Amalgams Which Involve Sporadic Simple Groups I

193, 441]491 Ž1997. JA966957

JOURNAL OF ALGEBRA ARTICLE NO.

Amalgams Which Involve Sporadic Simple Groups I Christopher Parker Department of Mathematics and Statistics, Uni¨ ersity of Birmingham, Edgbaston, Birmingham B15 2TT, United Kingdom

and Peter Rowley Department of Mathematics, UMIST, PO Box 88, Manchester M60 1QD, United Kingdom Communicated by Gernot Stroth Received August 26, 1994

1. INTRODUCTION This paper and its successor wPR3x study subgroup configurations which involve groups belonging to one of the following classes: L s L i e1 Ž even . j D j  S5 4 and S s  L ¬ F U Ž L . rZ Ž F U Ž L . . is a sporadic simple group and Z Ž F U Ž L . . has odd order 4 , where L i e1Ževen. s  L2 Ž2 n ., U3 Ž2 n ., SU3 Ž2 n ., Sz Ž2 2 nq1 . ¬ n g N4 ; D s  D 2 p ¬ p an odd prime4 Ž D 2 n is the dihedral group of order 2 n.; S5 is the symmetric group of degree 5. The specific situation we shall be investigating is described in HYPOTHESIS 1.1. Suppose that G is a group generated by proper finite subgroups P1 and P2 for which the following hold: Ži.

X

X

X

X

O 2 Ž P1 .rO 2 Ž O 2 Ž P1 .. g L and O 2 Ž P2 .rO 2 Ž O 2 Ž P2 .. g S ; 441 0021-8693r97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.

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Žii. Syl 2 Ž B . : Syl 2 Ž P1 . l Syl 2 Ž P2 ., where B [ P1 l P2 ; X Žiii. Pi s O 2 Ž Pi . B for i s 1, 2; Živ. CP Ž O 2 Ž Pi .. F O 2 Ž Pi . for i s 1, 2; i Žv. B contains no nontri¨ ial normal subgroup of G. X

For i s 1, 2, we put L i s O 2 Ž Pi . and Q i s O 2 Ž L i .. Our main result which is concerned with the possible structure for P1 and P2 is as follows: THEOREM A. Assume that Hypothesis 1.1 holds. Then the chief factor structure of the pair Ž P1 , P2 . is known and is tabulated in Table 1. In particular, Ži. Žii. Žiii.

P1rQ1 g  L2 Ž2., S54 ; P2rQ2 g  M22 , AutŽ M22 ., 3 AutŽ M22 ., M23 , M24 , Co 2 , Co14 ; < B < F 2 46 .

In Table 1 we present information about the 12 possible pairs of subgroups Ž P1 , P2 . which satisfy Hypothesis 1.1. The first column presents a nomenclature for the amalgams. Column 2 gives the critical distance b associated with each amalgam Ž b is defined in Section 3.. In column 3 we give a chief series for P1; here, for example, 2 1 4q2 5q1 3 L2 Ž2. is a compressed form of 2 1q 1q1q1q2q2q2q2q2q1q1q1 L2 Ž2. and Žfollowing wMSx. 4 represents the natural GF Ž2. S5-module. Column 4 describes the chief series structure for P2 ; here again we follow wMSx by representing the 11-dimensional Žrespectively, 10-dimensional. M23 , M24 Žrespectively, M22 , AutŽ M22 .. Golay-code module by 11 Žrespectively, 10. and Todd-module by 11 Žrespectively, 10.. The chief factors of dimension 12 for 3 AutŽ M22 ., 24 for Co1 , and 22 for

TABLE 1 b

P1

A1

1

A2 A3 A4 A5 A6 A7 A8 A8U A9 A10 A11

1 1 1 1 1 1 1 1 2 2 2

2 1 4q2 5q1 3 L2 Ž2. 2 1 4q2 6q1 4 L2 Ž2. 2 1 5q2 4q1 3 L2 Ž2. 2 1 5q2 4q1 4 L2 Ž2. 2 1 6q2 4q1 3 L2 Ž2. 2 1 6q2 4q1 6 L2 Ž2. 2 1 7q2 4q1 6 L2 Ž2. 2 1 3q4 3q1 3 S5 2 1 3q4 3q1 3 S5 2 2 1q1 5q2 5q1 3 L2 Ž2. 2 2 1q1 10q2 10q1 8 L2 Ž2. 2 2 1q1 11q2 11q1 10 L2 Ž2.

P2

G

210 AutŽ M22 .

Co 2

211 M24 2 10 M22 2 10 AutŽ M22 . 2 11 M23 2 11 M24 2 11q1 M24 2 11 M24 2 11 M24 1q12 2q 3 AutŽ M22 . 1q22 2q Co 2 1q24 2q Co1

Co1 Fi 22 AutŽ Fi 22 . Fi 23 X Fi 24 Fi 24 J4 J4 B M

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443

Co 2 come from the embedding of 3 M22 in SU6 Ž2., the action of Co1 on the Leech lattice reduced mod 2, and the embedding of Co 2 in Co1 , respectively. The last column of the table gives, in all but case A8U , a finite example for G which contains the given configuration. As we see from Table 1, 8 of the 26 sporadic simple groups satisfy Hypothesis 1.1. However, the principal motivation of this work is to produce results applicable to the revision of the simple group classification. For more details of the relevance of Theorem A to such matters we refer the reader to wPR1x and wSt2x. In order to enhance the usefulness of these papers for revision purposes, we have sought to make them as self-contained as possible. Our major sources for pertinent module data are wMSx and wDSx with some additional facts from wAx. While for group theoretic information about the sporadic simple groups we rely upon the Atlas wAx. At certain points in the proof of Theorem A, the case L1rQ1 ( S5 gets singled out for special treatment. One such instance of this occurs in the proof of Theorem 4.2; there we need to make use of what is a special case of Theorem A which we now state as THEOREM B. Suppose that Hypothesis 1.1 holds with P1rQ1 ( L2 Ž2 n .. Then P1rQ1 ( L2 Ž2., B [ P1 l P2 g Syl 2 Ž P1 . l Syl 2 Ž P2 ., and the pair Ž P1 , P2 . is one of the amalgams A1 , . . . , A7 , A9 , . . . , A11 of Table 1. In particular, one of the following holds: Ži. Q2 is elementary abelian and core P Ž Q1 . s 1; 2 Žii. Q2 is extraspecial of q-type, core P Ž Q1 . s V 1Ž ZŽ B .. ( Z 2 and, 2 for any x g P1 _ B, core P 2Ž Q1 . l core P 2x Ž Q1 . s 1. The proofs of Theorems A and B lead us to the basic subdivision in amalgam problems, namely, the noncommuting case and the commuting case Ždefined below.. The main result of this paper addresses the noncommuting case. THEOREM C. Suppose that the noncommuting case holds for Hypothesis 3.1. Then LarQa ( L2 Ž2., Qb is extraspecial of q-type and one of the following holds: 1q 12 Ži. LbrQb ( 3 AutŽ M22 ., Qb ( 2q , h Ž La , Qa . s 6, and La has 2q 1 5q2 5q1 3 a chief series described by 2 L2 Ž2.. 1q 22 Žii. LbrQb ( Co 2 , Qb ( 2q , h Ž La , Qa . s 11, and La has a chief 2q 1 10 q2 10 q1 8 series described by 2 L2 Ž2.. 1q 24 Žiii. LbrQb ( Co1 , Qb ( 2q , h Ž La , Qa . s 12, and La has a chief 2q 1 11 q2 11 q1 10 series described by 2 L2 Ž2..

In wPR3x we prove a companion result for the commuting case.

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Since L1rQ1 is isomorphic to S5 or an element of D j L i e1Ževen., Hypothesis 1.1Žii. and Žiii., together with G / P1 , imply that either P1 l P2 contains a unique Sylow 2-subgroup of both P1 and P2 or that L1rQ1 ( S5 and Ž L1 l P2 .rQ1 ( S4 . We note a further point relating to this latter possibility in the case of A8 Žof Table 1.. This configuration in J4 has P1rQ1 ( S5 and P2rQ2 ( M24 with Ž P1 l P2 .rQ1 ( S4 , and using the embeddings of P1 l P2 into P1 and P2 we may construct a pair Ž P1U , P2U . which satisfies Hypothesis 1.1 and for which P1U l P2U g Syl 2 Ž PiU . Ž i s 1, 2.. We may think of this procedure as ‘‘pulling apart’’ the pair Ž P1 , P2 .; we have denoted this pulled-apart amalgam by A8U . The remainder of this section gives an overview of the proof of Theorem A as well as introducing certain notation ŽwPR1x also offers a discussion of the proof of Theorem A.. We observe that G, in Hypothesis 1.1, is a homomorphic image of the free amalgamated product of P1 and P2 over B. Since Theorem A focusses only upon the structure of P1 and P2 we may assume, without loss, that G s P1 )B P2 . In Section 3 the coset graph G of G is introduced. This yields a very useful geometric framework which enables us to analyse exhaustively the action of G upon G. It is well known in these so-called amalgam arguments that certain GF Ž2.-modules for the groups in L and S play a prominent, and often decisive, role. For example, the frequent occurrence of AutŽ M22 . and 3 M22 in our arguments is due to these groups possessing modules which have small centralizer indices for involutions or large quadratically acting 2-groups. Accordingly, in Section 2 we amass an extensive catalogue of module information for later use. In Section 3, after introducing G, we define, for each vertex a of G, a normal subgroup Za of the vertex stabiliser Ga , the critical distance b of G, and the set C of critical pairs Žof G .. Let Ž a , a 9. g C . Then we have either w Za , Za 9 x / 1 Žthe noncommuting case. or w Za , Za 9 x s 1 Žthe commuting case.. These two possibilities lead to different considerations. The balance of Section 3 is concerned with the proof of Theorem 3.20, which itself depends heavily on the solution of certain pushing-up problems. Among these we single out for mention Proposition 3.13 and Corollary 3.14, which are of independent interest. With Theorem 3.20 to hand, in Section 4 we deal with the case when G has noncommuting critical pairs. As is always the case in this situation we find that Za is a failure of factorisation module for LarQa and, as the groups in S , the Suzuki groups, and the unitary groups have no such modules, we almost immediately know that LarQa ( S5 or L2 Ž2 n .. We then employ Theorem 3.20 to guarantee that Zt F ZŽ Lt . for all vertices t in G with LtrQt g S . After this the bulk of the section is directed to showing that b s 2, LarQa ( L2 Ž2., and thus that Ž G, P1 , P2 , B . is an amalgam of symplectic type Žsee wPR2x.. Most of the work here, as

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445

suggested above, is in eliminating the LarQa ( S5 configurations. It is at this point that we apply Theorem B, which we always assume to be proven when analysing the amalgams with L1rQ1 ( S5 . We emphasize here that the proofs of Theorem A Žfor the S5 case. and Theorem B are independent. Having shown that Ž G, P1 , P2 , B . is an amalgam of symplectic type the main result of Section 4, Theorem 4.2, follows from wPR2, Main Theoremx. Thus we find that the noncommuting critical pairs give rise to the amalgams A9 , A10 , and A11. Sections 5 and 6 in wPR3x examine, respectively, the case of commuting critical pairs Ž a , a 9. when LarQa g L and LarQa g S , respectively. In this case we know that b is odd and thus La 9rQa 9 g S or L , respectively. In Section 5 after a short preliminary study we may apply two results of Stellmacher wSt2x, which we have recorded as Lemmas 3.9 and 3.10, to uncover the fact that LarQa ( L2 Ž2 n . or S5 . The main result of this section is Theorem 5.2, which states that there are no amalgams with commuting critical pairs and a g O Ž L .. At almost every turn in the analysis of this situation we are confronted by the groups 3 M22 , AutŽ M22 ., and S5 . Finally we show that in a counterexample to Theorem 5.2 we must have b s 3, and it is the final elimination of this configuration that completes the proof of the theorem. In Section 6 Lemma 3.10 can be applied again to show that if LtrQt g L and d is a neighbor of t in G, then Vt [ ² ZdtG : contains two noncentral Lt-chief factors. This enables us once again to prove, in Lemma 6.7, that Žso long as b G 3. LtrQt ( L2 Ž2 n . or S5 and more that F U Ž LarQa .r ZŽ F U Ž LarQa .. ( M22 . The work in this section is then to demonstrate that the amalgams involving S5 do not exist. This done we quickly extract Theorem 6.2. Thus, by the end of Section 6 we have shown that if Ž a , a 9. g C is a commuting critical pair then b s 1 and LarQa g S . In Section 7 we pursue this configuration relentlessly, eventually concluding Žsee Theorem A8 or A8U . Two facts of 7.2. that Ž G, P1 , P2 , B . is one of the amalgams A1 ]A importance in this section are that in a sporadic simple group H with Sylow 2-subgroup S, V 1Ž ZŽ S .. has order 2 or H g  J1 , Fi 23 4 and an odd order automorphism of S has order dividing 3 or H ( J1. These two facts are proven in Lemma 2.22 and wPR4x and are used, in conjunction with bounds on the 2-rank of S, to prove in Lemma 7.10 that, if b s 1, then La 9rQa 9 ( SU3 Ž2., U3 Ž2., Sz Ž2., L2 Ž2., or S5 . When La 9rQa 9 ( SU3 Ž2., U3 Ž2., Sz Ž2., or L2 Ž2. we find, in Lemma 7.11, that Za is an Ž FF q 1.module for LarQa and F U Ž LarQa .rQa g  M22 , M23 , M24 4 Žsee below for the definition of an Ž FF q 1.-module.. The next three lemmas show that, if La 9rQa 9 \ S5 , then La 9rQa 9 ( L2 Ž2. and Proposition 7.15 reveals the A7 . We then move on to the La 9rQa 9 ( S5 case. In Lemma amalgams A1 ]A 7.16 we apply Theorem B again to show that a maximal subgroup of La 9

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containing a Sylow 2-subgroup normalizes Qa ; this foreshadows the ‘‘pulled apart’’ configuration mentioned earlier. Finally in Lemma 7.17 we unveil our last two amalgams A8 and A8U . In our last section, Section 8, we complete the proofs of Theorems A and B. Our group theoretic notation is standard as given in either wGorx, wHux, or wSuzx, with the following additions. We use EŽ p n . to denote the elementary abelian group of order p n. If H is a group acting on another group X, then the number of noncentral chief factors that H induces on an H-chief series in X is denoted by h Ž H, X .. For H a group we write mŽ H . for the 2-rank of H, AŽ H . s  A F H ¬ F Ž A. s 1 and mŽ A. s mŽ H .4 , and J Ž H . s ² AŽ H .: is the elementary abelian version of the Thompson subgroup. If V is a nontrivial GF Ž2. H-module and A is a subgroup of H with w V, A, A x s 0 and w V, A x / 0, then we say A acts quadratically on V. If < ACH Ž V .rCH Ž V .< G 4, then A is called a quadratic subgroup and V is called a quadratic module for H. Assume that A acts quadratically on the GF Ž2. H-module V; if, furthermore, 1 / w V : C V Ž A.x F < ACH Ž V .rCH Ž V .<, then V is called an FF-module for H Žfailure of factorisation module.. While if < ACH Ž V .rCH Ž V .< is elementary abelian, though not necessarily quadratic, and 1 / w V : C V Ž A.x F 2 < ACH Ž V .r CH Ž V .<, then V is called an Ž FF q 1.-module for H. If V is an FF-module Žrespectively, Ž FF q 1.. and A satisfies 1 / w V : C V Ž A.x F < ACH Ž V .r CH Ž V .< Žrespectively, 1 / w V : C V Ž A.x F 2 < ACH Ž V .rCH Ž V .<., then A is called an offending Žrespectively, an Ž FF q 1. offending. subgroup. For a particular sporadic simple group we use the Atlas names for conjugacy classes.

2. GF Ž2.-REPRESENTATIONS This section gathers together and proves results about the structure and GF Ž2.-representation of the elements of L j S . Our first theorem is from wMSx, given in a shortened form. We recommend that the reader, while reading this section, have a copy of this article as well as a copy of the atlas wAx to hand. THEOREM 2.1 wMS, Theorems 1 and 3x. Suppose that H g S is a faithful irreducible quadratic module for H. Set G s ² F < F is a quadratic subgroup of H :. Then Ži. Žii. Žiii.

G ( M12 and dim G F Ž2. V s 10; G ( AutŽ M12 . and dim G F Ž2. V s 10; G ( AutŽ M22 . and dim G F Ž2. V s 10;

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G ( 3 M22 and dim G F Ž2. V s 12; G ( M24 and dim G F Ž2. V s 11; G ( J 2 and dim G F Ž2. V s 12; G ( Co1 and dim G F Ž2. V s 24; G ( Co 2 and dim G F Ž2. V s 22; G ( 3Suz and dim G F Ž2. V s 24.

Živ. Žv. Žvi. Žvii. Žviii. Žix.

The structure of the above mentioned quadratic modules will be revealed more fully later in this section. LEMMA 2.2. Suppose that G g S . Then an upper bound for the 2-rank, m s mŽAutŽ GXrZŽ GX ... is as indicated in the following table: X

X

X

X

X

X

G rZŽ G .

m

G rZŽ G .

m

G rZŽ G .

m

M11 M12 M22 M23 M24 J1 J2 J3 J4

2 4 5 4 6 3 4 4 11

McL Ly HS He Suz Ru X ON Co 3 Co 2

4 4 5 6 6 6 4 4 10

Co1 Fi 22 Fi 23 X Fi 24 HN Th B M

11 10 11 12 6 5 22 24

Proof. See wA, Table 11.2x. Later in this section we shall need to draw on properties of the following ‘‘small’’ GF Ž2.-modules which we now describe. Recall that A 6 , respectively, S6 contains two conjugacy classes of subgroups isomorphic to A 5 , respectively, S5 . Thus A 6 and S6 possess two nonisomorphic permutation modules of dimension 6, each of which contains a composition factor of dimension 4. In either case we call the four-dimensional module arising in this way a natural A 6 , respectively, S6 module; see wLPRx for further details. Properties of the six-dimensional irreducible modules for 3 A 6 and 3S6 may be gleaned from wRx. For A 7 and A 8 , the permutation module is the six-dimensional composition factor of the seven- Žrespectively, 8-. dimensional permutation module. By the four-dimensional natural A 8module we mean either the module or its dual obtained via the isomorphism A 8 ( GL4Ž2.. The relevant calculations for these A 7- and A 8-modules are omitted and left to the reader. We now turn to examine the irreducible GF Ž2. S5-modules. Recall that S5 has three nonisomorphic modules over GF Ž2. Žby wHB, 3.11x.: the trivial module, the natural module, and the orthogonal module. Both the natural

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module and the orthogonal module are four-dimensional over GF Ž2.. The natural module arises from the natural two-dimensional GF Ž4.-module for L2 Ž4. ( A 5 viewed as a GF Ž2.-module with the transpositions from G induced field automorphisms of GF Ž4.. The orthogonal module is the ‘‘natural’’ module obtained from the isomorphism S5 ( GO4yŽ 2.; however, for calculations it is usually best to consider it as the space spanned by the vectors of even weight in the five-dimensional permutation module. We observe that the orthogonal module remains irreducible on restriction to A 5 ( L2 Ž4., this restricted module is not isomorphic to a natural L2 Ž4.module, rather it is projective and can be identified with the L2 Ž4. Steinberg module, and we will call it the orthogonal L2 Ž4.-module. The next result is well known and may be verified by straightforward calculations. LEMMA 2.3. Suppose that G ( S5 and set H s GX . Assume that V is a nontri¨ ial irreducible GF Ž2.G-module, S g Syl 2 Ž G ., F1 s S l H, F2 is the fours subgroup of S not in H, C4 ( C F S, and t is an in¨ olution in S. Then Ži. < C V Ž S .< s 2; Žii. if t g H, then w V : C V Ž t .x s 4; Žiii. if w V : C V Ž t .x s 2, then V is an orthogonal module and t f H; Živ. C V Ž S . F w V, t x and C V Ž t . / w V, S x for all in¨ olutions t in G; Žv. if V is an orthogonal module, then Ža. w V, F2 x s C V Ž F2 . s w V, S; 2x ( EŽ2 2 .; Žb. w V, S x s w V, F1 x s w V, C x ( EŽ2 3 . and C V Ž S . s C V Ž F1 . s w V, F1; 2x s w V, S; 3x ( EŽ2.; Žc. CG Ž C V Ž S .. s NG Ž F1 . ( S4 ; Žd. CG Žw V, F2 x. s F2 ; Že. if g g G and w V, F2 x l w V, F2g x / 0, then either w V, F2 x s g w V, F2 x and F2g s F2 , or ² F2 , F2g : ( S4 and C V Ž F2 . l C V Ž F2g . s C V Ž² F2 , F2g :.; Žf. the elements of order 3 in G ha¨ e two-dimensional fixed point space on V; Žvi. if V is a natural module, then Ža. C V Ž F1 . s w V, F1 x s w V, S; 2x ( EŽ2 2 .; Žb. w V, S x s w V, F2 x s w V, C x ( EŽ2 3 . and C V Ž S . s C V Ž F2 . s w V, F2 ; 2x s w V, S; 3x ( EŽ2.; Žc. CG Ž C V Ž s .. s NG Ž F2 . s S; Žd. CG Žw V, F1 x. s F1; Že. if g g G, then w V, F1 x s w V, F1g x or w V, F1 x l w V, F1g x s 0; Žf. G operates transiti¨ ely on the nonzero ¨ ectors of V and the elements of order 3 in G act fixed point freely on V a; Žg. for all 1 / X F S, w V, X x G C V Ž S ..

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LEMMA 2.4. Suppose that G ( S5 , EŽ2 2 . ( F F S g Syl 2 Ž G ., V is a GF Ž2. S5-module, with h Ž G, V . s 1 and w V, F, F x s 0. If V s ² Z G :, where Z F C V Ž F . is S-in¨ ariant, then either < Z < F 2 2 or Z l C V Ž G . / 0. Proof. Supposing that < Z < G 2 3 and Z l C V Ž G . s 0 we derive a contradiction. Then without loss of generality we may assume that C V Ž G . s 0. Let V0 be a minimal G-invariant subspace of V. Since h Ž G, V . s 1 and V s ² Z G :, V s ZV0 and < V0 < s 2 4 . Choose g g G, such that ² F, F g : G O 2 Ž G .. From Lemma 2.3Žv. and Žvi., w V0 : C V 0Ž F .x s 2 2 and hence w V : C V Ž F .x s 2 2 with < C V Ž F .< G 2 3. But this then forces C V Ž O 2 Ž G .. / 0, whence C V Ž G . / 0, a contradiction. Our next lemma is useful for providing quadratically acting subgroups. LEMMA 2.5. Suppose that V is a faithful GF Ž2.G-module and suppose that 1 / E is a quadratically acting subgroup of G. Ži. If t g CC Ž E .Ž VrC V Ž E .._ E is an in¨ olution, then ² E, t : is a G quadratic subgroup of G. Žii. If F is a maximal quadratically acting subgroup of G and T G V 1Ž CT Ž F .. ) F, then w V, F, T x / 0 and w V, T, F x / 0. Proof. Ži. By assumption w V, t x F C V Ž E ., and so w V, t, E x s 0. Since we also have w t, E, V x s 0, the three subgroup lemma gives w V, E, t x s 0 and thus

w V , t xw V , E x F C V Ž t . l C V Ž E . . Hence Žnoting that <² E, t :< G 4. Ži. follows. Now suppose F is maximal and choose t g CT Ž F ._ F an involution. If w V, F, T x s 0, then w V, F, t x s 0 whence, using w F, t, V x s 0 and the three subgroup lemma, w t, V, F x s 0. So t g CC G Ž F .Ž VrC V Ž F .._ F and now part Ži. contradicts the maximality of F. Thus w V, F, T x / 0 and a similar argument gives w V, T, F x / 0. LEMMA 2.6. Suppose that G is a finite group which contains a subgroup H ( A 6 in which the distinct conjugacy classes of elements of order 3 in H are fused in G. If t g H is an in¨ olution and V is a GF Ž2.G-module with CG Ž V . F O Ž G ., then w V : C V Ž t .x G 2 4 . Proof. Suppose that G, H, t, and V are as in the statement of the lemma. Furthermore, assume that h Ž H, V . s 1 and that this noncentral H-chief factor is a natural GF Ž2. H-module. Then the different classes of elements of order 3 in H have different commutator sizes on the module V. Thus, since the elements of order 3 in H are fused in G, we have a contradiction. Hence either h Ž H, V . s 1 and the noncentral chief factor is

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not a natural GF Ž2. H-module or h Ž H, V . G 2. Using the fact that the centralizer in a four-dimensional Žrespectively, 16-dimensional. GF Ž2. A 6module of an involution in A 6 is two-dimensional Žrespectively, eight-dimensional., we deduce that Lemma 2.6 holds. PROPOSITION 2.7. Suppose that G g S is not perfect and that V is a GF Ž2.G-module with CG Ž V . F O Ž G .. If t g G_GX is an in¨ olution, then w V: C V Ž t .x G 2 4 or V is a quadratic module for G and t is contained in a quadratic fours group on V. Proof. Suppose that G, t, and V are as in the statement of the proposition and suppose that w V : C V Ž t .x F 2 3 ; we will show that V is a quadratic module for G. Since t is an involution in G_GX and, by wAx, w G : GX x s 2, CG Ž t . s CGX Ž t . = ² t :. Now, by wAx, either < CGX Ž t .< 2 G 2 4 or GrOŽ G . ( AutŽ M12 . and CG Ž t . ( S5 = 2, or GrOŽ G . ( AutŽ OX N . and CG Ž t . ( J1 = 2. Thus, as CG Ž t .rCC G Ž t .Ž VrC V Ž t .. must embed into GL3 Ž2., we have in all cases that < CC GX Ž t .Ž VrC V Ž t ..< 2 / 1; therefore, Lemma 2.5Ži. gives the result. PROPOSITION 2.8. Suppose that G g S and that V is a GF Ž2.G-module with CG Ž V . F O Ž G .. If t g GX is an in¨ olution, then either w V : C V Ž t .x G 2 4 , or V is a quadratic module for GX and t is contained in a quadratic fours group on V. Proof. We suppose that w V : C V Ž t .x F 2 3 and put L s F U Ž G .rOŽ G . and K s CC G Ž t .Ž VrC V Ž t ... Then CG Ž t .rK embeds in GL3 Ž2.. We seek to show that V must be a quadratic module for GX . Observe that this will follow from Lemma 2.5Ži. whenever we have mŽ CLŽ t .. G 4. If L contains a subgroup H such that w L : H x is odd and H contains a normal elementary abelian 2-subgroup of rank greater than or equal to 5, then it follows that mŽ CLŽ t .. G 4. Hence we deduce that mŽ CLŽ t .. G 4 when L is isomorphic to one of M24 , He, Co 2 , Fi 22 , Th, Fi 23 , Co1 , J4 , FiX24 , B, M Žwhere H is, respectively, 2 6 : 3S6 , 2 6 : 3S6 , 2 10 : AutŽ M22 ., 2 10 : M22 , 2 5. L5 Ž2., 2 11. M23 , Ž .. 2 11 : M24 , 2 10 : L5 Ž2., 2 11. M24 , 2 9.2 16 .S8 Ž2., 2 10q16. Oq 10 2 . If L is isomorphic to one of J 2 , HS, Ru, Suz, Co 3 , HN, then L has two conjugacy classes of involutions one of which is not a 2-central class. Since mŽ CLŽ s .. G 4 for a non-2-central involution s Žby virtue, respectively, of the following subgroups: 2 2 = A 5 , 2 = A 6 2 2 , 2 2 = Sz Ž8., 2 2 = L3 Ž4., 2 = M12 , 2 6. U4 Ž2.., again we get mŽ CLŽ t .. G 4. If L is isomorphic to one of M22 , M23 , J 3 , McL, Ly, then L has just one conjugacy class of involutions and, therefore, mŽ CLŽ t .. G 4 Žbecause, respectively, of the following subgroups: 2 4 : A 6 , 2 4 : A 7 , 2 4 : Ž3 = A 5 ., 2 4 : A 7 , 3 ? AutŽ McL... It remains to examine the possibilities M11 , M12 , J1 , and O’N. For L isomorphic to M12 Žwith t g 2 A., J1 , O’N we have, respectively, CLŽ t . (

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2 = S5 , 2 = A 5 , 4 ?L3 Ž4. : 2. Since CG Ž t .rK embeds in GL3 Ž2., Lemma 2.5 applies to give that V is a quadratic module. Finally when L ( M11 or M12 Žwith t g 2 B . we may appeal to Lemma 2.6 and obtain w V : C V Ž t .x G 2 4 , contrary to our assumption w V : C V Ž t .x F 2 3. This completes the verification of Proposition 2.8. PROPOSITION 2.9. nontri¨ ial irreducible subgroup of G. Then w V : C V Ž F .x G 2 5.

Suppose that G ( M12 or AutŽ M12 . and that V is a quadratic module for G with F a maximal quadratic < F < s 4, w V : C V Ž t .x G 2 4 for all in¨ olutions t in F and

Proof. From wMS, Theorem 2Ža., Žb.x, there are two possibilities for F. In both cases < F < s 4. We consider first the case when F F GX . Then, by wMSx and wAx, the involutions are all of class 2 B. Thus Lemma 2.6 applies and we have, for all involutions of t g F, w V : C V Ž t .x G 2 4 . Now if C V Ž t . s C V Ž F ., then C V Ž F . is invariant under ² NG Ž F ., CG Ž t .: G GX , which is a contradiction. Thus the proposition holds in this case. Now suppose that F g GX . Then, by wMS, Theorem 2x and wAx, the involutions in F _GX are both of class 2C and invert an element of order 11 in an L2 Ž11. subgroup of M12 . Thus Žas dim G F Ž2. V s 10. for t g F _GX , w V : C V Ž t .x G 2 5 and so w V : C V Ž F .x G 2 5. So it remains to consider the case when t g F l GX . Then t g 2 A and CGX Ž t . ( 2 = S5 . If w V : C V Ž t .x F 2 3, then Proposition 2.8 yields that GX contains a quadratic fours group which contains a 2 A involution, contrary to wMS, Theorem 2Ža.x. Therefore, w V : C V Ž t .x G 2 4 , and we have the proposition. PROPOSITION 2.10. Suppose that G ( M24 and that V is a nontri¨ ial irreducible quadratic module for G with F a maximal quadratic subgroup of G. Then Ži. < F < s 4; Žii. if t is an in¨ olution of class 2 A in G, then w V : C V Ž t .x s 2 4 ; Žiii. if t is an in¨ olution of class 2 B in G, then w V : C V Ž t .x s 2 5 ; Živ. w V : C V Ž F .x G 2 5. Proof. That Ži. holds is straight from wMS, Theorem 2x. Since V ( 11 or 11, it suffices to prove Žii. and Žiii. for V ( 11 Žthe Todd module.. Now suppose that t is an involution in G of class 2 A. Then CG Ž t . ( 2 1q Ž3q3. : L3 Ž2. is a maximal subgroup of H ( 2 4 : A 8 and we may assume without loss of generality that t g O 2 Ž H .. From wMS, Lemma 3.5x, dim G F Ž2. C V Ž O 2 Ž H .. s 6 and the chief factors of CG Ž t . in V have dimensions 3, 3, 3, 1, and 1. Also VrC V Ž t . contains at least one central chief factor. Using VrC V Ž t . ( w V, t x Žas CG Ž t .-modules. and w V, t x F C V Ž t . we conclude that either w V : C V Ž t .x s 2 or w V : C V Ž t .x s 2 4 . The former

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possibility, as O 2 Ž H . is generated involutions of class 2 A, yields w V : C V Ž O 2 Ž H ..x F 2 4 against dim G F Ž2. C V Ž O 2 Ž H .. s 6. Thus w V : C V Ž t .x s 2 4 , proving part Žii.. Now suppose that t is of class 2 B and suppose that w V : C V Ž t .x F 2 4 . Then, because CG Ž t . ( 2 Ž1q1q4. : S5 F 2 6 : 3S6 Žhere O 2 Ž CG Ž t .. ( 2 Ž1q1q4. is a uniserial S5-module and CG Ž t .X s O 2 Ž CG Ž t .. G O 2 Ž CG Ž t ... either ww V, t x, O 2 Ž CG Ž t ..x s 0, or
< F < s 4; F g GX and the elements of F _GX are of class 2 B; for t g F l GX , w V : C V Ž t .x s 2 4 ; for t g F _GX , w V : C V Ž t .x s 2 3 ; C V Ž F . s w V, F x and
Proof. Using wMS, Theorem 2x and wAx we have Ži. and Žii.. Now let t g F l G9. Then Lemma 2.6 applies and so w V : C V Ž t .x G 2 4 . If w V : C V Ž t .x s 2 5, then C V Ž t . s w V, t x and h Ž CG Ž t ., V . s 2 ? h Ž CG Ž t ., w V, t x. s 2 or 4; however, wMS, Theorems 2Žc. and 3x indicate that h Ž CG Ž t ., V . s 3, which is a contradiction. Therefore, Žiii. holds. Next, if t g F _G9, then P s CG Ž t . ( 2 = Ž2 3 L3 Ž2.. and wMS, Lemma 3.3Žb.x describes the restriction of V to P. Because VrC V Ž t . ( w V, t x as a GF Ž2. P-module, we deduce that Živ. holds. Now we embark on a proof of Žv.; by Žiii., we have, for x g F l GX , E Ž 2 6 . ( CV Ž x . G CV Ž F . G w V , F x G w V , x x ( E Ž 2 4 . .

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If C V Ž F . s w V, F x ( EŽ2 5 ., then as, by wMS, Theorem 2Žc.x, F is normalized by CG Ž x . and h Ž CG Ž x ., V . is even, we reach the same contradiction as in part Žiii.. Hence, if Žv. is false then we must have C V Ž x . s C V Ž F . ( EŽ2 6 . and w V, x x s w V, F x ( EŽ2 4 .. Let t 1 and t 2 be the distinct elements of F _GX , C1 s CG Ž t 1 . and C2 s CG Ž t 2 .. Then C1 l C2 is a maximal subgroup of both C1 and C2 of shape 2 1q 1q2 S4 . Now, w V, F x s w V, t 1 xw V, t 2 x and C V Ž F . s C V Ž t 1 . l C V Ž t 2 ., so, because
Žii. for t g F a , w V : C V Ž t .x s 2 4 . Žiii. w V : C V Ž F1 .x s 2 6 , C V Ž F1 . s w V, F1 x, and CG Ž C V Ž F1 .. s F, where F is the maximal quadratic group containing F1. Živ. w V, S, S, F1 x / 0. Žv. Assume that F1 ( EŽ2 3 . and that W is an F1-in¨ ariant subspace of V of codimension 4. Then
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w V : C V Ž t .x G 2 4 follows from the structure of V restricted to 2 4 3 A 6 given in wMS, Theorem 3x Žusing w W : CW Ž s .x s 2 2 for an involution, s, in 3 A 6 acting on its six-dimensional irreducible module W .. Now, if w V : C V Ž t .x ) 2 4 , then, since G preserves a GF Ž4.-structure via the action of ZŽ G ., we get w V : C V Ž t .x s 2 6 . However, using wMS, Lemma 3.4 and notationx and choosing t g Q1 l Q, we find that C V Ž t . is invariant under ²2 4 3 A 6 , 2 3 : L3 Ž2. = 3: s G, which is a contradiction. Hence Žii. is true. Now suppose that t 1 , t 2 g F1a with t 1 / t 2 . If w V : C V Ž F1 .x F 2 5, then, as before, w V : C V Ž F1 .x s 2 4 and C V Ž F1 . s C V Ž t 1 . s C V Ž t 2 . by Žii.. But then, from wAx, C V Ž F1 . is invariant under

² CG Ž t1 . , CG Ž t 2 .: g  3 M22 , 2 4 : 3 A6 , 2 4 S5 = 3 4 . All these cases are impossible by wMS, Lemma 3.4x and so w V : C V Ž F1 .x G 2 6 . Note that

w V , t1 xw V , t 2 x F w V , F1 x F w V , F x F C V Ž F . F C V Ž F1 . . From part Žii. and the action of ZŽ G . on V, either w V, t 1 x s w V, t 2 x ( EŽ2 4 . or w V, t 1 xw V, t 2 x s C V Ž F1 .. The former possibility, just as before, is contradicted by wMS, Lemma 3.4x. Therefore, C V Ž F1 . s C V Ž F . s w V, F x s w V, F1 x. By wAx, 2 3 : L3 Ž2. = 3 and 2 4 3 A 6 are maximal subgroups of G and so NG Ž F . is the stabilizer in G of C V Ž F . s C V Ž F1 .. Hence CG Ž C V Ž F1 .. F NG Ž F . and then CG Ž C V Ž F1 .. F O 2 Ž NG Ž F .. s F, so proving part Žiii.. Next suppose that w V, S, S, F1 x s 0. Then, because C V Ž F1 . s C V Ž F ., w V,S, S, F x s 0, whence, as the three subgroup lemma implies that w V, w S, S xx F w V, S, S x, w V, w S, S x, F x s 0. Therefore w S, S x F CG Ž VrC V Ž F .. s F. This, however, implies that the Sylow 2-subgroups of NG Ž F .rF are abelian, which is contrary to Ži.. Hence Živ. holds. Now we prove Žv.. Suppose that < F1 < s 2 3, W is an F1-invariant subspace of V of codimension 4, and that
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455

Žvi.,
for F e S, where S g Syl 2 Ž G ., w V, CS Ž F .x has codimension 1 in

V. Živ. If G ( 3Suz, then w V : C V Ž t .x s 2 8 if t is of class 2 A, and w V : C V Ž t .x G 2 6 if t is of class 2 B. Žv. w V : C V Ž F .x G 2 5 and
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2 8. Next assume that t has class 2 B. In the case that G ( J 2 wMS, Theorem 3x implies that w V : C V Ž t .x G 2 6 , for t an involution in 4 CG Ž2 A._ O 2 Ž CG Ž2 A.. Žsuch t exist since CG Ž2 A. ( 2 1q ] : A 5 .. Since 4 w V : C V Ž s .x s 2 for s of class 2 A, we deduce that w V : C V Ž t .x G 2 6 for t of class 2 B, which proves Žii.. Next we assume that G ( Co1 Žrespectively 3Suz . and that t is of class 2 B with w V : C V Ž t .x F 2 11 Žrespectively, F 2 5 .. Then, by wAx, CG Ž t . ( Ž2 2 = G 2 Ž4.. : 2 Žrespectively, ( Ž2 2 = SL 3 Ž4.. : 2. and O 2 Ž CG Ž t .. ( G 2 Ž4. Žrespectively, ( SL 3 Ž4..; in particular, as the minimal nontrivial GF Ž2.-representation of G 2 Ž4. is 12-dimensional and the minimal nontrivial GF Ž2.representation of SL 3 Ž4. is 6-dimensional, we observe that O 2 Ž CG Ž t .. centralizes VrC V Ž t .. Hence, by Lemma 2.5, ² t, t 1 : is quadratic for all involutions t 1 in O 2 Ž CG Ž t ... Thus wMS, Theorem 3x implies that every involution of ² t : = O 2 Ž CG Ž t .. is of class 2 B. Now if G ( 3Suz, then <² t : = O 2 Ž CG Ž t ..< 2 s 2 7 and, using the notation of wMS, Theorem 3x and wAx, ZŽ O 2 Ž P2 .. l Ž² t : = O 2 Ž CG Ž t ... s 1. Since a Sylow 2-subgroup of SL 3 Ž4. is isomorphic to a Sylow 2-subgroup of 2 4 : A 5 , a Sylow 2-subgroup of ² t : = O 2 Ž CG Ž t .. is generated by involutions. Thus there are involutions from ² t : = 0 2 Ž CG Ž t .. in P2 and not in O 2 Ž P2 .. But then, using wMS, Theorem 3x again, w V : C V Ž t .x G 2 8 , which is a contradiction. This completes the proof of Živ.. So we now assume that G ( Co1. Then since < O 2 Ž CG Ž t ..< 2 s 2 12 and < M24 < 2 s 2 10 , we may choose P a subgroup of G with P ( 2 11 M24 such that O 2 Ž CG Ž t .. l O 2 Ž P . / 1. But wAx tells us that O 2 Ž P . contains only involutions of class 2 A and 2C, so we have a contradiction here as well. Hence Žiii.Žb. holds. Now we suppose that F is a maximal quadratic subgroup of G and that w V : C V Ž F .x s 2 4 or
< F < s 4 and w V : C V Ž t .x G 2 6 for each t g F a ;

Žii.

suppose that F e S; then w V, CS Ž F .x has codimension 1 in V.

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Proof. From wMS, Theorem 2x, < F < s 4 and F contains one element of class 2 A and two of class 2 B Žsee the beginning of the proof of wMS, Lemma 3.8x.. Now, by wMS, Theorem 3x and using VrC V Ž t . ( w V, t x as GF Ž2.CG Ž t .-modules, the 2 A elements have centralizer index at least 2 6 . Now assume that t is a 2 B element. We refer to wMS, Lemma 3.8Žb.x where the restriction of V to P s CG Ž t . is given Žwrongly.: O - CV Ž Q . - CV Ž Z Ž Q . . - V , Z Ž Q . - w V , Q x - V . 4

4

6

4

4

Hence, as VrC V Ž t . ( w V, t x as a GF Ž2.CG Ž t .-module, the centralizer index of an element of class 2 B is 2 8 , and this completes Ži.. Let G G P1 G S with P1 of shape 2 10 AutŽ M22 . and Q1 s O 2 Ž P1 .. Since EŽ2 2 . ( F e S,
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Next, Živ. follows from wMS, part b. of Lemmas 3.2]3.9x. Finally, Propositions 2.7 and 2.8 together with part Žii. and Proposition 2.11Živ. give Žv.. PROPOSITION 2.16. Suppose that G g S , S g Syl 2 Ž G ., M is a maximal subgroup of S, z g V 1Ž ZŽ M .., and V is a nontri¨ ial irreducible GF Ž2.Gmodule. Then w V, M, z x / 0. In particular, we ha¨ e Ži. w V, S, z x / 0 and Žii. w V, S, t x / 0 for all in¨ olutions t g G. Proof. Let G, S, M, and z be as in the statement of the proposition and assume that w V, M, z x s 0. Suppose that t g M is an involution of M other than z. Then, by Lemma 2.5Ži., F s ² t, z : is quadratic on V. Moreover, using the three subgroup lemma, we have w V, z, M x s 0 and so w V, z x F C V Ž M .. Now S acts as an involution on C V Ž M . and thus, employing Theorem 2.15Živ.,

w V , z x F CV Ž M . F 2 CV Ž S . F 2 3 . Then G ( AutŽ M22 . by Theorem 2.15Žii.. Now Theorem 2.15Živ. gives < C V Ž S .< s 2, which gives the untenable
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Assume now that G ( Fi 23 , t is of class 2 A, t 1 is of class 2 B or 2C, and S g Syl 2 Ž G .. Then CG Ž t . s O 2 Ž CG Ž t .. ( 2 Fi 22 . If h Ž CG Ž t ., VrC V Ž t .. s 0, then w V, CG Ž t .x F C V Ž t .. Hence, using Lemma 2.5Ži. and the fact that CG Ž t ._² t : contains involutions we have a contradiction to Theorem 2.1. Therefore, h Ž CG Ž t ., VrC V Ž t .. G 1. This makes VrC V Ž t . into a nontrivial GF Ž2. Fi 22 -module and, as Fi 22 contains an extraspecial 3-subgroup of 3 order 37, wA, 7.1x implies that w V : C V Ž t .x G 2 2.3 s 2 54 . Now, without loss of generality we may assume that t 1 g V 1Ž ZŽ S .. ( EŽ2 2 . and so in H s CG Ž t .r² t : ( Fi 22 , t 1 projects to an involution s of class 2 B with CH Ž s . ( Ž2 1 = 2 1q8 : U4Ž2.. : 2. Assume that W is a nontrivial GF Ž2. H-module. Set X [ O 2 Ž CH Ž2... Then using the Atlas and Lemma 2.2, w H : X x F 4, < O 2 Ž X .< G 2 9 , and mŽ Sr² t :. s 10. Therefore, O 2 Ž X ._² s : contains involutions. From Lemma 2.5Ži. and Theorem 2.1 we see that O 2 Ž X . does not centralize W [ WrCW Ž s . and arguing as above we get h Ž X, W . G 1. Thus at least one of h Ž X, w W, O 2 Ž X .x. G 1 or h Ž X, Wrw W, O 2 Ž X .x. G 1 holds. Hence, as X contains an extraspecial 3-subgroup of order 3 3 we get from wA, 7.1x that either < Wrw W, O 2 Ž X .x< G 2 2.3 s 2 6 or
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that A is an FF q 1 offender. From w V : C V Ž A.x s 2 < A < and Proposition 2.10Žii. and Žiii., < A < G 2 3. If < A < s 2 3, then w V : C V Ž A.x s 2 4 which, using Proposition 2.10Žii. and Žiii., implies that C V Ž A. s C V Ž t . for all t g Aa . But then A acts quadratically on V, contradicting Proposition 2.10 and so < A < / 2 3. Thus by Lemma 2.2, < A < g  2 4 , 2 5, 2 6 4 . Assuming that < A < s 2 4 we aim to show that Ži.Ža. holds. Thus assume that A f  Q3G 4 . Suppose that A contains only 2 B elements. Then Proposition 2.10Žiii. implies that C V Ž A. s C V Ž t . for all t g Aa and so A is a quadratic subgroup of G, which is against Proposition 2.10Ži.. Therefore, we may assume that t g Aa is a 2 A element. So after replacing A and S by conjugates if necessary we may assume that Z [ ZŽ S . F A F S; in particular, A l Q3 / 1 / A l Q1. Employing Proposition 2.10Žii. we see that w C V Ž Z . : C V Ž A.x s 2 and so Ž2.18.1. w C V Ž Q i . : CC ŽQ .Ž A.x F 2 for i s 1, 3. V i If V is the Todd module, then, by wMS, Lemma 3.5x, C V Ž Q3 . is isomorphic to the six-dimensional A 8 permutation module, which does not admit transvections. Therefore, as A g Q3 , Ž2.18.1. implies that V is the Golay code module. Thus, by wMS, Lemma 3.5x, C V Ž Q1 . is isomorphic to a natural S6-module. So, again using Ž2.18.1., < AQ1rQ1 < F 2 and < A l Q1 < G 2 3. Now, if A l Q1 contains a 2 B element t then we have C V Ž Q1 . F C V Ž t . s C V Ž A. and we get A F Q1. Assuming that A l Q1 contains a fours group E which contains only 2 A elements we either get C V Ž Q1 . F C V Ž E . s C V Ž A. and A F Q1 , or C V Ž E . s C V Ž e . for each e g E a . In the latter case E is quadratic and w V : C V Ž E .x s 2 4 , which is against Proposition 2.10Živ.. Therefore we infer that A F Q1. Set P0 s CG Ž Z . and Q0 s O 2 Ž P0 .. Now, h Ž P0 , C V Ž Z .. s 2, and so w C V Ž Z . : C V Ž A.x s 2 implies that A F Q0 . Then A s Q1 l Q0 and C V Ž A. is invariant under P1 l P0 ; 2 9 S3 . If A contains a 2 B element t then C V Ž A. s C V Ž t . is invariant under CP1Ž t . ; 2 6 S5 as well, which forces C V Ž A. to be invariant under P1 , which is against wMS, Lemma 3.5x. Hence A contains only 2 A elements. So Ž2.18.2.

A s Q1 l O 2 Ž CG Ž t .. is normal in CG Ž t . l P1 for all t g Aa .

With the help of Ž2.18.2. we can now deduce a contradiction. We know that P1 l P0 ; 2 9 S3 is a maximal subgroup of P1. Thus, as A is not normal in P1 , Ž2.18.2. implies CG Ž t . l P1 s P1 l P0 for all t g Aa ; but then A F ZŽ S ., which is absurd. With this contradiction we conclude that A g  Q3G 4 and Ži.Ža. holds by wMS, Lemma 3.5x. Next assuming that < A < s 2 5, we aim for a contradiction. Evidently A g Q3 and A l Q3 / 1. As Q3a contains only 2 A elements we may assume A G Z. Suppose first that V is the Todd module. If < A l Q3 < G 2 2 , then we select a fours group E F A l Q3 and as above we find that w V : C V Ž E .x G 2 5. But then w C V Ž E . : C V Ž A.x F 2 and so w C V Ž Q3 . :

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CC V ŽQ 3 .Ž A.x F 2, which again forces A to act as a transvection on the six-dimensional A 8 permutation module, which is absurd. Thus A l Q3 s Z and < AQ3rQ3 < s 2 4 , and in this case calculation on the six-dimensional A 8 permutation module shows that CC V ŽQ 3 .Ž A. has order 2 2 , a fact which is at variance with w C V Ž Z . : C V Ž A.x s 2 2 , which forces w C V Ž Q3 . : CC V ŽQ 3 .Ž A.x F 2 2 . Therefore V is the Golay code module. Because SrQ1 ( D 8 = 2, < Q1 l A < G 2 2 . Then, as above, A operates as a transvection on C V Ž Q1 . and so < A l Q1 < G 2 4 . If C V Ž A l Q1 . ) C V Ž A., then A l Q1 is an FF q 1 offender on V and by Ži.Ža. V is the Todd module, a fact which we have disproven. Hence C V Ž Q1 . F C V Ž A l Q1 . s C V Ž A. and so A F Q1. Now, h Ž P0 , C V Ž Z .. s 2, and as VrC V Ž Z . ( w V, Z x as P0-modules the noncentral chief factors in C V Ž Z . are not isomorphic P0rQ0 ( L3 Ž2.modules. Note that A g Q0 as < Q0 l Q1 < s 2 4 and A F Q1. Therefore, since w C V Ž Z . : C V Ž A.x s 2 2 , we must have < AQ0rQ0 < s 2. Thus < A l Q0 < s 2 4 and, because A F Q1 , A l Q0 s Q1 l Q0 . In particular, A l Q0 e P0 l P1 ; 2 9 S3 . Now, by Ži.Ža., C V Ž A l Q0 . s C V Ž A. is invariant under P0 l P1 and so C V Ž A. s C V Ž² A P 0 F P1 :. s C V Ž Q1 ., and this is against wMS, Lemma 3.5x. Thus we have show that < A < / 2 5. Lastly, for this case, assume that < A < s 2 6 . Then A g  Q1G 4 j  Q2G 4 . Using wMS, Lemma 3.5x then immediately gives A g  Q1G 4 , and Ži.Žb. holds. Assuming that G ( M23 , we note that the M23 Todd module is the restriction of the M24 Todd module. Thus, any offender in M23 fuses in M24 to a subgroup in  Q3G 4 . So Žii. holds. Next suppose that G ( AutŽ M22 . and set H s F U Ž G .. As in the M24 case, V is a quadratic module for AutŽ M22 .; thus we may use Proposition 2.11. If < A < s 2 2 , then as A l H / 1, Proposition 2.11Žiii. delivers a contradiction. So Proposition 2.11Živ. and Lemma 2.2 imply < A < g  2 3 , 2 4 , 2 54 . Assume that < A < s 2 3. Then A 0 s A l H has order at least 2 2 and C V Ž A 0 . s C V Ž A. s C V Ž t . for all t g Aa0 by Proposition 2.11Žiii.. But then A 0 is a quadratic fours group in H, which is prohibited by Proposition 2.11Žii.. Therefore, < A < G 2 4 . Assuming now that < A < s 2 4 , we aim to prove Žiii.Ža.. As H contains a unique class of involutions Ž2 A. we may assume that A G Z [ ZŽ S .. Suppose first that V is the Todd module. Then Proposition 2.11 implies that w C V Ž Z . : C V Ž A.x s 2. Assume that A g Q1. Then w C V Ž Q1 . : CC V ŽQ 1 .Ž A.x F 2 and so, using wMS, Lemma 3.3x, < AQ1rQ1 < s 2 and < A l Q1 < s 2 3. As C V Ž A. h C V Ž Q1 ., we have C V Ž A l Q1 . ) C V Ž A., which is to say A l Q1 is an FF q 1 offender; this being already contradicted we have a contradiction to the assertion A g Q1. Hence Žiii.Ža. holds then V is the Todd module. Now assume that V is the Golay code module. Assume that A g Q2 . Then as above A operates as a transvection on C V Ž Q2 ., which is an orthogonal S5-module. This implies that < A l Q2 < s

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2 3. Again C V Ž A l Q2 . s C V Ž A. or A l Q2 is an FF q 1 offender, a situation already shown to be false. Thus C V Ž A. s C V Ž A l Q2 . G C V Ž Q2 . and hence A F Q2 . Set R 2 s H l Q2 . Then EŽ2 4 . ( R 2 e P2 . If A s R 2 , then C V Ž R 2 . has dimension 5 and contains C V Ž Q2 ., which is isomorphic to an orthogonal S5 ( Ž P2 l H .rR 2-module. But then C V Ž R 2 . splits as a Ž P2 l H . R 2-module which then gives the impossible C V Ž R 2 . s C V Ž Q2 .. So we conclude that A / R 2 . Because Q2 is an indecomposable P2rQ2-module and P2rQ2 ( S5 has six Sylow 5-subgroups and ten Sylow 3-subgroups we deduce that Q2 _ R 2 consists of six 2C elements and ten 2 B elements. Consequently there exists t g A with t of class 2 B. Put P s CG Ž t . Ž( 2 1q 3 L3 Ž2.. and Q s O 2 Ž P .. From wMS, Lemma 3.3x, A / Q. By Proposition 2.11Žii. and Živ. and wMS, Lemma 3.3x, h Ž P, C V Ž t .. s 2 with the noncentral PrQ-chief factors in C V Ž t . being nonisomorphic. Thus, because w C V Ž t . : C V Ž A.x s 2 2 , < AQrQ < s 2 and hence < A l Q < s 2 3. Finally, as h Ž P, C V Ž Q .. s 1, C V Ž Q . g C V Ž A. and so C V Ž A.C V Ž Q . is a subspace of codimension at most 4 centralized by A l Q of order 2 3, which has been ruled out. Lastly assume that < A < s 2 5. Then, as Q2 is not an FF-module for S5 , A g  Q2G 4 and Žiii.Žb. follows using wMS, Lemma 3.3x. LEMMA 2.19. Assume that G g  M23 , M24 4 , S g Syl 2 Ž G ., and W is a GF Ž2.G-module with W ) w W, G x s V, where V is irreducible of dimension 11 and CW Ž G . s 0. Then G ( M24 , V is the Todd-module and dim G F Ž2. W s 12. Moreo¨ er, if we select P2 and P3 subgroups of G Ž( M24 . containing S with P2 ; 2 3.2 Ž L3 Ž2. = S3 . and P3 ; 2 4A 8 and set Q i s O 2 Ž Pi ., i s 2, 3, then w W : CW Ž Q2 .x s 2 9 and w W : CW Ž Q3 .x s 2 5 ; in particular, Q3 is an FF q 1 offender on W. Proof. Since V s w W, G x - W we can pick a G-submodule W1 with V F W1 F W and dim G F Ž2. W1 s 12. We prove the result for W1 and then show that W1 s W. Observe that by Gaschutz’s theorem wHu, 17.4x and ¨ CW Ž G . s 0, C V Ž S . s CW1Ž S .. Let x g G be an element of order 23. Then W1 s CW1Ž x . [ w W1 , x x and CW1Ž x ._V is nonempty. Select ¨ g CW1Ž x ._V and put T [ StabG Ž ¨ .. Then < ¨ G 4< s w G : T x F 2 11 s 2048. Now 23 < < T <, so utilizing the Atlas wAx we see that if G ( M23 , there is a unique maximal subgroup which has order divisible by 23 and it has index 40,320 ) 2048. Thus if G ( M23 , then CW1Ž G . / 0, a contradiction. So we deduce that G ( M24 . Again consulting wAx, in this case there are exactly two maximal subgroups which have order divisible by 23. Arguing as in the last case, we find that the only possibility is that T ( M23 . Let R 3 be the maximal subgroup of T with shape 2 4A 7 .

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Then Žafter conjugating if needed. O 2 Ž R 3 . s Q3 . Thus w CW1Ž Q3 . : C V Ž Q3 .x s 2. Hence, if V is the Golay code module, wMS, Theorem 3x implies that < CW Ž Q3 .< s 2 2 and, as P3 is perfect, CW Ž Q3 . s CW Ž P3 . F CW Ž S ., which is 1 1 1 1 a contradiction. Therefore, V is the Todd-module and w W1 : CW1Ž Q3 .x s 2 5. Next suppose that CW1Ž Q2 . ) C V Ž Q2 .. By wMS, Lemma 3.5x, C V Ž Q2 . is an irreducible three-dimensional space for O 3 Ž P2 ., so < CW1Ž Q2 .< s 2 4 . Now Q2 Q3 s O 2 Ž O 3 Ž P2 .., whence Q3 induces an involution on CW1Ž Q2 . and w CW Ž Q2 ., Q3 x is a O 3 Ž P2 .-invariant subspace of C V Ž Q2 . of order at most 4. 1 We conclude that CW1Ž Q2 . F CW1Ž Q3 .. Now, by wMS, Lemma 3.5x again, C V Ž Q3 . is an irreducible six-dimensional GF Ž2. P3-module and CW1Ž Q3 . is a nonsplit extension of C V Ž Q3 . as a GF Ž2. A 8-module. We claim that P3 has orbits of length 8 and 56, with stabilizers T1 s 2 4A 7 and T2 s 2 4 Ž A 5 = 3. : 2, on CW1Ž Q3 ._C V Ž Q3 .. Clearly we only need to consider the action of A 8 on CW1Ž Q3 ._C V Ž Q3 .. Let F1 F A 8 with F1 a Frobenius group of order 21. Then F1 F StabŽ ¨ 1 . for some ¨ 1 g CW1Ž Q3 ._C V Ž Q3 .. By wAx, StabŽ ¨ 1 . is contained in a maximal subgroup of A 8 isomorphic to either A 7 or 2 3 L3 Ž2.. Since < CW1Ž Q3 ._C V Ž Q3 .< s 64 we see that StabŽ ¨ 1 . ( A 7 or 2 3 L3 Ž2. are the only possibilities, with the latter ruled out by CW Ž S1 . s C V Ž S . F V. So StabŽ ¨ 1 . ( A 7 . Choosing F2 F A 8 with F2 cyclic of order 15 a similar argument gives that StabŽ ¨ 2 . ( Ž A 5 = 3. : 2 for some ¨ 2 g CW1Ž Q3 ._C V Ž Q3 .. However, recalling that Q2 Q3rQ3 ( EŽ2 3 ., we see that Q2 Q3rQ3 cannot be conjugate to a subgroup of either A 7 or Ž A 5 = 3. : 2 Žas m 2 Ž A 7 . s 2 s m 2 ŽŽ A 5 = 3. : 2... Thus we have a contradiction to our supposition that CW Ž Q2 . ) C V Ž Q2 .. Therefore w W1 : CW1Ž Q2 .x s 2 9. Finally assume that W ) W1. Then, as G is simple, WrV is a direct sum of trivial modules. Therefore there exists a G-submodule of dimension 12 with W2 ) V and dim G F Ž2.ŽW1W2 . s 13. Applying the results above to W2 , we find that CW1W 2Ž Q3 . has dimension 8. Now, by wBex, H 1 Ž A 8 , 6. ( GF Ž2. and so < CW1W 2Ž S .< G 2 2 , a contradiction. Therefore we conclude that W1 s W and this completes the proof of Lemma 2.19. LEMMA 2.20. Assume that G ( M22 Ž respecti¨ ely, AutŽ M22 .., S g Syl 2 Ž G ., P1 ; 2 4A 6 Ž respecti¨ ely, 2 4 S6 ., P2 ; 2 4 S5 Ž respecti¨ ely, 2 4q 1 S5 ., Q i s O 2 Ž Pi ., i s 1, 2, and W is a GF Ž2.G-module with W ) w W, G x s V, where V is irreducible of dimension 10 and CW Ž G . s 0. Then w W : CW Ž Q1 .x G 2 6 and w W : CW Ž Q2 .x G 2 7. In particular, Q1 and Q2 are not FF q 1 offenders on W. Proof. It suffices to prove the lemma for a G-submodule W1 , where W1 has dimension 11 and W1 ) V. Assume first that V is the 10-dimensional Golay code module. Then w V : C V Ž Q2 .x s 2 6 and Žusing wMS, Lemma 3.3x. C V Ž Q2 . is an orthogonal S5-module. If CW1Ž Q2 . ) C V Ž Q2 ., then Žas the

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orthogonal S5-module is projective. < C V Ž S .< G 2 2 and so CW1Ž G . ) 0, by Gaschutz’s theorem wHu, 7.14x, which is a contradiction. Therefore, ¨ w W1 : CW Ž Q2 .x G 2 7 and, by wMS, Lemma 3.3x, w W1 : CW Ž Q1 .x G 2 9. So the 1 1 result holds in this case. Now we assume that V is the 10-dimensional Todd module. In this case w V : C V Ž Q1 .x s 2 5, again, if CW Ž Q1 . ) C V Ž Q1 ., then CW Ž Q1 . is an A 61 1 module Žrespectively, S6-module. of dimension 6 with Žusing wMS, Lemma 3.3x. w CW1Ž Q1 ., O 2 Ž P1 .x of dimension 4. There are no such indecomposable P1rQ1-modules and so once more we obtain < CW1Ž S .< G 2 2 , which is again a contradiction. Hence, w W1 : CW1Ž Q1 .x s 2 6 and, by wMS, Lemma 3.3x, w W1 :CW Ž Q2 .x G 2 9 and the lemma is proven. 1 To finish with the sporadic groups in this section we present two structural lemmas. LEMMA 2.21. following holds:

Suppose that G g S and S g Syl 2 Ž G .. Then one of the

Ži.
Ži. Žii. Žiii.

V 1Ž ZŽT .. s V 1Ž ZŽ S .. has order 2. G s O 2 Ž G . ( J1 and < V 1Ž ZŽ S ..< s 2 3. G s O 2 Ž G . ( Fi 23 and < V 1Ž ZŽ S ..< s 2 2 .

Proof. Clearly there is no loss in assuming that O Ž G . s 1. And as the lemma obviously holds for J1 , we assume that G \ J1. By checking the character tables in the Atlas wAx, G has exactly one conjugacy class of 2-central involutions or G ( Fi 23 and there are three conjugacy classes of 2-central involutions; also we observe that in any case the 2-central involutions are in O 2 Ž G .. Therefore in a counterexample to the lemma there are involutions s and t g ZŽ S . with s and t conjugate in G. But then s and t are already conjugate in NG Ž S . and hence NG Ž S . ) S. Thus Lemma 2.21 implies that G ( J 2 , J 3 , Suz, or HN. However, in each of these cases there exists G G K G S with CK Ž O 2 Ž K .. F O 2 Ž K . and O 2 Ž K . an extraspecial 2-group Ž K being, respectively, 2 1q 4A 5 , 2 1q4A 5 , 2 1q 6 U4Ž2., 2 1q 8 Ž A 5 = A 5 . : 2., and this implies V 1Ž ZŽ S .. s V 1Ž ZŽ O 2 Ž K ... ( EŽ2., which contradicts our assumption that there are at least two involutions in V 1Ž ZŽ S ... Hence Lemma 2.22 holds.

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LEMMA 2.23. Let X be a finite group, p a prime, V a GF Ž p . X-module, and S g Syl p Ž X .. Suppose that V s ²U X : for some subspace U of C V Ž S .. Then V s w V, X x q C V Ž X .. Proof. See wCD, 2.5x. LEMMA 2.24. Suppose that V is a GF Ž2. X-module. If H is a GF Ž2.-hyperplane of V, then ww V, t x : w H, t xx F 2 for all in¨ olutions t in X. Proof. If w V, t x F H, then H is t-invariant and we are done. Otherwise V s H q w V, t x and w V, t x s w H, t x, so Lemma 2.24 is true. We now state some results about the rank-1 Lie-type groups over GF Ž2 n .. We begin with some details of the structure of these groups and then continue with some GF Ž2.-module results. Thus until notified further we assume that G g L i e1Ževen., S g Syl 2 Ž G ., K is a complement to S in NG Ž S ., and V is is a nontrivial GF Ž2.G-module. Recall that a GF Ž2. L2 Ž2 n .-module V is called a natural module if C V Ž G . s 0, dim G F Ž2. V s 2 2 n, and w V, S, S x s 0. LEMMA 2.25. The following are true: Ži. G operates doubly transiti¨ e on  NG ŽT .< T g Syl 2 Ž G .4 and NG Ž S . is a maximal subgroup of G; Žii. if T g Syl 2 Ž G . and S / T, then S l T s 1; Žiii. if x is an in¨ olution of G and x f S, then G s ² S, x :; Živ. K is cyclic, ZŽ G . F K, and K normalizes exactly two Sylow 2-subgroup of G. Proof. See wDS, 5.1x. LEMMA 2.26.

Suppose that G ( L2 Ž2 n .. Then

Ži. S is elementary abelian, < S < s 2 n, < K < s 2 n y 1, and K operates transiti¨ ely on S _ 14 ; Žii. if A F G is a fours group and t g Aa , then there exists g g G such that G s ² t, A g :. Proof. See wDS, 5.2x. LEMMA 2.27.

Suppose that G ( U3 Ž2 n . or SU3 Ž2 n .. Then

Ži. ZŽ S . s F Ž S . s V 1Ž S . is elementary abelian of order 2 n and < SrZŽ S .< s 2 2 n ; Žii. all the in¨ olutions of S are conjugate; Žiii. if S / T g Syl 2 Ž G ., then ² ZŽ S ., ZŽT .: ( SL 2 Ž2 n .; Živ. if U is a fours group in G and x g U a , then there exists g, h g G, such that G s ² x, U g , U h :;

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Žvi. < KrZŽ G .< s Ž2 2 n y 1.rŽ2 n q 1, 3. and, if n G 2, then KrZŽ G . acts irreducibly on SrZŽ S . and ZŽ S .. Proof. See wDS, 5.4 and the table before 5.2x. LEMMA 2.28.

Suppose that G ( Sz Ž2 n ., n ) 1. Then

Ži. n is odd and 3 does not di¨ ide the order of G; Žii. S is special of order 2 2 n with < ZŽ S .< s 2 n, all the in¨ olutions of S are contained in ZŽ S . and all the in¨ olutions of G are conjugate; Žiii. if S / T g Syl 2 Ž G ., then ² ZŽ S ., ZŽT .: s G; Živ. < K < s 2 n y 1 and CS Ž k . s 1 for all k g K a ; Žv. if U is a fours group in G and x g U, then there exists g g G such that G s ² x, U g :. Proof. See wDS, 5.3 and the table before 5.2x. Recall that Sz Ž2. ( FrobŽ20. ( C5 : C4 , so the Sylow 2-subgroup S is cyclic of order 4. But V 1Ž ZŽ S .. has order 2 and contains the involution of S. LEMMA 2.29. Suppose that A F S and w V, A, A x s 0. Then A F V 1Ž ZŽ S .. and < A < F < V 1Ž ZŽ S ..< F q s 2 n. Proof. Since A acts quadratically, it is elementary abelian. Thus, A F V 1Ž S . s V 1Ž ZŽ S .. by Lemmas 2.26Ži., 2.27Ži., and 2.28Žii.. LEMMA 2.30. Ži. Žii. Žiii.

Suppose that V is a nontri¨ ial irreducible GF Ž2.G-module.

If G ( L2 Ž2 n ., then < V < G q 2 s 2 2 n. If G ( Sz Ž2 n ., then < V < G q 4 s 2 4 n. If G ( U3 Ž2 n . or SU3 Ž2 n ., then < V < G q 6 s 2 6 n.

Proof. See wDS, 5.7x. LEMMA 2.31. Suppose that n G 2, V is a nontri¨ ial irreducible GF Ž2.Gmodule, and F is a fours subgroup of G with t g F a. Then the following statements hold: Ži. If G ( L2 Ž2 n ., then w V : C V Ž t .x G 2 n. Žii. If G ( Sz Ž2 n ., SU3 Ž2 n ., or U3 Ž2 n ., then w V : C V Ž F .x G 2 2 n. Žiii. If G ( Sz Ž2 n ., SU3 Ž2 n ., or U3 Ž2 n . and w V, F, F x s 0, then < V : C V Ž t .< G 2 2 n. Živ. If G ( Sz Ž2 n ., SU3 Ž2 n ., or U3 Ž2 n . and n G 2, then w V : C V Ž t .x ) n 2 .

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Proof. See wCD, 2.9x for Ži. and wDS, 5.10 b.x for Žiii.. That leaves Žii. and Živ.. Suppose that G ( Sz Ž2 n ., SU3 Ž2 n ., or U3 Ž2 n ., and < V : C V Ž F .< 2 2 n. Then, by Lemmas 2.27Živ. and 2.28Žv., if G ( Sz Ž2 n ., < V < - 2 4 n, and, if G ( SU3 Ž2 n . or U3 Ž2 n ., then < V < - 2 6 n. This, of course, is against Lemma 2.30. Therefore Žii. is true. Because, by Lemma 2.27Živ. and Lemma 2.28Žv., SU3 Ž2 n . and U3 Ž2 n . are generated by five involutions and Sz Ž2 n . is generated by three involutions, Živ. follows from Lemma 2.30. LEMMA 2.32. Suppose that A F S, C V Ž O 2 Ž G .. s 1, w V, A, A x s 0, and < A < 2 ) w V : C V Ž A.x. Then G ( L2 Ž2 n . and V is a natural GF Ž2. L2 Ž2 n .-module. In particular, if V is FF-module with offending subgroup A, then G ( L2 Ž2 n . and A s S. Proof. See wDS, 5.12 b.x. LEMMA 2.33. Suppose that G ( L2 Ž2 n ., V s ² C V Ž S . G :, and C V Ž G . s 0. If w V, S, S x s 0, then V is a direct sum of natural GF Ž2.G-modules. Proof. This is given in wCD, 2.11x. LEMMA 2.34. Assume that x is an in¨ olution in G. Then C V Ž S . l C V Ž x . / 0 if and only if x g ZŽ S .. Proof. Put X s C V Ž S . l C V Ž x . and assume that X / 0. Then X is centralized by ² S, x :. Since V is irreducible we conclude from Lemmas 2.25Žiii., 2.26, 2.27Ži., and 2.28Žii. that x g ZŽ S .. Conversely, if x g S, then x g V 1Ž ZŽ S .., so C V Ž x . is S-invariant and therefore, X / 0. LEMMA 2.35. Suppose that G ( L2 Ž2 n ., 1 / t g S g Syl 2 Ž G ., and V is a natural GF Ž2.G-module. Then C V Ž t . s C V Ž S . s w V, S x s w V, t x. Proof. This is immediate as V is a two-dimensional module over GF Ž2 n .. LEMMA 2.36. Suppose that G ( SU3 Ž2. or U3 Ž2., S g Syl 2 Ž G ., and V is a faithful irreducible GF Ž2.G-module. Then Ži. dim G F Ž2. V G 6; Žii. if G ( U3 Ž2., then V is the eight-dimensional irreducible permutation module obtained from its permutation action on the Sylow 2-subgroups of G; Žiii. suppose that w V : C V Ž t .x s 2 2 for some in¨ olution t g G; then dim G F Ž2. V s 6 and G ( SU3 Ž2.; Živ. suppose that dim G F Ž2. V s 6; then V is a three-dimensional Ž GF 4.-module, dim G F Ž4.w V, S x s 2.dim G F Ž4.w V, S, S x s 2, and w V, S, ZŽ S .x s 0; Žv. If dim G F Ž2. V ) 6, then dim G F Ž2. V G 8.

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Proof. From wHu, p. 245x we recall that < U3 Ž2.< s 2 3 3 2 , < SU3 Ž2.< s 2 3 3 3, and U3 Ž2., SU3 Ž2. are both 3-closed with S ( Q8 . For R g Syl 3 Ž G ., RrZŽ G . ( EŽ3 2 . and S acts transitively Žby conjugation. on Ž RrZŽ G ..a . Part Ži. follows from Lemma 2.30Žiii.. Employing a result of Berman wHB, 3.11x shows that if G ( U3 Ž2., then G has exactly two irreducible GF Ž2.Gmodules. Since the eight-dimensional permutation module obtained from the action of G on Syl 2 Ž G . is irreducible, we have Žii.. Since the involutions of G invert all the elements of Ž RrZŽ G ..a and SU3 Ž2. is a nonsplit central extension of U3 Ž2. by C3 , it follows that we can find three involutions x, y, z of G such that <² x, y, z :< s 2 < R < and ² x, y, z :eG. So C V Ž² x, y, z :. is G-invariant, whence, using Ži., dim V s 6. Then, by Žii., G ( SU3 Ž2., so Žiii. holds. Part Živ. holds because the action of ZŽ G . on the six-dimensional GF Ž2.-module, V, imposes a GF Ž4. structure. If Žv. is false, then dim G F Ž2. V s 7, and, by Žii., ZŽ G . ( C3 with ZŽ G . acting fixed-point freely in V. But then 3 divides 2 7 y 1, a contradiction. LEMMA 2.37. Suppose that G ( Sz Ž2. ( FrobŽ20. and V is a faithful, irreducible GF Ž2.G-module. Then dim G F Ž2. V s 4, w V : C V Ž t .x s 2 2 for all in¨ olutions t g G and e¨ ery nontri¨ ial ¨ ector in V is centralized by exactly one in¨ olution in G. Proof. It follows from wHB, 3.11x that G possesses, up to isomorphism, exactly two irreducible GF Ž2.G-modules. Therefore, as the action of G on Syl 2 Ž G . provides an irreducible module of dimension 4 on which the elements of order 5 operate fixed-point freely and because two involutions generate a dihedral group of order 10, we have the result. LEMMA 2.38. Suppose that G ( D 2 p , where p is an odd prime, t is an in¨ olution in G, and V is a nontri¨ ial faithful irreducible GF Ž2.G-module. Then w V : C V Ž t .x G < V < 1r2 . Moreo¨ er, if V is an FF module for G, then < V < s 2 2 , and G ( S3 ( SL 2 Ž2.. Proof. Because G s ² t, t g : for any g g G_² t :, the result is immediate.

3. PRELIMINARY RESULTS AND THE PUSHING-UP CASE In this section we start the proof of Theorem A. Thus from now on we assume that the following hypothesis holds: HYPOTHESIS 3.1. G is the free amalgamated product of two proper finite subgroups P1 and P2 o¨ er P1 l P2 Ž here groups are identified with their

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images in the free amalgamated product . which satisfy the following conditions: Ž1. Ž2.

X

X

X

X

O 2 Ž P1 .rO 2 Ž O 2 Ž P1 .. g L and O 2 Ž P2 .rO 2 Ž O 2 Ž P2 .. g S ; B [ P1 l P2 contains a Sylow 2-subgroup of both P1 and P2 ; X Pi s O 2 Ž Pi . B for i s 1, 2;

Ž3. Ž4.

CP iŽ O 2 Ž Pi .. F O 2 Ž Pi . for i s 1, 2;

Ž5.

no nontri¨ ial normal subgroup of G is contained in B.

As we remarked in the Introduction, any group satisfying the hypotheses of Theorem A is a homomorphic image of the group satisfying Hypothesis 3.1. Therefore, to prove our theorem it suffices to classify the amalgams fulfilling Hypothesis 3.1. In the situation described by Hypothesis 3.1 we can construct a graph G, the coset graph, which has vertices V Ž G . s  Pi g ¬ g g G, i s 1, 24 with edge set EŽ G . s Ž Pi g, Pj h. ¬ Pi g l Pj h / B, i / j4 . For d , l g V Ž G . we shall say that d , l are adjacent if, and only if, Ž d , l. g EŽ G .. Clearly G acts on the graph G by right multiplication. Since G is the free amalgamated product P1 )P1 l P 2 P2 and P1 and P2 are finite groups, G is a locally finite tree Žsee Serre wSex.. We draw the following result from wDSx: PROPOSITION 3.2. The following hold: Ži. G acts faithfully on G; Žii. if a g V Ž G ., then Ga [ stabG Ž a . is conjugate in G to one of P1 or P2 ; Žiii. If Ž a , b . g EŽ G ., then Gab [ stabG ŽŽ a , b .. is conjugate in G to P1 l P2 . We next introduce the notation that will be maintained throughout the rest of this paper and wPR3x. Suppose that d g V Ž G . and Ž d , l. g EŽ G .. Then DŽ d . s g g V Ž G . ¬ Žg , d . g EŽ G .4 ; O Ž L . s g g V Ž G . ¬ Gg is G-conjugate to P1 4 ; O Ž S . s g g V Ž G . ¬ Gg is G-conjugate to P2 4 ; X Ld s O 2 Ž Gd .; Qd s O 2 Ž Ld .; Kd s core L Ž Ql .; d

Sdl is a Sylow 2-subgroup of Gdl ;

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Zd s ² V 1Ž ZŽ Sdl .. Ld :; Vd s ² ZlLd :; Ud s ² VlLd :; Wd s ²UlLd :; Dd s CLdŽ Vdrw Vd , Qd x.; Td s O 2, 2X Ž Ld .. Further, for d g O Ž S . we let Fd G Qd be such that FdrQd s F U Ž LdrQd .. Finally, if d g O Ž L . and LdrQd f  S5 4 j D, then qd is the order of the field of definition of LdrQd ; if LdrQd ( S5 , then qd s 4; if LdrQd g D, then qd s 2. LEMMA 3.3. Suppose that d g V Ž G .. Then Ld is transiti¨ e on DŽ d .. Furthermore, if d g O Ž S ., then Fd is transiti¨ e on DŽ d .. Proof. Assume that d g V Ž G .. Then by Proposition 3.2Žii. we may suppose that d g  P1 , P2 4 . Since DŽ d . consists of cosets of B in P1 Ž i g  1, 24., Gd is transitive on DŽ d ., whence, by Hypothesis 3.1Ž3., Ld is also transitive on DŽ d .. Moreover, if d g O Ž S ., then from the Atlas wAx w Ld : Fd x F 2 and so Ld s Sgd Fd . Therefore Fd is transitive on DŽ d .. The following result will be important in our later arguments. LEMMA 3.4. Suppose that Ž d , l. g EŽ G . and N F Gdl . If NGdŽ N . is transiti¨ e on DŽ d . and NGlŽ N . is transiti¨ e on DŽ l., then N s 1. Proof. See wDS, 2.6x. LEMMA 3.5.

Suppose that d g V Ž G .. Then

Ži. Zd F V 1Ž ZŽ Qd ..; Žii. if Zd / V 1Ž ZŽ Ld .., then CL Ž Zd . F Td and Qd g Syl 2 Ž CL Ž Zd ..; d d Žiii. if X is a 2-subgroup of Ld with X g Qd and w X, Zd x s 1, then Zd s V 1Ž ZŽ Ld ... Proof. From Hypothesis 3.1Ž4., V 1Ž ZŽ Sdl .. F Qd for any l g DŽ d . and hence Zd F V 1Ž ZŽ Qd ... Žii. Since CL Ž Zd . is a proper normal subgroup of Ld Žand Ld s Fd Sdl d when d g O Ž S . for any l g DŽ d .., we deduce from the structure of LdrQd given in Hypothesis 3.1Ž1. that CLdŽ Zd . F Td . By part Ži., Qd F CLdŽ Zd ., so Qd g Syl 2 Ž CLdŽ Qd ... Thus Žii. holds and Žiii. follows directly from Žii.. We define dŽ , . to be the standard distance function on G. Then the critical distance b is defined by: bs

min

t , r gV Ž G .

 d Ž t , r . ¬ Zt g Qr 4 .

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If Žt , r . g V Ž G . = V Ž G . is a pair of vertices with dŽt , r . s b and Zt g Qr , then Žt , r . is called a critical pair. The set of all critical pairs will be denoted by C . Observe that, since Zd / 1 for d g V Ž G ., Proposition 3.2Ži. implies that b is finite. Further, by Lemma 3.5Ži., b G 1. Whenever we fix Ž a , a 9. g C , we will denote the vertices of the unique path joining a and a 9 as follows:

a , b , a q 2, . . . , a 9 y 1, a 9. The proof of the following proposition is straightforward and is left to the reader. PROPOSITION 3.6. The following hold: Ži. Žii. Žiii. particular,

if if if b

Ž a , a 9. g C , then CL Ž Za . F Ta ; a Ž a , a 9. g C and w Za , Za 9 x / 1, then Ž a 9, a . g C ; Ž a , a 9. g C and w Za , Za 9 x s 1, then Za 9 s V 1Ž ZŽ La 9 ..; in is odd and CZaŽ La . s 1.

LEMMA 3.7. Let Ž d , l. g EŽ G . and suppose that Z is a nontri¨ ial normal subgroup of Ld contained in Zd . Put V s ² Z Ll :, U s ² V Ld :, and W s ²U Ll :. If b G 3 Ž respecti¨ ely, b G 5, b G 7., then V Ž respecti¨ ely, U, W . is elementary abelian. Furthermore, Ži. Žii. Žiii.

if b G 2, then V F Ql and lŽ Ll , Vrw V, Ql x. G 1; if b G 3, then U F Qd and h Ž Ld , UrwU, Qd x Z . G 1; if b G 4, then W F Ql and h Ž Ll , Wrw W, Ql xV . G 1.

Proof. The first part follows easily from the definition of b, V, U, and W. As the same type of argument establishes each of parts Ži., Žii., and Žiii., we just prove Žiii.. Again it is immediate from its definition that W F Ql. Now suppose that h Ž Ll, Wrw W, Ql xV . s 0. Then W s w W, Ql xVU s w W, Ql xU and hence

w W , Ql x s w W , Ql x U, Ql s w W , Ql x , Ql w U, Ql x . Therefore, W s w W , Ql x U s w W , Ql x , Ql w U, Ql x U s w W , Ql x , Ql U. By repeatedly commutating with Ql it follows that W s U, whence, by Lemma 3.3 and Lemma 3.4, W s U s 1. But this is against Z / 1. Hence h Ž Ll, Qrw W, Ql xV . G 1. Our next lemma is a variant of Lemma 3.7Žii. in the case when l g O Ž S ..

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LEMMA 3.8. Let Ž d , l. g EŽ G . and let Z, V, and U be as in Lemma 3.7. If l g O Ž S ., Sdl s Qd Ql , and b G 3, then either h Ž Ld , UrZ . G 2, or CLlŽ V . l Qd F Kd . Proof. By Lemma 3.7, h Ž Ll, Vrw V, Ql x. G 1 and h Ž Ld , UrZ wU, Qd x. G 1. Assume that h Ž Ld , Z wU, Qd xrZ . s 0. Then wU, Qd x Z s w V, Qd x Z. Put X s CLlŽ V . l Qd , and let t g DŽ d .. Clearly X F Lt and t g O Ž S .. By Lemma 3.3, we may choose g g Ld so as l. g s t . Setting V1 s V g we observe that X centralizes w V1 , Qd x since w V1 , Qd x F wU, Qd x Z s w V, Qd x Z F V. Put V1 s V1rw V1 , Qt x. From Std s Qt Qd we get w V1 , Std x s w V1 , Qd x is centralized by X. Now, X e Qd implies that XQtrQt e StdrQt . Since h Ž Lt , V1 . G 1, Lemma 2.16 forces X F Qt and hence, as t was an arbitrary element of DŽ d ., X F Kd . This proves the lemma. The next two results are drawn from Stellmacher’s article wSt2x; first we establish his notation. Suppose H is a group, S is a 2-subgroup of H and V is a GF Ž2. H-module. Then Q Ž S, V . s  A F S ¬ A acts quadratically on V 4 .

¡min log Ž VrC Ž A . . ¬ A g Q Ž S, V . 4 , ~ q Ž S, V . s when Q S, V / B, ¢0, when Q ŽŽ S, V .. s B. ArC AŽ V .

V

Now suppose that Ž a , a 9. g C and let 1 s V0 F V1 F ??? F Vn s Vb be a chief series of GF Ž2. Lb-submodules. Set c s h Ž Lb , Vb . ; q s q Ž Qb , Za . ; r s min  q Ž Sab , VirViy1 . ¬ q Ž Sa b , VirViy1 . / 0 and 1 F i F n4 . LEMMA 3.9. Suppose that Ž a , a 9. g C , b G 3, w Za , Za 9 x s 1, and Va 9 g Qb . If q y 1 G 0 and rc y 1 G 0, then Ž q y 1.Ž rc y 1. F 1. Proof. Since b G 3, Va 9 acts quadratically on Vb and Vb acts quadratically on Va 9. Also, because the situation is symmetric we may assume that < VbrC V Ž Va 9 .< F < Va 9rC V Ž Vb .<. Thus wSt2, 3.1 c.x is applicable and yields the b a9 result. LEMMA 3.10 wSt2x. Suppose that Ž a , b . g EŽ G ., b G 2, and h Ž Lb , Vb . s 1. Set Q s w O 2 Ž Lb ., Qb x and assume that Q g CLaŽ Za .. Then Q operates quadratically on Za and
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Proof. By Lemma 3.7Ži., h Ž Lb , Vb rw Vb , Qb x. G 1. Therefore, as h Ž Lb , Vb . s 1, w Vb , Qb , O 2 Ž Lb .x s 1. Because Q F O 2 Ž Lb ., we thus have w Vb , Q, Q x s 1, and so, in particular, Q operates quadratically on Za . Further, as w Q, O 2 Ž Lb .x s Q, the three subgroup lemma gives Vb , O 2 Ž Lb . , Q s Vb , Q, O 2 Ž Lb .

s Vb , Q .

Ž ).

Now set V0 s C VbŽ O 2 Ž Lb .., and select ¨ g Za _V0 . Then w Vb , O 2 Ž Lb .x F 2 2 ² ¨ O Ž Lb . :V0 and Vb s ² ¨ O Ž Lb . :V0 Za . Therefore, using Ž). we have 2 w Za , Q x F Vb , Q s Vb , O 2 Ž Lb . , Q F ² ¨ O Ž Lb . :V0 , Q

s ²¨ O

2

Ž Lb . :

, Q s w¨ , QxO

2Ž

Lb .

s w ¨ , Q x F w Za , Q x .

Hence w Za , Q x s w ¨ , Q x. Since w ¨ , Q x F ZŽ Q . the map x ¬ w ¨ , x x is a surjective group homomorphism from Q to w ¨ , Q x and consequently w ¨ , Q x ( QrCQ Ž ¨ .. In particular,
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NH Ž O 2 Ž X ... If y g H is such that w Z, Z y x / 1, then N s NH Ž O 2 Ž X . y . and <² y : NrN < s 2; in particular, y has e¨ en order. Proof. Suppose that y g H satisfies w Z, Z y x / 1, and set Q s O 2 Ž X . and R s Q y l Q. It follows from w Z, Z y x / 1 that Z y g Q, and hence y f N. Without loss of generality we may assume that < Z y QrQ < G < ZQ yrQ y <. Then, noting that CS Ž Z y . s Q y , < Z y QrQ < G < ZQ yrQ y < s w Z : Z l Q y x s Z : CZ Ž Z y . .

Ž ).

Thus Z y QrQ is an offending subgroup on Z and hence equality holds in Ž). by Lemma 2.32, with Z y Q s S. From w Z : Z l Q y x s w S : Q x s w S : Q y x we also get S s ZQ y. Using Dedekind’s modular law gives Q s Q l ZQ y s ZR and Q y s Q y l Z y Q s Z y R. Therefore,

Ž 3.13.1.

F Ž Q . s F Ž ZR . s F Ž R . s F Ž Z y R . s F Ž Q y . .

Set S s SrF Ž R .. Then Q and Q y are elementary abelian subgroups of S. From a lemma of Burnside’s wGor, Theorem 5.1.4x and Lemma 3.12, we have h Ž X, Q . s 1 with the X-noncentral chief factor in Q being a natural L2 Ž2 n .-module. Note that S s QQ y gives S s QQ y and w Q : R x s w Q : R x s w S : Q y x s 2 n. Using Lemma 2.35 we infer that R s CQ ŽQ y. s CQ Ž x . for all x g S _ Q. Consequently, by wLPR, Lemma 3.12x, Q and Q y are the only maximal elementary abelian subgroups of S. Since N normalizes Q and hence normalizes F Ž Q . s F Ž R ., N acts upon S. Because N normalizes Q, it must then normalize Q y and hence N normalizes Q y. Thus we conclude that N s NH Ž O 2 Ž X . y .. Observe that, using Ž3.13.1., F Ž R . y s F Ž Q . y s F Ž Q y . s F Ž R . and, therefore, y also acts upon S. Hence y 2 normalizes Q, whence y 2 g N and this completes the proof of the proposition. The next result is similar in spirit to that in wSt1, Theorem Cx; however, we do not require that NH Ž O 2 Ž X .. is normal in H. COROLLARY 3.14. Suppose that XrO 2 Ž X . ( L2 Ž2 n ., that no nontri¨ ial characteristic subgroup of S is normal in X, and that H F AutŽ S . with w H : NH Ž O 2 Ž X ..x odd. Then ² Z H : is a normal subgroup of X contained in O 2 Ž X .. Proof. Suppose there exists h g H such that Z h g O 2 Ž X .. Then, by Proposition 3.13, h 2 g NH Ž O 2 Ž X .. and ² h: NH Ž O 2 Ž X .. is a subgroup with w² h: NH Ž O 2 Ž X .. : NH Ž O 2 Ž X ..x s 2. This is against our hypothesis that w H : NH Ž O 2 Ž X ..x is odd. Therefore, as Z F ² Z H : F O 2 Ž X ., Lemma 3.12 implies that ² Z H : is normal X. The next two lemmas allow us to by-pass certain pushing-up configurations which involve S5 .

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LEMMA 3.15. Suppose that XrO 2 Ž X . ( L2 Ž2 n ., or XrO 2 Ž X . ( S5 and ZrCZ Ž O 2 Ž X .. is a natural GF Ž2. S5-module. If J ŽT . g O 2 Ž X ., then B ŽT . g Syl 2 Ž² B ŽT . X :. and ² B ŽT . X :rO 2 Ž² B ŽT . X :. ( L2 Ž2 m ., where m g  n, 24 . Proof. Compare with wB1, 2.11.1.4x and wB2, Ž7.x. Set Q s O 2 Ž X ., X 0 s O 2 Ž X . Q, S0 s S l X 0 , and R s ² V 1Ž ZŽ J ŽT ... X :. Put q s 2 n if XrQ ( L2 Ž2 n . and q s 4 if XrQ ( S5 . Then, since J ŽT . g Q, we can pick A g AŽT . such that A g Q. Suppose that XrQ ( S5 . Then, as ZrCZ Ž O 2 Ž X .. is a natural S5-module by hypothesis, we infer, from Lemma 2.3, that J ŽT . Q s AQ s S0 . While if XrO 2 Ž X . ( L2 Ž2 n ., then Lemma 2.32 implies that J ŽT . Q s AQ s S0 . Now, because A g AŽT ., Lemmas 2.32 and 2.3Žvi.Ža. imply that < AQrQ < s < ZrCZ Ž A.< s < ZrŽ Z l A.< s q. Hence Ž3.15.1. Ži. Ž A l Q . Z g AŽT . l AŽ Q . and Žii. w Z : A l Z x s < AQrQ < s q with h Ž X, Z . s 1. Moreover, for any AU g AŽT . with AU g Q we also have Ž AU l Q . Z g A Ž T . and consequently A l Z F V 1Ž Z Ž J Ž T ... . Hence Z F ² V 1Ž ZŽ J ŽT ... X : s R. Notice that, by Ž3.15.1.Ži., J Ž Q . F J ŽT ., so V 1Ž ZŽ J ŽT ... F V 1Ž ZŽ J Ž Q ... and, in particular, R F V 1Ž ZŽ J Ž Q .... Therefore we have R F Ž A l Q . Z, and so, Z F R and Dedekind’s modular law then give R s R l ZŽ A l Q . s ZŽ A l R .. Thus Ž3.15.2.Žii. implies that < Rr Ž A l R . < s < Z Ž A l R . r Ž A l R . < s < Zr Ž Z l A . < s q. It follows that h Ž X, R . s 1 and hence R s ZV 1Ž ZŽ J ŽT .... Therefore Žas A l Z F V 1Ž ZŽ J ŽT ...., R l A s ZV 1 Ž Z Ž J Ž T . . . l A s Ž Z l A . V 1 Ž Z Ž J Ž T . . . s V 1Ž Z Ž J Ž T . . . . Now choose x g X such that ² A, A x : Q G X 0 . Then R 0 s R l A l A x s V 1Ž ZŽ J ŽT ... l V 1Ž ZŽ J ŽT x ... is normalized by X 0 and w R : R 0 x s q 2 . Since h Ž X 0 , RrR 0 . s 1, R s ZR 0 which then gives V 1 Ž Z Ž J Ž T . . . s R l A s ZR 0 l A s Ž Z l A . R 0 , and so V 1Ž ZŽ J ŽT ... s V 1Ž ZŽ S0 .. R 0 . Set C s C X 0Ž R 0 . and S1 s S0 l C g Syl 2 Ž C .. Then C e X 0 and S1 g Syl 2 Ž C .. Clearly, S1 centralizes V 1Ž ZŽ J ŽT ... and so S1 F B ŽT .. Equally clearly B ŽT . F C and thus we conclude that B ŽT . s S1 g Syl 2 Ž C . with Žas B ŽT .e S . C s ² B ŽT . 0X : s ² B ŽT . X :. Since CrO 2 Ž C . ( X 0rQ, the lemma is proven.

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LEMMA 3.16. Suppose that XrO2 Ž X . ( S5 , ZrCZ Ž O 2 Ž X .. is an orthogonal module, and J ŽT . g O 2 Ž X .. Then there exists a subgroup R of X such that B ŽT . g Syl 2 Ž R ., RrO2 Ž R . ( S3 ( L2 Ž2., and ² R, S : s X. Proof. Put X s XrO 2 Ž X .. Then, since J Ž T . g O 2 Ž X . and ZrCZ Ž O 2 Ž X .. is an orthogonal module, Lemma 2.3Žiii. and Žv.Ža. and the normality of T in S imply that J Ž T . is a fours group which is not contained in O 2 Ž X .. Without loss of generality, we assume that B Ž T . s J Ž T . s ²Ž1, 2., Ž1, 2.Ž3, 4.:. Choose R1 G O 2 Ž X . such that R1 s C X ŽŽ1, 2.. G B Ž T . . Then R1rO 2 Ž R1 . ( L2 Ž2., ² R1 , S : s X, and, for S1 s S l R1 g Syl 2 Ž R1 ., J Ž S1 . s J ŽT . g O 2 Ž R1 .. The result now follows directly from Lemma 3.15 with R s ² B ŽT . R 1 :. We now revert to our standard notation. LEMMA 3.17.

Suppose that Ž d , l. g EŽ G . and Qd l Ql e Ld . Then

Ži. h Ž Ld , QdrŽ Qd l Ql .. s 0; L : Žii. Zd F Vd s ² Zld F Qd l Ql ; in particular, b G 2. Proof. First observe that Ql g Qd , for otherwise Ql e² Ld , Ld :, which then contradicts Lemma 3.4. Thus, since w Qd , ² dlLd :x F Qd l Ql , we infer that h Ž Ld , QdrŽ Qd l Ql .. s 0. We next prove part Žii.. Evidently, w Qd l Ql, Zl x s 1, and so, if Zl g Qd , then h Ž Ld , Qd l Ql . s 0. This together with Ži. then implies the untenable h Ž Ld , Qd . s 0. Hence Zd F Vd s ² ZlLd : F Qd l Ql , as stated. LEMMA 3.18. Suppose that Ž d , l. g EŽ G . and Qd l Ql e Ld . Set Z s V 1Ž ZŽ Qd l Ql ... Then Ži. LdrQd ( L2 Ž qd . or S5 ; Žii. w Sdl : J Ž Ql . Qd x s 1 Ž respecti¨ ely, 2. if LdrQd ( L2 Ž qd . Ž respecti¨ ely, LdrQd ( S5 . and so, in particular, w Sdl , Qd Ql x F 2; Žiii. h Ž Ld , Z . s 1 with ZrCZ Ž O 2 Ž Ld .. being either a natural LdrQdmodule or an orthogonal S5-module. Proof. First we note that J Ž Ql . F Qd would imply that J Ž Ql . s J Ž Qd l Ql ., which is a contradiction to Lemma 3.4. Therefore, there exists A g AŽ Ql . such that A g Qd . Since Ž A l Qd . V 1Ž ZŽ Qd l Ql .. is elementary abelian, <Ž A l Qd . V 1Ž ZŽ Qd l Ql ..< F < A <. Hence AQdrQd acts as an offending subgroup on any noncentral chief factor within V 1Ž ZŽ Qd l Ql ... Now Lemma 3.17Žii. Žas Vd F V 1Ž ZŽ Qd l Ql ... implies that h Ž Ld , V 1Ž ZŽ Qd l Ql ... / 0, so Lemmas 2.3, 2.32, and 2.38 imply that LdrQd ( L2 Ž qd ., or S5 and also yield part Žiii. Žrecall that the orthogonal S5-module is projective.. If LdrQd ( L2 Ž qd ., then AQd s Sdl , by Lemma 2.32, and so J Ž Ql . Qd s Sdl . Now assume that LdrQd ( S5 . Because neither the natural

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nor the orthogonal S 5 -modules possess 2-central transvections < J Ž Ql . QdrQd < / 2, so using Lemma 2.3 again, we see that < J Ž Ql . QdrQd < s 2 2 . This completes the verification of the lemma. We now set L s ² QlLd :. LEMMA 3.19. Suppose that Ž d , l. g EŽ G ., Qd l Ql e Ld , and Zl g Ž Z Ll .. Then O 2 Ž L. s Qd l Ql . Proof. Together Lemmas 3.17Žii. and 3.18Žiii. give h Ž Ld , Vd . s 1, and thus w Vd , Qd x s w Zl , Qd x is centralized by L. In particular, w Zl , Qd x is centralized by O 2 Ž L.. By Lemma 3.18Ži., l g O Ž S .. Thus, since h Ž Ll , Zl . / 0, by hypothesis, combining Proposition 2.16 and Lemma 3.18Žii. we get that O 2 Ž L. F Ql . Therefore, Qd l Ql s O 2 Ž L., as required. THEOREM 3.20. Suppose that Ž d , l. g EŽ G . and Qd l Ql is normal in Ld . Then Zl F ZŽ Ll .. Proof. Assume by way of contradiction that Zl g ZŽ Ll .. Then, by Lemma 3.19, O 2 Ž L. s Qd l Ql . Put S s Sdl l L. Then O 2 Ž L. F Ql e S. If J Ž Ql . F O 2 Ž L., then J Ž Ql . s J Ž O 2 Ž L.. is normal in ² Ll , L: s ² Ll, Ld :, which is against Lemma 3.4. Therefore J Ž Ql . g O 2 Ž L.. Also observe that, if C is a normal subgroup of L which is characteristic in B Ž Ql ., then, as B Ž Ql . is characteristic in Ql , C is normal in Ll , in which case Lemma 3.4 implies that C s 1. We consider the two cases which arise in Lemma 3.18Žiii. separately. Set Z s V 1Ž ZŽ Qd l Ql .. and E s ² B Ž Ql . L :. Assume first that ZrCZ Ž O 2 Ž L.. is a natural LrO2 Ž L.-module. Then, by Lemma 3.15, B Ž Ql . g Syl 2 Ž E ., ErO2 Ž E . ( L2 Ž2 n ., and EO2 Ž L. G O 2 Ž L.. Moreover, we have O 2 Ž E . s O 2 Ž L. l B Ž Ql . and, since Z F B Ž Ql ., CE Ž O 2 Ž E .. F O 2 Ž E .; therefore, we may apply Corollary 3.14 with H s Ll F AutŽ B Ž Ql ... Then because NLlŽ O 2 Ž E .. G Sdl , we get ² Z Ll : F O 2 Ž E .. By Lemma 3.12, we have h Ž E, O 2 Ž E .. s h Ž E, Z . s 1. Hence ² Z Ll : is simultaneously normalized by Ll and E, which contradicts Lemma 3.4. This concludes the proof of Theorem 3.20 in this case. Next assume that ZrCZ Ž O 2 Ž L.. is an orthogonal S5-module. In this situation we apply Lemma 3.16 to find a subgroup R of L such that O 2 Ž L. F R, B Ž Ql . g Syl 2 Ž R ., RrO2 Ž R . ( L2 Ž2., and ² R, S : s L. Once again we notice that any nontrivial normal subgroup of R better not be characteristic in B Ž Ql .. Set R1 s RQd . Then O 2 Ž R . s B Ž Ql . l O 2 Ž R1 . is a normal subgroup of B Ž Ql . Qd , where B Ž Ql . Qd has index 2 in Sdl . Put Z0 s ² V 1Ž ZŽ B Ž Ql ... R :. Then, by Lemma 3.12, h Ž R, O 2 Ž R .. s h Ž R, Z0 . s h Ž R, Z . s 1, and so by the argument just presented in the prior case we have Ž3.20.1. ² Z0Ll : g O 2 Ž R ..

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It follows from Corollary 3.14 and Ž3.20.1. that N [ NLlŽ O 2 Ž R .. does not contain a Sylow 2-subgroup of Ll. Since B Ž Ql . Qd F N we deduce that Ž3.20.2.

B Ž Ql . Qd g Syl 2 Ž N . and that w Ll : N x s 2 n, where n is odd.

Fix y g Ll such that Z0y g O 2 Ž R .. Then, by Proposition 3.13, N normalizes O 2 Ž R y . and N1 [ ² y : N satisfies w N1 : N x s 2. Notice that by Ž3.20.2., N1 contains a Sylow 2-subgroup of Ll . Choose w g N1 such that y g A s ² w : B Ž Ql . Qd g Syl 2 Ž N1 .. If A s Sdl , then, as w s ny for some n g N, O 2 Ž R . y s O 2 Ž R . w G O 2 Ž L. w s O 2 Ž L. s Qd l Ql G Z. Thus, as Z0 s ZV 1Ž ZŽ B Ž Ql ..., w Z0 , Z0y x s 1, which is against our choice of y. Thus A / Sdl . Set D s ² Sdl , A: and put D s DrB Ž Ql . Qd . Then, by Ž3.20.2., D is a dihedral group of order 2 m, where m is odd. Therefore, we have O 2 Ž R . w s O 2 Ž R . y s O 2 Ž R . x , where x is an element of odd order in D. Now apply Corollary 3.14 with H s D. Then x 2 g ND Ž O2 Ž R .. G B Ž Ql . Qd , whence, as ² x : B Ž Ql . QdrB Ž Ql . Qd has odd order x g ND Ž O 2 Ž R .. and so O 2 Ž R . s O 2 Ž R . x s O 2 Ž R . y , which contradicts the original choice of y. Therefore, Zl F ZŽ Ll . and the proof of the theorem is complete. Our next result is also related to pushing-up. LEMMA 3.21. Suppose that Ž d , l. g EŽ G ., d g O Ž L .. Then Ql QdrQd contains a nontri¨ ial subgroup normal in NLdŽ Sdl .rQd . Proof. Notice that, by Lemma 3.18Ži., Qd l Ql is not normal in Ll , whence Ql g Qd . Set D [ NLdŽ Sdl .. Put H s ² D, Ll : and R s core H Ž Ql ., and suppose the result is false. First we consider the case when h Ž Ll, QlrR . s 0. Then R g Qd , for otherwise we obtain Qd l Ql e Ll , which is contrary to Lemma 3.18Ži.. But then RQdrQd is a nontrivial subgroup of Ql QdrQd which is normalized by D. Thus h Ž Ll , QlrR . / 0, which yields 1 / V 1Ž Z Ž Sdl rR .. F V 1Ž Z Ž Ql rR .. and then, as V 1Ž ZŽ SdlrR ..e DrR, h Ž Ll , V 1Ž ZŽ QlrR ... G 1. Since J Ž SdlrR . g QlrR we conclude that V 1Ž ZŽ QlrR .. admits an FF-action by LlrQl ; however, this contradicts Theorem 2.15Žiii.. Therefore the lemma holds. The last result of this section is designed so as to deal with some of the exceptional S5 problems that we will encounter later on. LEMMA 3.22. Assume that Ž d , l. g EŽ G ., LdrQd ( S5 , and b G 2. Let Z be a minimal normal subgroup of Ld contained in Zd and assume that h Ž Ld , Z . s 1 with CZ Ž Ld . s 0. Put V s ² Z Ll :. Then one of the following holds: Ži.

Qd Ql F Fl s CLlŽw Z, Ql x. - Ll and w V, Ql x s w Z, Ql x ( EŽ2 2 .; 2

Žii. h Ž Ll, V . G 2 and h Ž Ll , w V, Ql x. / 0.

SPORADIC SIMPLE GROUPS

479

Proof. Put Y s core LlŽ Z .. Note that the assumptions on Z imply that Z is either a natural or an orthogonal LdrQd-module Žand so < Z < s 2 4 .. Supposing that h Ž Ll, V . F 1 or h Ž Ll , w V, Ql x. s 0 we show that Ži. holds. By Lemma 3.7Ži., h Ž Ll, Vrw V, Ql x. G 1, and so in either case we obtain w V,Ql x s w Z, Ql x F Y. If Ql F Qd , then we have a contradiction to Lemma 3.18. So we have that Ql g Qd . Also observe, as SdlrQd ( D 8 and LlrQl g S , that Qd g Ql . Suppose < Y < G 2 3 , and let g g Ll . Now Qd l Ql centralizes Y s Y g Žs w Z g , Sd Ž l? g . x. and so Lemma 2.3Živ. forces Qd l Ql F Qdg . Hence Qd l Ql s Kl e Ll , which is again contrary to Lemma 3.18. Therefore < Y < F 2 2 . Using Lemma 2.3Živ. Žas Ql g Qd . gives
4. NONCOMMUTING CRITICAL PAIRS In this section we assume that the following hypothesis is fulfilled: HYPOTHESIS 4.1. 1.

Hypothesis 3.1 holds and for Ž a , a 9. g C , w Za , Za 9 x /

The goal of this section is the proof of the following theorem: THEOREM 4.2. Suppose that Hypothesis 4.1 is satisfied. Then LarQa ( L2 Ž2., Qb is extraspecial of q-type, and one of the following holds: 1q 12 Ži. LbrQb ( 3 AutŽ M22 ., Qb ( 2q , h Ž La , Qa . s 6, and La has 2q 1 5q2 5q1 3 a chief series described by 2 L2 Ž2.. 1q 22 Žii. LbrQb ( Co 2 , Qb ( 2q , h Ž La , Qa . s 11, and La has a chief 2q 1 10 q2 10 q1 8 series described by 2 L2 Ž2.. 1q 24 Žiii. LbrQb ( Co1 , Qb ( 2q , h Ž La , Qa . s 12, and La has a chief 2q 1 11 q2 11 q1 10 series described by 2 L2 Ž2..

We prove the theorem via a series of lemmas. We begin with LEMMA 4.3. The following hold: Ži. b is e¨ en; Žii. LarQa ( L2 Ž qa . or S5 ; Žiii. ZarCZ Ž O 2 Ž La .. is a natural LarQa-module or LarQa ( S5 and a ZarCZaŽ O 2 Ž La .. is an orthogonal module;

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Živ. Za Qa 9rQa 9 is an offending subgroup on Za 9 and Za 9 QarQa is an offending subgroup on Za ; moreo¨ er, < Za Qa 9rQa 9 < s < Za 9 QarQa < s qa or ZarCZaŽ O 2 Ž La .. is an orthogonal S5 -module and < Za Qa 9rQa 9 < s < Za 9 QarQa < s 2. Proof. Suppose that Ž a , a 9. g C . Then, as Ž a 9, a . g C , by Proposition 3.6Žii., we may assume, without loss of generality, that < Za 9 QarQa < G < Za Qa 9rQa 9 <. Thus, by Lemma 3.5Žii. and Proposition 3.6Ži., Za : CZa Ž Za 9 . s w Za : Za l Qa 9 x s < Za Qa 9rQa 9 < F < Za 9 QarQa < . Ž ) . Thus Za is an FF-module for LarQa . Applying Theorem 2.15Žiii. and Lemmas 2.3, 2.32, and 2.38, we deduce that LarQa ( L2 Ž qa . or S5 and that ZarCZaŽ O 2 Ž La .. is a natural LarQa-module or an orthogonal S5module. We now use Lemmas 2.3 and 2.32 and Ž)., to find that < Za Qa 9rQa 9 < s < Za 9 QarQa <. Thus the configuration is symmetric, and so La 9rQa 9 ( L2 Ž qa ., or S5 ; in particular, b is even and we have proven all parts of the lemma. LEMMA 4.4.

Zb F ZŽ Lb ..

Proof. Suppose that Zb g ZŽ Lb . and pick a y 1 g DŽ a . such that ² Say1 a , Za 9 : s La . Then, by Lemma 4.3Žii., Zay1 F Qa 9y1. Assume that Zay1 Za F Za Qa 9. Then

w Zay1 Za , Za 9 x s w Za , Za 9 x F Za . Therefore Zay1 Za e La . But then CQaŽ Zay1 Za . s Qay1 l Qa is normal in La , so Theorem 3.20 implies that Zb F ZŽ Lb ., which is against our assumption. Hence Zay 1 Za Qa 9rQa 9 ) Za Qa 9rQa 9. Since b G 2, Zay1 Za Qa 9rQa 9 is elementary abelian and hence we obtain, using Lemma 4.3Živ., LarQa ( S5 ( La 9rQa 9 , < Zay1 Za Qa 9rQa 9 < s 2 2 , < Za Qa 9rQa 9 < s 2, and Za 9rCZa 9Ž O 2 Ž La 9 .. is an orthogonal S5-module. From Lemma 2.3Žv.Ža., we get that Za 9 l Qa s CZa 9Ž Za . does not centralize Zay1 and so Za 9 l Qa g Qay1. Since w Zay1 : CZay1Ž Za 9 l Qa .x F 2 2 , this contradicts Theorem 2.15Žv.. Hence the result is proven. The next two results are easy consequences of Lemmas 4.3 and 4.4. LEMMA 4.5.

Suppose that LarQa ( L2 Ž qa .. Then

Ži. Sab s Qa Qb ; Žii. Za is a natural LarQa-module; Žiii. w Za , Za 9 x s w Za , Qb x s w Za 9 , Qa 9y1 x s Zb s Za 9y1.

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481

Proof. From Lemma 4.4, Zb s V 1Ž ZŽ Sa b .. and hence CZaŽ O 2 Ž La .. s 1. Thus, by Lemma 4.3, Za and Za 9 are natural modules for LarQa and La 9rQa 9 , respectively. Now Ži., Žii., and Žiii. follow from Lemma 4.3Živ. and Lemma 2.35. LEMMA 4.6. Suppose that LarQa ( S5 . Then Za is either a natural or an orthogonal S5-module. Furthermore, if < Za Qa 9rQa 9 < s 4, then w Za , Za 9 x G Za 9y1 Zb . Proof. Again we have CZaŽ O 2 Ž La .. s 1, and the lemma follows using Lemma 4.3Žiii. and Lemma 2.3Žv.Ža. and Žvi.Žb.. LEMMA 4.7. Suppose that b G 4 and Za is a natural LarQa-module. Then Ua F Sa 9y2 a 9y1 l Sa 9y2 a 9y3 F La 9y1 and Ua 9 F Saq2 b l Saq2 aq3 F Lb . Proof. From Lemmas 4.5, 4.3Živ., and 4.6, Zb Za 9y1 F w Za , Za 9 x F Za l Za 9 Žnote that when LarQa ( S5 we have < Za Qa 9rQa 9 < s 4 as Za is a natural module.. Hence, as b G 4, Ua 9 centralizes Zb and then Lemma 2.34 implies Ua 9 F CLaq2Ž Zb . F Saq2 b F Lb . Similarly, we deduce that Ua F Sa 9y2 a 9y1 l Sa 9y2 a 9y3 F La 9y1. LEMMA 4.8.

Suppose that b G 4. Then qa s 2 or LarQa ( S5 .

Proof. We suppose to the contrary that qa G 4 and LarQa \ S5 . Then LarQa ( L2 Ž qa . with qa G 4, by Lemma 4.3Žii.. Note that Zb s Za 9y1 Žby Lemma 4.5. and CZa 9Ž Za . s Za 9y1 imply that Vb l Za 9 s Za 9y1 s Zb . Also from Lemma 4.5 we deduce that w Vb , Qb x s Zb and that wUa , Ua x F Za . Employing Lemma 4.7 gives Vb , Ua 9 , Ua 9 F Vb l w Ua 9 , Ua 9 x F Vb l Za 9 s Zb . Thus Ua 9 acts quadratically on VbrZb . Since w Za 9 : CZa 9Ž Za .x s qa by Lemma 4.3Živ., Lemma 3.7Žii. forces < Ua 9 QbrQb < G 2 2 . Consulting Theorem 2.15Ži. gives < Ua 9 QbrQb < F 2 4 and hence, as qa G 4, Lemma 2.31 gives Ž4.8.1. either Ži. h Ž La , UarZa . s 1 or Žii. h Ž La , UarZa . s 2, qa s 4, FbrDb ( 3 M22 , and < Ua 9 QbrQb < s 2 4 . Suppose that Ž4.8.1.Ži. pertains. Then w Vb , Qaq2 x Zaq2 s w Vaq3 , Qaq2 x Zaq2 , which yields w Vb , Qaq2 x F Vaq3 F Qa 9q1 for all a 9 q 1 g DŽ a 9.. Hence Vb , Qaq2 , Ua 9 F Za 9 l Vb s Zb . Thus, as Qaq2 Qb s Sbaq2 , we have a contradiction to Proposition 2.16Žii..

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PARKER AND ROWLEY

Finally, considering Ž4.8.1.Žii. we have here that w Vb , Qaq2 , Qaq2 x Zaq2 s w Vaq3 , Qaq2 , Qaq2 x Zaq2 , from which we similarly infer that Vb , Qaq2 , Qaq2 , Ua 9 F Zb . This situation is impossible by Proposition 2.12Živ. and so we have shown that either qa s 2 or LarQa ( S5 . LEMMA 4.9. Suppose that b G 4. Then LarQa ( S5 and Za is an orthogonal S5-module. Proof. Suppose the lemma is false. Thus, by Lemmas 4.3Žii. and 4.8, our job here is to show that LarQa ( L2 Ž2. as well as the situation when LarQa ( S5 and Za is a natural S5-module lead to a contradiction. So choose a y 1 g DŽ a . such that ² Say1 a , Za 9 : s La . From Lemma 4.7 we have Ž4.9.1. Vay1 F Sa 9y2 a 9y3 F Sa 9y2 a 9y1 F La 9y1. Assume that Vay1 F Ga 9. Then, by Lemma 4.3Živ., Vay1 Qa 9 s Za Qa 9 , which implies that w Vay1 , Za 9 x s w Za , Za 9 x F Za F Vay1. But then Vay1 is normalized by ² Say1 a , Za 9 : s La , a contradiction. Thus Vay1 g Ga 9. Also, if w Vay1 , Va 9y1 l Qay1 x / 1, then, by Lemma 2.3Žii. and Žv.Ža. and Lemma 2.35, for some a y 2 g DŽ a y 1., w Vay1 , Va 9y1 l Qay1 x G w Zay2 , Va 9y1 l Qay1 x G Zay1 , which gives the contradiction Zay1e ² Say1 a , Za 9 : s La . Thus Ž4.9.2. Ži. Vay1 g Qa 9y1 and Žii. w Vay1 , Va 9y1 l Qay1 x s 1. Our next aim is to prove Ž4.9.3. If FbrDb \ 3 M22 , then h Ž Lb , Vb . s 1 and < Vay1 Qa 9y1r Qa 9y1 < s 2. Suppose that FbrDb \ 3 M22 . Using Ž4.9.1., Va 9y1 l Qa acts quadratically on Vay1 , so Theorem 2.15Ži. implies that <Ž Va 9y1 l Qa . Qay1rQay1 < F 2 2 . This, together with Ž4.9.2., yields Va 9y1 : C Va _ y 1Ž Vay1 . F

½

23 2

4

if LarQa ( L2 Ž 2 . , if LarQa ( S5 ,

Ž ).

Now Theorem 2.15Žv. implies that h Ž Lb , Vb . s 1. Furthermore, in the case when LarQa ( S5 , Lemma 3.22 implies that Qa 9y1 Qa 9y2 F Fa 9y1. Now we assume that < Vay1 Qa 9y1rQa 9y1 < G 2 2 and argue for a contradiction. First we claim that Va 9y1 s Va 9y1rw Va 9y1 , Qa 9y1 x is a quadratic La 9y1rDa 9y1-module. Indeed if <Ž Va 9y1 l Qa . Qay1rQay1 < s 2 2 , then we

SPORADIC SIMPLE GROUPS

483

are done by Ž4.9.1.. Hence we may assume that <Ž Va 9y1 l Qa . Qay1rQay1 < F 2. But then w Va 9y1 : C Va 9y 1Ž Va y1 .x F 2 3 by Ž4.9.2.Žii., and the claim follows from Theorem 2.15Žv.. If LarQa ( L2 Ž2., then Ž)., Theorem 2.15Žii., Proposition 2.11Žii., and < Vay1 Qa 9y1rQa 9y1 < G 2 2 yields a contradiction. While LarQa ( S5 implies, using Ž). and Theorem 2.15Žii., that La 9y1rDa 9y1 ( AutŽ M22 .. Recalling that Qa 9y2 Qa 9y1 F Fa 9y1 , we see that Vay1 g Qa 9y2 by Proposition 2.11Žii.. Hence, there exists a y 2 g DŽ a y 1. such that Ž a y 2, a 9 y 2. g C . Therefore, as Za 9y2 is a natural La 9y2rQa 9y2-module, Vay1 Qa 9y2rQa 9y2 s Sa 9y3 a 9y2rQa 9y2 l O 2 Ž La 9y2 . rQa 9y2 s Qa 9y3 Qa 9y2rQa 9y2 . Hence, employing Ž4.9.1., Vay1 Qa 9y2rQa 9y2 s Sa 9y2 a 9y1rQa 9y2 l Sa 9y3 a 9y2rQa 9y2 s Qa 9y1 Qa 9y2rQa 9y2 , and Vay1 F Qa 9y1 Qa 9y2 F Fa 9y1 , which is a contradiction, so verifying Ž4.9.3.. Ž4.9.4. If h Ž Lb , Vb . s 1, then < Vay1 Qa Xy1rQa 9y1 < / 2. Suppose that h Ž Lb , Vb . s 1 and < Vay1 Qa Xy1rQa 9y1 < s 2. Then either LarQa ( L2 Ž2. and w Va 9y1 , Qa 9y1 x s Za 9y1 ( EŽ2., or LarQa ( S5 and w Va 9y1 , Qa 9y1 x ( EŽ2 2 ., by Lemma 3.22. Since w Va 9y1 : Va 9y1 l Qa x F 2 2 , Ž4.9.2. and Theorem 2.15Žv. force Va 9y1 l Qa g Qay1. Let 1 / t g Ž Va 9y1 l Qa ._ Qay1. Then we have
FbrDb ( 3 M22 , and h Ž Lb , Vb . s 1.

Combining Ž4.9.2.Žii., Ž4.9.5., and Theorem 2.15Žv., we see that <Ž Va 9y1 l Qa . Qay1rQay1 < G 2 2 and so, by Ž4.9.1., the noncentral Lay1-chief factor in Vay1 is a quadratic module. Since < Vay1 Qa 9y1rQa 9y1 < G 2 2 by Ž4.9.4. and Ž4.9.5., Proposition 2.12Žii. demands that <Ž Va 9y1 l Qa . Qay1rQay1 < s 2 4 and < Va 9y1 QarQa < s 2 2 . Thus LarQa ( S5 . By Lemma 4.3Živ. we have

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w Sa 9y2 a 9y1 : Qa 9y1 Qa 9y2 x F 2 and so appealing to Lemma 3.22 we deduce that Qa 9y1 Qa 9y2 g Syl 2 Ž Fa 9y1 .. Employing Lemma 3.22 again gives E Ž 2 2 . ( w Za 9 , Qa 9y1 x s w Va 9y1 , Qa _ y1 x s w Za 9y2 , Qa 9y1 x , and hence w Za , Za 9 x s Za 9 l Za 9y2 s w Za 9 , Qa 9y1 x. Now, by Lemma 4.7, Ua F La 9y1 , and so

w Va 9y1 , Ua , Ua x F w Ua , Ua x l Va 9y1 F Za l Va 9y1 s w Za , Za 9 x s w Za 9 , Qa 9y1 x s w Va 9y1 , Qa 9y1 x . Thus Ua acts quadratically on Va 9y1rw Va 9y1 , Qa 9y1 x and hence < Ua Qa 9y1rQa 9y1 < F 2 4 by Proposition 2.12Ži.. Therefore, since wUa , Za x s 1, wUa : CUaŽ Za 9 .x F 2 6 and so, as < Za 9 QarQa < s 2 2 , h Ž La , Ua . F 3. Consequently,

w Ua 9y2 , Qa 9y2 ; n x s w Va 9y3 , Qa 9y2 ; n x s w Va 9y1 , Qa 9y2 ; n x , where n s h Ž La , Ua . y 1. Because, for Ž d , d 9. g C , we have w Zd , Zd 9 x F Zd l Zdq2 we obtain

w Va 9y3 , Qa 9y2 ; n x , Ua F Va 9y1 l Za s w Za , Za 9 x F w Va 9y1 , Qa 9y1 x . Note that Va 9y1rw Va 9y1 , Qa 9y1 x is an Fa 9y1rDa 9y1-module with Qa 9y2 acting as a Sylow 2-subgroup of Fa 9y1rDa 9y1. From Ua G Vay1 , < Ua Qa 9y1rQa 9y1 < G 2 2 and so if n s 1 we have a contradiction to Proposition 2.16Žii.. In the case n s 2 Žso h Ž La , Ua . s 3., we then have < Ua Qa 9y1rQa 9y1 < s 2 4 , which is at variance with Proposition 2.12Živ.. This completes the proof of the lemma. LEMMA 4.10.

b s 2.

Proof. Suppose that G is a counterexample to Lemma 4.10 and let Ž a , a 9. g C . Then, by Lemma 4.9, LarQa ( S5 and Za is an orthogonal S5-module. If t g O Ž L . and Žt , z . g EŽ G ., then the unique subgroup of Lt properly containing Stz is denoted by Mtz ; note that EŽ2 2 . ( O 2 Ž Mtz .rQt F O 2 Ž LtrQt . and MtzrO 2 Ž Mtz . ( L2 Ž2.. Our first result follows from Lemmas 2.3Žiii. and 4.3Živ.. Ž4.10.1.

Za 9 g O 2 Ž Ma b ..

Now we define H s ² Mab , Lb : and Q H s core H Ž Lb l Ma b .. Ž4.10.2. h Ž Lb , QbrQH . G 1.

485

SPORADIC SIMPLE GROUPS

Suppose that h Ž Lb , QbrQH . s 0. Then Qb l O 2 Ž Ma b . s Qb l O 2 Ž Maq2 b . G Qb l Qaq2 G Za 9 , which contradicts Ž4.10.1.. Ž4.10.3. One of the following holds: Ži. core L Ž O 2 Ž Ma b .. s Q H ; b Žii. core L Ž O 2 Ž Maq2 b .. l core L Ž O 2 Ž Maq2 b .. s Q H , where g is any g b vertex in DŽ a q 2._ b 4 with Maq2g s Maq2 b . If h Ž Mab , O 2 Ž Ma b .rQH . s 0, then clearly core LbŽ O 2 Ž Ma b .. is normalized simultaneously by Lb and Ma b , whence we obtain Ži.. While h Ž Mab , O 2 Ž Ma b .rQH . / 0 together with Ž4.10.2. yield the hypothesis of Theorem B, and then Žii. holds by Theorem B. Since, by Lemma 3.18, Qa l Qb is not normal in Lb , the following fact follows from Lemma 2.3Živ.. Ž4.10.4.

Yb [ core LbŽ Za . s w Za , Sa b ; n x ( 2 4y n, for n s 2 or 3.

For the next few steps we will be concerned with the following subgraph of G: (v ( vq 1 ( a

(

(

(

b aq2

aq3

???

(

a_

( gq 1 (g where Maq2 b s Maq2 v q1 s Maq2gq1. Ž4.10.5. Suppose that < Yb < s 2 2 . Then, up to a relabelling of vertices, we have Ži. Za 9 F core Lgq1Ž O 2 Ž Maq2gq1 ..; and Žii. Ž a , a 9., Ž v , a 9. g C , < Zv Qa 9rQa 9 < s < Za Qa 9rQa 9 < s 2, and Za Qa 9rQa 9 / Zv Qa 9rQa 9 both act as transvections on Za 9. Suppose that there exist t 1 , t 2 g DŽ b . j DŽ v q 1. j DŽg q 1. with Ž d t 1 , t 2 . s 4 and 1 / w Zt 1, Za 9 x F w Zt 2 , Za 9 x. Without loss of generality we assume that t 1 g DŽ b . and t 2 g DŽg q 1.. Then, by Lemma 2.3Žv.Že. Žapplied at a q 2., Zt 1 , Za 9 F Zaq2 , Saq2 b ; 2 l Zaq2 , Saq2gq1 ; 2 s Zb s Zgq1 , which is against Lemma 2.3Živ. Žapplied at t 1 ..

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Now suppose that we find Ž a , a 9., Ž v , a 9., Žg , a 9. g C Žsee the diagram following Ž4.10.4... Then, since b G 4 and Sa 9y1 a 9rQa 9 contains exactly two transvections, we have to conclude that there exists a pair t 1 , t 2 4 :  a , v , g 4 with 1 / w Zt , Za 9 x F w Zt , Za 9 x, which we have already shown is 1 2 impossible. It follows that we may choose notation so that Za 9 F core Lgq1Ž O 2 Ž Maq2gq1 .. and Ži. is true. Fix this g . Now if, for all v g DŽ v q 1. Za 9 F Qv , then, by Ž4.10.3.Žii., Za 9 F core Lgq1Ž O 2 Ž Maq2 b . . l core Lv q 1Ž O 2 Ž Maq2 b . . s Q H F O 2 Ž Ma b . , which is against Ž4.10.1.. Thus Ž4.10.5.Žii. holds. Ž4.10.6.

< Yb < s 2.

Suppose that < Yb < s 2 2 . Then Ž4.10.5.Ži. and Theorem B imply that 1 / Za 9 Q H rQH F core Lgq1Ž O 2 Ž Maq2gq1 . . rQH ( E Ž 2 . . Hence, by Ž4.10.5.Žii., Za 9 l Qa s Za 9 l Q H s Za 9 l Qv . But then the transvections Za Qa 9rQa 9 and Zv Qa 9rQa 9 centralize the same hyperplane in Za 9 , which contradicts Ž4.10.5.Žii. and Lemma 2.3Žv.Ža.. Hence < Yb < s 2. Now define Hb [ ²w Za , Sa b ; 2x Lb :. Then one of the consequence of Ž4.10.6. is that h Ž Lb , HbrZb . G 1. Because of Lemma 2.3Živ., Qb QarQa is either the quadratic fours group on Za or Sa brQa and so w Za , Qb x G w Za , Sa b ; 2x, whence w Vb , Qb x G Hb . Therefore, h Ž Lb , Vb . G 2 and Za g Hb . We next put Xa s ² HbLa :. Ž4.10.7. h Ž La , Xa . G 3. Suppose that h Ž La , Xa . F 2. Then, using Lemma 3.7Žii., w Xa , Qa x Za s w Hb , Qa x Za e La and, as Za g Hb , w Hb , Qa , Qa x s 1. Hence Qa acts quadratically on HbrZb . Since w Sa b : Qa Qb x F 2, we have a contradiction to Theorem 2.15Ži.. Ž4.10.8. The contradiction. Pick a y 2 g DŽ a y 1._ a 4 and set Jay2 ay1 s w Zay2 , Say2 ay1; 2x. So Jay2 ay1 is a typical generator of Xa . Notice that, by Lemmas 4.3Živ. and 2.3Žv.Ža., w Jay2 ay1 , Za 9y2 x s 1, and thus Xa F Qa 9y2 and Xa is elementary abelian.

SPORADIC SIMPLE GROUPS

487

Now Xa g La 9 , for otherwise h Ž La , Xa . F 2, which is against Ž4.10.7.. Since Va 9y1 l Qay1 F Say2 ay1 ,

w Jay2 ay1 , Va 9y1 l Qay1 x F w Zay2 , Say2 ay1 ; 3 x , and so
La

¦ V , Q ;. b

a

LEMMA 4.11. h Ž La , WarZa . G 1 and Qb l Qa is not normal in La . Proof. Suppose first that Wa s w Vb , Qa x Za . Then Wa F Qa l Qb and so Qa l Qb is normal in La . So to prove Lemma 4.11 it suffices to show that Xa [ Qa l Qb being normal in La leads to a contradiction. Clearly, as b s 2, we have Xa F La 9 and Xa Qa 9rQa 9 G Za Qa 9rQa 9. If LarQa ( L2 Ž qa ., then Xa Qa 9 s Za Qa 9 and thus w Xa , Za 9 x s w Za , Za 9 x F Za from which we infer that h Ž La , XarZa . s 0. Hence h Ž La , Qa . s 1. Now suppose that LarQa ( S5 . If Za is the natural S5-module, then EŽ2 2 . ( w Za , Za _ x F CZ _ Ž Xa .. For the case Za is the orthogonal S5-module, a Lemmas 4.3Živ. and 2.3Živ. and Zb F Za l Za 9 imply that EŽ2 2 . ( ² Zb , w Za , Za 9 x: F CZ Ž Xa . l Za . So in either case we deduce that w Xa , Za 9 x a9 F Za and, therefore, h Ž La , Qa . s 1 when LarQa ( S5 . Since h Ž La , Qar F Ž Qa .. G 1 and CQaŽ La . s 1, Qa must be elementary abelian and so Qa acts quadratically on Vb . If LarQa ( L2 Ž qa ., then Sa b s Qa Qb , while if LarQa ( S5 , then w Sa b : Qa Qb x F 4 and in both cases Theorem 2.15Ži. provides the desired contradiction. Set Ra s core LaŽ Vb .. Because of Lemma 4.11, Vb l Qa ) Ra . Hence we may choose a nontrivial element x in V 1Ž ZŽ Sa brRa .. l Ž Vb l Qa .rRa and we then define Na s ² x La : Ra . Observe that we have h Ž La , NarRa . / 0.

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PARKER AND ROWLEY

L EMMA 4.12. NarR a V 1Ž ZŽ Qa QbrQb ...

is elementary abelian and Na Qb rQb F

Proof. Since xRarRa g V 1Ž ZŽ QarRa .., it follows that Na rRa is elementary abelian and that w Na , Qa x F Ra F Vb F Qb . Hence Na Qb rQb F V 1Ž ZŽ Qa QbrQb .. and so Lemma 4.12 holds. LEMMA 4.13.

Suppose that FbrTb \ J1 or Fi 23 . Then LarQa ( L2 Ž2..

Proof. Put Nˆa s NarRa . Suppose first that Sa b s Qa Qb holds. We show that it then follows that LarQa ( L2 Ž2.. Since Za 9 Qa e Qa Qb s Sa b , Lemmas 2.3Živ. and 4.3Živ. imply that when LarQa ( S5 , Za 9 QarQa ( EŽ2 2 .. Thus Žnote that w Na , Za x s 1 for the LarQa ( S5 case. Na l Qb F Za Qa 9 and so Na l Qb , Za 9 F w Za , Za 9 x F Za . Combining Qa Qb s Sa b , Lemmas 4.12 and 2.22 gives w Nˆa : CNˆaŽ Za 9 .x F 2. Now h Ž La , Nˆa . / 0 and Za 9 QarQa ( EŽ2 2 . Žwhen LarQa ( S5 . rules out LarQa ( S5 . Then an appeal to Lemmas 4.3Žii. and 2.31Ži. yields LarQa ( L2 Ž2.. Now to complete the proof of the lemma we show that Qa Qb / Sa b is untenable. So suppose that Qa Qb / Sa b . Then, using Lemma 2.3Živ., Ž4.13.1. LarQa ( S5 and we may suppose Ž a , a 9. g C chosen so that Vb Qa s ² Za 9 QaSab : s Qa Qb has index 2 in Sa b and Vb acts quadratically on Za . Ž4.13.2. h Ž Lb , w Vb , Qb x. / 0. If Ž4.13.2. is false, then, by Lemma 3.22 and Ž4.13.1., Qa Qb g Syl 2 Ž Fb . and Vb Qa s Za 9 Qa . Then we may argue as above to get w Nˆa : CNˆaŽ Za 9 .x F 2. Hence, by Ž4.13.1., h Ž La , Nˆa . s 0, a contradiction. So Ž4.13.2. holds. Ž4.13.3. w Vb , Qb x F K a . First observe from Ž4.13.1. that w Vb , Qb x F Qa and w Za , Qb , Qb x s 1. Hence w Vb , Qb x F V 1Ž ZŽ Qb ... Now suppose that w Vb , Qb x g K a . Then we may select an a y 1 g DŽ a . such that w Vb , Qb x g Qay1. A standard argument now shows that w Vb , Qb x is an FF-module for LbrCLbŽw Vb , Qb x., contrary to Theorem 2.15Žiii.. Therefore, w Vb , Qb x F K a and we have Ž4.13.3.. Put Ma s ²w Vb , Qb x La :. Clearly, by Ž4.13.3., Ma F K a Qa 9 F Qb Qa 9 F Sba 9 and so w Ma : CMaŽ Za 9 .x F 2 2 , from which we see that h Ž La , Ma . F 2. In particular, we have Za w Vb , Qb , Qa xe La . Therefore, w Vb , Qb , Qa , Qa x e La . Since Za g w Vb , Qb x, we infer that w Vb , Qb , Qa , Qa x s 1. But this, as w Sab : Qa Qb x s 2, is against Ž4.13.2. and Proposition 2.16Ži.. Thus Qa Qb / Sab is untenable, and Lemma 4.13 is proven.

489

SPORADIC SIMPLE GROUPS

LEMMA 4.14.

If LbrQb ( J1 or Fi 23 , then LarQa ( L2 Ž2..

Proof. Aiming for a contradiction we suppose that LbrQb ( J1 or Fi 23 and qa G 2 2 . Again put Nˆa s NarRa . First we examine the case when LarQa ( L2 Ž qa .. Fix l g DŽ a ._ b 4 so that ² Qb , Ql : QarQa s LarQa . Then combining Lemma 4.12 with Lemma 2.22 and w Na l Qb , Za 9 x F Za we obtain qa F Nˆa : CNˆa Ž Za 9 . F

½

23 2

2

if LbrQb ( J1 , if LbrQb ( Fi 23 .

Ž ).

Suppose that LbrQb ( Fi 23 . Then, as qa G 2 2 Žand using Lemma 2.22., Ž). implies that Na QbrQb s V 1Ž ZŽ Sa brQb .. ( EŽ2 2 . and w Na , Vl xQb s Na Qb . Select x g w Na , Vl x so that xQbrQb is of class 2 A in LbrQb . Then, on the one hand, we have w Vb l Qa l Ql , Vl x F Zl has order at most 2 2 and, by Lemma 2.2, w Vb : Vb l Qa l Ql x F 2 2q 11 giving w Vb : C VbŽ x .x F 2 15 , while on the other hand, we have from Lemma 2.17Žii. that w Vb : C VbŽ x .x G 2 54 , these two statements being incompatible we deduce that LbrQb ( J1. Since w Vb , K a x F w Vb , Qb x s Zb F K a , h Ž La , K a . s h Ž La , Za . s 1, which then forces K a to be elementary abelian. Note that Vb l Qa g K a else we obtain w Vb , Qa x F K a , against h Ž La , QarK a . / 0 by Lemma 4.11. So we select x g Ž Vb l Qa ._ K a . Then, as
490

PARKER AND ROWLEY

Now we consider the remaining case LarQa ( S5 . Since Lb s Fb , applying Lemma 3.22 gives h Ž Lb , Vb . G 2. Choose n maximal so that h Ž Lb , w Vb , Qb ; n x. / 0. Set Rb s w Vb , Qb ; n x. From the maximal choice of n, w Rb , Qb x F Zb and so
Vay1 : Vay1 l Qa l Qb F

½

2 3q 3

if LbrQb ( J1 ,

2 3q18

if LbrQb ( Fi 23 .

Ž ).

Since
V : CV Ž y . F

½

24

if LbrQb ( J1 ,

2 12

if LbrQb ( Fi 23 .

Both these possibilities are ruled out by Lemma 2.17. We therefore deduce that Rb l Qa F K a . If Rb g Qa , then it follows that h Ž La , QarK a . s 0, against Lemma 4.11. So Rb F Qa and thus Rb F K a . Also, observe that Rb F ZŽ Vb .. Now select m g DŽ a . so that La s ² Sm a , Za 9 :. Since Rm F K a , Rm F Qb F La 9. If Za 9 is the natural S5-module, then Rm F Za Qa 9. While if Za is the orthogonal S5-module, then either Rm Qa 9rQa 9 s Za Qa 9rQa 9 ( EŽ2. or CZa _ Ž Rm . s ² Zb , w Za , Za 9 x:. So in any case w Rm , Za 9 x F Za . Therefore, Rm Za e La . Commutating with Qa reveals, as Za g Rm , that Rm F V 1Ž ZŽ Qa ... But then h Ž Lm , Rm . s 0, a contradiction. This contradiction concludes the proof that if LbrQb ( J1 or Fi 23 , then qa s 2. Using the language of wPR2x, Lemmas 4.10, 4.11, 4.13, and 4.14 show that any group G, satisfying Hypothesis 4.1, is an amalgam of symplectic type o¨ er GF Ž2.. Consequently the assertions in Theorem 4.2 can be read from wPR2, Main Theoremx, and Theorem C is proven.

ACKNOWLEDGMENTS Both authors express their gratitude to the Sonderforschungsbereich 343 at Bielefeld.

SPORADIC SIMPLE GROUPS

491

REFERENCES wAx

J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson, ‘‘Atlas of Finite Groups,’’ Clarendon, Oxford, 1985. wAx M. Aschbacher, GF Ž2.-representations of finite groups, Amer. J. Math. 104 Ž1982., 683]771. wB1x B. Baumann, Endliche nichtauflosbare Gruppen mit einer nilpotenten maximalen ¨ Untergruppe, J. Algebra 38 Ž1976., 119]135. ¨ wB2x B. Baumann, Uber endliche Gruppen mit einer zu L2 Ž2 n . isomorphen Faktorgruppe, Proc. Amer. Math. Soc. 74 Ž1979., 215]222. wBex D. Benson, The Loewy structure of the projective indecomposable modules for A 8 in characteristic 2, Comm. Algebra 11Ž13. Ž1983., 1395]1432. wCDx A. Chermak and A. Delgado, J-modules for local BN-pairs, Proc. London Math. Soc. Ž 3 . 63 Ž1991., 69]112. wDSx A. Delgado and B. Stellmacher, Weak Ž B, N .-pairs of rank 2, in ‘‘Groups and Graphs: New Results and Methods,’’ DMV Seminar, Vol. 6, Birkhauser, Basel, 1985. ¨ wGorx D. Gorenstein, ‘‘Finite Group,’’ Harper & Row, New York, 1968. wHux B. Huppert, ‘‘Endliche Gruppen I,’’ Springer-Verlag, BerlinrNew York, 1968. wHBx B. Huppert and N. Blackburn, ‘‘Group Theory II,’’ Springer-Verlag, BerlinrNew York, 1982. wLPRx W. Lempken, C. Parker and P. J. Rowley, Ž S5 , S6 .-amalgams I, No¨ a J. Algebra Geom. 3 Ž1994., 209]270. wMSx U. Meierfrankenfeld and G. Stroth, Quadratic GF Ž2.-modules for sporadic simple groups and alternating groups, Comm. Algebra, 18Ž7. Ž1990., 2099]2139. wNx R. Niles, Pushing up in finite groups, J. Algebra 57 Ž1979., 26]63. wPR1x C. Parker and P. J. Rowley, Amalgams of minimal local subgroups and sporadic simple groups, in ‘‘Groups ’93 GalwayrSt. Andrews,’’ Vol. 2, pp. 495]506, Cambridge Univ. Press, Cambridge, UK, 1995. wPR2x C. Parker and P. J. Rowley, Amalgams of symplectic type, unpublished manuscript, 1996. wPR3x C. Parker and P. J. Rowley, Amalgams which involve sporadic simple groups II, J. Algebra, to appear. wPR4x C. Parker and P. J. Rowley, Odd automorphisms of Sylow 2-subgroups of sporadic simple groups, Manuscripta Math. 89 Ž1996., 49]60. wRx P. J. Rowley, On the minimal parabolic system related to M24 , J. London Math. Soc. Ž 2 . 40 Ž1989., 40]57. wSex J. P. Serre, ‘‘Trees,’’ Springer-Verlag, BerlinrNew York, 1980. wSt1x B. Stellmacher, Pushing-up, Arch. Math. 46 Ž1986., 383]487. wSt2x B. Stellmacher, On the 2-local structure of finite groups, in ‘‘Groups, Combinatorics and Geometry’’ ŽM. W. Liebeck and J. Saxl, Eds.., London Mathematical Society Lecture Note Series, Vol. 165, Cambridge Univ. Press, Cambridge, UK, 1992. wSuzx M. Suzuki, ‘‘Group Theory I,’’ Die Grundlehren der Mathematischen Wissenshaften, Band 247, Springer-Verlag, BerlinrNew York, 1982.