Journal of Algebra 233, 309᎐341Ž. 2000 doi:10.1006rjabr.2000.8429, available online at http:rrwww.idealibrary.com on
The Radical 2-Subgroups of the Sporadic Simple
Groups J4 , Co2, and Th
Satoshi Yoshiara CORE Metadata, citation and similar papers at core.ac.uk Di¨ision of Mathematical Sciences, Osaka Kyoiku Uni¨ersity, Kashiwara, Provided by Elsevier - Publisher Connector Osaka 582-8582, Japan E-mail: [email protected]
Communicated by Gernot Stroth
Received December 13, 1999
1. INTRODUCTION
For a finite group G and a prime divisor p of its order, a nontrivial s p-subgroup R of G is called radical,ifONpGŽŽ.. R R. With the poset BpŽ.G of radical p-subgroups of G with respect to inclusion we naturally ⌬ associate the simplicial complex of its chains, denoted ŽŽ..Bp G , which is known to be G-homotopy equivalent to the complex associated with the poset of nontrivial p-subgroups as well as that of nontrivial elementary abelian p-subgroups of G Žsee, e.g.,wx 5, 6.6. . The latter are important tools to investigate the structure of G concerning the prime p Žsee, e.g.,w 5, x ⌬ Chap. 6.Ž . Thus the smaller complex BpŽG..has enough information to understand behaviors of G on p. The following fact, a corollary of the Borel᎐Tits theoremwx 5, 6.8.4 , ⌬ shows the further importance of ŽŽ..Bp G : for a finite group of Lie type ⌬ defined over a field in characteristic p, the complex ŽŽ..Bp G is the barycentric subdivision of the building for G. Thus for an arbitrary finite ⌬ group G we may think of the complex ŽŽ..Bp G as a generalization of the concept of buildings: in fact, as is observed inwx 14 , for each sporadic simple ⌬ group G of characteristic-p type, ŽŽ..Bp G is homotopy equivalent to some smaller simplicial complex, previously constructed by an ad hoc method and referred to as a ‘‘p-local geometry’’ of G. This suggests the importance of examining minimal complexes which are homotopy equiva- ⌬ lent to ŽBp ŽG ..for every Ž specifically sporadic simple.Ž group G see also the introduction ofwx 17. .
309 0021-8693r00 $35.00 Copyright ᮊ 2000 by Academic Press All rights of reproduction in any form reserved. 310 SATOSHI YOSHIARA
To investigate such complexes, we first need to determine BpŽ.G for every sporadic simple group G and a prime divisor p of < This paper is the first one in a series of papers determining BpŽ.G for every sporadic simple group G and a prime divisor p of < B2Ž..HN ; and for all sporadic simple groups G and odd primes p, respectively. Thus at the time of writing, the only remaining cases are the Monster M and the Baby Monster BM for p s 2. The principle of the classification is the inductive method proposed in wx14 and discussed in wx 16Ž see alsowx 12. , which is efficient if we know the list of maximal p-local subgroups of G as well as inclusions of some p-local subgroups up to conjugacy. Our method is purely group theoretic and computer-free, while the method presented inwx 3 requires a permuta- RADICAL 2-SUBGROUPS 311 tion representation of the quotient group of each maximal p-local sub- group by its Op-part to examine all p-subgroups of that quotient group. In Section 2, we prepare some basic technical tools which are frequently used inwx 10, 18, 19 as well. Some of them already appeared in wx 12 in a slightly different form. The most convenient one is Lemma 1Ž. 4 saying that r the inverse images of radical subgroups of M OMpŽ.are radical if a maximal p-local subgroup M has extraspecial OMpŽ.. The geometric descriptions of the radical 2-subgroups of the Mathieu groups M24 and wx M22 Žwith the correction of the results in 16. are included in this section as well as that of their actions on the Golay code and cocode. They are of fundamental importance in Sections 3, 4 and the paperwx 10 . In Sections ᎐ 3 5, the groups J4 , Co2, and Th are treated independently. Each of these sections gives brief descriptions of maximal 2-local subgroups of the simple group in question. 2. PRELIMINARIES 2.1. Some Lemmas In this subsection, the author collects some lemmas which will be frequently used later. LEMMA 1. Let G be a finite group and p a prime di¨isor of<< G . Ž.1wx 14, Lemma 1.9 If a radical p-group R of G is contained in a F r subgroup M of G, then OppŽ. M R and R OŽ. M is a radical p-subgroup of r ¨ M OpŽ. M or the tri ial group. Ž.2 ŽA ¨ariant of wx12, Lemma 2.1 . Assume that a p-subgroup R contains OpŽ. M as a normal subgroup for some subgroup M of G and that r r F R OppŽ. M is a radical p-subgroup of M OMŽ.. If NG Ž. R M, then R is a radical p-subgroup of G. Ž.3 Let Z be a nontri¨ial p-subgroup of G and M be a maximal p-local [ r subgroup of G containing NGŽ. Z . Set R0 OpipŽ. M and let R OMŽ. s ¨ r Ž.i 1,...,m be the complete representati es of M OMpŽ.-classes of radical r [ ¬ s s p-subgroups of M OMpjjŽ.. Then the set S Ä R j 0,...,m, ZRŽ. Z4 is a system of complete representati¨es of radical p-subgroups R whose centers ZŽ. R are conjugate to Z. s Ž4 .Assume that OpG Ž C Ž z .. is non-abelian and ² z :ZO ŽpG Ž C Ž z ... ¨ ¨ for a p-element z. Then the in erse images in CGŽ. z of the representati es of r classes of radical p-subgroups of CGpGŽ.ŽŽ.. z O C z together with OpGŽŽ.. C z form a set of complete representati¨es of radical p-subgroups R with ZŽ. R generated by a conjugate of z. 312 SATOSHI YOSHIARA r Proof. Ž.2 The group R OMp Ž .is trivial or a radical p-subgroup of r s F M OMppŽ.exactly when ONŽMG Ž.. R R.As NRŽ. M, we have s s g ONpGŽ Ž R ..ON pM Ž Ž R .. R, which implies that R BpŽ.G . F Ž.3 Each member R of S has the center Z, so that NRGŽ. s F NZRGGŽŽ..NZ Ž. M. Then it follows from the claimŽ. 2 that R is a radical p-subgroup of G with center Z. Conversely, let R be a radical g s g g F p-subgroup of G with ZRŽ. Z for some g G. Then NRGŽ . g s F 1 g NZRGGŽŽ ..NZ Ž . M. It follows fromŽ.Ž. 1 that OMpy R and either g s gr r R OMppŽ.or R OMŽ.is a radical p-subgroup of M OMpŽ.. Then R g is conjugate to exactly one member of S under M. Distinct members Rijand R of S are not conjugate under G, because s h g s s if RijR for some h G then h normalizes ZRŽ.iZR Ž. jZ, and g F s so h NZGŽ. M and i j. Hence the claim is verified. Ž.4 This is just a corollary of the claimŽ. 3 , because ZR Ž .s ²z :for every subgroup R of CzGpŽ.containing OCŽG Ž.. z. LEMMA 2wx 12, Lemma 3.2 . E¨ery radical p-subgroup of the direct product = = A B of two finite groups A and B is of the form RABR , where R A and ¨ RB are radical p-subgroups of A and B, respecti ely, allowing at most one of them tri¨ial. Con¨ersely, e¨ery subgroup of this form is a radical p-subgroup of A = B. LEMMA 3. Let A be a finite group ha¨ing a normal subgroup G of index p. ¨ Assume that U1,...,Us are the representati es of G-conjugacy classes of g radical p-subgroups U of G with NAsŽ. U G, and that U q1,...,Ut are the representati¨es of A-conjugacy classes of radical p-subgroups of G with F s NUAŽ. G. We also set U0 1. s r r For each i 0,...,s, consider subgroups P UiA of order p of NŽ. UiUi r r r outside NGiŽ. U U i such that the centralizer of P UinNUi GiŽ.U i has the ¨ Ž j.r ¨ r tri ial Opi-part, and let P Ui be the representati es of NGŽ. UiUi-classes of those subgroupsŽŽ. j s 1,..., i .. If such a subgroup P does not exist, set Ž.i s 0. Then Ž j. s / Ž.1 The set of p-subgroups ² Uii, P:Ž for i 0,...,s with i.0 and j s 1,..., Ž.igi¨es a set of complete representati¨es of A-conjugacy classes of radical p-subgroups of A not contained in G. s q s Ž.2 The set of UjiŽ j s 1,...,t.Ž. and U i 0,...,s with r s ¨ ¨ ONUpAiŽŽ..U i 1 gi es a set of complete representati es of A-conjugacy classes of radical p-subgroups of A contained in G. RADICAL 2-SUBGROUPS 313 Proof. Ž.2 Let R be a radical p-subgroup of A which is contained in G. s s q If NRAGŽ.NR Ž., then R is conjugate to exactly one of Ujj Ž s 1,...,t.Žunder A the A-conjugacy class of R splits into p classes under / G.Ž..If NRAGis not contained in NR Ž.Ž., then R 1 is conjugate to s exactly one of Uii Ž.1,...,s .As R is a radical p-subgroup of A,we s s r s have ONUpAiŽŽ.. U i. Conversely, UiiŽ1,...,s .with ONUpAi ŽŽ..U i 1 s q and Ujj Ž.s 1,...,t are radical p-subgroups of A contained in G. Ž j. [ Ž j. [ Ž j. ¬ s s Ž.1 We set Riii²U , P : and R Ä Rii 0,...,s, j 1,..., Ž.i 4. Let R be a radical p-subgroup of A not contained in G. Then it follows fromwx 12, Lemma 3.1 that R l G is either trivial or a radical subgroup of G.As R normalizes R l G but is not contained in G, R l G is conjugate s l under G to exactly one of Uii Ž.0,...,s . Thus we may assume R G s s _ Uiiand R UP for some subgroup P of A G. Then the factor group r s r r r R UiiiPU U is a subgroup of order p of NUAiiŽ.U outside NU GiiŽ.U . F l s s F F We have NRAAŽ.NR ŽG .NU Ai Ž.and A GP with P R s s g ¬ NRAAGŽ.. Thus NR Ž.NRP Ž.. Observe that NRGAŽ. Ä g NRŽ. wxF g wxF wxF l s P, g Ui4, because for g G we have R, g R iff R, g G R r r r UiGi. Thus NRŽ.U coincides with the centralizer of R UiGiiin NUŽ.U , r r r and NRAiŽ.U is the direct product of NRGiŽ.U with R U i. Since R is a s r radical p-subgroup of A, we have ONRpAŽŽ.. R and thus ONRpAŽŽ..U i s r r s R Uip. The last condition is equivalent to ONŽŽ..G R Ui1. Hence r taking conjugation under NUAiŽ., R U iis conjugate to exactly one of Ž j.r s Ž j. PiiPjŽŽ.1,..., i .. Thus R is conjugate to a member Ri of R under A. Ž j. g Conversely, any Ri R is a radical p-subgroup of A not contained in Ž j. l s G, because its normalizer is contained in NRAiŽ G.Ž.NUAiand Ž j. r s = Ž j.Žr j.r NRAiŽ .Ž.U iNRU G iR iU ihas the Op-part R iU iby the defining Ž j. Ž j. a s Žl. g property of Pii. Furthermore, if Ž R . Rkfor some a A, then we g Ž j. Ž j. may take a G,as Pi is a subgroup of A outside G normalizing Ri . s Ž j. l s Žl. l Furthermore, a sends UiiR G to UkkR G. Then we have s g s Ž j. i k and a NUGiŽ., and hence k l by the definition of Pi. Thus R forms a complete set of representatives of A-conjugacy classes of radical p-subgroups of A not contained in G. LEMMA 4. Let p be a prime, and let G be a finite group with nontri¨ial center of order prime to p. Assume that A is a group containing G as a normal subgroup of index p but ZŽ. A s 1. Then Ž.1 The set of complete representati¨es of conjugacy classes of radical p-subgroups of G corresponds bijecti¨ely to that of GrZGŽ.. Ž.2 E¨ery radical p-subgroup of G is a radical p-subgroup of A. 314 SATOSHI YOSHIARA Proof. It is immediate to see the claimŽ. 1 . For a radical p-subgroup R of G, the normalizer NRGpŽ.always contains ZG Ž.. Suppose ONR ŽA Ž.. w x F l s contains an element x outside G. Then x, ZGŽ. ONRpAŽŽ..Ž.ZG X 1, as ZGŽ.is a p -group. But this implies that 1 / ZGŽ.lies in ZA Ž., as A s Gx²:, which contradicts the assumption ZA Ž .s 1. Hence F s s ONRpAŽ Ž ..G and ONRpA Ž Ž ..ON pG Ž Ž R .. R is also a radical p-sub- group of A. To determine the radical 2-subgroups of the alternating groups, the following observation is useful. We first prepare some notation. We denote by A⍀⍀Ž.resp. S the alternating Ž resp. symmetric . group naturally acting on a set ⍀ of letters. For a subset ⌬ of ⍀, we identify A⌬ Ž.resp. S⌬⍀with the subgroup of A Ž.resp. S⍀fixing all letters of ⍀_⌬. For a subgroup U of A⍀⍀, let ⌽ Ž.U be the set of elements of ⍀ fixed by every element of U, and set ⌬ ⍀⍀Ž.U [ ⍀_⌽ Ž.U . For a nonnegative wx integer i, we denote by B2Ž.Ai⍀⍀the set of radical 2-subgroups U of A <⌽ < s wx with ⍀Ž.U i. The subset B2Ž.Ai⍀ of B2Ž.A⍀ is a union of conju- gacy classes under S⍀ for each i. LEMMA 5. With the abo¨e notation, the following hold. Ž.g Ž .wx F F <<⍀ g Ž.wx 1 If U B2 A⍀ i for some 0 i , then U B2 A⌬ ⍀ ŽU . 0. wxs л Ž.2 B2 ŽA⍀ .4 . Ž.3 Let i be an integer with 0 F i F <<⍀ distinct from 2 and 4, and let << << ⌬ be a subset of ⍀ with ⍀ s ⌬ q i. Then the identification of A⌬ with a subgroup of A⍀ induces a bijecti¨e correspondence of the conjugacy classes of wx ⌬ ⌬ G B2Ž.A⌬⍀0 under the stabilizer of in AŽ which induces S⌬ on if i 2 s wx but A⌬ if i 0 or 1.Žwith the conjugacy classes of B2 A⍀⍀. i under A . Proof. Let U be a nontrivial 2-subgroup of A⍀ . For brevity we write ⌽ Ž.s ⌽ ⌬ Ž.s ⌬ Ž. ⌽ ⌬ ⍀⍀U and U .As NUA ⍀ preserves both and , we have s = l s = ²: NUA ⍀⌬Ž.Ž.Ž.S⌽ NUS Ž. A⍀⌽A NUA⌬Ž. g, where either g is trivial of g induces odd permutations both on ⌽ and ⌬. Ž. Ž. Ž Ž..F In particular, A⌽ and NUAA⌬⍀⌬are normal in NU. Thus ONU2 A ŽŽ.. Ž. Ž. ON2 A ⍀ U, from which Claim 1 follows. Claim 2 also immediately Ž.= Ž. follows, as OA2 ⌽ U is a normal subgroup of NUA ⍀ properly contain- ing U,if <<⌽ s 4. <<⌽ / Ž.r Ž. Now assume 2, 4. From the above equation, NUAA⍀⌬NUis ŽŽ.. isomorphic to either A⌽⌽or S . Thus in this case, ON2 A ⍀ U lies in a Ž. Ž. Ž Ž..F Ž Ž.. normal subgroup NUAA⌬⍀of NU, and hence ON2 A ⍀ U ONU2 A ⌬. Thus U is a radical 2-subgroup of A⌬ Ž.which does not fix any letter of ⌬ if and only if U is a radical 2-subgroup of A⍀ fixing ⌽. This establishes RADICAL 2-SUBGROUPS 315 ClaimŽ. 3 except the statement on conjugacy. That part follows from the observation that two radical 2-subgroups U and V of A⍀⍀with ⌽ Ž.U s ⌽⍀⍀Ž.V are conjugate under A if and only if they are conjugate under the stabilizer of ⌬ ⍀⍀Ž.U s ⌬ Ž.V in A ⍀. LEMMA 6. There are just 10 classes of radical 2-subgroups of the alternat- ⍀ s ing group A9 of degree 9, naturally acting on the set Ä40,...,8 : with the ¨ wx notation abo e, B2Ž.A⍀ 5 forms a single class represented by ²Ž01 .Ž 23 . , wx Ž.Ž.:Ž02 13 ; B2 A⍀ .3 has two classes represented by ²Ž01 .Ž 23 . , Ž 01 .Ž 45 .: and ( wx ¨ ²Ž01 .Ž 23 . , Ž 02 .Ž 13 . , Ž 01 .Ž 45 .: D82; and B Ž.A⍀ 1 has 7 representati es ( corresponding to the classes of unipotent radicals of A84SL Ž.2. < << < Proof. For a radical 2-subgroup U of A⍀⍀⍀, we have Ž ⌽ Ž.U , ŽŽ.⌬ U . s Ž.Ž.1, 8 , 3, 6 , or Ž. 5, 4 . From Lemma 5Ž. 3 , the conjugacy classes of wx wx B2Ž.A⍀⍀5 under A bijectively correspond to those of B2Ž.A⌬ 0 under ⌬ s s wx S⌬, where Ä40, 1, 2, 3 . It is easy to see that B2Ž.A⌬ B2 Ž.A⌬ 0 forms a single class represented by²Ž 01 .Ž 23 . , Ž 02 .Ž 13 .: . wx The conjugacy classes of B2Ž.A⍀⍀3 under A bijectively correspond to wx ⌬ s those of B2Ž.A⌬⌬0 under S , where Ä40, 1, 2, 3, 4, 5 . Observing a Sylow [ ( ( 2-subgroup D ²Ž01 .Ž 23 . , Ž 02 .Ž 13 . , Ž 01 .Ž 45 .: D8 of A⌬ A6 , a 2-sub- group U of A⌬ without fixing any letter of ⌬ is conjugate to²Ž 01 .Ž 23 . , Ž.Ž.:01 45 , the cyclic subgroup of order 4 of D or D itself. It is easy to see that the first and the last are radical 2-subgroups, but the second is not. wx wx Thus B2Ž.A⌬ 0 and hence B2Ž.A⍀ 3 are represented by²Ž 01 .Ž 23 . , Ž.Ž.:01 45 and D. ⌬ s wx Let Ä40, . . . , 7 . The classes of B2Ž.A⍀⍀1 under A bijectively wx correspond to those of B2Ž.A⌬⌬0 under A . In the following, we will wxs ( verify that B2Ž.A⌬ 0 B2Ž.A⌬ . This establishes the claim, because A8 L4Ž.2 is a group of Lie type of rank 3 in characteristic 2, and hence it has 2 3 y 1 s 7 classes of radical 2-subgroups. ( Suppose V is a radical 2-subgroup of A⌬ A8 fixing a letter. Then < < ⌽⌽Ž.V s 2 or 4. In the latter case, V is not a radical 2-subgroup of ( <⌽ < s A⌬ A8 by Lemma 5Ž. 2 . In the former case, we have ⍀Ž.V 3. As we saw in the above paragraph, we may then assume that V is conjugate to ( ²Ž01 .Ž 23 . , Ž 01 .Ž 45 .: , the cyclic subgroup of D of order 4 or D D8.Aswe Ž.r ( = = have NVA⌬ V S3 2, 2 2 or 2, respectively for those candidates, none of them is a radical 2-subgroup of A⌬. 2.2. The Radical 2-Subgroups of M24 wx The radical 2-subgroups of M22 and M24 are classified in 1 , but more geometric description is required to examine radical 2-subgroups of simple groups which contain 2-local subgroups with quotients containing M22 or M24 . In the rest of this section, we give a description of radical 2-subgroups 316 SATOSHI YOSHIARA of M24, M 22 and related groups in terms of the Golay code or cocode, which will be frequently used in the later sections. Following the standard conventionwx 6, Chaps. 11, 10 , we arrange the 24-set ⍀ [ Ä40, 1, . . . , 9, A, B,...,N in the 4 by 6 rectangular form shown in Table I, called the MOG arrangement.Ž Note that we adopt the newer version by Conway and Sloane, but not the original version of MOG invented by R. T. Curtis.. A subset of ⍀ will sometimes be specified by putting asterisks at the corresponding entries, and a permutation on ⍀ is presented pictorially, e.g.,Ä4 0, 1, 8, 9, G, I, K, N and a permutationŽ.Ž.Ž.Ž.Ž.Ž. 45 67 CD EF GI HJ Ž.Ž.KN LM are described in Table II. ⍀ The count of a subset X of is a sequence Ž.c161,...,c , r of integers, where ci Ž.resp. r1 is the number of asterisks in the ith columnŽ resp. the first row. when we represent X as in Table II. The score of X is a vector 62s q q s Ž.s16,...,s of F 4determined by si x2 i x3i xi4 i Ž.1,...,6 , = s where is a generator of F4 and xij 0Ž. resp. 1 iff the asterisk for X appearsŽ. resp. does not appear at the Ži, j.-entry of the MOG arrange- ment. A subset of ⍀ is called a Golay code word, whenever the integers of its count have the same parity and its score lies in the hexacode H, that is, 6 q s the subspace of F41of vectors Ž.x ,..., x6with the condition x1x2x3 q s q q q s q ⍀ x456x x and x 135x x Ž.x 12x . The subsets of form a ⍀ vector space over F2 under symmetric sum, which is denoted PŽ.. The Golay code words form a subspace G of PŽ.⍀ of dimension 12, which is called the Golay code. RADICAL 2-SUBGROUPS 317 U The factor space G [ PŽ.⍀ rG is called the Golay cocode, which can be thought of as a set of the 0, 1, 2, 3-subsets and equivalence classes on 4-subsets of ⍀, where two 4-subsets are equivalent exactly when their union lies in G. A 2-Ž. resp. 3- and 4- subset of ⍀ is also referred to as a duad Ž.resp. triple and tetrad . Each equivalence class of tetrads, called a sextet, consists of exactly six tetrads. The hyperplane PqŽ.⍀ of P Ž.⍀ consisting of subsets of ⍀ of even cardinality contains G, and the factor space PqŽ.⍀ rG is called the e¨en part of the Golay cocode. An 8-subset of ⍀ is called an octad, if it is a Golay code word. A triple of octads is a trio, if they partition ⍀. Typical examples of octads areÄ4 0, 1, . . . , 7 , Ä4Ä48, 9, A,...,F , and G, H,...,N ; the left, middle, and right ‘‘bricks,’’ respectively: The triple of them is called the standard trio, and the six columns of the MOG arrangement form a sextet, called the standard sextet. The Mathieu group of degree 24, denoted M24 , is the group of permuta- ⍀ tions of preserving G. The group M24 acts naturally on the Golay ⍀ cocode and its even part. It is known that M24 acts 5-transitively on and transitively on the set of sextetsŽ. resp. trios and octads . To explicitly describe the radical 2-subgroups of M24 , we choose the permutations of ᎐ M24 shown in Tables III V. We use the fairly standard notation UO , etc., to denote the following subgroups, which is modeled after that of parabolics for Lie-type groups and continued for sporadic groupsŽ e.g., in the work of Ronan and Smith on 2-local geometrieswx 11 , especially with the ‘‘square node’’. : [ UO ²:a1212, a , r , r , the pointwise stabilizer of the left-brick-octad O; [ UT ²:a123413, a , a , a , r , r , the octadwise stabilizer of the standard octad T; 318 SATOSHI YOSHIARA [ ¬ s USi²:a i 1, . . . , 6 , the tetradwise stabilizer of the standard sex- tet S; [ [ UO I UaO ²:²:34512, aa, x , UT I UaT 512, x ; [ [ [ UOTUU O T, U OSUU O S, U TSUU T S; [ [ s [ UOTSUUU O T S, U OTI UUOTI UOO I TOS, U I UUSOI ; [ [ UTSI UUSTI , UOTSI UUSTO U I . wx The following result is obtained in 16 , but two representatives UT I and UST I are missing there. To describe the structures of 2-subgroups, we follow the ATLAS notationwx 4 , e.g., N . Q means a group having a normal subgroup isomorphic to N with quotient by that normal subgroup isomor- RADICAL 2-SUBGROUPS 319 phic to Q; the symbol 2 n is used to denote the elementary abelian group of nnqm 1q2 m n order 2 ; 2Ž resp. 2 . is a special group with center 2 and central quotient 2m Ž resp. the extraspecial group of order 21q2 m of type s "1;. andw 2n x means a 2-group of order 2n . LEMMA 7. The radical 2-subgroups of M24 consist of 13 conjugacy classes ¨ ¨ s I I with representati es UX abo eXŽ O, T, S, O , T , OT, OS, TS, OTS, OT I, OSI, TSI, OTSI.Ž. The structures of U and N U .rU are XMXX24 described in Table VI, where the third and fourth columns gi¨e generators of ¨ s ZŽ. UXX and a brief description of the shape of U , respecti ely, where I Ä41, 2, 3, 4 and J s Ä41,...,6 . The group M24 has two classes of involutions: a 2 A-Ž. resp. 2 B- involution has a permutation type 188 2Ž resp. 212 . on ⍀, where the set of U fixed points is an octad. Let G be the Golay cocode on which M24 U U naturally acts. It is easy to verify that the subspace CaG Ž.of G fixed by a 2 A-involution a is of dimension 8 consisting of the following elements, where C is the octad consisting of the points fixed by a: the empty set, the 8 points of C, Ž.8 s ⍀ the 2 28 duads of C together with 8 2-cycles of a on , Ž.8 s = ⍀ the 3 56 triples of C together with 8 8 triples of , each of which consists of a point of C and a 2-cycle of a, Ž.8 r s the 4 2 35 sextets determined by complementary pairs of tetrads of C together with 56 sextets determined by tetrads of ⍀, each of which consists of a 2-cycle of a and a duad of C Žfour of them determine the same sextet, and a induces a transposition on the six tetrads of such a sextet. . 320 SATOSHI YOSHIARA Then it is immediate to see the following resultsŽ though they are standard in the various places in the literature; the above description provides more explicit information. : U LEMMA 8. Under the natural action of M24 on the Golay cocode G and U U U its e¨en part Gqq, the subspaces of G and G ¨ectorwise fixed by each ¨ representati e UX of the radical 2-subgroups of M24 Ž. Lemma 7 are as follows: Ž.UU Ž.s Ž. 1 CUG X CGq UX is a 1-subspace spanned by the standard sextet S s 0123 for e¨ery X containing SŽ that is, S, OS, TS, OTS, OSI, TSI, and OTSI.. Ž.UU Ž .Ž Ž .. Ž . 2 CG UO resp. CGq UO is a 7-subspace resp.6-subspace consist- ing of i-subsets of O for i s 0,...,4 Ž.resp. i s 0, 2, 4 , where a 4-subset of O is identified with its complement in O. Ž.UU Ž.s Ž .s UU Ž.s Ž .s ² 3 CUG T CUG OT CUGqqT CUG OT 0123, 0145, 0246: ( 2.3 3 Ž.UU Ž .s Ž .s ² :( 4 CUG O I CUGq O I 0123, 0145, 0247 2. Ž.Ž.UUs Ž .s UU Ž.s Ž .²s 5 CUG T I CUG OTI CUGqqT I CUG OTI 0123, 0145: ( 2.2 2.3. The Radical 2-Subgroups of M22 We use the notation in the previous subsection. We may identify the Mathieu group G [ M of degree 22 with the stabilizer in M of two 22 X 24 points 0, 1 of ⍀. Then M acts triply transitively on ⍀ [ ⍀_Ä40, 1 . A 22 XX hexad Ž.resp. heptad and octad of ⍀ is the intersection C l ⍀ for an octad C of the Golay code G with ¨ LEMMA 9. E ery 2-local subgroup of M22 is contained in the stabilizer of a hexad, a quintet, or an octad. wx Proof. In 16, 4.4 , it is proved that every 2-local subgroup of M24 stabilizes an octad, a trio, or a sextet. Assume that a 2-local subgroup M of s G M22 stabilizes a sextet Ä4T1,...,T 6 . We may assume that it is not a g g quintet, so that 0 T12and 1 T , say. Then M stabilizes the hexad j _ Ž.T12T Ä40, 1 . Assume that M fixes a trio ÄO123, O , O 4. If there is an _ octad Oi containing both 0 and 1, then M stabilizes the hexad Oi Ä40, 1 . Otherwise there is a unique octad Oi disjoint fromÄ4 0, 1 , which is stabi- lized by M. Assume that an octad C of G is stabilized by M.If XX[ s [ s UQOUU Q OUx Q²:56 , UHQUU H QUa H ²56, a :, XX[ s XX[ s UHOUU H OUa H²:556, x and UHQOUUU H O QUa H²56, a , x56 :. LEMMA 10. The radical 2-subgroups of M22 consist of 7 conjugacy classes ¨ л / : X ¨ with the abo eUX Ž X Ä H, Q, O4. as representati es. The structures of r UXMXX and NŽ. U U are described in Table VII, where the second and third ¨22 columns gi e generators of ZŽ. UX and a brief description of the shape of R, respecti¨ely. 322 SATOSHI YOSHIARA s Proof. Let R be a radical 2-subgroup of G M22 . It follows from F X Lemma 9 that NRGGHGQGOŽ.NU Ž ., NU Ž .,orNU Ž .up to conjugacy, as 4 4 3 NUŽ.( 2:A , NUŽ.( 2:S and NUŽ.X ( 2:L Ž.2 are stabiliz- GH 6 GQX 5 GO 3 F X ers of H, Q, and O , respectively. Assume that NRGGOŽ.NU Ž .but / X r X R UOO. Then R U is one of 3 representatives of classes of unipotent ( XXr X radicals of L3Ž.2 NUGO Ž . U O, acting naturally on the 3-space UO.If it corresponds to a point p or a point-line flag Ž.p, l , then ZR Ž.s X CRXŽ.rU is a 1-space corresponding to p, so that NR Ž.ŽŽ..F CZR. UOO GG But the involutions of G form a single conjugacy class and the centralizer of an involution is contained in the stabilizer of the hexad, the set of points ⍀X F of fixed by that involution. Thus NRGGHŽ.NU Ž .up to conjugacy. If X XX RrU corresponds to a line, ZRŽ.s CRX ŽrU .is a 2-subspace of U . OUO OO s X We may take ZRŽ. ²a15, a :up to conjugacy under NUGOŽ., so that ⍀X NRGŽ.stabilizes the set of points of fixed by ²:a15, a , that is, the duad 23. Thus NRGŽ.stabilizes the corresponding standard quintet Q, that is, F NRGGQŽ.NU Ž .up to conjugacy. F / r Assume that NRGGQQŽ.NU Ž .but R U . Then R U Qcorresponds to one of three representatives of classes of radical 2-subgroups of ( r ( 2 S5 NUGQŽ .U Q. We may take ²ŽQQ34 .Ž QQ 56 ., ŽQQ 35 .Ž QQ 46 .: 2, ( ( ²ŽQQ34 .Ž QQ 56 ., ŽQQ 35 .Ž QQ 46 ., ŽQQ 56 .: D 8and ²ŽQQ56 .: 2 as the s representatives, represented as permutations of the ith columns Qii Ž 1, . . . , 6. of the MOG arrangement, which form the standard quintet Q.In r s j _ the first two cases, R UQ fixes the hexad H Ž.Q12Q Ä40, 1 , which is F F stabilized by NRGGQGGHŽ.ŽNU Ž ... Thus NR Ž.NU Ž .. In the last case, s s X we have R UxQ ²:56 UQO ,as x56induces Ž.QQ 5 6 . Conversely, we X have ZUŽ.X s Dx Ž.²s a , a , a :and the set of points of ⍀ fixed QO UQ 56 1 5 6 X X F by ZUŽ.QO is 23, corresponding to the quintet Q. Thus NUGŽ. QONU G Ž. Q , X and so UQO is a radical 2-subgroup of G by Lemma 1Ž. 2 . F / r Assume finally NRGGHHŽ.NU Ž .but R U . Then R U Hcorresponds ( to one of three representatives of classes of radical 2-subgroups of A6 r ( 2 ( NUGHŽ. U H. We may take²Ž 45 .Ž 67 . , Ž 46 .Ž 57 .: 2 ,²Ž 45 .Ž 67 . , Ž 23 .Ž 67 .: 2 ( 2 and²Ž 45 .Ž 67 . , Ž 46 .Ž 57 . , Ž 23 .Ž 67 .: D8 as the representatives, repre- sented as permutations on H s Ä42, 3, . . . , 7 . It is easy to see that those s s three representatives correspond to UaH ²:56, a HHQ, Ua H ²556, x : RADICAL 2-SUBGROUPS 323 XXs UHO and UaH²:5556, a , x UHQO , respectively. It is not difficult to see that their normalizers are contained in NUGHŽ., and hence they are radical 2-subgroups of G by Lemma 1Ž. 2 . Thus we verified that every radical 2-subgroup of G is conjugate to one л / : X of UX for X Ä H, Q, O 4. Observing their orders, centers, and nor- malizers, we see that no two of them are conjugate under G. ( LEMMA 11. The automorphism group AutŽ.M22M 22 :2 has exactly 7 classes of radical 2-subgroups with the representati¨es in Table VIII, among which UH is the unique one contained in M22 Ž. in the notation of Lemma 10 . [ s s Proof. We set A AutŽ.M22 Ga ²:3 with G M22 . The proof is a typical example of applications of the methods described in Lemma 3. s : X Observe first that NUAXŽ.NU GX Ž.²: a3 for every X Ä H, Q, O 4 with s r convention Uл 1. The image of a subgroup L of NUAXŽ.in NU AX Ž.U X is denoted L. As for X s л, there are two classes of involutions in A_G with X representatives a3233and aa: a fixes the octad O pointwise, while aa23 ⍀ does not fix a point on . The centralizer CaGŽ.324is a subgroup of M X X stabilizing the octad O and fixing two points 0, 1 outside O , and thus ( 3 CaGŽ.332.L Ž.2 , which has the nontrivial O 2-part. The centralizer CaaŽ.lies in the stabilizer of the standard sextet S, which is the M 24 23 F unique sextet tetradwise fixed by aa23. Then CaaGŽ.23 NUGQ Ž.and g / a12OCŽŽG aa23 ..1. Hence there is no radical 2-subgroup R of A with R l G s Uл s 1 by Lemma 3Ž. 1 . s 3 For X H, a3 induces a permutation of type 2 on the hexad H,so ( ( s that NUGHŽ.A6 and NUAH Ž.S62.AsONUŽ.AHŽ. 1, U His a radical 2-subgroup of A by Lemma 3Ž. 2 . There are three classes of 2-elements in _ S66A with representativesŽ.Ž.Ž.Ž. 12 , 12 34 56 , and Ž 1234 . with centralizers in A64isomorphic to S , S4, and Z4, respectively. As none of them has the trivial O2-part, it follows from Lemma 3Ž. 1 that there is no radical subgroup of A containing UH of index 2. 324 SATOSHI YOSHIARA ( Since NUGQŽ.S5 induces the full permutation group on the 5 columns of the standard quintet Q, while a3 acts trivially on them, we have ( = NUAQŽ.S532, where a corresponds to the central involution. For every nontrivial 2-element g of S5, the centralizer CgS Ž.has nontrivial 5 _ trivial O2-part. Thus there is a single class of 2-elements in NUAQŽ. NUGQŽ.whose centralizer in NUGQŽ.has the trivial O2-part, represented by a3. Thus it follows from Lemma 3Ž. 1 that a radical subgroup of A containing UQQof index 2 is conjugate to Ua²:3 . ( As for the other possible X, NUGXŽ.L33Ž.2, S or 1 by Lemma 10. ( = Thus NUAXŽ.NU GX Ž.2 and hence UXis not a radical subgroup of A. Furthermore, there is exactly one class of 2-elementsŽ. in fact, involutions _ in NUAXŽ.NU GX Ž.whose centralizers in NUGXŽ.have the trivial O2-parts ŽŽthat is, isomorphic to L332,.S , or 1. . For each X, it is easy to see that a3 corresponds to the central involution of NUAXŽ.. Thus every radical subgroup containing UXXof index 2 is conjugate to Ua²:3 by Lemma 3. ( и LEMMA 12. Let G 3 M22 be a nonsplit triple central extension of M22 , and A be a finite group ha¨ing G as a normal subgroup of index 2 with an in¨olution which in¨erts the center of G. Then there are exactly 13 classes of radical 2-subgroups of A:7of them are contained in G and correspond to the ¨ л / : X representati es UX Ž X Ä H, Q, O4. of radical 2-subgroups of M22; the ¨ ¨ other 6 classes ha e representati es corresponding to those of M22 :2 not : X / л contained in M22 Ž that is, U˜X for X Ä4H, Q, O with X , H .. Proof. By Lemma 4Ž. 1 , the radical 2-subgroups of 3 . M22 bijectively л / : X correspond to those of M22 . Thus we may take UX Ž X Ä H, Q, O 4. [ as the representatives of radical 2-subgroups of G 3M22 . Then the arguments given in the proof of Lemma 11 for examining radical sub- groups of M22 : 2 not contained in M22 go through without any change to [ determine the radical 2-subgroups of A 3.M22 . 2 not contained in G. The difference is that every UX is a radical 2-subgroup of A, because of Lemma 4Ž. 2 . Hence we obtain the lemma. X The truncated Golay code G Ž.at two points is a subspace of the Golay code G consisting of codewords which trivially intersect withÄ4 0, 1 . It is a X 10-subspace of G.As X g G with < X U Ž.22 s ⍀ Ž . 2 231 duads of corresponding to tetrads of G containing 0, 1 , X Ž.22 r s ⍀ Ž and 3 2 770 pairs of triples of corresponding to equivalent pairs U of tetrads of G containing exactly one of 0 and 1. . s XXU The group AutŽ.M22 Ga ²:3 naturally acts both on G and G ,as a3 flips 0 and 1. We have the following on those actionsŽŽ. the claim 2 is not needed later, but is given for completeness. . л / : X ¨ LEMMA 13. Let UX Ž X Ä H, Q, O4. be the representati es of 7 ¨ classes of radical subgroups of M22 gi en in Lemma 10. X X ¨ Ž.1 The subspaces CG Ž. UX of the truncated Golay code G ectorwise X fixed by UX are as follows: CG Ž. UQ is a 4-subspace consisting of the empty l ⍀X F - F set, the 5 hexads Q1iij, and the 10 octads Q Ž.2 i j 6, where g [ j Ä4Q16,...,Q is the standard quintet with 0, 1 Q1 and QijQ iQ j; CX Ž. U is the 1-subspace spanned by the left-brick hexad H if H g X, and G X XX X g CG Ž. UX is the 1-subspace spanned by the right-brick octad O if O X. XU XU ¨ Ž.2 The subspaces CG Ž. UX of the truncated Golay cocode G ector- XU wise fixed by UX are as follows: CG Ž. UH is a 5-subspace consisting of the 0, 1, 2, and 3-subsets of H, where 3-subsets are identified with their complement; XU X ¨ CG Ž. UO is a 3-subspace spanned by 23, 45, and 89 Žall nontri ial elements XU s XU s ( 2 are duads.Ž.Ž.²:; CUG Q CUG HQ 2, 3 223Ž.is the unique duad ; XU X s ( 2 XU X s XU X s ( CUG Ž.²:HO 23, 45 2;and CG Ž. UQO CUG ŽHQO .²:23 2. Proof. Ž.1 The stabilizer of a dodecad in M22is M 10 , which does not 3 X contain a 2 -subgroup. Thus none of the UX ’s fixes a dodecad of G ,as ( ( 43X ( ( r ( UQHU 2 and UO2. As A6 NUGHŽ.U Hand L3Ž.2 3 NUŽ.XXrU do not contain 2 -subgroups, if U Ž.resp. U X fixes a hexad GO O X HO or octad other than H Žresp. O ., then it contains an involution which X pointwise fixes H Žresp. O . together with that hexad or octad, which is impossible. Thus for UXQother than U , the claim follows. As for UQ, each hexadŽ. resp. octad stabilized by UQ is the union of two points 2, 3 fixed by s UQiand one of 5 orbits QiŽ.2, . . . , 5 of length 4 Ž resp. two orbits Qij, Q .. Conversely, such hexads and octads are stabilized by UQ , and X CUG Ž.Q is of the claimed shape. Ž.2 It follows from the remark in the paragraph preceding Lemma 8 XU g that CaG Ž.for an involution a UH of M22 is a 6-subspace consisting of Ž.6 the empty set, 6 points of H,2 duads in H together with 8 2-cycles of a, Ž.6 = and3 triples in H together with 6 8 triples, each of which consists of a point of H and one of 8 2-cycles of a. Then the claim follows from straightforward examination of subspaces fixed by generators of UX . 326 SATOSHI YOSHIARA 3. RADICAL 2-SUBGROUPS OF J4 s Let G J4 be the largest sporadic simple group of Janko. It follows fromwx 9, 2.1, Proposition 2.2.3 and Corollary 2.3.7; 14, 2.12, Step 1 that every 2-local subgroup of G is contained in one of the following groups up to conjugacyŽ to applywx 14 , note that every p-local group is contained in the normalizer of a radical p-group, which is easily verified. : ( 1q12 и CzGŽ. 2:3ŽŽM22 .: 2 . , the centralizer of a 2 A-involution z; U ( 3q12 и = U s U NVGŽ . 2 ŽŽ..S53L 2 , where V ZOŽŽ 2 NG Ž V ... is a 2 A- pure 2 3-subgroup; ( 11 NVGŽ. 2:M24, where the action of a complement on ONV 2ŽŽ..G is equivalent to the action of M24 on the even part of the Golay cocode; ( 10 and NAGŽ. 2:L5Ž.2. wx [ ( 1q12 [ Following the notation in 7 , we set E OC2ŽŽ..G z 2,L U ( 3q12 ( ( 11 ONV2ŽŽG .. 2 , and let K M24 be a complement to V 2in s s U NVGGGGGŽ.. Then NE Ž.Cz Ž.and NL Ž.NV Ž .. We follow the nota- ( tion in the previous section when we work with subgroups of K M24 and U V, identified with the even part Gq of the Golay cocode. Note that a sextetŽ. resp. duad of V is a 2 A- Ž. resp. 2 B- involution of G and F wx CwGGŽ.NV Ž.for a 2 B-involution w of V 7, Propositions 9, 19 . Let UX be one of the representatives of 13 classes of radical 2-subgroups of M24 given in Lemma 7. Note that the subspace CUVXŽ.of V fixed by UXis determined in Lemma 8, which is used without further reference in the later arguments. We may take z s 0123, a 2 A-involution of G. It follows from the proof wx of 7, Proposition 13 that the normalizer of CUVTŽ.in NV G Ž.is V : KT ( 11 6 = 2:2:ŽŽ..S33L 2 , where KT is the stabilizer of the standard trio T, 3q12 и = and that it extends to the full normalizer in G of the shape 2 ŽS5 U s s L3Ž..Ž2 non-split . . Thus we may take V CUVTŽ. ²0123, 0145, 0246: . Moreover, it follows from the proof ofwx 7, Proposition 21 that we may s = take A CUVOŽ.U O. The normalizer NAG Ž.is a split extension of ( 10 L5Ž.2byA 2 . We use LA to denote a complement and represent s elements of LAias matrices with respect to the natural basis eiŽ.1,...,5 F / F for the natural module W of L5Ž.2 . Let ij Ž1 i j 5. be the matrix of s s LA whose Ž.k, l -entry is 1 Ž resp. 0 . iff Ž.Ž.k, l i, j or k l, and let [ ( 4 [ ( 6 Vp ²:21, 31, 41, 51 2,Vl ²:31, 41, 51, 32, 42, 52 2, [ ( 6 [ ( 4 V ²:41, 51, 42, 52, 43, 53 2,VH ²:51, 52, 53, 54 2, and [ л / : VYiŁ g YiV for Y Ä4p, l, , H . RADICAL 2-SUBGROUPS 327 Then the VY ’s for Y ranging over the nonempty subsets of Ä4p, l, , H ( form a set of complete representatives of 15 unipotent radicals of LA s s L5Ž.2 , corresponding to subflags of Ž.p, l, , H , where p ²e1 :, l s s ²:e12, e , ²e123, e , e :, and H ²e1,...,e 4 :. The representation of LA afforded by A is equivalent to the alternating square ⌳2 Ž.W of W with [ n s F / F ⌳2 basis eije ie je ji Ž.1 i j 5 . We identify A with Ž.W . Then the subspaces CVAYŽ.of A fixed by VYare easily calculated: s ( 4 s g : CVApŽ. ²e12, e 13, e 14, e 15: 2, CVAYŽ.²:e12 for l Y Ä p, l, , H4, s s ( 3 CVAŽ. CVA Ž H . ²e12, e 13, e 23 : 2, s ¬ F - F ( 6 CVAHŽ.²e ij1 i j 4:2, s ( 3 s s CVApHŽ.²e12, e 13, e 14 :2 , and CVApŽ. CVAp Ž H .²e12, e 13 : ( 2 2. [ ( 6 F [ s ( 2 We set VOVOCUŽ. 2 Ž.A , Z CUVT Ž.²I 0123, 0145 :2, [ s ( 3 and F CUVOŽ.²I 0123, 0145, 0247 :2. LEMMA 14.Ž. 1 The subgroup CAŽ V .Ž resp. CApH Ž V . and C Ap Ž V .. is U ¨ conjugate to VŽ. resp. F and Z under the action of NG Ž A.. Moreo er, s CVAHŽ.V O. ¨ s F ( 4 Ž.2 We ha eNGOŽ. V V : K ONV GŽ., where K O 2:L4Ž.2 is the stabilizer in K of the octad O. ¨ 2 F U s 2 Ž.3 We ha eOCŽGG Ž Z ..CV Ž .Ž, L OOC2 ŽG Ž Z ..., and F U NZGGŽ.NV Ž .. ¨ s F Ž.4 We ha eCGOŽ. F V:UI and NGGŽ. F NV Ž., but N G Ž. F is not U conjugate to any subgroup of NGŽ V. under G. [ = Ž.5 The subgroup CApŽ. V is conjugate to Fp F ²a1 : under NG Ž A .. ¨ s F We ha eCGpŽ. F AU OI and NGpŽ. F NA G Ž.. U Proof. We first show the claimŽ. 3 . The normalizer NVGŽ . contains ( ( wxF w x subgroups S S53and C L Ž.2 with S, C L 7, Proposition 13 , U ( 3 where C is a subgroup of KT acting faithfully on V 2 . Then we have U s U ( 2 CVGCŽ . LS.AsZ is a 2-subspace of V , CZŽ. 2 is the correspond- и = ing unipotent radical of C. Thus CZGCŽ.contains L ŽS CZ Ž... In view w of the orders of centralizers of noncentral involutions in CzGŽ.7, Proposi- x = tion 1, 3, 6 , we see that CZGCŽ.coincides with LS ŽCZ Ž... Then the 2 normal subgroup OCŽŽ..G Z generated by elements of odd order lies in X ( 3q12 X LS 2 A5. As a 5-Sylow subgroup of LS acts fixed-point-free on r U wxws 2 x F 2 L V by 7, Proposition 13 , we have L L, OCŽŽ..GG Z OCŽŽ.. Z X s 2 2 F U s and so LS OCŽŽ..GG Z. In particular, OCŽ Ž Z ..CVG Ž . and L 2 OOC2Ž ŽGG Ž... Z . Thus NZ Ž.normalizes CZ GŽ.and so L, and hence F s U NZGGGŽ.NL Ž.NV Ž .. 328 SATOSHI YOSHIARA Next we show the claimŽ. 4 . As F contains Z, CFG Ž .is a subgroup of s = CZGCŽ.LS ŽCZ Ž... We claim that there is no element of order 5 in CFGŽ.. For otherwise, there is an element x of order 5 contained in both 2 F U s CFGGGŽ.and OCŽ Ž Z ..CV Ž ., and x centralizes a 2 B-involution 67 q U F 0246 0247 in ²F, V :Ž. However, as CGG67.NVŽ.for a 2 B-involu- tion 67 in V wx7, Proposition 19 , this leads us to the contradiction x g l U s ( 3q12 = NVGGŽ.NV Ž . V : K T2 ŽŽ..S33L 2. F w l x s This implies that CFGGŽ.NV Ž.,as CZ GGGŽ.: NV Ž.CZ Ž. 5 F F 11 and CFGGŽ.CZ Ž.. Since V CFGŽ.and there is a unique 2 -subgroup of each Sylow 2-subgroup of G wx7, Proposition 7 , V is the unique 11 F 2 -subgroup of CFGGGŽ.and NF Ž.NV Ž.. Then it is easy to see s ( 11 1q6 s ( 11 1q6 CFGOŽ. V : U I 2 : 2q and NFGOŽ. V : U I : LO I 2:2: ( L3Ž.2 , where LO I L3Ž.2 is a complement to UO I in the subgroup ( I CaK Ž.124of K M , the parabolic subgroup corresponding to O . U Suppose NFGGŽ.is conjugate to a subgroup of NVŽ .. In view of their 3q12 = orders, that subgroup is the shape 2 . ŽŽ..D83L 2 , and hence the U s center of its O22-part is V , but it is also conjugate to ZOŽ Ž NG Ž F ... F. Observing their normalizers, this is impossible. Then the claimŽ. 2 is verified as follows. As V0 contains F, CVGOŽ.is the s subgroup of CFGOŽ. V : U I fixing the duads of O. Examining the s F F actions of generators of UO I , we see CVGOŽ.VU ONV G Ž..AsV 11 F CVGOŽ., V is the unique 2 -subgroup of CVGOŽ., and hence NVGOŽ. s ( 11 4 NVGGŽ.and NV ŽOO . V : K 2:2:L4Ž.2 , where K O is the stabilizer ( of the octad O in K M24 . Now we prove the claimŽ. 1 . In the stabilizer K O , a complement ( ( 4 s = LO L4Ž.2ofUOOOO2 acts faithfully on U . Then L acts on A V s ( 6 r VVOOŽŽ.CU VO 2.Ž , and the image of NALVOG.in NAŽ.A is a 4 subgroup isomorphic to 2 : L4Ž.2 . Furthermore, the unipotent radical r ( 4 NAVOŽ.V 2 of this parabolic subgroup of L5Ž.2 fixes VO . Hence it corresponds to the unipotent radical VH of L5Ž.2 in the notation above. s Thus we may take CVAHŽ.V O. The 6-subspace VOaffords the alternat- ( ing square representation of LO L4Ž.2 , which is also thought of as the ⍀q ( orthogonal module for 64Ž.2 L Ž.2 . The explicit quadratic form is given Ž.s q q Ž8 .r s by q Ý1F i- jF 4 xeij ij xx12 34xx 13 24xx 14 23. Then 4 2 35 sextets ŽŽ.8 s . Ž . resp. 2 28 duads of O correspond to singular resp. nonsingular points. As sextetsŽ.Ž. resp. duads are 2 A- resp. 2 B- involutions, a subspace of VO is totally singular iff it is 2 A-pure. As there is a single class of totally singular lines under LOA, totally singular lines Z and CVŽ.p are conjugate under LO . There are two classes of totally singular 3-subspaces under LO s ⌳2 s with representatives CVAŽ. ²e12, e 13, e 23 :Žthe subspace of Ž.W A corresponding to a set of lines lying on a plane ²:e123, e , e of W under the s Klein correspondence .Ž.²:Žand CVApH e12, e 13, e 14 the subspace of A RADICAL 2-SUBGROUPS 329 corresponding to a set of lines through a point ²:e1 of W under .. Note ( that CVAŽ. is centralized by an element x of order 3 of LA L5Ž.2, s s s which fixes every point of , e.g., Ž.exiiei Ž1, 2, 3 .Ž. , ex45e , and s q Ž.ex545e e . On the other hand, it follows from the claimsŽ. 3 , Ž. 4 U 3 above that V and F are 2 A-pure 2 -subgroups of VO , which are not conjugate even under G.AsCFGŽ.is a 2-group, the above remark shows U that F is conjugate to CVApHŽ., and hence V is conjugate to CVAŽ. under LO . ( 4 Finally we show the claimŽ. 5 . Take the unipotent radical Vp 2of ( s ¬ s ( 4 [ LA L5Ž.2 with CVAp Ž . ²e1ipApi 1,...,5: 2 . We set F CVŽ.. By the claimŽ. 1 , the subspace ²e12, e 13, e 14 :of Fp is conjugate to F under NAGppŽ.. Then we may assume F contains F by replacing V by a suitable ( conjugate under NAGpŽ.. There is a Levi subgroup L L4Ž.2ofLA normalizing Vpp, which acts faithfully on F . Thus Fpis 2 A-pure. As F is a s = l s maximal singular subspace of VOOOpOand A V U , we have F V F s = s and FpOF ²:a for an involution a of U . Since CFGŽ. V : UOI by g F s s the claimŽ. 4 and a UOOU I , we have CFGpŽ.Ž.CF G , a CaŽ.: Ca Ž.. It follows from the remark in the paragraph previous to VUO I Lemma 8 that CaVOŽ.is a 7-dimensional space generated by V and a sextet determined by a tetrad consisting of two points of O and a 2-cycle of ⍀ s a on . Since FpApCVŽ.and A is abelian, CFGp Ž.contains a subgroup A : V of order 214 . Then s r s then ZRŽ.CRA ŽA .is a 1-subspace spanned by a 2 A-involution z s s 0123, and thus NRGGŽ.is contained in CzŽ..If Y H, then ZRŽ. VO F F s and NRGGOGŽ.NV Ž .NV Ž.by Lemma 14Ž. 2 . If Y or H, then U ZRŽ.is conjugate to V by Lemma 14Ž. 1 , and so NRG Ž .is conjugate to a U s subgroup of NVGŽ ..If Y p or p H, then it follows from Lemma F U s 14Ž.Ž. 1 3 that NRGG Ž .is conjugate to NZŽ.ŽNVG..If Y pH, then F Lemma 14Ž.Ž. 1 4 implies that NRGGŽ.is conjugate to NFŽ.Ž.NVG.In s s F the remaining case where Y p, we have ZRŽ. FpGand hence NRŽ. NAGŽ.. Ž p. Conversely, if we take the inverse image A in NAGŽ.of a radical r Ž p. s Ž p. F 2-subgroup VpGof NAŽ.A, we have ZAŽ .ŽFpand NA G. F Ž p. NFGpŽ.NA G Ž.. Thus A is in fact a radical 2-subgroup of G ŽLemma Ž p. U 1Ž.. 2 . Furthermore, NAGGŽ .Žis not conjugate to a subgroup of NV. U nor CzGŽ., as it involves L4Ž.2 but neither NVGGŽ .Ž.nor Czdoes. If Ž p. Ž p. NAGGŽ .Žwould be conjugate to a subgroup of NV., then A contains a conjugate V g of V ( 211 by Lemma 1Ž. 1 . But then < A l V g < G 210q11r< AŽ p. < s 27 , which contradicts the facts that there are exactly two conjugates of A in a Sylow 2-subgroup T of G Žthe proof ofw 7, Proposi- tion 20x. and that they intersect with the unique 211 -subgroup of T in exactly a 26 -subspaceŽ see the proof ofwx 7, Proposition 21. . Observing the normalizers, it is easy to see that A is a radical 2-sub- U group of G and NAGGŽ.is not contained in CzŽ., NVG Ž .,orNVG Ž.. F LEMMA 16. If NGGŽ. R NV Ž., then one of the following occurs up to F F U s s ŽO . [ conjugacy: NRGGGGŽ.Cz Ž., NR Ž.NV Ž .; R V, R V V : U0 ( 11 4 ( 6 s ŽO I . [ ( 11 1q6 2 :2 with center VOO2 , or R V V : U I 2 :2 with center F ( 2 3. The subgroups V, V ŽO ., and V ŽO I . are radical 2-subgroups of G with r ( ŽO . r ŽO . ( ŽO I . r ŽO I . ( NVGŽ.V M24, NVGŽ . V L4Ž.2, and NG Ž V . V L3Ž.2. None of the normalizers of those radical groups are conjugate to a U subgroup of CGGŽ. z , NV Ž .Ž., or N G A . F / r Proof. Assume that NRGGŽ.NV Ž.but R V. Then R V corre- sponds to one of the 13 radical 2-subgroups UX of a complement K for r ( r NVGŽ.V M24 up to conjugacy. As the action of NVGŽ.V on V is equivalent to the natural action of M24 on the even part of the Golay s r cocode, it follows from Lemma 8 that ZRŽ.CRV ŽV .is a 1-subspace of s r s V spanned by the 2 A-involution z 0123, if R V UX for seven X F s s containing S. Thus in these cases NRGGŽ.NZR ŽŽ..Cz G Ž.. For X T s r s U F U s I or OT, ZRŽ.CRVG ŽT . V and NRŽ.NVG Ž .. For X T or I s r s F F U OT , ZRŽ.CRVG ŽT . Z and hence NRŽ.NZG Ž.NVG Ž . by Lemma 14Ž. 3 . RADICAL 2-SUBGROUPS 331 s I s F F If X O , then ZRŽ.F and thus NRGGG Ž.NF Ž.NV Ž.by Lemma 14Ž. 4 . Thus it follows from Lemma 1Ž. 2 that the subgroup V ŽO I . [ ŽO I . F V : UO I of NVGGŽ.is a radical 2-subgroup of G.As NV Ž . F ŽO I . s NFGGŽ.NV Ž., it is easy to see NVGGŽ .Ž.NF, and thus ŽO I . U ŽO I . NVGGGŽ .Žis not a subgroup of NV.Žby Lemma 14 4.Ž . As NV . s ( 11 1q6 VUO I . LO I 2.2.L3Ž.2 contains a Sylow 2-subgroup of G,itis < < s 20 s < s 14Ž. 3 , we may take R2 LCC Ž Z .by taking suitable conjugation under C. ŽZ . [ ŽZ .r - 2 We set L LCC Ž. Z , so that L L . Then up to conjugacy we have R s LŽZ ., LŽ¨ .LŽZ ., LŽd.LŽZ .,orLŽt.LŽZ .. Since they have the center Z, their U normalizers are contained in NVG Ž .Žby Lemma 14 3. . Then it follows from Lemma 1Ž. 3 that those four radical groups form a set of complete representatives of classes of radical 2-subgroups with centers conjugate to ŽZ . Ž¨ . ŽZ . Žd. ŽZ . Žt. Ž Z . Z. The normalizers of L , L L , L L , and L L are L . S4 , = = = = LSŽ.Ž.44S , LD 84S , and L . ŽS32 S 4 ., respectively. F s r If NRGGŽ.Cz Ž., then R E or R E is conjugate to one of the r representatives of 13 conjugacy classes of radical 2-subgroups of CzGŽ.E ( 3.M22 . 2Ž. Lemma 12 . It follows from Lemma 1Ž. 4 that the inverse images of those 13 groups as well as E are the representatives of conjugacy classes of radical 2-subgroups with centers conjugate to ²:z . Hence we have obtained in total 2 q 3 q 8 q 14 s 27 radical 2-sub- groups: ( 10 Ž p. ( 4 A 2,A Žwith center Fp 2;. ( 11 ŽO . ( 6 ŽO I . ( 3 V 2,V Žwith center VO 2,. V Žwith center F 2;. U L, LŽ¨ .Ž, L d.Ž, and L t. Žwith center V ( 2;3. LŽZ .Ž, LL¨ .ŽZ .Ž, LLd.ŽZ .Ž, and LLt.ŽZ . Žwith center Z ( 2;2 . E and 13 radical groups corresponding to the representatives of the r ( radical 2-subgroup of CzGŽ.E 3.M22 . 2 Ž with center ²:.z . No two of those radical groups are not conjugate, in view of their centers and by Lemma 1Ž.Ž. 3 4 . Thus we obtained THEOREM 17. There are exactly 27 radical 2-subgroups of J4 with repre- ¨ sentati es shown in Table IX, where E . RX denotes the radical 2-subgroup of ¨ CGXŽ. z corresponding to the representati e R in Lemma 12 of the classes of r ( radical 2-subgroups of CGŽ. z E 3M22 2. The second column of Table IX describes a brief structure of R, in which, e. g., E .2w 7 x means that R has a normal subgroup E with RrE a group of order 27. 4. RADICAL 2-SUBGROUPS OF Co2 Let G s Co2 be one of the simple groups of Conway. This is the stabilizer in Co0, the automorphism group of the Leech lattice ⌳,ofa vector of length 4. It follows fromwx 14, 2.13, Step 1 that every 2-local subgroup of G is contained in one of the following groups up to conjugacy: RADICAL 2-SUBGROUPS 333 ( 1q8 CzGŽ. 2:q S6Ž.2, z a2A-involution of G, ( 1q64 CwGŽ. 2.2q A8, w a2B-involution of G, ( 4q10 = ( 4q10 NKGQŽ.2 Ž.S35S , K Q 2, ( 10 NWGŽ. 2:Ž.M22 .2 . To describe those maximal 2-local subgroups more precisely, we prepare the notation, which is fairly standardwx 6, Chap. 10, 3 . Explicitly, the Leech lattice is presented as follows. We take the MOG arrangement in Subsec- ⍀ tion 2.2, and let Q be the vector space over the rationals Q with basis ex indexed by the elements x g ⍀ s Ä40,1,...,9,...,N . Then the Leech ⌳ lattice is defined to be the set of vectors Ý x g ⍀ aexxwith the following conditions: g ⍀ Ž.i axx Ž .are all integers with the same parity, ' Ž.ii Ý x g ⍀ ax 4a0 Ž.mod 8 , s g ⍀ ¬ ' Ž.iii for each m 0, 1, 2, 3, the subset Ä4x ax m mod 4 is a Golay code word. Then ⌳ forms an integral lattice which is even unimodular with respect s 2 r to the quadratic form q defined by qŽ.ŽÝ x g ⍀ aexxÝ xg ⍀ ax . 8. There is no vector x of ⌳ with qxŽ.s 2. The automorphism group of ⌳ is defined to be AutŽ.⌳ s Ä4 g OV Ž, q .¬ ⌳ s ⌳ . Its isomorphism type is denoted Co0. The group AutŽ.⌳ acts transitively 334 SATOSHI YOSHIARA on the set ⌳Ž.i of vectors x of ⌳ with qx Ž .s 2i for each i s 2, 3, 4. The stabilizer of a vector of ⌳Ž.2 in Aut Ž⌳ .is known to be simple and its isomorphism type is denoted Co2. For explicitness, we choose a vector y ⌳ ⌳ ( 4e014e of Ž.2 and denote its stabilizer in Aut Ž .by G Co2. The action of AutŽ.⌳ on ⌳ Ž.i is not primitive: Ä4x, yxxŽŽ..g ⌳ i is a maximal nontrivial block of imprimitivity for i s 2, 3, but for i s 4, each maximal nontrivial block of imprimitivity in ⌳Ž.4 consists of 24 pairs of " " s y mutually orthogonal vectors x12,..., x 4with Ž.xij, x 0 and xix j g 2⌳ s Ä42 x ¬ x g ⌳ Ž.Ž.Ž1 F i / j F 24 , where a, b s qaŽ.Žq b y qa.y qbŽ..r2. Such a block is called a cross, and each pair of vectors of ⌳Ž.4 consisting of a cross is called its axis. Under the natural homomorphism : ⌳ 2 x ¬ x q 2⌳ g ⌳r2⌳, each cross is sent to a vector of the 24-space ⌳r2⌳ over ⌳r ⌳ F2 . Conversely, if a vector x of 2 is the image Ž.x of a vector x of ⌳Ž.4 , then y1 Žx .l ⌳ Ž.4 is a cross. Thus we may identify a cross with ⌳r ⌳ [" ¬ g ⍀ some vector of 2 . For example, C˜ Ä48ex x is a cross, called ⌺ s ⌺ ⌺ the standard cross. With each sextet Ä416,..., , we may associate the crosses C˜q ⌺ ["4 Ý g ⌺ ae , ¬ a s "1, Ł g ⌺ a s 1, k s 1,...,6 , Ž. Ä4Ž.x kkxx x x x C˜y ⌺ ["4 Ý g ⌺ ae , ¬ a s "1, Ł g ⌺ a sy1, k s 1,...,6 . Ž. Ä4Ž.x kkxx x x x Observe that the corresponding vectors of these crosses in ⌳r2⌳ satisfy Ž.C˜˜q ŽŽ..ŽŽ..Cqy⌺ q C ˜⌺ s 0. The stabilizer of the standard cross C˜ in AutŽ.⌳ has a normal subgroup EŽ.G of sign-changes wxX for X g G, the Golay code, where wxX is a diagonal matrix with y1Ž. resp. 1 in the Ž.x, x -entry iff x g X Žresp. x f X .Ž.. The group E G is isomorphic to the Golay code G via wxX ¬ X, and so EŽ.G ( 212 . The group P of permutation matrices which preserves ( ⌳ G Ž.so P M24 is a subgroup of Aut Ž.stabilizing the standard cross C˜. Then we are allowed to use the same notation as in Subsections 2.2, 2.3 to denote the permutation matricesŽ. or the subgroups of P corresponding to ⍀ the specific permutationsŽ. or the subgroups of M24 on . We will do so from now on. It is known that EŽ.G : P coincides with the stabilizer of C˜ in AutŽ.⌳ . Then the stabilizer GC˜ in G of the standard cross C˜ is the semidirect X X X product of EŽ G . ( 210 with PŽ G .Ž.( Aut M , where E ŽG . [ ÄwxX ¬ X X 22 X X g G 4 for G the truncated Golay codeŽ.Ž see Subsection 2.3 and P G . is the stabilizer ofÄ4 0, 1 in P ( M . Clearly the action of a complement X X 24 PŽ G .Žon E G .Žis the natural action of Aut M22 .on the truncated Golay code. This describes a maximal 2-local subgroup isomorphic to 210 : Ž.M22 .2 . RADICAL 2-SUBGROUPS 335 There are three classes of involutions of G: the centralizers of a 2 A- 1q8 и 1q6 = Ž.resp. 2 B- and 2C- involution are isomorphic to 2q S6Ž.Ž2 resp. Ž 2q 4 10 2 .Ž.L412 and 2 . M 0.. They can be easily distinguished by traces as matrices of AutŽ.⌳ :a2A- Ž resp. 2 B- and 2C-. involution has trace y8, 8, X and 0 on Q ⍀. For example, ywxX for a hexad X of G Žan octad of G X containing 0 and 1. is a 2 A-involution, wxC for an octad C of G Žan octad of G not containing 0 nor 1. is a 2 B-involution, and wxD for a X dodecad D of G is a 2C-involution. The 2-subspace of ⌳r2⌳ is called a sextet space if all of its nonzero vectors are crossesŽŽ.. the images of elements of ⌳ 4 . From the observation above, ² ŽC˜˜ ., ŽCq Ž⌺ ..: is a sextet space for every sextet ⌺. Since we y ⌳ work with the stabilizer G of 4e014e in AutŽ., we only consider quintets ⌺ Ž.sextets with a tetrad containing both 0 and 1 . The sextet space Q˜˜˜[ ² ŽC ., ŽCQq Ž ..: for the standard quintet Q, consisting of the columns of the MOG arrangement, is referred to as the standard quintet space. We describe the structure of the stabilizer GQ˜ in G of the standard X quintet space. Let be a linear transformation of Q ⍀ defined by X Ž.e [ Ž.Ž1r2 Ý g e .y e if Q is the tetrad of Q containing x g ⍀. xyQyi x i [ X wx ⌳ wx Then Q1 is a matrix of AutŽ.6, Chap. 10, 3.3 . It is easy to see y that fixes 4e014e and the cross CQ˜˜˜qyŽ.but flips C and CQŽ.. Thus g _ [ GQC˜˜G . Let E Ä40, 1, 3, 7, B, F, G, K be an octadŽ. a hexad in fact < ⌳ X ⌳r ⌳ image Ž.O in 2 under the natural projection . In the intersection X ⌳ X l ⌳ Ž.O ŽŽ..4 , we can verify that there is a unique 4-subspace O˜ with X q y : ⌳ the property that O˜ Ž.ŽŽ..4e014e 2 . It is straightforward to X X see that O˜ is generated by ŽC˜˜ ., ŽCQqq Ž .., ŽCA ˜ Ž .., and ŽCA ˜ q Ž .., X where A and A are the quintets given in Table X where the tetrad Ai or X X XX AiOis indicated by i. The kernel K ˜˜of the action of GOon O˜ is X calculated inside GCO˜˜. The subgroup L of G acting trivially on the ⌳ X X X s sublattice OOlies in K ˜˜. It is straightforward to verify that LO X wxwxwx y wx ( 4 XXXs = ² Q23, Q 24, A 46 , Oa3: 2 and K OOO˜˜˜M L , where X wx wx MO˜ is the central product of three D8455646-subgroups ² Q , x :², A , w X x w X x X ( a53:², and A 5, a1:²with the common center O :Ž. Inside P G . X X M22 . 2, there is a subgroup L3Ž.2 stabilizing O , which acts on O˜ fixing X ywx X Ž.C˜˜.As ², E : acts also on O , we see that the stabilizer GO˜ X includes the full automorphism group L4Ž.2onO˜ . This also implies that XXr X X GOO˜˜K acts faithfully on LOO˜. Thus G ˜, is an extension of L4Ž.2bya X X w x X normal subgroup K OO˜˜. In particular, O lies in the center of G , and we X X s w x have GOG˜ C Ž O . comparing their orders. The structure of the centralizer of the 2 A-involution ywxH is similarly described as the stabilizer of the hexad space H˜, the image of the sublattice ⌳ [ ⌷ l y HxÝ g HxZe by modulo the 1-subspace ²Ž4e014e .:.Aswe ywxs do not require the detailed information about CGHŽ H .ŽG ˜ espe- ( 1q8 cially on K H˜ 2q . later, it is omitted here. The only information we need is the claimŽ. 1 below, which is well known. Summarizing, we have the claimsŽ.Ž. 1 ᎐ 4 below. ¨ X LEMMA 18. In the notation abo e, the stabilizers GHO˜˜, G , G Q ˜, and G C ˜ in X G of the standard hexad space H˜˜, the standard octad space O , the standard quintet space Q˜˜, and the standard cross C ha¨e the following structures: ¨ ywxs ¨ ywx Ž.1 We ha eCGH Ž H . G˜ for the 2 A-in olution H and 1q8 G rK ( S Ž.2 with kernel K s CHŽ.˜ ( 2.q HH˜˜ 6 HG˜ H˜ RADICAL 2-SUBGROUPS 337 X X ¨ w x s X ¨ w x Xr Ž.2 We ha eCGQ Ž O . G˜˜ for the 2 B-in olution O and GO X XX( s X K O˜˜L4Ž.2,where the kernel K OGCXŽ O. is the direct product of M O˜ q X O˜ ( 1 6 XXs w x s wxwxwxwx 2q ŽŽZMOO˜˜ . ² O:. with L ² Q23, Q 24, A 24, A 46 , X yw x ( 4 XXr Xs Oa3: 2,on which GOO˜˜K naturally acts. In particular, ZKŽ. O˜ X w x = X ² O : LO˜ . ¨ s ywx s Ž.3 We ha eGQQ˜˜K ² , E:Ž with kernel K Q˜OG2 QQ ˜˜. P, where [ 14 y wx¬ RQ˜˜O2Ž. GQ is a special group of order 2 with center ² Q1i s ( 4 ( r ( = i 2,...,6: 2 and PQ˜˜S5. Furthermore, GQQR ˜S35S and s y wx ( 4q10 ( CZRGQŽ Ž˜˜ ..R Q ² E , : 2 S3. The representation of PQ˜S5 ¨ ¨ gi en by ZŽ. RQ˜ is equi alent to that of S5 afforded by the permutation module modulo the tri¨ial submodule. X ( 10 X ( Ž.4 GC˜ is a semidirect product of EŽG . 2 with PŽ G .Ž.Aut M22 , X X where the action of PŽ G .Žon E G . is equi¨alent to the natural action of X AutŽ.M22 on the truncated Golay code G Ž.see Subsection 2.3 . We start the classification of radical 2-subgroups of G s Co2. Let R be X a radical 2-subgroup of G. Then NRGHŽ.is contained in G ˜˜, GO, GQ ˜,or GC˜ up to conjugacy. F / s Assume that NRGCŽ. G ˜˜but R W OG2Ž.C . Then up to conjugacy RrW is one of the representatives of 7 conjugacy classes of radical ( r 2-subgroups of AutŽ.M22 GC˜ W ŽLemma 11 . . As W is the truncated s Golay code for AutŽ.M22 , it follows from Lemma 13Ž. 1 that ZR Ž. r CRW Ž.W is a 1-space generated by either a 2 A-or2B-involution r unless R W corresponds to U˜Q . Thus up to conjugacy X F s ywx F X s yw x r NRGHGŽ.G ˜˜C Ž H .Ž.or NR GOGG C ŽO ., unless R W s r s s U˜˜QQ. In the exceptional case where R W U UaQ²:33, since a fixes every tetrads of Q, ZRŽ.is a 4-subspace of W spanned by 5 2 A-involu- _ s s tions corresponding to hexads Q1iQÄ40, 1 , i 2, . . . , 6. Thus ZRŽ.ZR Ž˜ . F F and NRGGŽ.NZR ŽŽ.. G Q˜ in this case. F r ( = Assume that NRGQŽ. G ˜˜. We have G QQR ˜S35S , where y wxr r RQQ˜˜² E :ŽŽ..R is a Sylow 2-subgroup of the S3-factor CZRGQQ˜R ˜; r r r and the subgroups RrQ˜˜˜˜˜˜²:12, r RQQ, Rr ²12, r , x 56 :RQQ, and Rx ²:56 RQ r of the S5-factor RPQQ˜˜R Q ˜correspond respectively to the representatives ( 2 ( ( ²Ž34 .Ž 56 . , Ž 35 .Ž 46 .: 2 ,²Ž 34 .Ž 56 . , Ž 35 .Ž 46 . , Ž 56 .: D8, and²Ž 56 .: 2of three classes of radical 2-subgroups of S5, considered as the permutation s group on 5 tetrads QiiQŽ.Ž.2, . . . , 6 . Since ZR˜ is the permutation r ( module modulo the trivial submodule for RPQQ˜˜R Q ˜ S5, it is easy to see r r r that if the S5-part of R RQQ˜˜is Rr²:12, r RQQ ˜or Rr ˜ ²12, r , x 56 :RQ˜up to conjugacy, then ZRŽ.s CR ŽrR .is a 1-space generated by a 2 A-in- ZŽ R Q˜. Q˜ ywx F r volution H , and thus NRGHŽ. G ˜˜. If the S5-part of R RQ is 338 SATOSHI YOSHIARA Rx²:rR up to conjugacy, then ZR Ž.s CR ŽrR .is a 3-space Q˜˜56 QZŽ R Q˜. Q˜ ywxs generated by three 2 A-involutions Qi1i Ž.2, 3, 4 . Since they ex- haust the 2 A-involutions of ZRŽ.and their product is a 2 B-involution w X x F w X x s r O , we have NRGGŽ.C Ž O . G O˜˜. If the S5-part of R RQ is r trivial, then ZRŽ.ZR ŽQ˜ ., and it follows from Lemma 1Ž. 3 that there are exactly two classes of radical 2-subgroups of G with centers conjugate to y wx ZRŽ.QQ˜˜, with representatives R and RQ˜ ² E :. Hence we obtained 2 classes of radical 2-subgroups, including RQ˜. F XX/ s X= XXr Assume that NRGOŽ. G ˜˜˜˜˜, but R K OOOM L . Then R K Ois ( r one of the 7 representatives of unipotent radicals of L4Ž.2 GQQ˜˜K , X which correspond to flags of proper subspaces of the 4-space LO˜ . Since X X XXXs w x = s l r Xs w x = ZKŽ.²OOOO˜˜˜˜O :Ž.Ž.Ž.²L , ZR ZK CR K O : XX CRXŽ.rK .If RrK corresponds to a flag containing a point, then LOO˜ ˜˜ O r X wx CRLOXŽ.K ˜ is a 1-space, which we may take ²Q34:Ž.because L 4 2 acts O˜ X X s w xw x transitively on the 1-subspaces of LO˜ . Then ZRŽ. ² O , Q34 : ywx F ywx contains a unique 2 A-involution H , and so NRGGŽ.C Ž H ..If r X Ž.r X R K O˜ corresponds to a line or a line-plane flag, then CRLO˜X K ˜ is a wxwx O 2-space, which we may take ² Q24, Q 34 :Ž.by the transitivity of L4 2on X X s w xw xw x the 2-subspaces of LO˜ . Then ZRŽ. ² O , Q24, Q 34 :, which has ywxy wxy wx exactly 2 A-involutions QH , Q13, and Q 14 . Thus NRGŽ. w X x F w X x centralizes their product O , and NRGGŽ.C Ž O .. Then it follows from Lemma 1Ž. 3 that there are exactly two classes of radical 2-subgroups w X xw xw x with centers conjugate to ² O , Q24, Q 34 :. In the remaining case r X where R K O˜ corresponds to a plane, the same argument shows that there X is a unique class of radical 2-subgroups with centers conjugate to ²wO x, wxwxwx Q24, Q 34, A 24 :, because this group has exactly 5 2 A-involutions X w x X whose product is O . Hence together with K O˜ we found in total 4 new classes of radical 2-subgroups. F s ywx Then we may assume that NRGHGŽ.G ˜ C Ž H . up to conjugacy, and we obtain 1 q 7 classes of radical 2-subgroups with centers conjugate y wx r to ² H :Ž.Ž.by Lemma 1 4 , as OG2 HH˜˜is extraspecial and G OG2 Ž.H˜ ( S6Ž.2 is a group of Lie type of rank 3 in characteristic 2. Hence we obtained in total 15 s 1 q 2 q 4 q 8 classes of radical 2-sub- groups from the representatives of maximal 2-local subgroups, and no two of them are conjugate in view of their orders and centers and by Lem- ma 1Ž.Ž. 3 4 . THEOREM 19. There are in total 15 classes of radical 2-subgroups of Co2. The brief descriptions of structures of the representati¨es and their normalizers X are gi¨en in Table XI, where sywxE . RADICAL 2-SUBGROUPS 339 5. RADICAL 2-SUBGROUPS OF Th It follows fromwx 15, Theorem 2.2 that every 2-local subgroup of G [ Th is contained in one of the following groups up to conjugacy: ( 1q8 CzGŽ. 2.A9 for an involution z of G, ( 5 и 5 NAGŽ. 2 L5Ž.2 for some 2 -subgroup A. There is a single class of involutions of G. For an involution z, CzGŽ.has a unique orbit of length 9 over the set of totally singular 4-spaces of r OC2ŽŽ..²:G z z . Two subspaces in that orbit intersect trivially. The inverse images of subgroups of that orbit are 2 5-subgroups of G, called nice with respect to z. The second maximal 2-local subgroup is described as the 5 normalizer of a nice 2 -subgroup A.As NAGŽ.acts transitively on the involutions of A, the notion of nice subgroup does not depend on the choice of an involution contained in A. For an i-subspace F of A Ž.i s 2, 3, 4 , A is the unique nice 25 -subgroup containing F, because in the r orthogonal space OC2ŽŽ..²:G z z , the images of nice subspaces intersect trivially, where z is any involution of F. As there are only two maximal 2-local subgroups, the radical 2-sub- groups of G are easily determined as follows. Let R be a radical 2-sub- F / ( 5 r group of G. Assume NRGGŽ.NA Ž.but R A 2 . Then R A is conjugate to one of the 15 representatives of classes of unipotent radicals r ( r of NAGŽ.A L5 Ž.2 , which correspond to flags of the 5-space A.If R A s r corresponds to a flag containing a point, then ZRŽ.CRA ŽA .is a r 1-space so that NRGGŽ.lies in Cz Ž.up to conjugacy. If R A corresponds to one of 7 other flags, ZRŽ.is an i-space of A for some i s 2, 3, 4. By the remark in the last paragraph, A is a unique nice 2 5-subgroup containing F ZRŽ., and therefore NRGG Ž.NA Ž.. Then it follows from Lemma 1 Ž. 3 340 SATOSHI YOSHIARA that there are exactly 4Ž. resp. 2 and 1 classes of radical 2-subgroups of G with centers conjugate to a specific 22 -Ž resp. 234 - and 2 -. subgroup of A. 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