ~ ( xThen ) . by 48.9, C M ( X 1 ) *= C M ( z ) * ,where z is the involution in ( X ,X I ) . Then z centralizes xy and X , so z centralizes Y . Then Y E CA(x)(z) = CA(x,)(z), contradicting x y E C ( X 1 ) . So Xl,Yl E A ( X , Y ) . But now from 48.9, x l y l 4 XY,contradicting XY = XlYl. So the normalizers of our three classes of subgroups of order 3 are as claimed; it remains to show that each subgroup of order 3 is in one of these three classes. So let B be of order 3. As M contains a Sylow 3-subgroup P of G we may take B 5 M. Now NG(X)* Z2/U4(3) has three classes of elements of order 3 corresponding to elements with Jordon block structure Jz, J?; 5:; and J3, J1 on the natural module for SU4(3).Here J: indicates i Jordon blocks of size j.Elements of the first two types are fused under M* into J ( P * ) = J(P)* and by 48.5, each subgroup of J ( P ) of order 3 is in 3~ or 3B. Thus we may take B* of type J3,J1. But then the three subgroups of order 3 in B X - X are fused in M , so all such subgroups of order 3 must be in 3c, and our proof that G has just three classes of subgroups of order 3 is complete. Next we observe -

-

51. Subgroups of Col of prime order

Chapter 17 Subgroups of Prime Order

Lemma 50.1: Let g E Col be of order 13. Then - -0)CA(g)= 0, and (2) g centralizes no element of order 11 i n Col.

Proof: By 23.3, Col has three orbits on the points of with stabilizers Co2, Coa, and Mz4/E211. Hence, using, for example, 22.15, 13 does not divide the order of the stabilizer of a point of A, so (1)holds. Next the minimal dimension of a nontrivial GF(2)(g)-module is 12,

;

1

295

so d is the sum of two irreducible (9)-modules and hence CGLIAl(g) G L ~ ( whose ~ ~ ~order ) , is not divisible by 11. Thus ( 2 ) also holds. We have seen that the order of centralizers of elements of prime order p < 11 in G are not divisible by 11 or 13, and by 50.1 there is no element of order 13 11 in G, so elements of order 11 and 13 are self-centralizing. Therefore the normalizer of a Sylow psubgroup P of G is a Frobenius group with kernel P for p = 11 and 13; further Sylow's Theorem gives us the order of that normalizer, completing the verification of Table Suz.

51. Subgroups of Gol of prime order i

1

1

In this section we assume G is Col and continue the hypotheses and notation of Sections 46 and 49. By 49.6, G has order 221 ~ 3 ' .54 7 2 .11 13 23. By 49.4, G has three classes of involutions and the centralizers of these involutions are described in 46.1, 46.13, and 49.5. We next consider the subgroups of order 3. First A = 3~ and the normalizer of a member of A is described in 46.6. Next 3~ consists of the elements ( x y ) ,where X = ( a ) E A and Y = ( y ) E: A(X). We argue as in the previous section that NG((xy)) _< N G ( ( X , Y ) ) ,so NG((xy)) is as described in 49.8.1. Namely if M = C G ( X ) ,M* = M / X , and xy = x l y l with X I = ( 2 1 ) f A and Yl = ( y l ) E A ( X 1 ) then x y E C M ( X l ) . Now if X 1 E A i ( X ) for i = 3 or 4 then (xy)* = y* E C M ( X 1 ) * % J2 or G2(4). But then Y 5 C M ( X l ) whiie X n C M ( X 1 ) = 1, contradicting x y E C M ( X 1 ) .Then we complete the proof as in the previous section. The subgroups of type 3 0 are described in 49.12. Thus it remains to show there exists a subgroup of type 34, and that each subgroup of order 3 is in one of these four classes. Let PH E Sy13(H). Then A n PH = { X I ,. . ,X4). Let a = x l . x4 with Xi = (xi). Then CQ(a) = ( z ) and CH.(a) r SU4(2), with a inverted in H . F'urther if PH < P E Syl3(G) and J = J ( P ) then as a is the product of generators of four members of A n J and since as a module for N G ( J ) / J Mlz/Z2, J is an image of the module induced from the sign character for 2 2 x Mll on J n A, a is 3-central in N G ( J ) , so we can take a E Z ( P ) . Indeed z is a square in N G ( J ) n CG(a),so by Exercise 16.6.3, C H ( a )E Sp4(3) x Z3. Let T E Sylz(CH(a));by Exercise 16.6, T QswrZ2, so ( z ) = Z(T) and hence T E S y l a ( C ~ ( a ) )We . claim z is weakly closed in T with respect to CG(a). Each involution in T - ( z ) is fused into Z 2 ( T ) under C H ( a ) ,while if z # t E Z2(T) then To = CT(t) % Q8 x Q8.Thus if

.

.

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296

Chapter 17 Subgroups of Prime Order

51. Subgroups of Col of prime order

h E CG(a) with th = z then CT(t) = CTh (z) and conjugating in CH(a),

in J. So as in the previous paragraph, we conclude NG(J) is transitive 0 n 5 ~ n J a n d 1 5 ~ n J I rmod10. 5 Similarly 15AnJl 1 6, so as J has 31 points we conclude /!jBftJI = 10 and then as ING(J)nNG(B)I = 24.3-53, we have ING(J)I = z5-3.S4. Then as I NG(J) n NG(D)I = z5 . 53 we get I J n 5 ~ = 1 15. This leaves J n 5~ to be an orbit of NG(J) of order 6. Now NG(J) n NG(D) contains an element of order 4 inducing scalar action on J , so NG(J) induces a 2-transitive group of order 120 on the six points of J n 5,4, which must then be PGLz(5). Therefore

we may assume h E NG(To).

Thii is impossible as (z) is characteristic

in To. Therefore z is weakly closed in T. So by Glauberman's Z*-Theorem [Gl], CG(a) = KCH(a), where K = O(CG(a)). Now z inverts K/(a) so by a Fkattini argument CH(a)" S Sp4(3) is faithful on f i E Sylp(K) for each prime divisor p of 1KI. However, if p # 3 then lPll < p4 with equality only when p = 5, so Sp4(3) is not faithful on PI. That is, K = 03(CG(a)). Therefore as P 5 CG(a), 1KI = 35, and then as CH(a) is faithful on K , K r 31+4. Therefore NG((a)) has the structure described for a member of 3.4 and we take 3.4 = (a)G. It remains to show G has just four classes of subgroups of order 3. First from the structure of NG((a)), NG(J) f l NG((a)) is of order 24 . 3', so I ( u ) ~ G (=~220. ) I But by 49.7, [ A n JI = 12 and 1 3 n~JI = 2)(: = 132, so all 364 subgroups of order 3 of J are accounted for and hence the subgroups of J of order 3 are in 3,4, 3 ~or, 3 ~ . Next each element of order 3 in CG(a) is fused into J K under CG(a). F'urther each subgroup of order 3 in J K / J is 3-central in N c- (. J ).,I J , and hence is fused under NG(J) into CG(X). Then from the discussion in Section 50, eafh of CG'(X) of order 3 is fused into J or to a subgroup V with V* of type 3 c in M* = CG(X)/X 2 Suz. But all subgroups of order 3 in VX-X are conjugate in CG(X) and from 49.12.3 each is in 3 ~ So. our treatment of subgroups of order 3 is complete. Next we consider subgroups of order 5. First @(a) has three classes of subgroups of order 5 permuted transitively by triality with centralizer Z5 x A5. Thus H also has three classes, which we write as 5.4, sB,and 5c. We choose notation so that CQ(B) Q8D8 for B € sB.Then by 49.1, there is a s u b p u p (L1 x L ~ ) ( Q with ) L1 A59 L2 J2, a inducing an outer automorphism on L1 and L2 with B 5 L1, and Nc(B) ILlL2(a). That is, B has the normalizer described in Table Col for a subgroup of type sB. This leaves (z) = CQ(D) for D in 5.4 and 5c, so CH(D) 2 Z5 x (A5lZ2)Let B J E Sy15(LlL2). Then J % Elz5. Now NL,(J) contains an element of order 3 whose unique fixed point on J is B. Further J 2 P E Sy15(G) and J = Cp(B). As the same holds for each B1 E 5B n J, we conclude NG(J) is transitive on 5 8 fl J and n JI z 10 mod 15. Next by 26.5 we may choose notation so that for D E 5 c , CG(D) % Z5 X (A~wrZ2).Then we take J E Syi5(CG(D)) and a Sylow 2-group of NG(J) n Nc(D) is isomorphic to Zq x D8 and fixes only the point D

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<

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297

Lemma 51.1: Let P E Sy15(G). Then (1) J = J ( P ) E E125, NG(J) is the split extension of J by Zq x PGL2(5), and J has the stntcture of a 3-dimensional orthogonal space over GF(5) presented by NG(J)/J acting as its group of similarities. (2) J n 5,4 is the set of six singular points of 3 and IJ fl 5 ~ = 1 10 and IJ n 5 ~ =1 15. (3) NG(P) is the split extension of P by Z4 x Z4.

Indeed an element of order 5 in P - J has one Jordon block of size 3 on J, J = J(P). Also NG(P) hap m o orbits on subgroups of order 3 in P J of length 25 and 100. Next we choose A E S A n J to be centralized by z and z to act on P. As NG(P)nC(A) 2 Zq, CH(A) S Z5 x SL2(5). Then by the Brauer-Suzuki

,

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,

,

I

Theorem(cf.12.1.1in[Go]),CG(A)=KC~(A),whereK=O(C~(A)). Now z inverts KIA, so each element k E K of prime order p # 5 is in the center of K. Further PK= P n K r 51+2, so PKis transitive on the five subgroups of order 5 in P r-7 J distinct from A. As J has the structure of an orthogonal s p x e this says J n K = A~ contains subgroups D in sC. But CG(AD) is of order 25 from the structure of CG(D), so K E 51+2 and NG(A) is transitive on subgroups of order 5 in K distinct from A. In particular the orbit of NG(P) on P - J of length 25 is contained in 5 ~We . claim the orbit of length 100 is in 5,4 or 5 ~This . will complete our analysis of subgroups of order 5 by showing each such subgroup is in 5.4, 58, or 5c. So assume the orbit of length 100 is not as claimed. For each F in this orbit, Np(AF) is transitive on the five points of A F - A, so as J . F has the structure of an orthogonal space, F is not of type 5 ~ Thus is in some fourth class, and FGn P = FNC('). We claim NG(F) 5 NG(P). By the Frobenius pcomplement Theorem (cf. 39.4 in [FGT]), CG(F) = (L x F)A, where L = O51(CG(F)). Notice CL(A) = 1. Hence

298

Chapter 17 Subg~oupsof Prime Order

52. Subgroups of prime order in He

as A = [A,t] for t an involution inverting AF, Sylow psubgroups of L are noncyclic for each prime divisor p of ILI. Thii forces p = 2,3. But elements of order 2 and 3 centralize only subgroups of order 5 in 51 for I = A, B, C. For example, each subgroup of order 5 centralizing an element of order 3 also centraliies an involution; we've seen the statement for z, and it is easy to check for the other two classes of involutions. So indeed NG(F) 5 NG(P) and hence F is contained in a unique Sylow &subgroup of G. To complete the proof let T = Z2(P)F, so that T &% 51+2 has a unique subgroup A in 5~ and five subgroups in 56 (those in Z2(P) distinct from A), and the remaining twenty-five subgroups of order 5 are conjugate to F under P. In particular as F is in a unique Sylow 5-subgroup of G, J is the unique member of JGfixed by T. Also ( ~ are )the conjugates of J satisfying the five members J' E ~ ~ c- {J) NT (J') = A. Next each D E T n 5~ is in 36 members of JG,so there are 5 35 E 50 mod 125 members of J~ containing a member of T n 5 0 but not A. Finally N*(J~) = D $ Jh if and only if Njh(D)is one of the 71 conjugates of A under NG(D) distinct from A. Then D 5 o~(cG(A~))and there are five such conjugates of J under CG((AD)~) &xed by D, so there are 71 .25 25 mod 125 such members of JG. Hence

' 1

-

But lJGl= lG : NG(J)I = 216 . 38 72 11 13.23 r 56 mod 125, a contradiction. This completes our discussion of subgroups of order 5. We next turn to subgroups of order 7. By 49.12 and 26.2 there is L G with E(L) = L1 x L2, L1 E L2(7), L2 Z A7, and L = E(L)(a) with a an involution inducing an outer automorphism on L1 and L2. Further if Pi E Sy17(Li) then NG(fi) 5 L. Let P = PlP2, so that P E Sy17(G); observe that NL(P)/P r z6x 23. Now ING(P) : PI r lG : PI r 2 mod 7, so ING(P) : NL(P)I zz 4 mod 7. On the other hand a subgroup Z of order 6 in NL(P) induces scalar action on P and NG(P)/PZ 5 PGL2(7) with NL(P)/PZ of order 3 and NG(P)/PZ a {2,3)-group, so we conclude NG(P)/PZ r A*. Therefore

<

Lemma 51.2: For P E Sy17(G), NG(P)/P r SL2(3) x

23.

In particular NG(P) has two orbits of order 4 on points of P , so Pl and P2 are representatives of the conjugacy classes of subgroups of G of order 7.

Next let I be a root A4-subgroup of G and S E Syl13(CG(I)). From 49.5 and the structure of G2(4), CG((u,S)) = US for 1 # u E U = 02(I). Hence by Exercise 16.6 in [FGT], either NG(S) 5 NG(I) or CG(S) 2 S x A5. The latter is impossible as no element of order 5 centralizes an element of order 13. This takes care of subgroups of order 13. Let X E A and V E SylI1(CG(X)). Then CG(XV) = XV and V is centralized by a dodecad involution d inverting X. From 46.13, (d)V = CG((d)V). Therefore CG(V) = K(d), where K = O(CG(V)), and d inverts K/V. In particular X 5 Z(K), so K 5 CG(XV) = XV. This completes our discussion of subgroups of order 13. Finally subgroups of order 23 can be handled using Sylow's Theorem.

52. Subgroups of prime order in He

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299

!

In this section G is He and we adopt the hypotheses and notation of Chapter 14. By 42.13, IGI = 21°. 33. 52 73 .17. By 40.6, G has two classes of involutions. The centralizers of involutions are described in 39.1, 42.5, and 42.7. By 42.16 there is X of order 3 in G with CG(X) Z A7/Z3 quasisimple and NG(X)/X c 2 S7. Thus xG= SA. By 42.16.2, the involutions in CG(X) are in 2j3 while an involution v E NG(X) with CG((v)X) r 2 2 x S4 is in 2.4. Next by 43.1.5 there is a subgroup L1 x L2 with L1 S4,L2 L3(2), and involutions in L2 in 2 ~ Thus . if Y E Sy13(L1), then Y $ x G , so we take yG= 3B. kt M = NG(Y) and K = Y(t) x L2, where t is an involution in L1 inverting Y. We may take z to be an involution in L2; then CH(Y) Z D8, so CM(z) 5 K and as tKis the set of involutions in CK(Y), zM n K = zK, so z fixes a unique point of M/K. Thus if M # K, we can apply Exercise 2.10. Now by the construction of 43.1, t E 2~ and NG((t)Y) = NK((t)Y). Let U be maximal in the set U of Exercise 2.10; as K has more than one class of involutions, U # 1, so up to conjugation in K, U = (tz). But then condition (2c) of Exercise 2.10 is violated. So M = K and NM(Y) has the desired structure. Next for P E Sy13(I$G(X)), 31+2 E P, and P E Sy13(G).As NG(X)/ X Z S7, there are three NG(X)-classesof subgroups of order 3 in P, and X is fused into one of these classes in the centralizer of a 2~-involution, so G has two classes of subgroups of order 3. Thii completes our analysis of subgroups of order 3. Next 42.14 supplies us with a subgroup K of index 2 in S5wrZ2. Let Kl and K2 be the components of K and P E Sy15(K). Then

300

Chapter 17 Subgroups of Prime Order

52. Subgroups of prime o d s r in He

NK(P)/P S2 Z4 * D8, SO INK(P)I G 1 mod 5. But ING(P) : PI = lG : PI G 3 mod 5, so ING(P) : NK(P)I = 3 mod 5. Next NK(P) has a subgroup Z r 2 4 inducing scalars on P, so NG(P)/PZ PGL2(5). As NK(P)/PZ Z Eq and ING(P) : NK(P)I EE 3 mod 5, we conclude NG(P)/P S Z4*SL2(3) or the multiplicative group of GF(25) extended by a field automorphism. The latter case is out as a Sylow 2-group is Z4 * D8. Thus

In particular NG(P) has orbits of length 1,14,21,21 on the subgroups of P of order 7. Now Z is centralized by X E 3A and hence by 42.16.2, X Z is inverted by a conjugate v of z. That is, NG(P) = PXY(v). Further subgroups of order 7 centralizing invo1utions in 2B are in yB, while subgroups of order 7 in H are not inverted in H, so CG(Z) = O(CG(Z)). As CG(XZ) = X Z , X E Sy13(CG(Z)). As elements of order 5 don't centralize elements of order 7, Cc(Z) has order prime t o 5. Finally as X doesn't centralize an element of order 17, CG(Z) has order prime to 17. Therefore NG(Z) = NG (P), and we take 7~ = Let W = Cp(v); then W(v) = CG(W(v)), so CG(W) = O(CG(W))(v) with v inverting O(CG(W))/W. Thus NG(W) 5 NG(ZW) NG(Z), and we set 7~ = wG. It remains to show G has just three classes of subgroups of order 7. Now Z, V, W are representatives for the orbits of NG(P) on subgroups of P of order 7 of length 1,14,21, respectively, so it remains to show the ) length 21 is in 7 ~Assume . not; as D is inverted second orbit D * G ( ~of ( ~n P. ) Elements of order 2, in NG(P), D 4 ?B or 7C, so D ~ ~= D~ 3 - 5 do not centralize members of DG, so NG(D) 5 NG(ZD) <_ NG(Z), and hence each member of DZ# is in a unique Sylow 7-group of G. Therefore lpGlr 1 mod 49. But IpG(= lG : NG(P)I 5 22 mod 49, completing our treatment of subgroups of order 7. The normalizer of a subgroup of order 17 can now be determined using Sylow's Theorem. This completes our discussion of subgroups of We of prime order.

<

Lemma 52.1: For P E Syls(G), NG(P)/P

24

* SL2(3).

In particular NG(P) is transitive on the points of P, so G has one class of subgroups of order 5. This completes our analysis of such subgroups. Recall our subgroup L1 x L2; by construction in 43.1, L1 = (U, u h ) for some root Csubgroup U of G. Then L2 5 CG(U) n cG(uh). Indeed if V E Sy17(L2) then NL,(V) = VX, where X E 3~ is faithful on V and VX = CG(S), where Sq 2 S with U = 02(S) and S contains a member of uh. Therefore J = (S, LI) CG(VX) 5 CG(X) r A7/Z3. Further, from the structure of NG(U), NG(U) n CG(V) = VS, so from the subgroup structure of A7 we conclude J X / X 2 As or L3(2). The former is impossible as the preimage of A6 in CG(X) is quasisimple whereas X $ CG(V). Therefore J 2 L3(2). Let M = CG(V) and M* = M/V. As U is weakly closed in a Sylow 2-subgroup of NM(U), that Sylow group is Sylow in M; notice it is D8.similarly if Y E Sy13(S) then Y € 3~ and YV = CG(YV), so Y E Sy13(M). Therefore \M*l = 23.3.7a.17b with a = 1 , 2 and b = 0 , l . But N = NM*(Y) S3 so 1= IM* : NI = 2b mod 3, and hence b = 0. That is, IM : JVI = 1or 7. The latter is impossible as 72 does not divide the order of A7. Therefore NG(V) = X V x J and we take vG = yB. Notice the involutions centralizing V are in 2B and the subgroups of order 3 are in 3 ~ . Let Z E Sy17(J); then V Z q P E Sy17(G) and V Z admits the subgroup XY with orbits of length 1, 1, 3, 3 on the points of VZ. NG(VZ) is not transitive on the eight points of V Z as Z(P) is not conjugate to V, so NG(VZ) NG(P). As V and Z are the fixed points of X Y it follows that Z = Z(P). Further there is i of order 3 in XY inducing scalars on VZ, so Cp(i) is a complement to V Z in P and P 2 7f+2. By a F'rattini argument, NG(VZ) = P(NG(VZ) fl NG(V)) = PXY. By Sylow's Theorem, ING(P) : PI = IG : PI 4 mod 7, so as NG(VZ) = P X Y , we conclude

I

1

zG.

<

<

Table MZ4 Normalizers of Subgroups of Prime Order 2A 2B 3A

<

=

Lemma 52.2: Let P E Sy17(G). Then P 2 extension of P by Z3 x S3.

and NG(P) is the split

38 I

~3(2>l~: s5/E64

S6/Z3 = ~ 2 / & S3 x L3(2)

5A Z2/(Dl0 x -44) 7A s 3 X (23/27) 11A Zl0/Z11 23A Zi1/Zz3

Chapter 17 Subgroups of Prime Order

52. Subgroups of prime order in He

Table Jz

Table He

G = J2; [GI = 27 3 3 . 5 2 . 7

G = He; IGI = 21°. 33 52. 7 3 . 17

Normalizers of Subgroups of Prime Order

Normalizers of Subgroups of Prime Order

Table Col G = C o l ; 1GI = 221 .3'

. 54 . 7 2 e11

+

13.23

Normalizers of Subgroups of Prime Order

Table Suz G = S U Z ; [GI= 213. 37 . s 2 . 7 . 1 1 . 1 3

Normalizers of Subgroups of Prime Order

List of symbols

List of symbols ci ( A ) C(l)

A1 C ( 4 1 m(v)1A,, eY 1 &I eY

Symbol H(w1L ) page Aut ( X ) a*, AutG(H),CG(H),NG(H),Sn, S y m ( X ) ,An, Alt(X) Fix(S),G y ,G ( Y ) ,xG 4 % 4 1 uL PG(V) r(G13) K ( A ) ,s t ~ ( s ) LinkK ( s ) SJ,EI 3) GL(V),GLn(F),S L ( V )1 SLn ( F ) ,r ( V ) Pr(V)1PGL(V)1GLn(q),SLn(q),PGLn(q),Ln(9) @(GI z p , Eprn 1 [ ~ l Y l[XI l YI D8, Q 8 , DgQr 1 mp(G),Inn(H) Op(G)lOn(G),O(GI1MG(Q, 2') F(G),F*(G),E(G),A wr B WQ),O(Q) M(Vi.9.. . ,Vn;V),L(Vll lVniV)lMn(V)lLn(V)lO(Vla) VIU ue1 Rad(f )1 R ( x ) L(e),0 par(e)le, Eu

go

HlIH2I -..IHn S(v, k, t)1D ( X ,X ) Im(X) M22 M23 M24 Cn(u> M12, M11, Mlo, M91 Ma

%

vc

xi 1 2 3 4 6

7 8 9 10 13 14 21 22 23 24

I

$

1

26 31 36 37 43 48 49 50 51 70

78 79

82 84 85 86 89 90 91

1

:

$

si(~)t~n,h$,~:,~$, C0l Cog,Co3,2A Mc HS Suz, J2 Fl,F2,F3, F5,F7, F24 P ( A ) ,w~(P), end@),k e ~ ( ~ ) r (T), I ( P ) D(P), , [PI, Bas, Basi, @(A) [PIs1 degs(p) bash) An(x),ASn(x),A,Cn(A) K ( A ) ,A(K)l Ln %A1 4 1 9 ( ~ 1 x ) Fzy,Fp AF~~FIDF G J ,Pj 1 JJ1A(3) F ( A )3 G ( A ) 3(@), r(@)l C(P), A(P, i ) N~

r(a)l r'i(a)l (ri) A(X),J ( X )

Bibliography

Bibliography I [All M. Aschbacher, Flag structures on Tits geometries, Gwm. Ded. 14 (1983), 21-32. [A21 M. Aschbacher, Overgroups of Sylow subgroups of sporadic groups, Memoirs AMS 342 (1986), 1-235. [A31 M. Aschbacher, Chevalley groups of type Gz as the group of a trilinear form, J. Alg. 109 (1987), 193-259. [A41 M. Aschbacher, The geometry of trilinear forms, Finite Geometries, Buildings, and Related Topics, Oxford University Press, Oxford, 1990, pp. 75-84. [A51 M. Aschbacher, A characterhation of some finite subgroups of characteristic 3, J. Alg. 76 (1982), 400-441. [AS11 M. Aschbacher and Y. Segev, Extending morphisms of groups and graphs, Ann. Math. 135 (1992), 297-323. [AS21 M. Aschbacher and Y. Segev, The uniqueness of groups of type J4, Invent. Math. 105 (1991), 589-607. (AS31 M. Aschbacher and Y. Segev, Locally connected simplicial maps, Israel J. Math. 77 (1992), 285-303. [ASe] M. Aschbacher and G. Seitz, Involutions in Chevalley groups over fields of even order, Nagoya Math. J. 63 (1976), 1-91. [B] R. Bruck, A Survey of Binary Systems, Springer-Verlag, New York, 1971. [BF] R. Brauer and K. Fowler, On groups of even order, Ann. Math. 62 (1955), 565-83. [Bu] F. Buekenout, Diagrams for geometries and groups, J. Corn. Th. Ser. A 27 (1979), 121-51. [Ch] C. Chevalley, Sur certains groupes simples, Tohoku Math. J. 7 (1955), 14-66. [CN] J. Conway and S. Norton, Monstrous moonshine, Bull. London Math. SOC.11 (1979), 308-39. [Col] J. Conway, Three lectures on exceptional groups, Finite Simple Groups, Academic Press, London, 1971, pp. 215-47. [Co2] J. Conway, A group of order 8,315,553,613,086,720,000, Bull. London Math. Soc. 1 (1969), 79-88. [Co3] J. Conway, A simple construction of the Fischer-Griess monster group, Invent. Math. 79 (1985), 513-40. [Cuj R. Curtis, On the Mathieu group M24 and related topics, Thesis: U. - -Cambridge (1972). - . . - .. [CW] J. Conway and D. Wales, Construction of the Rudvalis group of order 145,926,144,000, J. Alg. 27 (1973), 538-48. [F] B. Ficher, Finite groups generated by 3-transpositions, Invent. Math. 13 (1971), 232-46. [FGT] M. Aschbacher, Finite Group Theory, Cambridge University Press, Cambridge, 1986.

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[FLM] I. Renkel, J. Lepowski, and A. Meurman, Vertex Operator Algebras and the Monster, Academic Press, San Diego, 1988. [GI M. Golay, Notes on digital coding, Proc. IRE 37 (1949), 657. [GI] G. Glauberman, Central elements of core-free groups, J. Alg. 4 (1966), 403-20. [Go] D. Gorenstein, Finite Groups, Harper and Row, New York, 1968. [Grl] R. Griess, A construction of Fl as automorphisms of a 196,883 dimensional algebra, PTOC.Nut. Acad. Sci. USA 78 (1981), 689-91. [Gr2] R. Griess, The =endly Giant, Invent. Math. 69 (1982), 1-102. [GMS] R. Griess, U. Meierfrankenfeld, and Y. Segev, A uniqueness proof for the Monster, Ann. Math. 130 (1989), 567-602. [HW] M. Hall and D. Wales, The simple group of order 604,800, J. Alg. 9 (1968), 417-450. [Ha] K. Harada, On the simple group F of order 214 . 36 . 56 . 7 11 . 19, Proceedings of the Conference on Finite G~oups,Academic Press, New York, 1976, pp. 119-95. [He] D. Held, The simple group related to M24, J. Alg. 13 (1969), 253-79. [Hi] D. Higman, Finite permutation groups of rank 3, M. Zeit. 86 (19641, 145-56. [HiG] G. Higman, On the simple group of D. G. Higman and C. C. Sims, Illinois J. Math. 7 (1963), 79-96. [HM] G. Higman and J. MacKay, On Janko's simple group of order 50,232,969, Bull. London Math. Soc. 1 (1969), 89-94. [HS] D. Higman and C. Sirns, A simple group of order 44,352,00, M. Zeit. 105 (1968), 110-13. [Jl] Z. Janko, A new finite simple group with abelian ZSylow subgroups and its characterization, J. Alg. 3 (1966), 147-86. [J2] Z. Janko, Some new simple groups of finite order, I, Symposia Math. 1 (1968), 25-65. (541Z. Janko, A new finite simple group of order 86,775,571,046,077,562,880 which possesses M24 and the full cover of Mz2 as subgroups, J. Alg. 42 (1976), 564-96. [JT] Z. Janko and J. Thompson, On a class of finite simple groups of Ree, J. A@. 4 (1966), 274-92. [Lel] J. Leech, Some sphere packings in higher space, Canadian J. Math. 16 (1964), 657-82. [Le2] J. Leech, Notes on sphere packings, Canadian J. Math. 19 (1967), 251-67. [LS] J. Leon and C. Sirns, The existence and uniqueness of a simple group generated by {3,4)-transpositions, Bull. AMS 83 (1977), 1039-40. [Ly] R. Lyons, Evidence for a new finite simple group, J. Alg. 20 (1972), 540-69. [Ml] E. Mathieu, Mbmoire sur le nombre de d e w s que peut acqubrir une fonction quand on y permut sea variables de toutes la manikres possibles, J. de Math. Pure et App. 5 (1860), 9-42. [M2] E. Mathieu, Memoire sur l'btude des fonctions de plusieures quantitb, sur la manikre des formes et sur les substitutions qui les laissent invariablw, J. de Math. Pure et App. 6 (1861), 241-323.

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Bibliography [M3] E. Mathieu, Sur la fonction cinq fois transitive des 24 quantitb, J. de Math. Pure et App. 18 (1873), 25-46. [Mc] J. McLaughlin, A simple group of order 898,128,000, Theory of Finite Groups, Benjamin, New York, 1969, pp. 109-1 11. [Mil] G. Miller, On the supposed five-fold transitive function of 24 elements and 19!/48 values, Mess. Math. 27 (1898), 187-90. [Mi21 G. Miller, Sur plusieurs groupes simples, Bull. Soc. Math. de h n c e 28 (1900), 266-7. [N] S. Norton, The construction of J4, PTOC. Sym. Pure Math. 37 (1980), 271-8. [ON] M. O'Nan, Some evidence for the existence of a new simple group, Proc. London Math. Soc. 32 (1976), 421-79. [PI N. Patterson, On Conway's group . O and some subgroups, Thesis: U. Cambridge (1974). [Pa] N. Paige, A note on the Mathieu groups, Canadian J. Math. 9 (1957), 15-18. [Ph] K. Phan, A characterization of the finite simple group U4(3), J. Amtralian Math. Soc. 10 (1969), 77-94. [PWl] N. Patterson and S. Wong, A characterization of the Suzuki sporadic simple group of order 448,345,497,600, J. Alg. 39 (1976), 277-86. [PW2] N. Patterson and S. Wong, The nonexistence of a certain simple group, J. Alg. 39 (1976), 138-49. [Q] D. Quillen, Homotopy properties of the coset of nontrivial psubgroups, Adv. in Math. 28 (1978), 101-28. [RS]M. Ronan and S. Smith, 2-local geometries for some sporadic groups, Pmc. Sym. Pure Math. 37 (1980), 283-289. [Se] Y. Segev, On the uniqueness of the Harada-Norton group, J. Alg. 151 (1992), 261-303. [Sill C. Sims, On the isomorphism of two groups of order 44,352,000, Theory of Finite Groups, Benjamin, New York, 1969, pp. 101-8. [Si2] C. Sims, The existence and uniqueness of Lyons' group, Gainesville Conference on Finite Groups, North-Holland, Amsterdam, 1973, pp. 138-41. [Sm] S. Smith, Large extraspecial subgroups of width 4 and 6, J. Alg. 58 (1979), 251-280. [Sp] E. Spanier, Algebraic Topology, McGraw-Hill, New York, 1966. [St] B. Stellmacher, Einfache Gruppen, die von einer Konjugiertenklasse von Elementen der Ordnung drei erzeugt werden, J. Alg, 30 (1974), 320-54. [Su] M. Suzuki, A simple group of order 448,345,497,600, Theory of Finite Groups, Benjamin, New York, 1969, pp. 113-9. [TI] J. Tits, A local approach to buildings, The Geometric Vein, SpringerVerlag, 1981, pp. 517-47. [T2] J. Tits, On R. Griess' Friendly giant, Invent. Math. 78 (1984), 491-9. [Thl] J. Thompson, A simple subgroup of &(3), Finite Groups Symposium, Japan Soc. for Promotion of Science, Tokyo, 1976, pp. 113-6. [Th2] J. Thompson, The uniqueness of the Fischer-Griess monster, Bull. London Math. Soc. 11 (1979), 340-6.

I

[Tm] F. Timmesfeld, Finite simple groups in which the generalized Fitting group of the centralizer of some involution is extra-special, Ann. Math. 107 (1978), 297-369. [Toll J. Todd, On representations of the Mathieu groups as collineation groups, J. London Math. Soc. 34 (1959), 406-16. [To21 J. Todd, A representation of the Mathieu group M2.4 89 a collineation group, Annali di Math. Pure ed App. 71 (1966), 199-238. [Wall H. Ward, On Ree's series of simple groups, h n s . AMS 121 (1966), 62-89. [Wa2] H. Ward, Combinatorial polarization, Discrete Math. 26 (1979), 18597. [Wa3] H. Ward, Multilinear forms and divisors of codeword weights, Quart. J. Math. 34 (1983), 115-28. [Wl] E. Witt, Die 5-fach transitiven Gruppen von Mathieu, Abl. Math. Hamburg 12 (1938), 256-64. [W2] E. Witt, ~ b e Steinersche r Systeme, Abl. Math. Hamburg 12 (1938), 265-74.

309

Index

adjacent, 4 algebra, 37 algebra, commutative, 37 algebra, symmetric, 37 amalgam, 195 collinearity graph, 197 completion, 196 faithful completion, 196 free amalgamated product, 196 geometric complex, 197 morphism, 195 rank, 195 residually connected, 199 Baby Monster, 68 basic relation, 178 bilinear form, radical, 43 binary Golay code 41, 71 extended, 90 Brauer, R., 66 Brauer-Fowler Theorem, 26, 66 Buekenout, F., 17 Burnside, W., 66 central product, 23 centralizer, 2 chamber, 6 chamber system, 8 nondegenerate, 8 Chevalley, C., 66 clique, 8, 185 clique complex, 8, 185 code, 40 word, 40 (m,n), 40 distance, 40 doubly even, 40 even, 40 minimum weight, 40 perfect, 40 weight, 40 collinearity graph, 197 complement, 25 completion, 196

faithful, 196 components of a group, 26 connected component, 4 Conway group, 67, 71, 72, 74, 76, 116, 128, 130,290, 293 Conway, J., 67, 68, 108, 121, 139, 140, 170, 171, 174,290 Curtis, R., 105 cycle, 176 derived form, 41 degree, 42 hyperbolic subspace, 43 singular subspace, 43 subhyperbolic subspace, 43 diagonal orbital, 5 distance, 4 dodecad, 88 dodecad involution, 255 elementary abelian pgroup, 22 extension, 25 split, 25 extraspecial group, 22 width, 22 faithful, 2 fiber product, 15 fibering, 185 field automorphism, 13 Fischer, B., 67, 68 Fischer groups, 67,69 Fitting subgroup, 26 flag, 6 flag complex, 9 form, 36 %form, 44 alternating, 36 symmetric, 36 Frattini subgroup, 21 free amalgamated product, 196 free product, 196 Fkenkel, I., 72 Frobenius, 66

-

Index

Index F, type, 172, 173 fundamental groupoid, 180, 191 Galois, 65 general linear group, 13 generalized diagon, 8 generalized Fitting subgroup, 26 geometric complex, 9 basic diagram, 11 coset, 10 direct sum, 11 geometry, 6 coset. 7 direct sum, 7 edge transitive, 6 flag transitive, 6 rank, 6 string, 11 truncation, 206 global stabilizer, 3 Golay, M., 94 Golay code, 35, 41, 71 Golay code module, 91 graph, 4 closed subset. 176 closure of cycles, 178 coset, 5 covering, 75, 175, 185, 191, 192 cycle, 176 deletion, 178 edge transitive, 4 fundamental group, 180 insertion, 178 invariant relation, 176 morphism, 4 n-generated, 182 simply connected, 175, 185 triangulable, 182 Griess algebra, 35, 70, 72, 142, 151, 1 fia

standard basis, 162 Griess, R., 68, 70, 169, 170, 174 groupoid, 179 Hall, M., 67, 290

Hall, P., 32, 66 Hall-Janko group, 69, 74, 135 Harada group, 68, 69, 74 Harada, K., 68, 174 Held, D., 68, 212, 238 Held group, 26, 68, 69, 74, 174, 212, 293 root 4-subgroup, 225 Higrnan, D., 17,67, 140 Higrnan, C.,67,68, 140, 238 Higman-Sims group, 67, 69, 74, 119, 131, 140 inner automorphism, 23 invariant relation, 175, 176 basic, 178 kernel, 176 involution, 18 %central, 19 isometry, 36 Janko groups, 66, 67,69, 70, 73, 74, 76, 290, 293 Janko, Z., 32,66, 70 large extraspecial %subgroup, 23, 71 Leech, J., 108, 121 Leech lattice, 67, 71, 108 coordinate frame, 116 mod 2, 116 shape, 111 Leon, J., 68 Lepowski, J., 72 linear representation, 1 link, 9, 185 local bijection, 185 local subgroup, 66 local system, 187 loop, 46, 47 associator, 49 central isomorphism, 50 coboundary, 49 cocvcle. . 48 commutator, 49

diassociative, 47 diassociative cocycle, 53 even automorphism, 58 Moufang, 47 parameters, 49 power map, 49 symplectic, 48 Lyons group, 68, 69, 73, 74 Lyons, R., 68 MacKay, J., 67, 68, 238 Mathieu, E., 77, 94 Mathieu group, 26, 65,69, 74, 77, 82, 84, 85, 94, 212, 238, 293 2-local geometry, 96, 99 McLauglin group, 67,69, 71, 74, 118, 128, 130, 140 McLauglin, J., 67, 140 Meierfrankenfeld, U., 70 Meurman, A., 72 Miller, G., 94 modular function, 72 module, core, 42 monomial, 36 Monster, 35, 63, 68, 69, 70, 72, 74, 142, 169, 174 Monster type, 172, 173 n-linear form, 36 n-simplex, 185 n-skeleton, 185 Norton, S.,68, 70, 170, 174 octad, 85 involution, 255 collinear, 99 coplanar, 99 O'Nan group, 68 O'Nan, M., 68 orbit, 3 orbital, 4 paired, 4 Paige, N., 94

Parker loop, 63, 72, 142, 144, 170 path, 4 basic degree, 181 end, 176 origin, 176 T-gon, 183 reduced, 182 trivial, 182 Patterson, N., 140, 290 permutation rank, 4 permutation representation, 1 Phan, K., 249 pointwise stabilizer, 3 pregroupoid, 179 inversion, 179 morphism, 179 primitive, 3 projective geometry, 6 projective plane, 7, 71 quadratic form, singular vector, 43 quasiequivalent, 2 quasisimple, 26 Quillen complex, 76 rank 3 group, 5 parameters, 5 regular representation, 3 regular normal subgroup, 3 representation, 1 residually connected, 10 residue, 8 Ronan, M., 105 root Ar-subgroup, 269 root As-subgroup, 269 root As-subgroup, 279 root A,-subgroup, 288 root 4-involution, 260, 269 root 4-subgroup, 260, 269 root Jz-subgroup, 280 root U3(3)-subgroup, 288 Rudvalis, A., 67 Rudvalis group, 67,69,73,74

313

Index

-

Segev, Y., 70, 75, 210 self-paired orbital, 4 semilinear transformation, 13 sextet, 91 simplex, 8, 185 dimension, 185 simplicial complex, 8, 184 covering, 185 F-homotopy, 187 fundamental group, 186,191 graph of, 8, 185 link, 185 local system, 187, 188 n-skeleton, 185 star, 185 vertex, 184 simplicial map, 185 Sims, 67,68, 140 Smith, P., 69 Smith, S., 105, 170 special linear group, 13 split extension, 25 star, 8, 185 Steiner system, 70, 71, 78 block, 78 extension, 78 Extension Hypothesis, 79 independent subset, 78 point, 79 residual design, 78 Stellrnacher, B., 290 string diagram, 11 string geometry, 11, 204 collinearity graph, 205 line, 205 plane, 205 point, 205 string ordering, 11 strongly embedded subgroup, 20 Suzuki group, 67, 69, 71, 74, 76, 135, 290, 293 h&&i, M., 67, 290 symmetric form, 36

t-transitive, 3 Thompson group, 68,69, 74 Thompson, J., 32, 66, 68, 70, 140, 174, 290 Thompson Order Formula, 19 3-transposition, 67 Timmesfeld, F., 32 Tits, J., 17, 70, 72, 170, 171, 191 Todd, J., 94 Todd module, 92 totally singular subspace, 7 transitive, 3 transvection, 14 triangulable, 182 trio, 97 type Col, 250, 258 type He, 219 type Jz, 250,258 type J3, 258 type L5(2), 219 type Mz4, 219 type Suz, 250,258 type U4(3), 244 uniqueness system, 198 amalgam, 199 base, 200 equivalence, 199 geometry, 199 morphism, 200 similarity, 199 unitary group, 128, 241 vertex, 4 Wales, 67 Wales, D., 290 Ward, 45, 66 Witt, 71, 94 Wong, S., 290 wreath product, 26

)