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 an irreducible character of NRŽ.rR. Then ŽR, .is called a B-weight if G has p-defect 0 and BŽ. s B Ž.in the sense of Brauer , where BŽ.is the block of NRŽ.containing . 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
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0021-8693r97 $25.00 Copyright ᮊ 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 ⌽ 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 ⌽ G, p if 1 xHxyg⌽Ž.G,pfor some x g G. We shall follow the notation ofwx 7 . In 1q2␥ 1q2␥ 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 ގ, we use Ep n or simply p to denote the n elementary abelian group of order p , ޚn 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, :for some 2 10 3 2 3 g A _ G with g G. Thus < 0 ÝIrr Ž.NrCRRŽ. Ž.2.1 R is the number of B0-weights, where R runs over the set ⌽Ž.G, p such that the p-part 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 ޚ and complement ޚnfor Ž. Ž. positi¨e integers m, n. Moreo¨er, NRA rN,ޚ2 for all R g ⌽ 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 ޚ4o¨er Z SL2 3 s Ž. Ž. Ž. ⍀14ޚ.Moreo¨er, NRA rN,ޚ2for all R g ⌽ 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,ޚ2for all R g ⌽ 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. ⌿ Ä2Ž2. 1q6444 Ž. where S g Syl28G . Let s 2,2 *,D, 2q , 2 .2 , S . Then NA R rN Ž. Ž. ,ޚ2 for R g ⌿, and NA R s N for R g ⌽ G,2 _⌿. 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/Rg⌽G, 2 , then bywx 5, Theorem 9 , NR Ž Ž Ž ... Ž. F N ⍀1 ZR FGiM for some i, so that we may suppose R g ⌽ M i,2 and NRŽ.NR Ž.. s Mi Ž. Ž. Ž . Ä U 4 1 Set L s L342 . Then ⌽ L,2 s 1, E , E4, SЈ , where D8, SЈ 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 ⌽Ž.M18,2 sÄ42 ,D ,2 ,2 Ž.*,2 = SЈ, D8= E4, D8= E4, D8= SЈ , 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 ⌽Ž.M, 2 such that NR Ž.Ž.NR, then R Ä2,2D4, 41g s M1 g 8 2Ž. and so 2 , D8 gG ⌽ G,2 . Ž. Ž. 1q6 ⌽Ž. 2 Let Q s OM22s2.Ifq RgM2, 2 , then Q F R and RrQ Ž. gL ⌽L, 2 . Thus we may suppose ⌽ 1q6 1q6 2 1q6 2 Ž.M2,2 sÄ42qq,2 .2 ,2Ž. q.2 *, S ,2.3Ž. 38 JIANBEI AN 1q62 Ž1q62. U Ž. where 2q .2 s Q.E4 , 2q .2 * s Q.E42and S g Syl G . For each R ⌽ŽM,2 . , NR Ž.NR Ž., so we may suppose ⌽Ž.Ž.M ,2 ⌽ G,2 . g 2 s M2 2 : 1q62 Ž1q62. Moreover, 2qq .2 and 2 .2 * are normal in S. By a theorem of Burnsidewx 10, Theorem 7.1.1 , they are conjugate in G if and only if they Ž. 1q62 Ž1q62. are conjugate in NSG sS. Thus 2q .2 /G 2q .2 *, since they are non-conjugate in M2 . Ž.3 Now ⌽ ŽM , 2 . has 7 subgroups R, and NRŽ.NR Ž.. More- 3 s M3 over, ⌽ 6 2 6 4 4 1q6 2 2 3 4 2 3 4 Ž.M3,2 sÄ42 ,2 .2 ,2 .2 ,2q.2 ,2 .2 .2 ,2Ž.2 .2 .*, S ,2.4 Ž. Ž.Ž. and we may suppose ⌽ M3,2 :⌽ G, 2 . Similarly, ⌽ 6 2 6 4 4 1q6 2 2 3 4 2 3 4 Ž.Ž.Ž.M4,2 sÄ42 *, 2 .2 *,2 .2 ,2Ž.q.2 *,2 .2 .2 ,2Ž..2 .2 *, S , Ž.2.5 Ž.Ž. and we may suppose ⌽ M4,2 :⌽ G,2 . Ž. Ž.Ž 2.Ž. 4 Let Q s OM25s2 * and L s L34 . The subgroups of Ž. Ž. ⌽M5, 2 can be obtained as follows, and only two of them satisfy NRs NRMŽ.. 5 Ž.Ä U 4 Ž. U Suppose ⌽ L,2 s 1, E16, E 16 , SЈ , where SЈ g Syl2 L , and E16, E 16 are unipotent radicals of parabolic subgroups of L associated with compo- Ž. sitions 21 and 12, respectively. If R g ⌽ M5, 2 , then P s RrQ gM Ž.Ž. ŽŽ.. 5 ⌽M52rQ, 2 and P : P l L F 2. Thus P l L s ONPLlLand P l Lis radical in L. We may suppose P l L g ⌽Ž.L,2 . Suppose P l L s 1 and P / 1. If x g Q is an element of type 2 A, then Ž. ²: Ž. CxsQ.L.␥, where ␥ induces a field automorphism on L s L3 4. ² : Ž Ž ..² : Ž We may suppose P s ␥ . Thus R s OCx28␥,D cf.w 11, p. .Ž.Ž. Ž. 1877x , and NR,R=L3 2 , so that R gG ⌽ G, 2 . Bywx 11, Table , G Ž. has exactly one 2-block with a defect group D8, so that R gG ⌽ M1,2 . U Ž. Suppose P s E16or E 16. Then NQ.P is a maximal subgroup of G, CQŽ..P CQ Ž..P Q.Pand NQŽ..PQ.P L Ž.4.S. We may sup- s MM55s r ,23 6UŽ6.²:U²: pose Q.E16 s 2 and Q.E16 s 2 *. Suppose P s E16. ␥ or E16. ␥ . ²: 34 U²: Ž34. Then we may suppose Q.E16 . ␥ s 2 .2 and Q.E16. ␥ s 2.2*.In addition, CRŽ. 23 and NRŽ.R.L Ž.2. MM55, s 2 Suppose P SЈ. Then R 2.2,44CRŽ. 24 , and NRŽ.R.3 Ž= s , MM55, s .Ž44.Ž44.Ž . ² : 3 .2, since N 2.2 s2.2.S33=S . Suppose P s SЈ. ␥ . Then R s Ž. SЉgSyl25M . Thus we may take 2 6 6 3 4 3 4 4 4 ⌽Ž.Ž.M58,2 sÄ42 *, D ,2 ,2 Ž.Ž.*,2 .2 ,2.2 *,2 .2 , SЉ .2.6 Ž. Finally, it follows easily by the orders of radical subgroups that Ž. NRA rN,ޚ234for all R g ⌿. Bywx 7, p. 104 , M s M for some g A _ G THE ALPERIN AND DADE CONJECTURES 39 6 6 26 26 and soŽ 2 . s Ž2.Ž.Ž.Ž *. By 2.4 and 2.5 , 2 .2 .s Ž2 .2. * for some Ž. gA_G. Similarly, L3 2 .2 has only one class of maximal subgroups 2 Ž. Ž . 1q6 Ž. Ž1q62. 2.L2 2 . Since NMA 2 s2.q L3 2 .2, it follows that 2q .2 s Ž 1q62.Ž. 2q .2 * for g NMA 22_M. The last assertion now follows easily. Given B g BlkŽ.G , let DB Ž.be a defect group of B, IrrŽ.B the set of irreducible ordinary characters of B, kBŽ.s 0 Ž.2E Let G s He and let Blk Ž.G, p be the set of p-blocks with a non-tri¨ial defect group. Ž. 0Ž.Ä4 Ž. aIf p s 7, 5, or 2, then Blk G, p s B01, B , where D B1, ޚ p Ž. except when p s 2, in which case D B1 sG D8. In the notation of w7, p. 105x , Ä4,,,,if p 7, ¡ 15 17 18 29 32 s IrrŽ.B1s~Ä4612132226,,,,if p s 5, ¢Ä412, 14, 15, 16, 22 if p s 2, Ž. Ž.ŽŽ. 0Ž.. and Irr B01s Irr G _ Irr B j Irr G . Moreo¨er, ¡3if p s 7, ¡10 if p s 7, lBŽ.10s~~4if p s 5, lBŽ.s 14 if p s 5, Ž.2.7 ¢¢3if p s 2, 11 if p s 2. Ž. 0Ž.Ä4Ž. bIf p s 3, then Blk G,3 s B01234,B ,B ,B ,B , where D B1sG Ž. E9 and D BiGs 3* for i s 2, 3, 4. In the notation of wx7, p. 105 , IrrŽ.B1s Ä4k: k g Ä42, 3, 7, 8, 10, 11, 12, 13, 16 , Ž. Ä4Ž. Ä4Ž. Ä4 Irr B2s 192732, , , Irr B 3s 41720, , , Irr B 4s 51821, , , Ž. Ž.Ž4 Ž. 0Ž.. Ž . Ž . and Irr B0 s Irr G _ Dis1Irr Bi j Irr G . Moreo¨er, lB01slB Ž. s7and l Bi s 2 for i s 2, 3, 4. 0Ž.Ä4 Ž. Proof. Let B g Blk G, p _ B0. We may suppose D s DBg Ž. Ž. ⌽G,p.If ps2, then bywx 11, p. 1877 , D sG D81and Irr B is given above. Moreover, bywx 4, Theorem 2 , lBŽ.s3. Suppose p is odd. If Ž.D, b is a Sylow B-pair and is the canonical 0 character of b, then CDDŽ. /Dand g Irr ŽŽCDD .rD ., so that D g Ä7, 5, 3*, 324 . Ž. Ž. Let D , ޚ p, so that CDsD=L, where L , L257orL,As Ž. Ä4 0Ž. Ä4 L2 5 according to whether p g 7, 3 or p s 5. Then Irr L s St or 0 Ž. berg character of L. Let s 1DD= and let b be the block of CD 0Ž. containing , where g Irr L and 1D is the trivial character of D. Then Ž. Ž. Ž. Ž. D,bD is a maximal Brauer pair of ND, since NDrCD is a Ž. Ž.Ž. pЈ-group. If D, bD is a B-pair for some B g Blk G , then D, bD is a Ž. Ž. Ž. Sylow B-pair, so that DBsG D. Since DB is cyclic, lB is the number of B-weights bywx 8, Theorem 9.1 . But NDŽ.rCDD Ž.is cyclic and N Ž. s 0 0 NDŽ., so< Irr ŽŽ..N , < 0 lGŽ.s lBŽ.qIrr Ž.G , D0 BgBlk Ž.G, p where lGŽ.is the number of p-regular conjugacy G-classes. This com- pletes the proof. Ž.2F Let G s He and B be a p-block of G. Then the number of B-weights is lŽ. B . Ž. Ž . Proof. We may suppose DB is non-cyclic. If B s B0 , then 2F Ž. Ž.Ž. follows by 2.1 and 2A ᎐ 2E . If B / B01, then p s 2 or 3 and B s B . Ž. 0ŽŽ . . If p s 3, then DBsG E9990and< Irr NE rE 0 0 Irr ŽŽ.NRrR .. Now Irr ŽŽ.NRrR .has exactly one character, which is an Ž. Ž. extension of the Steinberg character of CRrZR. It follows that B1 has 3 weights. ThusŽ. 2F follows. 3. DADE’S INVARIANT CONJECTURE We shall also follow the notation ofwx 8, 9 . Given a p-subgroup chain C : P01- P - иии - Pn Ž.3.1 Ž. Ž . Ž . of a finite group G, define < NCŽ.sNCG Ž.sNP Ž01 .lNP Ž .lиии l NP Žn ..3.2 Ž . The chain C ofŽ. 3.1 is said to be radical if it satisfies the following two conditions, k Ž.a P OG Ž . andŽ. b P ONP 0s pks pFŽ.j ž/j1 s for 1 F k F n. Denote by R s RŽ.G the set of all radical p-chains of G. Suppose ZGŽ.s1. Then we can identify G with its inner automor- Ž. Ž. Ž. phism group Inn G , so that Geᎏ A s Aut G and O s Out G s ArG. Let NCAAŽ.be the stabilizer of C in A, and NCŽ., the stabilizer of ŽŽ.. Ž. Ž. Ž . gIrr NC in NCAA, so that NC FNC,. Thus NCAGOŽ.Ž.Ž.,rNC,NC,sNC A Ž.,GrG. Given B g BlkŽ.G , C g R Ž.G , an integer d G 0, and U F O, let G BlkŽŽ.NC< B.sÄbgBlkŽŽ..NC : b sB4, and let kNCŽŽ.,B,d,U .be the number of characters in the set IrrŽ.NCŽ.,B,d,U G sÄ4gIrrŽ.NCŽ.:B Ž. sB,NCO Ž, .sU,d Ž.sd, Ž.3.3 where BŽ. is the block of NCŽ.containing and dŽ. is the p-defect of . The following is Dade’s invariant conjecture. Ž. Ž. Dade’sIn¨ariant Conjecture.IfZG sOGp s1 and B is a p-block of G with defect dBŽ.)0, then for any integer d G 0 and any subgroup 42 JIANBEI AN UFOutŽ.G , < 4. RADICAL p-CHAINS The notation and terminology of Sections 2 and 3 are continued in this Ž. Ž. Ž. Ž. Ž. section. Let G s He, C g R G , CC sCCGG, and NC sNC. Ž.4A In the notation of Ž.2A , the radical 7-chains C of HeŽ up to conjugacy. are CNŽ.CNAŽ.C CŽ.1:1 GA 36 CŽ.2:1-7 F72=LŽ.7 F 72=LŽ.7 33 63 CŽ.3:1-7-SЈ F77=FF 77=F 22 2 CŽ.4:1-7 7 .SL22Ž. 7 7 .SLŽ. 7 .2 21q21q21q2 CŽ.5:1-7-7qq 7 .6 7q .Ž. 2 = 6 1q21q21q2 CŽ.6:1-77.qqŽ.S3=37q Ž.S3=6, Ј Ž3Ž.. Ž Ž.. 1q2 where S g Syl77F = L 27. Moreo¨er, regarding NA C 6 r7q and ŽŽ.. 1q2 Ž1q2.Ž. NCA 5r7qqas subgroups of Out 7 s GL2 7 , we may suppose ŽŽ.. 1q2 ²: ŽŽ..1q2²: NCA 6r7q , a,b,c and NA C 5 r7q , b, c , where a s Ä4 Ž.01 ŽŽ..²1q22 : diag 2, 4 , b s 10, and c s 2 I2 . Thus N C 6 r7q , a, b, c and ŽŽ.. 1q2 ² 2: NC5 r7q , bc . Proof. The proof follows easily byŽ. 2A ,wxwx 7, p. 104 , or 5, p. 74 . Ž.4B In the notation of Ž.2B , the radical 5-chains C of He Ž up to conjugacy. are CNŽ.CNA Ž.C CŽ.1:1 GA 4 CŽ.2:1-55Ž.=A555.4 F = S 22 4 4 CŽ.3:1-5-55.4Ž.=2 F55=F 22 2 CŽ.4:1-55:4Ž.Ž.A445:4S . THE ALPERIN AND DADE CONJECTURES 43 Ž 2 .ŽŽ2.. Proof. Since NA 5 rON545 ,4S, it contains a Sylow 2-subgroup Ž2.Ž. 2 of Aut 5 , GL2 5 . Thus the fixed-point set of a line of 5 in Ž 2 .ŽŽ2.. Ž Ž .. 44 NA 5rON5 5is4=4, and so NCA 3 ,F55=F. Similarly, ŽŽ.. 4 NCA 2,F55=S. Suppose p s 3 and follow the notation ofŽ. 2C . Define the radical 3-chains CiŽ.for 1 F i F 6as CŽ.1:1 CŽ.2:1-3 CŽ.3:1-3-32 CŽ.4:1-3* CŽ.5:1-3* - SЈ CŽ.6:1-32,4Ž..1 ŽŽ.. qŽ. where SЈ g Syl3 N 3* . Let Blk G be the set of p-blocks with a non-cyclic defect group. 0 Ž.Ž.4C a Let R Ž.G be the G-in¨ariant subfamily of RŽ.G such that 0 RŽ.GrGsÄCiŽ.:1FiF6.4 Then < < for each B g BlkqŽ.G and all integers d G 0 and w G 0. Ž.bSuppose C is a chain gi¨en by Ž4.1 .with<< C G 1. Then CNCŽ. NCAŽ. CŽ.23.S773.S = 2 22 22 CŽ.Ž.Ž.Ž.Ž.33=2.2=S333=2.2=S=2 CŽ.4S32=L Ž.7S32=LŽ.7.2 CŽ.5S33=SS33=S=2 22 22 CŽ.63 Ž=2.GL3 .22 Ž. Ž 3=2 . .GL Ž. 3 .2. Ž. Ž. Ž.Ž. Proof. a Let S g Syl3 G such that ZS s3g⌽G,3 . If C:1- Ž. Ž. Ž. Ž. Ž. ZS -Sand CЈ :1-S, then NCAAsNCЈand NC sNCЈ. Thus the contributions of C and CЈ in the sumŽ. 3.4 are zero. Similarly, we may delete the chains C :1-32 -S and CЈ :1-ZSŽ.-32-Sin the sum Ž.3.4 . The remaining radical 3-chains have representatives Ci Ž.given by Ž.4.1 . This proves Ž. a . Ž. Ž. Ž. bIfRs3*, then bywx 5, p. 73 , NRFM,S42=L 7 . But NMŽ.S=L Ž.7 .2, so NC ŽŽ..5 S=H, where H NQŽ.for A ,42 A ,3 s L2Ž7..2 44 JIANBEI AN Ž Ž.. Ž. Ž. ² : some Q g Syl32L 7 . But L27.2sPGL2 7 , so H , x, y, z , S3= ŽŽ.. ŽŽ.. Ž. 2, where x s aZ GL22 7 and y s bZ GL 7 with a, b given by 4A , and Ä4ŽŽ.. Ž. Ž. zsdiag 1, y1 Z GL2 7 . The rest of 4C follows bywx 7, p. 104 , 2C , or the proof ofŽ. 2E . Suppose p s 2 and follow the notation ofŽ. 2D . Define radical 2-chains CiŽ.for 1 F i F 18 as 1q6 CŽ.1:1 CŽ.2:1-2q 61q62 6 CŽ.3:1-2-2.2q CŽ.4:1-2 CŽ.5:1- Ž2*644 . -2.2 CŽ.6:1- Ž2* 6 . 61q62 6 44 CŽ.7:1- Ž2* . -Ž.2.2*q CŽ.8:1- Ž2* . -2.2 -S CŽ.9:1-2644-2.2 CŽ.10 : 1 - 2644- 2.2 -S CŽ.11 : 1 - 2.244-SCŽ.12 : 1 - 2.244 CŽ.13 : 1 - Ž.2*244-2.2 CŽ.14 : 1 - Ž.2*2 CŽ.15 : 1 - Ž.2*26- Ž.2* C Ž.16 : 1 - Ž.2*2644-2-2.2 CŽ.17 : 1 - Ž.2*26-2 CŽ.18 : 1 - Ž.2*2- Ž.2* 644-2.2. Ž.4.2 We have the following proposition: 0 Ž.Ž.4D a Let R Ž.G be the G-in¨ariant subfamily of RŽ.G such that 0 RŽ.GrGsÄCkŽ.:ks1, 2, . . . , 184 . Then < < CNCŽ. NCCNCAA Ž. Ž. NC Ž. Ž. Ž.1q6 Ž. Ž Ž.. C1 He AC 22.q L32 NC2.2 Ž . Ž1q62 . Ž Ž .. Ž . 6 Ž Ž .. C 3 2q .2 .SNC363C4 2 .3.SNC4 44 6 CŽ5 . Ž 2 .2 . . ŽS33= SNC . Ž Ž5 ..C Ž6 . Ž 2 . *.3.SNC6 Ž Ž6 .. Ž. Ž1q62 . Ž Ž.. Ž. C7 2q .2 *.SNC37C8SS 44 CŽ.9 Ž 2 .2 .Ž . S33= SNC . Ž Ž..9 C Ž10 .SS 44 CŽ.11 SS.2 CŽ.Ž12 2 .2 .Ž . S33= SNC . ŽŽ..12 .2 44 2 CŽ13 . Ž 2 .2 . . Ž 3 = 3.2 .NC Ž Ž13 .. .2 C Ž14 . Ž 2 . *.L33 Ž4. .SNC Ž Ž14 .. .2 644 CŽ15 . Ž 2 . *.L23 Ž4. .SNC Ž Ž15 ..C Ž16 . Ž 2 .2 . . Ž 3 = 3.2 .NC Ž Ž16 .. 644 CŽ17 . 2 .L23 Ž4. .SNC Ž Ž17 ..C Ž18 . Ž 2 .2 . . Ž 3 = 3.2 .NC Ž Ž18 .. . THE ALPERIN AND DADE CONJECTURES 45 Moreo¨er, if P is the final subgroup of the chain C, then CŽ. C s Z Ž. P except Ž. Ž. Ž2.Ž. when C s C 14 , in which case C C , 2*.L3 4. Proof. Ž.aIfCgR ŽG .is given inŽ. 3.1 with < X X CЈ :1-P1- иии - Pm.4Ž..3 Ž. ⌽Ž. Ä 1q64 Case 1 . Let R g M2 ,2 _ 2q . Define G-invariant subfamilies MqŽ.Rand M 0Ž.R of R Ž.G , such that q X M Ž.R rG s Ä CЈ g RrG : P1s R4 , and 0 Ј X 1q6 X M Ž.R rG s Ä4C g RrG : P1s 2q, P2s R .4.4Ž. For CЈ g MqŽ.R given by Ž 4.3 . , the chain Ј 1q6 X X иии X gCŽ.:1-2q-P12sR-P - - Pm Ž.4.5 Ž. 0Ž. Ž . ŽŽ .. Ž . Ž. is radical, gCЈgM R, and NCЈ sNgCЈ .By2D,NRA F Ž1q6.Ž.ŽŽ.Ј Ј. 0Ž. NA 2q and so NCAAsNgC . For any B g Blk G and for any integers d, w G 0, kNCŽ.Ž.Ž.Ј,B,d,wskNgCŽ.Ž.Ј ,B,d,w.4.6Ž. 1q6 Since G has exactly one class of radical subgroups 2q , g is a bijection between MqŽ.R and M 0Ž.R . So we may suppose 0 C f DŽ.Mq Ž.R j M Ž.R . ⌽Ž.Ä41q6 RgM2,2 _ 2q Ä 1q62Ž1q62. 4 1q6 In particular, P1 f 2.2,2.2*,qq S, and if P1s 2q , then C sG CŽ.2. Ž. ÄŽ 26. 234Ž234.4 Ž. Case 2 . Let ⍀* s 2 .2 *, 2 .2 .2 , 2 .2 .2 * : ⌽ M4 , 2 and ⍀ Ä 264 Ž. s2.2 :⌽M3, 2 . Given Q g ⍀ j ⍀*, define G-invariant subfamilies 0 0 MqŽ.Qand M Ž.Q of R Ž.G such that MqŽ.Q rG and M Ž.Q rG are Ž. 1q66Ž6. given by 4.4 , with R replaced by Q and 2q by 2 or 2 * according to whether Q g ⍀ or ⍀*. The same proof as above shows that we may suppose 0 C f DŽ.Mq Ž.Q j M Ž.Q . Qg⍀j⍀* 46 JIANBEI AN Ž 6 . Thus P11f ⍀ j ⍀*. Moreover, if P s 2 * and < q X 6 X L Ž.W rG s Ä4CЈ g RrG : P12s 2 , P s W , and 0 X 6 X 4 4 X L Ž.W rG s Ä4CЈ g RrG : P123s Ž.2 *, P s 2 .2 , P s W .4.7Ž. A similar proof to above shows that we may suppose C f ŽLqŽ.W j L0Ž..W. 234 234 644 6 6 Let W s 2 .2 .2 orŽ 2 .2 .2. *. Replace 2 by 2 .2 andŽ 2. * by 2 in the definitionŽ. 4.7 of LqŽ.W and L 0Ž.W , and repeat the proof above. Ä 6 Ž 6 . 444 Then the remaining 2-chains C with P1 g 2 , 2 *, 2 .2 , up to conju- gacy are CjŽ.for 3 F j F 12. Now we may suppose 2 2 P18g Ä42 ,2Ž.*, D . q 0 Case Ž.3 . Let M Ž.D88and M Ž.D be given by Ž. 4.4 with R replaced 1q62 Ž. Ј qŽ. by D8 and 2q by 2 . Then 4.6 holds for C g M D8 and we may 2 Ž. suppose P18/ D . Moreover, if P1s 2 and < Ž. Ž 2.Ž. Case 4 . Suppose P12s 2 * and P g ⌽ M5, 2 . As shown in the proof ofŽ. 2.6 , N Ž2 6.2 6 L Ž.4.S.Thus⌽ŽŽN 26. 2,26 . M5 r , 23 M5 r s IJ: 4 ²: Ž . 1, ␥ , E42, X , where ␥ g Syl S3with ␥ acting as a field automor- Ž. Ž Ž.. Ž Ž. . phism on L24, E 4gSyl 22L 4 , and X g Syl22L 4.S 3. We may take ⌽ŽŽN 2,26.. Ä2,2.2,2.2,63444SЉ4⌽Ž.M, 2 , where 234 .2 2. 6²:␥ , M5 s : 5 s 2.2442. 6E, and SЉ Syl ŽŽN 26.. . As shown in the proofŽ. 2D , part s 42g M5 Ž.4, NY Ž.NY Ž.N Ž26.Ž for each Y ⌽ N Ž26.. , 2 . Moreover, by MM55s l g M5 ŽŽ 2 .. 2 Ž. wx7, p. 104 , NA 2*,2.L3124.D and by the structures of maximal subgroups of L Ž.4.D ,7,p.23,NR2 Ž.NR Ž.for 312wxNAŽŽ2 .*. sM5 6 6 3 4 3 4 R g Ä42 ,2Ž.*,2 .2 ,2 Ž.2 .* : ⌽ ŽM5,2 . . 34 0 6 6 For W s 2 .2 , define LqŽ.W and L Ž.W as Ž 4.7 . with 2 andŽ 2. * replaced byŽ 22 . * and 244 .2 by 2 6 . The same proof as above shows that we Ž qŽ. 0Ž.. 34 6 may suppose C f L W j L W . Thus P2 /G 2 .2 and if P2 s 2 Ä44 4 and < kNCŽ.Ž.Ј,Bii,d,wskNgCŽ.Ž.Ž.Ј ,B,d,w for any integers d, w G 0. So we may delete CЈ and gCŽ.Ј in the sum Ž. 3.4 . Using the same argument above, we can cancel each of the following pairs 48 JIANBEI AN ŽŽ..CЈ,gCЈ of chains, 2 CЈ s CЈŽ.2 gC ŽЈ .:1- Ž2 .*-D884-D =E 2 U CЈsCЈŽ.3gC ŽЈ .:1- Ž2* . -D884-D=E-D 8=SЈ 2 U CЈsCЈŽ.4gC ŽЈ .:1- Ž2* . -D884-D=E 2 CЈsCЈŽ.5gC ŽЈ .:1- Ž2* . -D8848-D=E-D=SЈ 2 CЈsCЈŽ.6gC ŽЈ .:1- Ž2* . -D88-D=SЈ, where CЈŽ.j for 2 F j F 6 are defined byŽ. 4.8 . This completes the proof of Ž.a. Ž.b The proof ofŽ. b follows by that ofŽ. a above or bywx 7, p. 104 . 5. THE PROOF OF DADE’S CONJECTURE The notation and terminology of Sections 2, 3, and 4 are continued in this section. In the proof of TheoremsŽ. 5A and Ž. 5B , we only give a detail proof for one case and sketch the proofs of other cases, since the proofs are similar. Ž.5A Let B be a p-block of the simple Held group G s He when defect dBŽ.G1. If p is odd, then B satisfies Dade’sin¨ariant conjecture. Ä4 Ž. Proof. We may suppose p g 7, 5, 3 , and B s B00s BGwith p s 7 or 5. Ž. Ž. Ž . 3 Ž. Suppose p s 7 and let C s C 2,CЈsC3 . Then NC sF72=L 7, Ž. 6 Ž. Ž . 33 Ž. 63 NCA sF72=L7,NCЈ sF 77=F, and NCAЈsF77=F.So kNCŽ.Ž.Ž.2,B00,d,wskNCŽ.Ž.Ž.3,B,d,w 15 if d s 2 and w s 2, s ½ 10 if d s 2 and w s 1, 0 otherwise. Ž. Ž. Ž . 2 Ž. Ž . Let C s C 4 and CЈ s C 5 . Then NC ,7 .SL2 7 , NCA , 2 Ž. ŽЈ . 1q2 Ž.Ј 1q2Ž.Ž 7 .SL2 7 .2, NC ,7q .6, and NCA ,7.2q =6 see Tables I and . Ž.Ž.²:Ž.²: II . In the notation of 4A , NCЈrRs and NCA ЈrRsb,c, 2 ŽŽЈ .. 1q2 where s bc and R s ONC7 ,7q . It follows by the character table of NCŽ.Ј that c stabilizes all linear characters and only two non-lin- ear characters, 14and 18 of IrrŽŽNCЈ ... Ž. Ž. Ž. Ž. Let g NCAA_NC. Then h g NCЈ for some h g NC and Ž.2 Ž. Ž. Ž. hgNCЈ, so that h g NCA Ј_NCЈ. Since we only consider the Ž. Ž. ŽŽ.. actions of an element of NCA Ј_NCЈ on Irr NCЈ or classes of Ž. Ž. NCЈ, we may suppose, for simplicity of notation that g NCA Ј. THE ALPERIN AND DADE CONJECTURES 49 TABLE I 2 The Degrees of Characters of IrrŽŽ.. 7 .SL2 7 123456789101112131415161718 1 3 34466 6 7 8 8 48484848484848 The characters 111,..., in Table I are the extensions of characters of IrrŽŽ..Ž. SL2 7 to NC. It follows by the character tables of GL2Ž. 7 and SL2Ž. 7 that interchanges characters in each of the pairs Ž.23, and Ž. 45,, and stabilizes the other characters jfor 1 F j F 11. If 12 F j F Ž NŽC.Ž.. Ž NŽC.Ž.. 18, then the inner product jN, Ind ŽCЈ. 18 s 1 and jN, Ind ŽCЈ.i/ 0 for exactly one i with 11 F i F 17. It follows that stabilizes exactly one ŽŽ.. character j g Irr NC of degree 48. Thus kNCŽ.Ž.Ž.4,B00,d,wskNCŽ.Ž.Ž.5,B,d,w ¡7ifds3 and w s 2, 10 if d s 3 and w s 1, s~ 1ifds2 and w s 2, ¢0 otherwise. ŽŽ.. 1q2Ž. The degrees of irreducible characters of NC6 ,7.q S3 =3 are Ä4 given by Table III. Since each character of ⌿ s j :1FjF9 covers the Ž 1q2 . Ž Ž.. Ž Ž.. trivial character of Irr 7q , an element g NCA 6_NC6 stabilizes Ž. Ž. each character of ⌿. Since g ZS3 =6 and since Res NŽCŽ5.. s 18 Ä4 for g 21, 22, 23 , it follows that s . Similarly, the restriction to NCŽŽ.5 of each character g IrrŽNC ŽŽ...6 of degree 9Ž. or 6 covers exactly oneŽ non-stabilized . character of Irr ŽNC Ž Ž5 ... of degree 3 Ž or 6 . , so /. It follows bywx 7, p. 105 that ¡10 if d s 3 and w s 2, 10 if d s 3 and w s 1, kGŽ.Ž.,B00,d,w skNCŽ.Ž.6,B,d,w.s~ 3ifds2 and w s 2, ¢0 otherwise. Thus Dade’s conjecture holds when p s 7. TABLE II Ž.1q 2 The Degrees of Characters of Irr 7q .6 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 1111113333666666642 50 JIANBEI AN TABLE III ŽŽ1q 2 Ž ... The Degrees of Characters of Irr 7q . S3 = 3 1 2 3 4 5 6 7 8 9101112131415 111111222666666 16 17 18 19 20 21 22 23 9 9 9 9 18 42 42 42 Suppose p s 5 and let C s CŽ.2 and CЈ s CŽ.3 . Then NC Ž ., Ž5= .Ž.2Ž. Ž.Ž. A5 .4, NCЈ ,5.4=2 . By the structures of NCAAand NCЈgiven by Ž.4B and character tables of NCŽ.and NC Ž.Ј, kNCŽ.Ž.Ž.2,B00,d,wskNCŽ.Ž.Ž.3,B,d,w 12 if d s 2 and w s 2, s ½ 2ifds2 and w s 1, 0 otherwise. It follows byw 2,Ž. 3.1 ; 7, p. 105x that 8ifds2 and w s 2, kGŽ.Ž.,B,d,w kNCŽ.Ž.4,B,d,w 8ifd 2 and w 1, 00s s½s s 0 otherwise. So Dade’s conjecture holds when p s 5. Suppose p s 3. It follows byŽ. 4C that kNCŽ.Ž.Ž.4,B00,d,wskNCŽ.Ž.Ž.5,B,d,w 9ifds2 and w s 2, s ½ 0 otherwise. Ž. Ž. Ž . Ž22.Ž . Ž . Let C s C 3 and CЈ s C 6 . Then NC , 3 =2.2=S3 and NCЈ Ž 22.Ž.Ž . ,3=2 .GL2 3 see Tables IV and V . Using the method of central ŽŽ. . ÄŽ.4 ŽŽ . . ÄŽ.4 characters, we have Blk NC< B11s bC, Blk NCЈ TABLE IV 22 The Degrees of Characters of IrrŽŽ 3 = 2.2 . Ž=S3 .. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 111122223333466666612 THE ALPERIN AND DADE CONJECTURES 51 TABLE V 22 The Degrees of Characters of IrrŽŽ 3 = 2 . .GL2 Ž 3 .. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 11222333333466688162424 Ž.Ž. 2 Ž . Ž. Ž. Ž. d,w/2, 2 . If R s 3 g ⌽ G, 3 , then NRA rCR,GL2 3 .2 s NQŽ.Q, where Q Syl ŽL Ž..4 . Using the character table of AutŽ L3Ž4.. r g 33 GL2Ž. 3 .2 and the restriction of each character of IrrŽNC ŽЈ ..to NC Ž .,we Ž. Ž. have that an element g NCA Ј_NCЈ interchanges characters in each of the pairs, Ž.Ž.Ž 7, 11, 8, 10 , and 13, 14 ., and stabilizes the other characters in IrrŽŽNCЈ ... Thus 3ifds2 and w s 2, kNCŽ.Ž.Ž.6,B,d,w kGŽ.,B,d,w 6ifd 2 and w 1, 11s s½s s 0 otherwise, and kNCŽ.Ž.Ž.3,B00,d,wskNCŽ.Ž.Ž.6,B,d,w 9ifds3 and w s 2, s ½ 2ifds2 and w s 2, 0 otherwise. Ž.ŽŽŽ... It suffices to show that kG,B00,d,w skNC2,B,d,w and ŽŽŽ.. . ŽŽŽ.. . Ž. kNC3,B11,d,wskNC2,B,d,wfor all d, w G 0 see Table VI . ŽŽŽ.. . Again using the method of central characters, we have Blk NC2 IrrŽ.bC1Ž.Ž.2 sÄ4j:jgÄ43,4,5,10,11,13,14,15,20 . TABLE VI The Degrees of Characters of IrrŽ. 3.S7 1234567891011121314 116612141414141515202121 15 16 17 18 19 20 21 22 30 30 35 35 42 42 48 48 52 JIANBEI AN TABLE VII 2 The Degrees of Characters of IrrŽŽ 2 . *.L3 Ž4 .. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 1 10101010101020282828282828353535 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 36 36 36 45 45 63 63 64 64 64 64 70 70 70 90 90 90 ŽŽ.. ŽŽ.. Ž ŽŽ.. . Since NCA 2sNC2 =2, it follows that kNC3,B1,d,ws ŽŽŽ.. . Ž. kNC2,B1,d,wfor all w, d G 0. Thus 5A follows by 9ifds3 and w s 2, kGŽ.Ž.,B,d,w kNCŽ.Ž.2,B,d,w 4ifd 2 and w 2, 00s s½s s 0 otherwise. Ž.5B Let B be a 2-block of the simple Held group G s He with defect dBŽ.G1. Then B satisfies Dade’sin¨ariant conjecture. Ž. Ž. Proof. 1 Let B s B1, so that DBsG D8. Bywx 8, Lemma 6.9 , ŽŽ. . ÄŽ. Ž .4 Ž. Ž. Blk NC< B1 sлunless C g C 1,C14 . If C s C 14 , then CC , Ž 2.Ž.Ž.Ž2.Ž.Ž . 2*.L334 and NC , 2*.L4.S3See Tables VII and VIII . Byw 4, ŽŽ.. Ä4ŽŽ. . Theorem 1x , Irr bC11s51,62,62,82,9, where Blk NC< B1s ÄbC1Ž.4. Ž. Ž. Now we consider the action of an element g NCA _NC on the Ž Ž .. Ž.Ž 2 .Ž. Ž. characters of Irr NC , where NCA ,2*.L3124.D . Set L s L 34, Ž. Ž . so that Aut L , L.D12 see Table IX . By character tables of L.D12 and L.S3 and by Clifford theory, interchanges two characters in each of the following pairs, Ž.Ž.Ž. 78, , 9101314, , , , and Ž 1718, ., and stabilizes every other character of IrrŽ.L.S3 . Since each character of IrrŽ.L.S3 can be extended to a character of NCŽ., it follows that stabilizes every character of ⍀sÄ4j:1FjF5, j g Ä410, 15, 16, 19, 20, 23, 26, 27 TABLE VIII 2 The Degrees of Characters of IrrŽŽ 2 . *.L33 Ž4. .S . 1 2 3 4 5 6 7 8 9 1011121314151617 1 1 2 2020303030304045454545646490 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 90 105 105 108 108 126 126 126 128 168 192 192 210 210 270 270 THE ALPERIN AND DADE CONJECTURES 53 TABLE IX The Degrees of Characters of IrrŽŽL33 Ž4. .S .. 1234 5 6 7 8 91011121314151617181920 1 1 2 20 20 40 45 45 45 45 64 64 90 90 105 105 126 126 126 128 and interchanges characters in each pair, Ž.Ž.Ž. 11, 12, 13, 14, 17, 18 , Ž. ŽŽ.. and 24, 25 . Let s 32 and Ј s 33 be characters of Irr NC , and Ä4ŽŽ.. let X s 32, 33, 34 : Irr CC . Then stabilizes X, and and Ј cover every character g X. Moreover, CCŽ.is a subgroup of index 2 in the stabilizer NŽ. of in NC Ž.. Since is an involution modulo NC Ž., Ž. Ž. stabilizes at least one character, say of X. Thus NA rCC ,ޚ2= Ž. Ž. Ž Ž. Ž.. ޚ2 , since NCA rCC ,D12 and g ZNA CrCC .So has 4 extensions to NAAŽ. and NC Ž.has 4 irreducible characters covering .In particular, stabilizes both and Ј. The same proof can be applied to each of the following pairs of characters in IrrŽŽ..NC ,Ä4Ä4Ä421, 22, 28, 29 , and 30, 31 . Extend each character of IrrŽL3 Ž..4toCC Ž.. Using the character table of CCŽ.,we Ä4 NŽC.Ž. NŽC.Ž. have 23g Y s , 5, 7. Since IndCŽC.289s q and IndCŽC. Ä4 s67qfor g Y, it follows that 6789, g , . Thus stabilizes Ä4 exactly each of the characters in ⍀ j 21, 22 , j :28FjF33 . The con- jecture for B1 follows by 4ifds3 and w s 2, kGŽ.Ž.,B,d,w kNCŽ.Ž.14 , B , d, w 1ifd 2 and w 2, 11s s ½s s 0 otherwise, and moreover, ¡4ifds9 and w s 2, 4ifds9 and w s 1, 6ifds8 and w s 2, kNCŽ.Ž.Ž.14 , B0, d, w s~8ifds8 and w s 1, Ž.5.1 4ifds7 and w s 2, 2ifds6 and w s 2, ¢0 otherwise. Ž. Ž . Ž . Ž . 2 Let B s B0 and let C s C 15 and CЈ s C 16 . Then NC , Ž 6.Ž.Ž.Ž44.Ž . Ž.Ž. 2*.L234.S, NCЈ , 2.2.3=3 .2. Since NCAsNC and Ž. Ž. Ž. Ž. NCA ЈsNCЈ, it follows by character tables of NC and NCЈ that 54 JIANBEI AN kNCŽ.Ž.Ž.,B00,d,wskNCŽ.Ј,B,d,w ¡8ifds9 and w s 1, 14 if d s 8 and w s 1, s~4ifds7 and w s 1, Ž.5.2 2ifds6 and w s 1, ¢0 otherwise. Since NCŽ Ž17 .., NC Ž Ž15 .. and NC Ž Ž18 .., NC Ž Ž16 .. , Ž 5.2 . still holds when C s CŽ.17 and CЈ s C Ž.18 . Ž.Ј Ž. Ž . Ž1q62.Ž.Ј Let C s C 3 and C s C 4 . Then NC , 2.2.q S3 and NC , 6 2 .3.S6. By character tables of NCŽ.and NC ŽЈ ., ¡16 if d s 10, 12 if d s 9, 10 if d s 8, kNCŽ.Ž.,B0,ds~ 6ifds7, 1ifds6, ¢0 otherwise, ¡16 if d s 10, 12 if d s 9, 2ifds8, kNCŽ.Ž.Ј,B0,ds~ 2ifds7, 1ifds6, ¢0 otherwise. Since NCŽ Ž7 ..,NC Ž Ž3 .. and NC Ž Ž6 ..,NC Ž Ž4 .. , the above equations still hold when C s CŽ.7 and CЈ s C Ž.6. Ž. Ž. Ž . Ž 44.Ž . Let C s C 5 and CЈ s C 8 . Then NC , 2.2.S33=S and NCŽ.Ј sS. By character tables of NCŽ.and NC Ž.Ј, ¡16 if d s 10, 20 if d s 9, 2ifds8, kNCŽ.Ž.,B0,ds~ 2ifds7, 1ifds6, ¢0 otherwise. ¡16 if d s 10, 20 if d s 9, 18 if d s 8, kNCŽ.Ž.Ј,B0,ds~ 6ifds7, 1ifds6, ¢0 otherwise. THE ALPERIN AND DADE CONJECTURES 55 Since NCŽ Ž9 ..,NC Ž Ž5 .. and NC Ž Ž10 .., NC Ž Ž8 .. , the above equations Ž. Ž . ŽŽ.. ŽŽ.. still hold when C s C 9 and CЈ s C 10 . Since NCjA sNC j for 3 F j F 10, it follows that 10 jy1 ÝŽ.y1 kNCjŽ.Ž. Ž.,B0,d,w js3 16 if d 8 and w 1, ys s Ž.5.3 s ½0 otherwise. Ž. Ž. Ž44.Ž . Ž . Ž 44.Ž If C s C 13 , then NC , 2.2.3=3 .2 and NCA ,2.2.3= 32.Ž=2 . . As shown in the proof ofŽ.Ž.Ž. 2D , part 4 , NQ AutŽ L3Ž4.. , Ž.Ž2.ŽŽ.. NCA r2*,whereQgSyl 23L 4 . Using character tables of NQŽ.and NC Ž.Ž22.Ž * and using a similar argument to that of 1. , AutŽ L3Ž4.. r we have that kNCŽ.Ž.Ž.13 , B00, d, w s kNCŽ.Ž.Ž.14 , B , d, w . 1q6Ž. Ј Ž. Ž.Ž. Let R s 2,q CsC2 , and C s C 11 , so that NC sNR, 1q6 Ž. ŽЈ . Ž . Ž Ž.. 2.q L3232 and NC sSgSyl G .If KsAut R.L 2 , then K , 6Ž. Ž . Ž .Ž . 2.L3 2.2,NCA rZR see Table X . Using character tables of K and Ž. Ž. Ž. NC and Clifford theory, we have that g NCA _NC interchanges characters in each of pairs, Ž.Ž.Ž.Ž.Ž. 2, 3, 5, 8, 6, 7, 12, 13, 15, 21 , Ž.Ž.Ž.16, 18, 17, 20 , and 22, 23 , and stabilizes the other characters. Ž. Ž . Ž. Similarly, if Q g Syl2 K , then Q , NCA ЈrZS. Using the character NŽ2 1q 6 . q Ž. Ž. table of Q and using the induction Ind S for g Irr S , we have Ž. that g NCA Ј_Sstabilizes exactly 4 linear characters, 2 irreducible characters of degree 2 or 4, 4 of degree 8, and 1 of degree 16 of IrrŽ.S .It TABLE X ŽŽ..1q 6 The Degrees of Characters of Irr 2q .L3 2 1 2 3 4 5 6 7 8 9 101112131415 1336777778814142121 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 21 21 21 21 21 21 24 24 28 28 42 48 56 56 64 56 JIANBEI AN follows that ¡4ifds10 and w s 2, 12 if d s 10 and w s 1, 2ifds9 and w s 2, ␣1 if d s 9 and w s 1, 2ifds8 and w s 2, kNCŽ.Ž.Љ,B0,d,ws~␣2if d s 8 and w s 1, 4ifds7 and w s 2, 2ifds7 and w s 1, 1ifds6 and w s 2, ␣3 if d s 4 and w s 2, ¢0 otherwise, Ž.Ž.Ž. Ž. where ␣123, ␣ , ␣ s 2, 0, 1 or 18, 16, 0 according to whether CЉ s C 2 or CŽ.1. Ž. Ž. Ž44.Ž . Finally, let C s C 12 , so that NC , 2.2.S33=S . Let H s 44 NCŽ Ž13 .., Ž2.2.3.Ž=3.2 .FNC Ž .. Using the results of IrrŽ.H , the in- duction from H to NCŽ., and the character table of NCŽ., we have that ¡4ifds10 and w s 2, 12 if d s 10 and w s 1, 2ifds9 and w s 2, 1 if d s 9 and w s 1, kNCŽ.Ž.Љ,B0 ,d,ws~2ifds8 and w s 2, 2ifds7 and w s 2, 1ifds6 and w s 2, 2 if d s 4 and w s 2, ¢0 otherwise, Ž.Ž.Ž. Ž. where 12,  s 18, 0 , or 2, 1 according to whether CЉ s C 12 or CŽ.1 . It follows that jy1 ÝŽ.y1 kNCjŽ.Ž. Ž.,B0,d,w jgÄ41,2,11,12,13,14 16 if d 8 and w 1, s s Ž.5.4 s ½0 otherwise. ThusŽ. 5B follows by Ž.Ž. 5.2 , 5.3 , and Ž. 5.4 . This completes the proof. THE ALPERIN AND DADE CONJECTURES 57 REFERENCES 1. J. L. Alperin, Weights for finite groups, in ‘‘The Arcata Conference on Representations of Finite Groups,’’ Proc. Sympos. Pure Math., Vol. 47, Amer. Math. Soc., Providence, RI, 1987. 2. Jianbei An, The Alperin and Dade conjectures for the simple Tits groups, New Zealand J. Math. 25 Ž.1996 , 107᎐131. 3. Jainbei An and M. 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