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The Alperin and Dade Conjectures for the Simple Held Group

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The Alperin and Dade Conjectures for the Simple Held Group JOURNAL OF ALGEBRA 189, 34]57Ž. 1997 ARTICLE NO. JA966856 The Alperin and Dade Conjectures for the Simple Held Group Jianbei An Department of Mathematics, Uni¨ersity of Auckland, Auckland, New Zealand Communicated by Walter Feit Received February 16, 1996 1. INTRODUCTION Let G be a finite group, p a prime, and R a p-subgroup of G. Then R ŽŽ.. ŽŽ.. is radical if ONRppsR, where ONR is the maximal normal Ž. Ž. p-subgroup of the normalizer NRsNRG . Let B be a p-block and w an irreducible character of NRŽ.rR. Then ŽR, w .is called a B-weight if w G has p-defect 0 and BŽ.w s B Ž.in the sense of Brauer , where BŽw.is the block of NRŽ.containing w. Alperinwx 1 conjectured that the number of B-weights should equal the number of irreducible Brauer characters of B. Here a weight is always identified with its G-conjugates. In this paper we verify the conjecture for the simple Held group He. In the paperswx 8, 9 , Dade has presented a conjecture exhibiting the number of ordinary irreducible characters of a fixed height in B, in terms of an alternating sum of similar integers for p-blocks of some local subgroups of the group G. In view of Dade's statementwx 9 , his final conjecture need only be verified for finite non-abelian simple groups. In this paper we also confirm the final conjecture for G s He. Bywx 9 , the final conjecture is equivalent to the invariant conjecture whenever a finite group has a trivial Schur multiplier and an outer automorphism group all of whose Sylow r-subgroups are cyclic. Thus it suffices to confirm the invariant conjecture for G. Most of the calculations were carried out using the CAYLEY computer systemwx 6 . In Section 2 we first classify the radical p-subgroups up to conjugacy and determine their local structures. The approach using CAY- LEY to find all radical p-subgroups of G is explained inwx 3 . If p s 2, then by Bulterwx 5 , it suffices to classify the radical 2-subgroups R in each Ž. Ž.Ž 2-local maximal subgroups M of G such that NRMGsNRand these 34 0021-8693r97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved. THE ALPERIN AND DADE CONJECTURES 35 2-subgroups will be used to classify radical 2-chains in Section 4. The fusions of these 2-subgroups in G can be obtained by applying a theorem of Burnside. After the classification we verify Alperin's weight conjecture. In Section 3 we state Dade's invariant conjecture and fix some notation. In Section 4 we determine radical p-chains of G up to conjugacy. If p s 2, then a lot of cancellations can be done in the alternating sum of radical chains. It turns out that each subgroup in the remaining radical 2-chains is a radical subgroup of G. The final section then verifies Dade's invariant conjecture for p odd and p s 2 separately, the latter being the more complicated case. The difficult part in this section is to determine the stabilized irreducible characters of the normalizer of a radical 2-chain C, under the actions of the stabilizer of C in the automorphism group of G. In order to find these characters we apply the automorphism group of the normalizer of C together with the help of Clifford theory and the induc- tion from a suitable subgroup. 2. RADICAL p-SUBGROUPS AND WEIGHTS Let BlkŽ.G be the set of all p-blocks of G, and F ŽG, p .a set of representatives for conjugacy classes of radical subgroups of G. For y1 Ž. H,KFG, we write H FGGK if xHxFK; and write H g F G, p if 1 xHxygFŽ.G,pfor some x g G. We shall follow the notation ofwx 7 . In 1q2g 1q2g particular, pq is an extra special group of order p with exponent p or plus type according to whether p is odd or even. If X and Y are groups, we use X.Y and X : Y to denote an extension and a split extension of X n by Y, respectively. Given n g N, we use Ep n or simply p to denote the n elementary abelian group of order p , Zn or simply n to denote the cyclic group of order n, and D2 n to denote the dihedral group of order 2n. Let G s He and A s AutŽ.G , so that G F A and A s ²G, t :for some 2 10 3 2 3 t g A _ G with t g G. Thus <<G s 2 ? 3 ? 5 ? 7 ? 17, and we may sup- pose p g Ä42, 3, 5, 7 , since both conjectures hold for a block with cyclic defect groupsŽ seewx 8, 9. Ž . Ž. Ž. Ž. Given R g F G, p , let CRsCRGGand N s NR. In addition, let Irr 0Ž.H be the set of ordinary irreducible characters of p-defect 0Ž seew 8, Ž. .Ž. 5.5x for the definition of a finite group H.If B00sBGis the principal p-block of G, thenŽ cf.w 3,Ž. 1.3x. , 0 ÝIrr Ž.NrCRRŽ. Ž.2.1 R is the number of B0-weights, where R runs over the set FŽ.G, p such that the p-part <CRRŽ.rR<<p of CRRŽ.rR<is 1. The character table of 36 JIANBEI AN NRŽ.rCRR Ž. can be created by CAYLEY, so that we can find the 0 number< Irr ŽŽ..NrCRR<. The proofs ofŽ. 2A and Ž. 2B follow easily bywx 5, Theorem 15 and 2.3 , and the proof ofŽ. 2C is straightforward by usingwx 7, p. 104; 5, 2.2 . Ž.2A The non-tri¨ial radical 7-subgroups R of HeŽ up to conjugacy. are 0 RCRŽ. N Irr Ž.NrCRRŽ. 3 77=L272Ž.7 F=L Ž.7 22 2 77 7.SL2Ž.71 1q21q2 777.qqŽ.S3=39, n where Fmmis the Frobenius group with kernel Z and complement Znfor Ž. Ž. positi¨e integers m, n. Moreo¨er, NRA rN,Z2 for all R g F G,7 . Ž.2B The non-tri¨ial radical 5-subgroups R of HeŽ up to conjugacy. are 0 RCRŽ. NrR Irr Ž.NrCRRŽ. 55=AA55.4 22 55SL3414,2Ž. Ž. Ž. Ž Ž.. where SL22 3 4 is the central product of SL 3 and Z4o¨er Z SL2 3 s Ž. Ž. Ž. V14Z.Moreo¨er, NRA rN,Z2for all R g F G,5 . Ž.2C The non-tri¨ial radical 3-subgroups R of HeŽ up to conjugacy. are 0 RCRNŽ. rRIrr Ž.NrCRRŽ. 33.AS77 3* 3* = L22Ž.72=L Ž.7 2222 33=2 2 .GL2Ž. 3 2 1q2 33q D8 5, where for a subgroup H F G, H* is another subgroup of G such that H* , H Ž. Ž . and H* /GAH. Moreo¨er, NRrN,Z2for all R g F G,3 . THE ALPERIN AND DADE CONJECTURES 37 Ž.2D The non-tri¨ial radical 2-subgroups R of He Ž up to conjugacy . are 0 RCRNŽ.rCRR Ž. Irr Ž.NrCRRŽ. 22 22=L33Ž.2S 22 Ž.2* Ž. 2*.L33 Ž.4 S D832=LŽ.21 66 223.S6 1 66 Ž.2* Ž. 2* 3.S6 1 1q6 22q L3Ž.21 26 2 2.2 2 S33=S 1 26 2 Ž.2.2* Ž. 2 * S33=S 1 44 4 2.2 2 S33=S 1 1q62 2.2q 2 S3 1 1q62 Ž.2.2*q 2 S3 1 234 2.2.2 2 S3 1 234 Ž.2.2.2* 2 S3 1 S 21 1, Ž. C Ä2Ž2. 1q6444 Ž. where S g Syl28G . Let s 2,2 *,D, 2q , 2 .2 , S . Then NA R rN Ž. Ž. ,Z2 for R g C, and NA R s N for R g F G,2 _C. Ä4 Proof. Given i g 1, 2, 3, 4, 5 , let Mi be a maximal subgroup of G such Ž. 1q6 Ž. 6 Ž6. that M14, S = L 32,M 2,2.qL332,M ,2:3.S 64,M,2*:3.S 6, 2Ž. Ž. Ž. and M533, 2.L 4.S.If1/RgFG, 2 , then bywx 5, Theorem 9 , NR Ž Ž Ž ... Ž. F N V1 ZR FGiM for some i, so that we may suppose R g F M i,2 and NRŽ.NR Ž.. s Mi Ž. Ž. Ž . Ä U 4 1 Set L s L342 . Then F L,2 s 1, E , E4, S9 , where D8, S9 g U Syl24Ž.L , and E , E4are unipotent radicals of parabolic subgroups of L associated with compositions 21 and 12, respectively. Thus we may suppose 2 4 4 2 U FŽ.M18,2 sÄ42 ,D ,2 ,2 Ž.*,2 = S9, D8= E4, D8= E4, D8= S9 , Ž.2.2 2 Ž . Ž. Ž. 42 Ž4. where 2 s OM21sOS 24,D 8gSyl 24S ,2 s2 =E4, and 2 * s 22=EU.If R FŽ.M, 2 such that NR Ž.Ž.NR, then R Ä2,2D4, 41g s M1 g 8 2Ž. and so 2 , D8 gG F G,2 . Ž. Ž. 1q6 FŽ. 2 Let Q s OM22s2.Ifq RgM2, 2 , then Q F R and RrQ Ž. gL FL, 2 . Thus we may suppose F 1q6 1q6 2 1q6 2 Ž.M2,2 sÄ42qq,2 .2 ,2Ž.
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