Counting Characters in Blocks ofM22

191, 1]75 Ž1997. JA966917

JOURNAL OF ALGEBRA ARTICLE NO.

Counting Characters in Blocks of M22 Janice F. Huang Department of Mathematics, Box 70663, East Tennessee State Uni¨ ersity, Johnson City, Tennessee 37614 Communicated by Walter Feit Received October 7, 1996

INTRODUCTION This paper will verify one case of a conjecture made by Everett C. Dade w11x in 1990 concerning the number of characters in certain blocks of finite groups. The conjecture is called the McKay]Alperin]Dade conjecture Žor the M-A-D conjecture. as it is based on an earlier conjecture of McKay as extended by Alperin w2x. It will be verified here for all primes p and all p-blocks B of all the covering groups of the Mathieu group M22 . In their 1989 paper w22x, R. Knorr ¨ and G. Robinson translated Alperin’s conjecture into a statement involving an alternating sum of ordinary irreducible characters of normalizers of normal chains of p-groups. Dade used this approach in formulating the M-A-D conjecture, which states that the number of complex irreducible characters with a fixed defect d, in a p-block B of a finite group G, can be expressed as an alternating sum of the number of complex irreducible characters with the same defect d in related blocks B9 of certain p-local subgroups of G. The alternating sums used in the conjecture are over conjugacy classes of radical p-chains C of G. The local subgroups involved are the normalizers NG Ž C . of these chains. Dade claimed that his conjecture, in its most general form, holds for arbitrary finite groups if it holds for all covering groups of finite, nonabelian, simple groups. Therefore, the verification of that conjecture is reduced to a verification of it for all such covering groups. As mentioned above, this paper provides the verification for the covering groups of M22 . The form of the conjecture which we verify is equivalent to the most general form for the covering groups of M22 because the outer automor1 0021-8693r97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.

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JANICE F. HUANG

phism group of M22 is cyclic. In Sect. 1 we introduce the notation and terminology used in the rest of the paper. Section 2 contains information about the group M22 . We include a description of the covering groups, as well as the outer automorphism group, of M22 . Dade has proven w13x that the form of the conjecture which we use here holds for all p-blocks with cyclic defect groups. This implies that the M-A-D conjecture holds for all p-blocks of covering groups of M22 when p ) 3. In Sect. 3, we verify that the M-A-D conjecture holds for the 3-blocks of any covering group, and in Sect. 4, we verify the conjecture for the 2-blocks.

1. DEFINITIONS AND TERMINOLOGY 1.1. Radical p-Chains and Radical p-Groups Let G by any finite group and p be any prime. Define H F G as ‘‘H is a subgroup of G.’’ Denote the center of G by ZŽ G .. We shall depict conjugation in G exponentially so that s t s ty1st and H t s ty1 Ht for s , t g G and H F G. If H, K F G, then NK Ž H . and CK Ž H . will be used to denote the normalizer and centralizer, respectively, of H in K. As usual, Op Ž G . will denote the largest normal p-subgroup of G. We use Syl p Ž G . for the family of all Sylow p-subgroups of G. If P is a p-group, then V Ž P . will denote its subgroup generated by all s g P satisfying s p s 1. The commutator of two elements g and h of G will be denoted by w g, h x s gy1 hy1 gh. We shall frequently follow the notation for groups used in the Atlas of Finite Groups w8x. For example, 2 4 : S4 will represent a split extension of a normal elementary abelian subgroup of order 2 4 by the symmetric group S4 on 4 letters. However, when we wish to identify the elementary abelian subgroup as E, we shall denote this same split extension by E i S4 . In general, Sn and A n will denote, respectively, the symmetric and alternating groups on n letters. The quaternion and dihedral groups of order 8 will be denoted by Q8 and D 8 , respectively. As usual, Zn will denote the cyclic group of order n. We use GLŽ n, q . to denote the general linear group of n = n matrices with entries in the field of order q. Also, SLŽ n, q . will denote the special linear subgroup of GLŽ n, q . which consists of all matrices with determinant 1. Further, PSLŽ n, q . will denote the factor group SLŽ n, q .rZŽ SLŽ n, q ... Following the convention used in the Atlas, L nŽ q . will also be used to denote the group PSLŽ n, q .. The symplectic group SpnŽ q . is the subgroup of GLŽ n, q . consisting of those elements which preserve a given nonsingular, alternating, bilinear form f Ž x, y .. Notice that any element of SpnŽ q . must have determinant 1, and, therefore, the general and special symplectic groups are the same. We use the

COUNTING CHARACTERS IN BLOCKS

3

Atlas notation w8x to denote the projective symplectic group PSpnŽ q . by SnŽ q .. We shall frequently use a superscript to identify a particular copy of a known group. For example, D Ž1. 8 will indicate a particular group isomorphic to D 8 . In w3x, Alperin and Fong defined radical p-subgroups as follows. DEFINITION 1.1.1. A radical p-subgroup of G is a subgroup P F G satisfying P s Op Ž NG Ž P . . . Since NG Ž P . F NG Ž V Ž ZŽ P ... for any p-subgroup P of G, it is clear that Ž1.1.2. P is a radical p-subgroup of G if and only if it is one of NG Ž V Ž ZŽ P .... Hence the nontrivial radical p-subgroups of G are among those of the NG Ž H . where H runs over the nontrivial elementary abelian p-subgroups of G. This fact is frequently useful in identifying radical p-subgroups of a group G. DEFINITION 1.1.3. A p-chain C of G is any nonempty, strictly increasing chain C : P0 - P1 - P2 - ??? - Pn of p-subgroups Pi of G. The length of such a chain is < C < s n G 0, the number of its inclusions. For a chain C of length n and an integer i s 0, 1, . . . , n we use Ci to denote the initial subchain P0 - P1 - ??? - Pi of length i. Denote by C s C Ž G . the family of all p-chains of G. Let K be a group acting as automorphisms of G. If s g K, then s sends the p-chain C of Ž1.1.3. to the p-chain C s : P0s - P1s - ??? - Pns . The stabilizer of C in any such group K is the ‘‘normalizer’’ NK Ž C . s NK Ž P0 . l NK Ž P1 . l ??? l NK Ž Pn . . In addition, NK Ž C . s NK Ž Cny1 . l NK Ž Pn . ,

if n G 1.

As Dade w11, p. 191x has mentioned, Knorr ¨ and Robinson w22x first introduced the idea of radical p-chains. The following definition uses Dade’s notation w11x.

4

JANICE F. HUANG

DEFINITION 1.1.4. A radical p-chain of G is a p-chain C: P0 - P1 ??? - Pn of G satisfying P0 s Op Ž G . and Pi is a radical p-subgroup of NG Ž Ciy1 . for i s 1, . . . , n.

Ž 1.1.5.

Note that Ž1.1.5. is equivalent to Pi s Op Ž NG Ž Ci . .

for i s 1, . . . , n,

Ž 1.1.6.

and that condition Ž1.1.6. also holds for i s 0. Denote by R s RŽ G . the family of all radical p-chains of G. Clearly R is invariant under the conjugation action of G on p-chains. We denote by RrG an arbitrary set of representatives for the G-orbits in R. 1.2. Notation for Characters and Blocks Let F be a finite algebraic extension of the field Q p of p-adic numbers such that F is a splitting field for all subgroups of G. Denote by R the integral closure in F of the ring Z p of all p-adic integers. Then R is a valuation ring by w9, Sect. 19x. We denote by p the unique maximal ideal of R. The expression IrrŽ H . will be used to denote the set of all irreducible F-characters of any subgroup H of G. If H F G, then a character f g IrrŽ H . lies under a character x g IrrŽ G . if f is an irreducible constituent of the restriction x H of x to an F-character of H. In this case we also say that x lies o¨ er f . By the Frobenius reciprocity law w20, 5.2x, this happens iff x is an irreducible constituent of the F-character f G of G induced by f . The expression IrrŽ G N f . will be used to denote the set of all x g IrrŽ G . which lie over f . Further, if K is any group acting on G, then we denote by Kx the stabilizer in the group K of any character x g IrrŽ G .. By a p-block B of G we mean a block of the group algebra RG of G over R. The set of all p-blocks of G will be denoted by BlkŽ G .. For x g IrrŽ G . we use B Ž x . to indicate the unique p-block to which x belongs. If B is in BlkŽ G . then IrrŽ B . denotes the set of all characters in IrrŽ G . which belong to B. Also IrrŽ B N f . will be the intersection IrrŽ B . l IrrŽ G N f .. The set of all defect groups of B will be denoted by DefŽ B ., and the defect of B will be denoted by dŽ B ., so that < D < s p dŽ B . for any D g DefŽ B .. We set k Ž X . s
Ž 1.2.1.

whatever the expression X might be. Thus, for example, k Ž B . s
COUNTING CHARACTERS IN BLOCKS

5

the unique nonnegative integer such that aŽ x Ž 1. . s aŽ < G <. y d Ž x . .

Ž 1.2.2.

If H is a subgroup of G and b g BlkŽ H . then b will represent the unique block of G induced by b, when it exists. It is well known w14, III.9.4x that the induced block b G is defined for any block b of any subgroup H F G satisfying G

PCG Ž P . F H F NG Ž P . for some p-subgroup P of G. We shall apply this when H is the normalizer NG Ž C . of some radical p-chain C: P0 - P1 - ??? - Pn of G. Since Pn CG Ž Pn . F NG Ž C . F NG Ž Pn ., the induced block b G is defined for any block b of NG Ž C . Žsee w22, 3.2x.. 1.3. Extensions by Central Subgroups We recall from the Atlas w8, 4.1x that a central extension or co¨ ering group of the group G is a pair Ž G, l. consisting of a finite group G and an epimorphism l of G onto G whose kernel N is a central subgroup of G. Thus l

1ªNªG ª Gª1 is an exact sequence. The group G is called a proper covering group of G if N is contained in the commutator subgroup G9 s w G, G x of G. The groups N that arise for proper covers of G are each quotients of a largest one called the Schur multiplier M Ž G . of G Žsee w8, 4.1x.. For perfect groups G, such as M22 , there is a unique proper universal covering group G of which any proper cover of G is a quotient and for which N s M Ž G . Žsee w8, 4.1x.. 1.4. The Conjecture We fix a finite group G and a prime p. Let AutŽ G . denote the automorphism group of G, and InnŽ G . be the normal subgroup of all inner automorphisms of G. Then the factor group OutŽ G . s AutŽ G .rInnŽ G . is the outer automorphism group of G. For a p-chain C of G the expression NOu tŽG.Ž C . will indicate the image in OutŽ G . of the subgroup NAutŽG.Ž C . of AutŽ G .. If, in addition, x g IrrŽ NG Ž C .. then NOu tŽG.Ž C .x denotes the image in NOutŽG.Ž C . of NAutŽG.Ž C .x , which is the stabilizer of both C and x in AutŽ G .. If s g G, we point out that NAutŽG.Ž C s .x s is the conjugate Ž NAutŽG.Ž C .x . s 9 of NAutŽG.Ž C .x by the image s 9 of s in InnŽ G .. It follows that NOutŽG. Ž C s . x s s NOutŽG. Ž C . x , for any s g G.

6

JANICE F. HUANG

The action of AutŽ G . on G induces an action of OutŽ G . as automorphisms of the center ZŽ G . of G. Ž1.4.1. We fix a cyclic central subgroup N of G and a faithful linear character n g IrrŽ N .. We denote by AutŽ G N N ., respectively OutŽ G N N ., the automorphisms of G, respectively outer automorphisms of G, which leave N invariant. Since N F ZŽ G ., the stabilizer OutŽ G N N .n of the character n g IrrŽ N . in OutŽ G N N . is defined. We shall count characters which lie over the fixed character n . Next Ž1.4.2. we choose a block B of G which lies over B Ž n ., an integer d G 0, and a subgroup O F OutŽ G N N .. Then, for any p-chain C of G, the expression IrrŽ NG Ž C ., B, d, O N n . will denote the set of all characters x such that

x g Irr Ž NG Ž C . N n . ,

G

B Ž x . s B,

d Ž x . s d, and

NOutŽG. Ž C . x s O.

Ž 1.4.3.

By Ž1.2.1., the number k Ž NG Ž C ., B, d, O N n . is then the order
Ž 1.4.4.

Cg RrG

Notice that every x g IrrŽ NG Ž C ., B, d, O N n . restricts to a multiple of n on N. Thus O must leave n fixed if it ‘‘fixes’’ x , that is, if O is contained in NOu tŽG.Ž C .x . Hence Ž1.4.5. the equation Ž1.4.4. holds whenever O g OutŽ G N N .n since k Ž NG Ž C ., B, d, O N n . is zero for all p-chains C in that case. Therefore, we need only compute the cases for which O F OutŽ G N N .n in verifying the conjecture. Because N is central in G, we have n s s n for all n g IrrŽ N . and all s g G. Further, conjugation by any s g G is a bijection of IrrŽ NG Ž C ., B, d, O N n . onto IrrŽ NG Ž C s ., B, d, O N n . for any C, B,

COUNTING CHARACTERS IN BLOCKS

7

d, O, and n . Hence k Ž NG Ž C s ., B, d, O N n . s k Ž NG Ž C ., B, d, O N n .. Thus the sum Ž1.4.4. is independent of the choice of the subfamily RrG in R s RŽ G .. Our goal is to prove that the M-A-D conjecture holds if G is the proper covering group n. M22 , for n s 1, 2, 4, 3, 6, and 12, of the Mathieu group M22 on 22 letters, N is its cyclic center Nn of order n, and p is any prime. We note again that Dade has verified w13x that the invariant projective form of his conjecture is true for any blocks with cyclic defect groups. So we only need verify the conjecture for the primes 3 and 2. We shall do so in Theorem 1, in Sect. 3 below, and Theorem 2, in Sect. 4 below, respectively.

2. DESCRIPTION OF M22 2.1. Structure and Properties We follow the notation of Huppert and Blackburn w19, XII.1.4x. We fix a field F4 of order 4 with distinct elements 0, 1, v , v 2 . As noted above, SLŽ3, 4. is the group of all three-by-three matrices over F4 with determinant 1. The center N3 s ZŽ SLŽ3, 4.. consists of the three scalar multiples fI, for nonzero f g F4 , of the identity matrix I. Then PSLŽ3, 4. is the factor group SLŽ3, 4.rN3 . Any element s g SLŽ3, 4. is a 3 = 3 matrix a ss d g

b e h



c f i

0

with entries a, b, . . . g F4 and with determinant 1. We denote by a d g

b e h

c f i

Ž 2.1.1.

the natural image s N3 of s in PSLŽ3, 4.. We use PGŽ2, 4. to denote the projective plane in two dimensions over F4 . The 21 points of PGŽ2, 4. are the one-dimensional subspaces of the vector space ŽF4 . 3 of all 1 = 3 row vectors Ž x, y, z . with x, y, z g F4 . If Ž x, y, z . / Ž0, 0, 0., then w x, y, z x will denote the corresponding point in PGŽ2, 4., i.e., the N3-orbit

w x, y, z x s  Ž x, y, z . , Ž v x, v y, v z . , Ž v 2 x, v 2 y, v 2 z . 4 of Ž x, y, z .. The natural action of SLŽ3, 4. on ŽF4 . 3 induces a faithful action of the group PSLŽ3, 4. as a doubly transitive permutation group on the points of PGŽ2, 4..

8

JANICE F. HUANG

Let P be a set consisting of the 21 points of PGŽ2, 4. and one additional point u. Let s2 be an involution acting on the 22 points of P as follows Žsee w19, XII.1.4x.: s2 : u l w 1, 0, 0 x Ž 2.1.2. and s2 : w x, y, z x ¬ x 2 q yz , y 2 , z 2 Ž 2.1.3. when x, y, z g F4 are not all zero and w x, y, z x / w1, 0, 0x. We extend the action of PSLŽ3, 4. on PGŽ2, 4. to one on P leaving u fixed. Since this action is faithful, Ž2.1.4. we may identify each element of PSLŽ3, 4. with the permutation of P which it induces. Then M22 s ² PSLŽ3, 4., s2 : is the permutation group on P generated by PSLŽ3, 4. and s2 . It is a simple group as well as a triply transitive permutation group on P with order 443,520 s 2 7 ? 3 2 ? 5 ? 7 ? 11, which is 22 ? < PSLŽ3, 4.< Žsee w8x.. We shall assume that PSLŽ3, 4. acts on the right of PGŽ2, 4., and that M22 acts on the right of P. For any subgroup H of M22 and any points p, q g P, we shall denote by Hp the stabilizer in H of the point p, by H p, q the pointwise stabilizer in H of p and q, and by H p, q4 the stabilizer in H of the set  p, q4 . Then it is clear from the above that PSLŽ3, 4. is equal to Ž M22 . u after the identification Ž2.1.4.. Further, the subgroup PSLŽ3, 4.w1, 0, 0x is equal to Ž M22 . u, w1, 0, 0x and is the set of all elements of the form 1 a d

0 b e

0 c f

Ž 2.1.5.

with a, b, . . . , f g F4 and bf y ce s 1. It is straightforward to verify that the element s2 of M22 has the following conjugation action on the element Ž2.1.5. of PSLŽ3, 4.w1, 0, 0x: 1 a d

0 b e

0 c f

s2

1 2 a q bc s 2 d q ef

0 b2 e2

0 c2 . f2

Ž 2.1.6.

The outer automorphism group OutŽ M22 . of M22 is cyclic of order 2 Žsee w8, p. 39x.. One automorphism of M22 whose image generates OutŽ M22 . is conjugation by the element s4 g M24 , identified in Huppert and Blackburn w19, XII.1.4x. The action of s4 on the elements of P can be described as follows: s4 : u ¬ u, Ž 2.1.7. s4 : w x, y, z x ¬ x 2 , y 2 , z 2 ,

Ž 2.1.8.

9

COUNTING CHARACTERS IN BLOCKS

for any w x, y, z x g PGŽ2, 4.. It follows that s2s4 s s2

Ž 2.1.9.

and that s4 takes the element Ž2.1.1. of PSLŽ3, 4. into a d g

b e h

c f i

s4

a2 s d2 g2

b2 e2 h

2

c2 f2 . i2

Ž 2.1.10.

We shall use the notation of the Atlas to denote by M22 . 2 the group AutŽ M22 .. 2.2. The Co¨ ering Group of M22 The accurate identification of the Schur multiplier M Ž M22 . of the group M22 as cyclic of order 12 was made by Pierre Mazet in 1979 w23x. We use the notation of the Atlas to denote by n. M22 the unique proper covering group of M22 having center Nn , Zn . We note that n. M22 is defined for each divisor n s 1, 2, 4, 3, 6, and 12 of 12. In this section we shall construct 3. M22 . In the following section we shall describe Mazet’s construction of 4. M22 . The maximal proper covering group 12. M22 is then just the residual product of 3. M22 and 4. M22 . We shall denote by Ž n. M22 . X the inverse image in n. M22 of Ž M22 . X whenever X s g or X s g, h or X s  g, h4 for some points g, h g P. By Ž2.1.4. we have Ž M22 . u s PSLŽ3, 4.. Further, M22 is the disjoint union M22 s Ž M22 . u j Ž M22 . u s2 Ž M22 . u ,

Ž 2.2.1.

where s2 is the involution described in Ž2.1.2. and Ž2.1.3.. Notice that

Ž Ž M22 . u .

s2

l Ž M22 . u s Ž M22 . u , w1, 0, 0x s PSL Ž 3, 4 . w1, 0, 0x . Ž 2.2.2.

The inverse image of ² s2 : , Z2 in 3. M22 must be cyclic of order 6. Thus there exists a unique involution sX2 g 3. M22 having s2 as its image in M22 . It follows that 3. M22 is the disjoint union 3. M22 s Ž 3. M22 . u j Ž 3. M22 . u sX2 Ž 3. M22 . u ,

Ž 2.2.3.

where X

Ž Ž 3. M22 . u .

s2

l Ž 3. M22 . u s Ž 3. M22 . u , w1, 0, 0x .

Ž 2.2.4.

Since Ž M22 . u s PSLŽ3, 4. contains a Sylow 3-subgroup of M22 , the group Ž3. M22 . u must be the unique nonsplit extension SLŽ3, 4. of N3 s

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JANICE F. HUANG

ZŽ SLŽ3, 4.. by PSLŽ3, 4. Žsee the Atlas w8x.. So Ž3. M22 . u, w1, 0, 0x is the subgroup SLŽ3, 4.w1, 0, 0x consisting of all matrices in SLŽ3, 4. of the form a b e



0 c f

0 d . g

0

Ž 2.2.5.

This subgroup is the direct product

Ž 3. M22 . u , w1, 0, 0x s SL Ž 3, 4 . w1, 0, 0x s N3 = K

Ž 2.2.6.

of the center N3 of SL Ž3, 4. with the derived group K s w SLŽ3, 4.w1, 0, 0x , SLŽ3, 4.w1, 0, 0x x. The latter consists of all matrices of the form Ž2.2.5. with determinant 1 and a s 1. Conjugation by sX2 must normalize SLŽ3, 4.w1, 0, 0x since conjugation by s2 normalized PSLŽ3, 4.w1, 0, 0x. Furthermore, the former conjugation must leave invariant both N3 s ZŽ SLŽ3, 4.. and the derived group K of SLŽ3, 4.w1, 0, 0x. From this and Ž2.1.6. it follows that sX2 acts on K so that X



1 b e

0 c f

0 d g

s2

1 s b 2 q cd e 2 q fg

0 

0 c2 f2

0 d2 g2

0

Ž 2.2.7.

whenever cg y df s 1. In view of Ž2.2.6., this and the fact that sX2 centralizes N3 s ZŽ3. M22 . completely determine the action of sX2 on SLŽ3, 4.w1, 0, 0x. The group SLŽ3, 4. is itself the disjoint union SL Ž 3, 4 . s SL Ž 3, 4 . w1, 0, 0x j SL Ž 3, 4 . w1, 0, 0x sX1 SL Ž 3, 4 . w1, 0, 0x , Ž 2.2.8. where sX1 is the involution 0 sX1 s 1 0

1 0 0

ž

0 0 . 1

/

Ž 2.2.9.

It is easily checked that s2 and the image s1 g PSLŽ3, 4. of sX1 generate a dihedral group ² s1 , s2 : of order 6. Because sX1 and sX2 both centralize N3 , it follows that ² sX1 , sX2 : is dihedral of order 6. In particular, sX2 sX1 sX2 s sX1 sX2 sX1 .

Ž 2.2.10.

It is straightforward to verify that the above decompositions and equations completely determine multiplication in the group 3. M22 . The action Ž2.1.9. and Ž2.1.10. of s4 on M22 can now be extended to a unique action of an involution sX4 on 3. M22 such that X

Ž sX2 . 4 s sX2 s

Ž 2.2.11.

11

COUNTING CHARACTERS IN BLOCKS

and X



a d g

b e h

c f i

s4

a2 s d2 g2

0 

b2 e2 h

2

c2 f2 . i2

0

Ž 2.2.12.

This is the only possible extension of the action of s4 to 3. M22 . Notice that Ž2.2.13.

sX4 inverts N3 and centralizes ² sX1 , sX2 :.

2.3. The Co¨ ering Group of M22 , II We nest look at the covering group 4. M22 of M22 . We begin by choosing a line L0 of PGŽ2, 4. which passes through the five points L0 : w 1, 0, 0 x , w 0, 1, 0 x , w 1, 1, 0 x , w 1, v , 0 x , w 1, v 2 , 0 x having last coordinate zero. We then set X s  u, w 1, 0, 0 x , w 0, 1, 0 x , w 1, 1, 0 x , w 1, v , 0 x , w 1, v 2 , 0 x 4 .

Ž 2.3.1.

The M22 -translates of X are called hexads. We point out that any 3 points in P belong to a unique hexad, that the hexads containing u are precisely the unions of u with the 21 lines in PGŽ2, 4., that there are exactly 5 hexads containing any given 2-element set, and that any 2 distinct, nondisjoint hexads intersect in exactly 2 points. ŽSee the Atlas w8, p. 39x for a description of the action of M22 on the system of hexads.. Following Mazet w23x we shall determine the structure of N4. M .2 Ž E ., where E is the 22 pointwise stabilizer in M22 of the hexad X in Ž2.3.1.. Using the structure Ž2.1.5. of the elements of PSLŽ3, 4.w1, 0, 0x s Ž M22 . u, w1, 0, 0x , it is straightforward to verify that Es

~¡ 10

¢b

0 1 c

¦¥ §

0 0 ; b, c g F4 , 1

Ž 2.3.2.

which is elementary abelian of order 2 4 . We observe that E acts regularly on the 16 points of P y X. We let AŽ1. 5 s

~¡ ac

¢0

b d 0

¦¥ §

0 0 ; ad y bc s 1 and a, b, c, d g F4 1

Ž 2.3.3.

denote a copy of A 5 , SLŽ2, 4.. This copy is clearly a complement to E in NP S LŽ3, 4.Ž E .. Then Ž1. : ² AŽ1. 6 s s2 , A 5

Ž 2.3.4.

12

JANICE F. HUANG

Ž is a complement to E in NM 22Ž E ., that is, NM 22Ž E . s E i AŽ1. 6 . We point out that Janko in w21, 2.6x proved that NM 22Ž E . , E i A 6 .. From Ž2.1.9. and Ž2.1.10. we have that s4 leaves AŽ1. 6 invariant. In addition, ² s4 : S6Ž1. s AŽ1. 6 i

Ž 2.3.5.

is a copy of S6 . Hence NM 22 .2 Ž E . s E i

Ž AŽ1. 6

i ² s4 : . s E i S6Ž1. .

Ž 2.3.6.

We pause here to consider the action of A 6 on E. We note that A 6 is isomorphic to the derived group S4Ž2.9 of the symplectic group S4Ž2. , S6 . We construct a nonsingular, alternating, bilinear form on the elements of E which is preserved by s2 as well as by the elements of NP S LŽ3, 4.Ž E .. For 1 xs 0 b

0 1 c

0 0 1

and ys

1 0 b9

0 1 c9

0 0 1

we define 2

f Ž x, y . s Ž b ? c9 q b9 ? c . q Ž b ? c9 q b9 ? c . g F 2 .

Ž 2.3.7.

The elements of NP S LŽ3, 4.Ž E ., in acting by conjugation on the elements of E, leave both terms on the right-hand side of Ž2.3.7. fixed, while the element s2 adds the same element to both terms and then interchanges them. Then Ž2.3.8. NM Ž E . acts on E as S4Ž2.9 leaving invariant the nonsingular, 22 alternating, bilinear form f described in Ž2.3.7.. Notice that the element s4 also preserves f on the elements of E, by interchanging the two terms on the right-hand side of Ž2.3.7., and that NM 22 .2 Ž E . acts as S4 Ž2. on E. We also point out that, since E s EX is the pointwise stabilizer of the hexad X, its normalizer NM 22Ž E . is also Ž M22 . X , the stabilizer in M22 of the set of points in X. In particular, AŽ1. acts 6 faithfully as A 6 on the six points of X and S6Ž1. acts on them as S6 . One additional piece of information follows from the fact that <Ž M22 . X < s < A 6 < ? < E < s 360 ? 16 s 2 7 ? 3 2 ? 5. Thus the index w M22 : Ž M22 . X x is 77, which must be the number of hexads Ž M22 conjugates of X .. It can be shown that E permutes regularly the 16 hexads which do not intersect X. The following lemma will be useful.

13

COUNTING CHARACTERS IN BLOCKS

LEMMA 2.3.9.

If H F EX and H / 1 then NM 22Ž H . F NM 22Ž EX ..

Proof. Since E is regular on P y X, all 15 involutions in E s EX have exactly the same 6 fixed points as EX . Thus if H F EX and H / 1 then the set of all points fixed by H is exactly the set X. It follows that NM 22Ž H . F Ž M22 . X s NM 22Ž EX . for any such H. This gives us the result of the lemma. Following Mazet’s approach, we next let Z be an arbitrary set of six elements, z1 , . . . , z 6 . We define a Clifford algebra, U s U Ž Z ., of dimension 2 6 over F, generated by elements e z , for z g Z, with defining relations e z2 s y1 and

e z e z 9 s ye z 9 e z

if z, z9 g Z with z / z9. Ž 2.3.10.

We shall denote by G 0 the subgroup of the unit group of U generated by the e z . ŽThis subgroup was called G by Mazet.. In addition, we use Mazet’s notation G for the subgroup of G 0 generated by the products of two elements e z e z 9 , with z, z9 g Z. We denote by N4 the cyclic subgroup generated by the product a s e z1 e z 2 e z 3 e z 4 e z 5 e z 6

Ž 2.3.11.

of all six elements e z , with z g Z. ŽThis group was called N by Mazet.. Using Ž2.3.10., we find that 2

a2 s Ž e z 1 e z 2 e z 3 e z 4 e z 5 e z 6 . s y1. Thus, we must have N4 , Z4 . In addition, the subgroup N4 is the center ZŽ G . of G, while GrZŽ G . is elementary abelian of order 2 4 , and w G, G x s ² a2 : has order 2. The symmetric group SZ on the set Z acts naturally as automorphisms of U, leaving both G 0 and G invariant, and we have a semidirect product G i SZ . Here we assume that the action of SZ on U is such that e zs s e z s for any z g Z and any s g SZ . We note that the alternating subgroup A Z of SZ centralizes N4 , while elements of SZ y A Z invert N4 . Mazet then identified Z s  z1 , z 2 , z 3 , z 4 , z 5 , z 6 4 as a hexad of P chosen to be disjoint from X. In particular, we choose Z to be the hexad consisting of z1 s w 0, 0, 1 x , z 4 s w 1, 1, v x ,

z 2 s w 1, 0, 1 x ,

z 3 s w 0, 1, v x ,

z 5 s w 1, v , 1 x , and

z 6 s w 1, v 2 , v x . Ž 2.3.12.

14

JANICE F. HUANG

This is clearly disjoint from X. For x, y g P y X, Mazet denoted by ² x, y : the unique element of E s EX which interchanges x and y. He fixed an element m g P y X and defined a homomorphism p of G 0 into EX by p Ž e z . s ² m, z :. Since p Ž e z e w . s ² z, w : for any distinct z, w g Z, the restriction p 9 of p to G is independent of the choice of m. Then p 9 is an epimorphism of G onto EX with kernel N4 . Thus GrN4 s GrZŽ G . , EX s E. ŽSee w23x.. Mazet defined a specific complement AŽ2. 6 s Ž M22 . X l Ž M22 . Z , A Z

Ž 2.3.13.

to E in NM 22Ž E . and proved in w23, IV.B.3x that N4. M 22Ž E . , G i A Z . Ž1. Ž . We note that AŽ2. 6 is not conjugate to A 6 of 2.3.4 . Suppose not, that is, suppose that these groups are conjugate. Then AŽ2. would fix some 6 point y g P y X, as well as fixing both disjoint hexads X and Z. Now y cannot belong to either X or Z, since AŽ2. 6 acts as A 6 on both these sets. If we fix a point x g X, then there are exactly three hexads containing x, y which intersect Z nontrivially. These three hexads intersect Z in three . disjoint subsets Z1 , Z2 , Z3 , each of order 2. The subgroup Ž AŽ2. 6 x , which is isomorphic to A 5 , must act faithfully on Z while permuting these three subsets among themselves. This is impossible for the elements of order 5 in this subgroup. Therefore, y does not exist and our statement follows. We consider the involution 1 s s s4 s 2 d s s4 s 2 0 1

0 1 0

0 0 , 1

Ž 2.3.14.

where s4 is the generator of AutŽ M22 . modulo InnŽ M22 . described in Ž2.1.7. ] Ž2.1.10. and s2 d g NM Ž E .. Then s is the odd permutation 22 Ž z1 , z 2 .Ž z 3 , z 4 .Ž z 5 , z6 . on Z and the odd permutation which fixes four ²s : points and interchanges u and w1, 0, 0x on X. Hence S6Ž2. s AŽ2. 6 i Ž2. acts faithfully on both X and Z, and we can identify S6 as a complement to E in NM 22 .2 Ž E . s E i S6Ž2. .

Ž 2.3.15.

There is some automorphism sˆ of 4. M22 inducing both the automorphism of conjugation by s on the factor group 4. M22 rN4 , M22 and some automorphism sˆN4 on N4 . We prove LEMMA 2.3.16. The abo¨ e automorphism sˆ is an in¨ olution. The natural epimorphism of 4. M22 onto M22 , with kernel N4 , extends to an epimorphism of 4. M22 .2 onto M22 .2, sending sˆ to s .

COUNTING CHARACTERS IN BLOCKS

15

Proof. Since AutŽ N4 . , Z2 , we have Ž sˆ 2 .N4 s Ž sˆN4 . 2 s 1. Therefore, sˆ is an automorphism of 4. M22 inducing the identity on both the central subgroup N4 and the factor group 4. M22rN4 . Any such automorphism has the form x ¬ xg Ž x . where g: 4. M22 ª N4 is a group homomorphism with N4 F kerŽ g .. Since M22 is perfect, the only such g is the trivial map. Therefore, sˆ 2 s 1 and sˆ is an involution. Then the natural epimorphism of 4. M22 onto M22 with kernel N4 extends to an epimorphism of 4. M22 .2 onto M22 .2 sending sˆ to s . This completes the proof of Lemma 2.3.16. 2

The inverse image of the normalizer NM 22Ž E . s E i AŽ2. in 4. M22 is 6 the semidirect product G i AŽ2. , contained in the semidirect product 6 G 0 i AŽ2. , where G was defined above as the subgroup of the Clifford 6 0 algebra U generated by the e z . The entire symmetric group S6Ž2. acts on G 0 , as well as on U, with Ž e z . f s e z f for all z g Z and f g S6Ž2.. So we can form the semidirect product G 0 i S6Ž2.. Note that the odd permutations in S6Ž2., including s , invert the generator a s e z 1 e z 2 e z 3 e z 4 e z 5 e z 6 of N4 . Evidently the subgroup G generated by all the e z e z 9 , for z, z9 g Z, is S6Ž2.-invariant. So G i S6Ž2. is a subgroup of G 0 i S6Ž2.. . The inverse image of NM 22 .2 Ž E . in 4. M22 .2 is the subgroup Ž G i AŽ2. 6 Ž2. i ² sˆ :, generated by sˆ and the inverse image G i AŽ2. of E i A 6 6 . Since s normalizes AŽ2. , the corresponding s must normalize the inverse ˆ 6 Ž2. image AŽ2. and N4 in this 6 = N4 of that group. Since the factors A 6 Ž2. . i decomposition are unique, ² sˆ : normalizes A 6 . Hence Ž G i AŽ2. 6 ² sˆ : s G i Ž AŽ2. ² :. i s . At this point we have two semidirect prodˆ 6 ² sˆ :. and G i S6Ž2., which agree on their common ucts, G i Ž AŽ2. 6 i Ž2. subgroup G i A 6 . We shall show that that they are isomorphic. In fact, we prove LEMMA 2.3.17. We can identify G i S6Ž2. with the in¨ erse image N4. M 22 .2 Ž G . of NM 22 .2 Ž E .. Proof. The first step is to prove that sˆ and s g S6Ž2. have the same action on G. Note that the epimorphism p 9: G ª E preserves the action of S6Ž2. on these two groups. In fact, if z, w g Z, then p 9 sends e z e w to the unique ² z, w : g E such that z ² z, w : s w. For any f g S6Ž2. we have f Ž z f . ² z, w :f s Ž z ² z, w : . s wf .

Therefore, ² z, w :f s ² z f , wf : s p 9 Ž e z f e wf . s p 9 Ž Ž e z e w . f . , which proves this result. Now both sˆ and s induce automorphisms sˆG and s G of G such that sˆG and s G induce the same automorphism on GrN4 s GrZŽ G . , E. It follows that s s Ž sˆG .y1 Ž s G . is an automorphism of G inducing the identity on GrN4 .

16

JANICE F. HUANG

Consider the action of s on GrG9 s Gr² " 1:. This action centralizes the 2-group N4rG9 and the factor group Ž GrG9.rŽ N4rG9. , GrN4 . If the action is nontrivial on GrG9, then there is some t g G such that t s s ta " 1. There is some j s 0, 1 such that ta j has order 2. Then a s s a e , where e s "1, and Ž ta j . s s ta j a " 1qjŽ ey1.. Here Ž ta j . s and ta j both have order 2. So the central element a " 1qjŽ ey1. must have order 2. This is impossible since "1 q jŽ e y 1. is congruent to 1 modulo 2. Therefore, s acts trivially on GrG9. Since s centralizes G9 s ² " 1: F ZŽ G ., we conclude that t s s thŽ t G9. for all t g G, where h: GrG9 ª G9 is a group homomorphism. Suppose that s / 1. Then h is nontrivial. So kerŽ h. is a subgroup of order 2 4 in the elementary abelian group GrG9 of order 2 5. Conjugation by both sˆ and s has the same effect on AŽ2. 6 . Therefore, conjugation by Ž2. . both sˆG and s G has the same effect on the image Ž AŽ2. in 6 G of A 6 y1 Ž Ž2. . Ž2. Ž . Ž . . Ž Aut G . It follows that s s sˆG s G centralizes A 6 G , A 6 . ThereŽ Ž2. . Ž . fore AŽ2. 6 and A 6 G both leave invariant the subgroup ker h of GrG9. 4 Ž . This is impossible, since ker h has order 2 while the only AŽ2. 6 -invariant proper subgroup of GrG9 is N4rG9, which has order 2. Thus s s 1, and sˆ and s induce the same automorphism sˆG s s G on G. Since they also induce the same automorphism on AŽ2. 6 , we conclude that there is an isomorphism ² s : , Ž G i AŽ2. ² sˆ : G i S6Ž2. s Ž G i AŽ2. 6 . i 6 . i which is the identity on G i AŽ2. ˆ . This completes the 6 and sends s to s proof of Lemma 2.3.17. 3. THE PRIME p s 3 3.1. Radical 3-Subgroups and Radical 3-Chains We fix 1 g9 s 0 0



0 v 0

0 0 v2

1 0 0

0 1 0

0

Ž 3.1.1.

and 0 t9s 0 1

ž

/

Ž 3.1.2.

COUNTING CHARACTERS IN BLOCKS

17

in SLŽ3, 4. and denote by g and t the corresponding cosets of N3 in PSLŽ3, 4.. Since M22 is a transitive extension of M21 , PSLŽ3, 4. with degree 22 s 2 ? 11, we have that the 3-part of < M22 <, which is equal to the 3-part of < PSLŽ3, 4.<, is 3 2 . Hence any Sylow 3-subgroup of PSLŽ3, 4. is also a Sylow 3-subgroup of M22 . Letting H1 s ²g : and H2 s ²g , t :, we see that H1 , Z3 and H2 , Z3 = Z3 g Syl 3 Ž M22 .. Notice that, from Ž2.1.10., the automorphism s4 of M22 , whose image is a generator of the outer automorphism group of M22 , fixes t and inverts g , while the extension sX4 acting on 3. M22 Žsee Ž2.2.12.. fixes t 9 and inverts g 9. We denote by H2X s ²g 9, t 9: the inverse image of H2 in 3. M22 . It is an extra special group of order 3 3 and exponent 3. It is clear that H2X is also a Sylow 3-subgroup of SLŽ3, 4. i ² sX4 :. The Atlas confirms that NP S LŽ3, 4.Ž H2 . , H2 i Q8 . It is straightforward to verify that H2 fixes no element of the projective plane PGŽ2, 4.. Hence it fixes exactly one element u of P. This forces NM 22Ž H2 . to fix u, and hence to be contained in Ž M22 . u s PSLŽ3, 4.. Therefore NM 22 Ž H2 . s NP S LŽ3 , 4. Ž H2 . , H2 i Q8 .

Ž 3.1.3.

One choice for the involution in Q8 is the element 1 0 0

0 0 1

0 1 0

Ž 3.1.4.

of NPS LŽ3, 4.Ž H2 ., which inverts both g and t . This implies that the quaternion group Q8 acts faithfully and transitively on the eight nontrivial elements of H2 . From the Atlas w8, p. 40x we determine that there is only one conjugacy class of elements of order 3 in M22 . Therefore, ²g :.

Ž3.1.5. any subgroup of M22 with order 3 is M22-conjugate to H1 s

We next prove LEMMA 3.1.6. The normalizer NM 22Ž H1 . has the form H1 i S4 , where the subgroup A 4 of S4 centralizes H1 and any element of S4 y A 4 in¨ erts H1. Proof. The group H1 is precisely the subgroup of M22 consisting of all elements which fix elementwise the set R s w1, 0, 0x, w0, 1, 0x, w0, 0, 1x, u4 of four elements in P. Furthermore, R is precisely the set of all points in P fixed by H1. It follows that NM 22Ž H1 . is precisely the stabilizer Ž M22 .R of this set, and that the action of NM 22Ž H1 . on this set defines a homomorphism of NM 22Ž H1 . into S4 Ž R . with kernel H1. Thus all we need to show is that the image I of this homomorphism is equal to S4Ž R ..

18

JANICE F. HUANG

We easily compute that NP S LŽ3, 4.Ž H1 . is the semidirect product H1 i S3Ž0., where S3Ž0. is the image in PSLŽ3, 4. of the group of all 3 = 3 permutation matrices in SLŽ3, 4.. It follows that I contains the maximal subgroup S4Ž R . u , S3 of S4 Ž R .. But, using Ž2.1.2. and Ž2.1.3., the involution s2 interchanges the two points u and w1, 0, 0x of R, while fixing the other two points w0, 1, 0x and w0, 0, 1x. Hence s2 lies in Ž M22 .R s NM 22Ž H1 . and its image in S4Ž R . does not lie in the maximal subgroup S4 Ž R . u . This is enough to prove that I s S4 Ž R .. We point out that the 3-Sylow subgroup of S4 can be chosen as ²t :, where t is a generator of H2 . This clearly splits over H1. By the extension theory of abelian groups Žsee w16x., this means that S4 splits over H1. Furthermore, the element Ž3.1.4. lies in S3Ž0. and inverts H1 s ²g :. This tells us that S4 acts nontrivially on H1. Hence A 4 centralizes H1 , while any element of S4 y A 4 inverts H1. This gives us the final result of the lemma. One consequence of Lemma 3.1.6 is that H1 is a radical 3-subgroup of M22 .

Ž 3.1.7.

The following propositions will summarize our results. PROPOSITION 3.1.8. The group M22 has three conjugacy classes of radical 3-subgroups, represented by 1, by H1 , and by H2 . We have immediately from Proposition 3.1.8 that, to within M22 -conjugacy, there is one radical 3-chain of length 0 and two of length 1 in M22 . Since H2 g Syl 3 Ž N Ž H1 .., we also have that 1 - H1 - H2 is a radical 3-chain of length 2. In fact, it is clear that any radical 3-chain of length 2 must be conjugate to 1 - H1 - H2 , since NM 22Ž H2 . is transitive on the subgroups of order 3 in H2 . We thus have PROPOSITION 3.1.9. The group M22 has exactly four conjugacy classes of radical 3-chains represented by the chains C0, 3 , C1, 3 , C2, 3 , and C3, 3 , gi¨ en by C0, 3 : 1 C1, 3 : 1 - H1 C2, 3 : 1 - H2 C3, 3 : 1 - H1 - H2 Notice that the second subscript j in ‘‘Ci, j ’’ represents the prime p s 3, to distinguish these chains from those found in Section 4 below.

19

COUNTING CHARACTERS IN BLOCKS

3.2. The M-A-D Conjecture and the Prime p s 3 This and the following four sections of the paper are devoted to the proof of THEOREM 1. The M-A-D conjecture holds when the group G is any co¨ ering group n. M22 of M22 , for some n s 1, 2, 3, 4, 6, or 12, the cyclic central subgroup N of G is the center Nn of n. M22 , the p-block B of G is any 3-block n. B of n. M22 lying o¨ er the block B Ž nn . containing a faithful linear character nn g IrrŽ Nn . and not ha¨ ing the Sylow 3-subgroup of Nn as a defect group, the defect d is any nonnegati¨ e integer, and the group O is any subgroup of Out Ž n. M22 N Nn .n n. In view of Proposition 3.1.9, the conclusion Ž1.4.4. of the M-A-D conjecture, in this case, is equivalent to the equation k Ž Nn . M 22 Ž C0, 3 . , n. B, d, O N nn . y k Ž Nn . M 22Ž C1, 3 . , n. B, d, O N nn . q k Ž Nn . M 22 Ž C3, 3 . , n. B, d, O N nn . y k Ž Nn . M 22 Ž C2, 3 . , n. B, d, O N nn . s 0.

Ž 3.2.1.

The bulk of the proof consists in verifying the entries in Table 1. The index for each column in the table is one of the radical 3-chains Ci, j given in Proposition 3.1.9. Underneath that index we write the sign Žq or y. of the coefficient Žy1.
1. B0 1. B0 1. B1 2. B0 2. B0 2. B1 4. B0 3. B0 3. B1 6. B0 6. B1 12. B0

2 2 1 2 2 1 2 2 2 2 2 2

Z2 Z1 Z2 Z2 Z1 Z2 Z1 Z1 Z1 Z1 Z1 Z1

C0, 3 q

C1, 3 y

C3, 3 q

C2, 3 y

4 2 3 4 2 3 6 5 3 5 3 5

4 2 3 4 2 3 6 2 3 2 3 2

4 2 0 4 2 0 6 2 0 2 0 2

4 2 0 4 2 0 6 5 0 5 0 5

20

JANICE F. HUANG

appearing in Table 1 are all the 3-blocks of n. M22 which lie over the block of Nn containing nn and have defect groups strictly larger than the Sylow 3-subgroups of Nn . These 3-blocks will be described explicitly in the next section. The table entry for the row with indices n. Bh , d, Zk and column with index Ci, j is the value of k Ž Nn. M 22Ž Ci, j ., n. Bh , d, Zk N nn .. These values will be computed in the next four sections. We shall also show that k Ž Nn. M 22Ž Ci, j ., n. Bh , d, Zk N nn . s 0 for any combination of n. Bh , d, Zk , and Ci, j not appearing in Table 1, where n. Bh satisfies the hypotheses of Theorem 1. Therefore, eq. Ž3.2.1. will hold if and only if the alternating sum of the values in any row of Table 1 is 0. Since this is obviously true, the verification that Table 1 is correct will complete the proof of Theorem 1. 3.3. The Chain C0, 3 Since the complete character table for N12. M 22 .2 Ž C0, 3 . s 12. M22 .2 is found in the Atlas w8, p. 39x, we shall not reproduce it here, but shall instead give information gained from the table. We let Cl Ž g . denote the conjugacy class of g g G. We make use of the fact Žsee w10, 11.16x. that x i and x j belong to the same p-block iff < Cl Ž g . < ? x i Ž g . rx i Ž 1 . ' < Cl Ž g . < ? x j Ž g . rx j Ž 1 .

Ž mod p . Ž 3.3.1.

for all elements g g G, where p is the maximal ideal of R. Since ZŽ M22 . s N1 is trivial, we let n 1 s 1. Using the notation from the Atlas, we determine that there are five 3-blocks of M22 , which we label 1. B0 , . . . , 1. B4 , where Irr Ž 1. B0 N n 1 . s Irr Ž 1. B0 . s  x 1 , x 5 , x 7 , x 10 , x 11 , x 12 4 , Ž 3.3.2. Irr Ž 1. B1 N n 1 . s Irr Ž 1. B1 . s  x 2 , x 8 , x 9 4 ,

Ž 3.3.3.

and Irr Ž 1. B2 N n 1 . s  x 3 4 ,

Irr Ž 1. B3 N n 1 . s  x4 4 ,

Irr Ž 1. B4 N n 1 . s  x6 4 .

We note that 1. B0 has H2 as a defect group, and that all the characters in 1. B0 have 3-defect d s 2. Furthermore, 1. B1 has H1 as a defect group, and all its characters have 3-defect d s 1. The blocks 1. B2 , 1. B3 , and 1. B4 all have the Sylow 3-subgroup 1 of N1 as a defect group, and hence do not satisfy the hypotheses of Theorem 1. The Atlas w8x indicates that Ž3.3.4. the two characters x 10 and x 11 of 1. B0 fuse under the action of OutŽ M22 .. No other characters of M22 fuse under this action. This gives us the values for the first three entries in the first column of Table 1. Furthermore, k Ž M22 , B, d, O N 1. s 0 for all other combinations of B, d, and O satisfying the hypotheses of Theorem 1.

COUNTING CHARACTERS IN BLOCKS

21

Next we let n 2 be a faithful character of the center N2 of 2. M22 . We determine, using Ž3.3.1., that there are four 3-blocks of 2. M22 lying over B Ž n 2 ., 2. B0 , . . . , 2. B3 , where Irr Ž 2. B0 N n 2 . s  x 13 , x 14 , x 15 , x 19 , x 20 , x 23 4 ,

Ž 3.3.5.

Irr Ž 2. B1 N n 2 . s  x 16 , x 21 , x 22 4 ,

Ž 3.3.6.

and Irr Ž 2. B2 N n 2 . s  x 17 4 ,

Irr Ž 2. B3 N n 2 . s  x 18 4 .

The defect group for the block 2. B0 is H2 and each character in 2. B0 has 3-defect d s 2, while the defect group for 2. B1 is H1 and each character has 3-defect d s 1. The blocks 2. B2 and 2. B3 both have the Sylow 3-subgroup 1 of N2 as a defect group, and hence do not satisfy the hypotheses of Theorem 1. The Atlas w8x indicates that Ž3.3.7. the pair of characters x 17 in 2. B2 and x 18 in 2. B3 , and the pair of characters x 19 and x 20 , both in 2. B0 , fuse under the action of OutŽ M22 .. No other characters of 2. M22 fuse under this action. This gives us the next three entries in the first column of Table 1 and tells us that k Ž2. M22 , B, d, O N n 2 . s 0 for all other combinations of B, d, and O satisfying the hypotheses of Theorem 1. We point out that the central elements of 3. M22 and 4. M22 are inverted by the outer automorphism group of M22 Žsee Atlas w8, p. 40x.. Therefore, for n G 3 the nontrivial outer automorphism of n. M22 inverts Nn and so does not fix nn . Hence k Ž N Ž Ci, j ., n. B, d, O N nn . s 0 when O s Z2 and n G 3, whatever the values of Ci, j , n. B, and d may be. We next let n4 be a faithful character of the center N4 of 4. M22 and determine, using Ž3.3.1., that there are three 3-blocks of 4. M22 lying over B Ž n4 ., 4. B0 , . . . , 4. B2 , where Irr Ž 4. B0 N n4 . s  x 24 , x 25 , x 28 , x 29 , x 30 , x 31 4

Ž 3.3.8.

and Irr Ž 4. B1 N n4 . s  x 26 4 ,

Irr Ž 4. B2 N n4 . s  x 27 4 .

The defect group for 4. B0 is H2 and each character in 4. B0 has 3-defect d s 2. The blocks 4. B1 and 4. B2 both have the Sylow 3-subgroup 1 of N4 as a defect group, and hence do not satisfy the hypotheses of Theorem 1. This gives us the value for the seventh entry in the first column of Table 1 and tells us that k Ž4. M22 , B, d, O N n4 . s 0 for all other combinations of B, d, and O satisfyng the hypotheses of Theorem 1.

22

JANICE F. HUANG

Now we let n 3 be a faithful character of the center N3 of 3. M22 . We determine, using Ž3.3.1., that there are five 3-blocks of 3. M22 lying over B Ž n 3 ., 3. B0 , . . . , 3. B4 , where Irr Ž 3. B0 N n 3 . s  x 36 , x 37 , x40 , x41 , x42 4 , Irr Ž 3. B1 N n 3 . s  x 32 , x 38 , x 39 4 ,

Ž 3.3.9. Ž 3.3.10.

and Irr Ž 3. B2 N n 3 . s  x 33 4 ,

Irr Ž 3. B3 N n 3 . s  x 34 4 ,

Irr Ž 3. B4 N n 3 . s  x 35 4 . The defect group for 3. B0 is the inverse image H2X of H2 in 3. M22 mentioned in Sect. 3.1. Each character in 3. B0 has 3-defect d s 2. The defect group for 3. B1 is the inverse image H1X of H1 in 3. M22 . It is isomorphic to the direct product N3 = H1. Each character in 3. B1 has 3-defect d s 2. The blocks 3. B2 3. B3 , and 3. B4 all have the Sylow 3-subgroup of N3 as a defect group and hence do not satisfy the hypotheses of Theorem 1. This gives us the values for the eighth and ninth entries in the first column of Table 1 and tells us that k Ž3. M22 , B, d, O N n 3 . s 0 for all other combinations of B, d, and O satisfying the hypotheses of Theorem 1. We now let n 6 be a faithful character of the center N6 of 6. M22 . We determine, using Ž3.3.1., that there are four 3-blocks of 6. M22 lying over B Ž n 6 ., 6. B0 , . . . , 6. B3 , where Irr Ž 6. B0 N n 6 . s  x43 , x44 , x49 , x 50 , x 52 4 ,

Ž 3.3.11.

Irr Ž 6. B1 N n 6 . s  x45 , x48 , x 51 4 ,

Ž 3.3.12.

and Irr Ž 6. B2 N n 6 . s  x46 4 ,

Irr Ž 6. B3 N n 6 . s  x47 4 .

H2X

The defect group for 6. B0 is and each character in 6. B0 has 3-defect d s 2. The defect group for 6. B1 is H1X and each character in 6. B1 has 3-defect d s 2. The blocks 6. B2 and 6. B3 both have the Sylow 3-subgroup of N6 as a defect group, and hence do not satisfy the hypotheses of Theorem 1. This gives us the values for the 10th and 11th entries in the first column of Table 1 and tells us that k Ž6. M22 , B, d, O N n 6 . s 0 for all other combinations of B, d, and O satisfying the hypotheses of Theorem 1. Finally, we let n 12 be a faithful character of the center N12 of 12. M22 . We determine, using Ž3.3.1., that there are three 3-blocks of 12. M22 lying over B Ž n 12 ., 12. B0 , . . . , 12. B2 , where Irr Ž 12. B0 N n 12 . s  x 53 , x 54 , x 57 , x 58 , x 59 4 , Irr Ž 12. B1 N n 12 . s  x 55 4 ,

Irr Ž 12. B2 N n 12 . s  x 56 4 .

Ž 3.3.13.

COUNTING CHARACTERS IN BLOCKS

23

The defect group for 12. B0 is H2X and each character has 3-defect d s 2. The blocks 12. B1 and 12. B2 both have the Sylow 3-subgroup of N12 as a defect group, and hence do not satisfy the hypotheses of Theorem 1. This gives us the value for the last entry in the first column of Table 1 and tells us that k Ž12. M22 , B, d, O N n 12 . s 0 for all other combinations of B, d, and O satisfying the hypotheses of Theorem 1. This completes the information about the normalizer N Ž C0, 1 ., as well as filling in the entries of the first column of Table 1. In Sects. 3.4, 3.5, and 3.6 we shall compute the corresponding entries in the remaining columns. 3.4. The Chain C1, 3 In Lemma 3.1.6, we determined that NM 22Ž C1, 3 . s NM 22Ž H1 . has the form H1 i S4 . Here H1 s ²g : was defined in the beginning of Sect. 3.1 as a subgroup of order 3. We determine next the structure of N4. M 22 .2 Ž C1, 3 .. We recall the definitions of Z and G from Sect. 2.3 above, as well as the fact that N4. M 22 .2 Ž G . , G i S6Ž2. , G i SZ Žsee Ž2.3.17... We note that M22 has only one conjugacy class of elements of order 3. Replacing 4. H1 by a conjugate, we can assume that it is generated by N4 and an element s 3 of order 3 in the complement S6Ž2. , SZ to G in N4. M 22 .2 Ž G .. We can even assume that s 3 corresponds to a cycle of order 3, when considered as a permutation of the set Z. In particular, we assume that s 3 s Ž z 4 , z5 , z 6 . and further define e 3 as Ž z1 , z 2 , z 3 .. Then there is some a g M22 such that g a s s 3 . The conjugate H2a of H2 s ²g , t : is then a Sylow 3-subgroup of CM 22Ž s 3 ., as well as a Sylow 3-subgroup of M22 . Hence there is some b g CM 22Ž s 3 . such that H2ab s ² e 3 , s 3 :. We can therefore replace H1 and H2 by their conjugates under ab to get H1 s ² s 3 : and

H2 s ² e 3 , s 3 : .

Ž 3.4.1.

Next we let i 2 s Ž z 4 , z 5 . and r 2 s Ž z1 , z 2 ., and note that these permutations on Z act naturally as automorphisms of G. We prove the following lemma. LEMMA 3.4.2. The normalizer N4. M 22 .2 Ž H1 . is isomorphic to Ž N4 ( Q8 . i Ž² e 3 , r 2 : = ² s 3 , i 2 :., where N4 ( Q8 is the central product of N4 and Q8 with V Ž N4 . amalgamated with V Ž Q8 .. Furthermore, N4. M 22Ž H1 . is isomorphic to Ž N4 ( Q8 . i Ž² e 3 : = ² s 3 :. i ² r 2 i 2 :. Here ² s 3 , i 2 : centralizes Q8 , while ² e 3 , r 2 : acts faithfully on Q8 . Proof. It is clear that N4. M 22 .2 Ž² s 3 :. contains NG i S 6Ž2. Ž ² s 3 : . s C G Ž s 3 . i NS 6Ž2. Ž ² s 3 : . .

Ž 3.4.3.

24

JANICE F. HUANG

Since z1 , z 2 , z 3 are fixed by s 3 , it follows that C G Ž s 3 . is generated by N4 s ² a:, where a was identified in Ž2.3.11., and the three products e z 1 e z 2 , e z 1 e z 3 , and e z 2 e z 3 . This implies that it is the central product of N4 and a quaternion group Q8 of order 8. The normalizer NS 6Ž2. Ž² s 3 :. is easily computed. It is the direct product ² e 3 , r 2 : = ² s 3 , i 2 : of ² e 3 , r 2 : , S3 with ² s 3 , i 2 : , S3 . From the order of CM 22Ž H1 . found in the Atlas, it follows that N4. M 22 .2 Ž H1 . s NG i S 6Ž2. Ž ² s 3 : . s Ž N4 ( Q8 . i Ž ² e 3 , r 2 : = ² s 3 , i 2 : . .

Ž 3.4.4. Furthermore, we can assume that Q8 s wŽ N4 ( Q8 ., ² e 3 :x, which is generated by the products ye z 1 e z 2 , ye z 2 e z 3 , and ye z 3 e z 1. In that case, Q8 is normal in N4. M 22 .2 Ž H1 ., centralized by ² s 3 , i 2 :, and acted upon faithfully by ² e 3 , r 2 :. We observe that N4. M 22 Ž H1 . s 4. M22 l N4. M 22 .2 Ž H1 . s Ž N4 ( Q8 . i s Ž N4 ( Q8 . i

 Ž ² e4 , r 2 : = ² s 3 , i 2 :. l A Z 4 Ž² e 3 , s 3 : i ² r 2 i 2 :. .

This holds because the permutations s 3 and e 3 are even, while r 2 and i 2 are odd. The second result of the lemma follows immediately. We pass next to the covering group N12. M 22 .2 Ž C1, 3 . of N4. M 22 .2 Ž C1, 3 ., where the Sylow 2-subgroups remain the same, while the Sylow 3-subgroup H2 becomes the group H2X , an M22-conjugate of the H2X mentioned in Sect. 3.1. Let N4 and Q8 be the Sylow 2-subgroups of the inverse images of N4 and Q8 , respectively, in N12. M 22 .2 Ž C1, 3 .. Let r 2 and i 2 be the unique involutions in N12. M 22 .2 Ž C1, 3 . having r 2 and i 2 , respectively, as their images in N4. M 22 .2 Ž C1, 3 .. Then we can choose elements e 3 and s 3 in the inverse images of e 3 and s 3 , respectively, such that r y1 Ž e3 . s Ž e3 . , 2

i Ž e3 . s e3 , 2

r Ž s3 . s s3 , 2

i y1 Ž s3 . s Ž s3 . . Ž 3.4.5. 2

We can then extend the results of Lemma 3.4.2 to N12. M 22 .2 Ž C1, 3 .. We prove LEMMA 3.4.6. The normalizer N12. M 22 .2 Ž C1, 3 . is isomorphic to Ž N4 ( Q8 . i Ž² e 3 , s 3 . i ² r 2 , i 2 :.. Furthermore, N12. M 22Ž C1, 3 . is isomorphic to Ž N4 ( Q8 . i Ž² e 3 , s 3 . i ² r 2 i 2 :.. Here Ž N3 = ² s 3 :. i ² i 2 : centralizes Q8 , while ² e 3 : i ² r 2 : , S3 acts faithfully on Q8 .

COUNTING CHARACTERS IN BLOCKS

25

Proof. In fact, eqs. Ž3.4.5., plus the observation that both r 2 and i 2 invert ZŽ H2X . s N3 , determine e 3 and s 3 uniquely. Now the structure of N12. M 22 .2 Ž C1, 3 . is clear. It has the form indicated in the lemma where ² e 3 , s 3 : s H2X and ² r 2 , i 2 : is a 4-group. Notice that Ž N3 = ² s 3 :. i ² i 2 : centralizes Q8 , while ² e 3 : i ² r 2 : , S3 acts faithfully on Q8 . The result for N12. M 22Ž C1, 3 . follows immediately. This completes the proof of the lemma. We can now proceed to identify the 3-blocks for each of the groups Nn. M 22 .2 Ž C1, 3 ., where n is a divisor of 12. In each case we shall determine the number of characters in the block, as well as the 3-defect of each character. We shall make use of the fact that the p-blocks B of the normalizer NG Ž P . of a p-subgroup P of a finite group G correspond one-to-one to the NG Ž P .-conjugacy classes of blocks b of CG Ž P ., with B corresponding to the conjugacy class  b1 s b, b 2 , . . . , bt 4 of b if and only if 1 B s 1 b 1 q 1 b 2 q ??? q1 b t g ZŽ R NG Ž P ... In that case, IrrŽ B . consists of all the x g IrrŽ NG Ž P .. such that x lies over some character f g IrrŽ b .. We shall also make use of the following well-known fact Žsee w14, 5x.. Ž3.4.7. Let P be a normal p-subgroup of G whose centralizer Ž CG P . has a normal p-complement K, while GrPCG Ž P . is a p9 group. Then there is a one-to-one correspondence between all p-blocks B of G and all G-conjugacy classes  f 1 , . . . , fn4 in IrrŽ K .. Here B corresponds to  f 1 , . . . , fn4 if and only if 1 B s 1f q ??? q1f . In that case, any Sylow 1 n p-subgroup of the stabilizer Gf i in G of any f i is a defect group of B. Since here NG Ž P .rCG Ž P . is a p9-group, the defect groups in CG Ž P . are also those in NG Ž P .. We begin with n s 1. We factor the groups in Lemma 3.4.6 by N12 s N4 = N3 . Recall from Ž3.1.6. that the normalizer NM 22Ž C1, 3 . s NM 22Ž H1 . is isomorphic to H1 i S4 . Then the centralizer CM 22Ž H1 . must have the form H1 = A 4 which has a normal 3-complement, the 4-group V4 in A 4 . Here V4 is the image in M22 of Q8 . Using Ž3.4.7., there must then be two 3-blocks, 1.b 0 and 1.b1 , of NM 22Ž C1, 1 ., corresponding to the two NM 22Ž H1 .conjugacy classes of characters in IrrŽ V4 .. The block 1.b 0 , which corresponds to the class of the trivial character in IrrŽ V4 ., has six irreducible characters, each with 3-defect 2. It has defect group H2 and, therefore, must induce the block 1. B0 of M22 . The block 1.b1 , which corresponds to the conjugacy class of nontrivial characters of the 4-group, has three irreducible characters, each with 3-defect 1. It has defect group H1 and, by Brauer’s first main theorem, must induce the block 1. B1 of M22 . In addition, we determine that the normalizer NM 22 .2 Ž C1, 3 . must fuse two characters in 1.b 0 and fixes every other character in IrrŽ1.b 0 . and IrrŽ1.b1 .. We let n 1 s 1. This gives us the values for the first three entries in the

26

JANICE F. HUANG

second column of Table 1 and tells us that k Ž NM 22Ž C1, 3 ., B, d, O N 1. s 0 for any other combination of B, d, and O satisfying the hypotheses of Theorem 1. We next let n s 2 and let n 2 be a faithful linear character of the center N2 of 2. M22 . Notice that the structure of N12. M 22 .2 Ž C1, 3 . given in Lemma 3.4.6 makes it clear that N2. M 22 .2 Ž C1, 3 . is isomorphic to Z2 = NM 22 .2 Ž C1, 3 ., where Z2 is the image of N4 in N2. M 22 .2 Ž C1, 3 .. Then N2. M 22 .2 Ž C1, 3 . has exactly two 3-blocks, 2.b 0 s B Ž n 2 . = 1.b 0 and 2.b1 s B Ž n 2 . = 1.b1 , lying over B Ž n 2 .. These two blocks have defect groups H2 and H1 , and induce 2. B0 and 2. B1 , respectively. Furthermore, IrrŽ1.bi . is isomorphic to IrrŽ2.bi N n 2 . as an N2. M 22 .2 Ž C1, 3 .-set, for each i s 0, 1. The isomorphism sends any f g IrrŽ1.bi . to n 2 = f g IrrŽ2.bi N n 2 .. Hence, it preserves both degrees and 3-defects. This gives us the next three entries in the second column of Table 1 and tells us that k Ž N2. M 22Ž C1, 1 ., B, d, O N n 2 . s 0 for any other combination of B, d, and O satisfying the hypotheses of Theorem 1. Next we let n s 4 and let n4 be a faithful linear character of the center N4 of 4. M22 . We factor the groups in Lemma 3.4.6 by N3 , giving us, as in Lemma 3.4.2, that N4. M 22 .2 Ž C1, 3 . is isomorphic to Ž N4 ( Q8 . i Ž² e 3 , r 2 : = ² s 3 , i 2 :.. Using Ž3.4.7., there must then be a single 3-block, 4.b 0 , of N4. M 22 .2 Ž C1, 3 ., corresponding to the unique irreducible character L of the normal 3-complement N4 ( Q8 in C4. M 22Ž H1 . such that L lies over n4 . The set IrrŽ4.b 0 N n4 . consists of six characters, each with 3-defect d s 2. It has defect group H2 and hence induces the block 4. B0 of 4. M22 . We recall that the outer automorphism group of M22 inverts the central elements of 3. M22 and 4. M22 . Therefore, we take O s Z1. This gives us the seventh entry in the second column of Table 1 and tells us that k Ž N4. M 22Ž C1, 3 ., B, d, O N n4 . s 0 for any other combination of B, d, and O satisfying the hypotheses of Theorem 1. We now let n s 3 and let n 3 be a faithful linear character of the center N3 of 3. M22 . We factor the groups in Lemma 3.4.6 by N4 . We represent by V4 the image Ž N4 ( Q8 .rN4 of Q8 in 3. M22 . Then V4 is precisely the normal 3-complement in C3. M 22Ž H1 . s V4 i H2X . Using Ž3.4.7., the normalizer N3. M 22Ž C1, 3 . then has two 3-blocks, 3.b 0 and 3.b1 , corresponding to the two conjugacy classes of characters of the 4-group V4 . Both 3.b 0 and 3.b1 lie over the unique 3-block B Ž n 3 . of N3 . The set IrrŽ3.b 0 N n 3 ., which corresponds to the conjugacy class of the trivial character of the 4-group, has two characters, each with 3-defect 2, while the set IrrŽ3.b1 N n 3 ., which corresponds to the conjugacy class of nontrivial characters of the 4-group, has three characters, each with 3-defect 2. The block 3.b 0 has defect group H2X , which implies that it induces the block 3. B0 of 3. M22 . On the other hand, block 3.b1 has defect group H1X and must induce 3. B1 , by Brauer’s first main theorem. Since O s Z1 in this case, we therefore have the

27

COUNTING CHARACTERS IN BLOCKS

eighth and ninth entries in the second column of Table 1. In addition, k Ž N3. M 22Ž C1, 3 ., B, d, O N n 3 . s 0 for any other combination of B, d, and O satisfying the hypotheses of Theorem 1. Next we let n s 6 and let n 6 be a faithful linear character of the center N6 of 6. M22 . Using an argument similar to the one we used for n s 2, we note that N6. M 22Ž C1, 3 . must be isomorphic to Z2 = N3. M 22Ž C1, 3 .. Then N6. M 22Ž C1, 3 . has two blocks, 6.b 0 and 6.b1 , lying over B Ž n 6 .. These two blocks have defect groups H2X and H1X , and induce 6. B0 and 6. B1 , respectively. The set IrrŽ3.bi N n 3 . is isomorphic to IrrŽ6.bi N n 6 . as an N6. M 22Ž C1, 3 .set, for each i s 0, 1. The isomorphisms preserve both degrees and 3-defects. This gives us the 10th and 11th entries in the second column of Table 1, since O s Z1 in this case, and tells us that k Ž N6. M 22Ž C1, 3 ., B, d, O N n 6 . s 0 for any other combination of B, d, and O. Finally, we let n s 12 and let n 12 be a faithful linear character of the center N12 of 12. M22 . Again using Ž3.4.7., the normalizer N12. M 22Ž C1, 3 . must have a single 3-block, 12.b 0 , which lies over B Ž n 12 .. It corresponds to the unique character L of the normal 3-complement N4 ( Q8 in C12. M 22Ž C1, 3 ., such that L lies over the restriction n4 of n 12 to N4 . The set IrrŽ12.b 0 N n 12 . has two characters, each with 3-defect d s 2, and has defect group H2X . It must induce the block 12. B0 of 12. M22 . We let O s Z1 and have the last entry in the second column of Table 1. In addition, k Ž N12. M 22Ž C1, 3 ., B, d, O N n 12 . s 0 for any other combination of B, d, and O satisfying the hypotheses of Theorem 1. This completes the information about the chain C1, 3 and the second column of Table 1. We proceed next to the third column, which contains information about the chain C3, 3 . 3.5. The Chain C3, 3 It is clear that N12. M 22 .2 Ž C3, 3 . is a subgroup of N12. M 22 .2 Ž C1, 3 ., since C3, 3 is the chain 1 - H1 - H2 , with normalizer N Ž H1 . l N Ž H2 .. In fact, Lemma 3.4.6 implies that N12 . M 22 .2 Ž C3, 3 . s N4 i ² e 3 , s 3 : i ² r 2 , i 2 :

ž

/

s N4 = ² e 3 , s 3 : i ² r 2 , i 2 : ,

ž

/

Ž 3.5.1.

and hence that N12 . M 22 Ž C3, 3 . s N4 = ² e 3 , s 3 : i ² r 2 i 2 : .

ž

/

Ž 3.5.2.

Following the same procedure as in Sect. 3.4, we begin with n s 1. We factor the groups in Ž3.5.1. and Ž3.5.2. by N12 and determine, using Ž3.4.7., that there must be one 3-block of characters, 1.bX0 , of NM 22Ž C3, 3 ., since

28

JANICE F. HUANG

CNM ŽC 3, 3 .Ž H1 . is the 3-group H2 . In fact, the normalizer NM 22Ž C3, 3 . is 22 isomorphic to H2 i Z2 , H1 i S3 . The set IrrŽ1.bX0 . consists of six characters, each with 3-defect d s 2, and has defect group H2 . Clearly the block 1.bX0 must induce the block 1. B0 of M22 . The group NM 22 .2 Ž C3, 3 . must fuse exactly two characters in this block. We let n 1 s 1 and we have the first two entries in the third column of Table 1. Further, k Ž NM 22Ž C3, 3 ., B, d, O N 1. s 0 for any other combination of B, d, and O satisfying the hypotheses of Theorem 1. Therefore, the third entry in this column of Table 1 is 0. Next we let n s 2 and let n 2 be a faithful linear character of the center N2 of 2. M22 . We identify N2. M 22 .2 Ž C3, 3 . with Z2 = NM 22 .2 Ž C3, 3 .. Then N2. M 22 .2 Ž C3, 3 . has exactly two 3-blocks, 2.bX0 and 2.bX1 , lying over B Ž n 2 .. These two blocks have defect groups H2 and H1 , and induce 2. B0 and 2. B1 , respectively. Furthermore, IrrŽ1.bi . is isomorphic to IrrŽ2.bi N n 2 . as an N2. M 22 .2 Ž C3, 3 .-set, for each i s 0, 1. The isomorphisms preserve both degrees and 3-defects. This gives us the fourth and fifth entries in the third column of Table 1 and tells us that k Ž N2. M 22Ž C3.3 ., B, d, O N n 2 . s 0 for any other combination of B, d, and O satisfying the hypotheses of Theorem 1. This gives us the sixth entry 0 in this column. We now let n s 4 and let n4 be a faithful linear character of the center N4 of 4. M22 . We identify N4. M 22Ž C3, 3 . with N4 = NM 22Ž C3, 3 .. Then N4. M 22Ž C3.3 . has exactly one 3-block, 4.bX0 s B Ž n4 . = 1.bX0 , lying over B Ž n4 .. This block 4.bX0 has H2 as a defect group, and hence induces 4. B0 . The map f ¬ n 2 = f is a defect-preserving bijection of IrrŽ bX0 . onto IrrŽ4.bX0 .. We recall that there is no fusion of characters when n s 4 and thus choose O s Z1. This gives us the seventh entry in the third column of Table 1. Further, k Ž N4. M 22Ž C3, 3 ., B, d, O N n4 . s 0 for any other combination of B, d, and O satisfying the hypotheses of Theorem 1. We next let n s 3 and let n 3 be a faithful linear character of the center N3 of 3. M22 . Again using Ž3.4.7. we find that there is one 3-block, 3.bX0 , of N3. M 22 .2 Ž C3, 3 ., since C3. M 22Ž C3, 3 . is the 3-group N3 . The set IrrŽ3.bX0 N n 3 . has two characters, each with 3-defect d s 2. It has defect group H2X and induces the block 3. B0 of 3. M22 . We choose O s Z1 and have the eighth entry in the third column of Table 1. Further, k Ž N3. M 22Ž C3, 3 ., B, d, O N n 3 . s 0 for any other combination of B, d, and O satisfying the hypotheses of Theorem 1. This gives us the ninth entry of the third column, which is 0. Now we let n s 6 and let n 6 be a faithful linear character of the center N6 of 6. M22 . Using an argument similar to the one for n s 2, we find that the group N6. M 22Ž C3, 3 . has a single block, 6.bX0 , lying over B Ž n 6 .. This block has defect group H2X and induces the block 6. B0 . The set IrrŽ3.bX0 N n 3 . is isomorphic to IrrŽ6.bX0 N n 6 . as an N6. M 22Ž C3, 3 .-set. The isomorphism preserves both degrees and 3-defects. This gives us the 10th entry in the third

COUNTING CHARACTERS IN BLOCKS

29

column of Table 1 and tells us that k Ž N6. M 22Ž C3, 3 ., B, d, O N n 6 . s 0 for any other combination of B, d, and O satisfying the hypotheses of Theorem 1. Therefore, the 11th entry of the third column must be 0. Finally, we let n s 12 and let n 12 be a faithful linear character of the center N12 of 12. M22 . We use Ž3.4.7. and determine, using an argument similar to the one for n s 2, that the normalizer N12. M 22Ž C3, 3 . must have a single 3-block, 12.bX0 , which lies over B Ž n 12 . and has defect group H2X . It must induce the block 12. B0 of 12. M22 . The set IrrŽ12.bX0 N n 12 . is isomorphic to IrrŽ6.bX0 N n 6 ., as well as to IrrŽ3.bX0 N n 3 .. The isomorphism preserves both degrees and defects. We choose O s Z1 and have the last entry in the third column of Table 1. Further, k Ž N12. M 22Ž C3, 3 ., B, d, O N n 12 . s 0 for any other combination of B, d, and O satisfying the hypotheses of Theorem 1. This completes the information about the chain C3, 3 and the third column of Table 1. We turn next to the last chain to consider, C2, 3 . 3.6. The Chain C2, 3 We recall from Ž3.1.5. that the normalizer NM 22Ž C2, 3 . s NM 22Ž H2 . has the form H2 i Q8 , where the quaternion group Q8 acts faithfully and transitively on the eight nontrivial elements of H2 . Here we are using, as in Sect. 3.1, the original definition of H2 as ²g , t :, while H2X s ²g 9, t 9:. We consider next the structure of the normalizer N3. M 22 .2 Ž C2, 3 . s N3. M 22 .2 Ž H2 .. We let 1 i9 s 1 1

1 v v2

1 v2 v

0

Ž 3.6.1.

1 j9 s v v

v2 v2 v

v2 v v2

0

Ž 3.6.2.



and



be elements of SLŽ3, 4. and denote by i and j the corresponding cosets of N3 in PSLŽ3, 4.. We recall that sX4 is the extension of s4 acting on 3. M22 and prove LEMMA 3.6.3. The automorphism sX4 normalizes the quaternion group s ² i9, j9:. Together they generate a semidihedral group Q8Ž1. i ² sX4 : of order 16, which normalizes H2X and acts faithfully on H2 s H2X rN3 . The semidirect product H2X i Ž Q8Ž1. i ² sX4 :. is the normalizer N3. M 22 .2 Ž H2 ., and its normal subgroup H2X i Q8Ž1. is the normalizer N3. M 22Ž H2 .. Q8Ž1.

30

JANICE F. HUANG

Proof. The normalizer N3. M 22 .2 Ž H2 . must equal NS LŽ3, 4. i Ž sX4 .Ž H2X .. In fact, this is an immediate consequence of Ž3.1.3.. It is straightforward to verify that Ž i9. 4 s 1, Ž i9. 2 s Ž j9. 2 , and i9 j9 s j9Ž i9. 3. Hence ² i9, j9: is a quaternion group Q8Ž1. of order 8. It is also not difficult to compute that ² i9, j9: normalizes H2X , since Žg 9. i9 s Žt 9. 2 , Žt 9. i9 s g 9, Žg 9. j9 s v 2 I ? g 9t 9, X X and Žt 9. j9 s v I ? g I ? g 9Žt 9. 2 . Furthermore, since Ž i9. s4 s Ž i9. 3 and Ž j9. s4 s j9i9, it follows that sX4 normalizes both ² i9, j9: and H2X . Then ² i9, j9: i ² sX4 : is semidihedral of order 16. Recalling our results in Ž3.1.3. and the remarks following it, we can see that this group acts faithfully on H2X rN3 , H2 . Furthermore, Ž3.1.3. makes it clear that H2X i Ž Q8Ž1. i ² sX4 :. is the normalizer N3. M 22 .2 Ž H2 ., and its normal subgroup H2X i Q8Ž1. is the normalizer N3. M 22Ž H2 .. This gives us the results of the lemma. Now we must determine the character theory of N12. M 22 .2 Ž C2, 3 .. Let H2X be the unique Sylow 3-subgroup of 12. M22 in the inverse image of H2X . The inverse image of the quaternion group Q8Ž1. in 12. M22 has the form of an extension N4 .Q8Ž1., where Q8Ž1. centralizes the Sylow 2-subgroup N4 of N12 . Then N12. M 22Ž C2, 3 . s H2X i Ž N4 .Q8Ž1. .. We let r be an arbitrary linear character of N4 , and let sX4 be an element in the inverse image of sX4 in 12. M22 .2. We prove LEMMA 3.6.4. IrrŽ N4 .Q8Ž1. N r . consists of four linear characters and one character of degree 2. If r Ž a2 . s 1, then conjugation by sX4 interchanges two of these four linear characters and fixes the three other characters. Proof. The group N4 .Q8Ž1. is generated by two arbitrary elements, i9 and j9, in the inverse images of i9 and j9, respectively, as well as by the generator a of N4 . Because Ž i9. 2 s Ž j9. 2 is the involution in the center Z9 of the quaternion group Q8Ž1., the squares Ž i9. 2 and Ž j9. 2 each generate the inverse image Z9 of that center modulo N4 . Since i9 centralizes both Ž i9. 2 and a, it centralizes Z9 s ²Ž i9. 2 , a:. Similarly j9 centralizes Z9 s ²Ž j9. 2 , a:. Hence Z9 is contained in the center of N4 . Q8Ž1.. But the image Z9 of Z9 is the center of the image Q8Ž1. of N4 .Q8Ž1.. Therefore 2

2

Z Ž N4 .Q8Ž1. . s Z9 s ² Ž i9 . , a: s ² Ž j9 . , a: has order 8. The factor group N4 .Q8Ž1.rZ9 is isomorphic to Q8Ž1.rZ9, which is elementary abelian of order 4. It follows that N4 .Q8Ž1. is a 2-group of class 2 and that commutation in N4 .Q8Ž1. induces a bilinear map from Ž N4 .Q8Ž1.rZ9. = Ž N4 .Q8Ž1.rZ9. to Z9. The group N4 .Q8Ž1.rZ9 is elementary of order 4, generated by the images i9Z9 and j9Z9 of i9 and j9. It follows that the image w N4 .Q8Ž1., N4 .Q8Ž1. x of this commutation map has order 2 and is generated by w i9, j9x. Since this generator does not lie in N4 , we conclude

COUNTING CHARACTERS IN BLOCKS

31

that Z9 is the direct product Z9 s N4 = N4 .Q8Ž1. , N4 .Q8Ž1. . If r is any linear character of N4 , and z 9 is the nontrivial character of w N4 .Q8Ž1., N4 .Q8Ž1. x, then r = z 9 and r = 1 are the two extensions of r to linear characters of Z9. The extension r = 1 is trivial on w N4 .Q8Ž1., N4 .Q8Ž1. x, and hence extends to four linear characters of N4 .Q8Ž1.. The extension r = z 9 is faithful on w N4 .Q8Ž1., N4 .Q8Ž1. x. So there is exactly one irreducible character Fr of N4 .Q8Ž1. lying over it. The character Fr is equal to 2Ž r = z 9. on Z9 and is zero on N4 .Q8Ž1. y Z9. We conclude that IrrŽ N4 .Q8Ž1. N r . consists of four linear characters and one character of degree 2, whatever r may be. Next we consider how sX4 acts on the above characters when a2 lies in the kernel of r . Because sX4 inverts a, its square Ž sX4 . 2 g ² a: must lie in ² a 2 :. We are interested in the structure of the group Q s N4 .Q8Ž1.r² a2 , w N4 .Q8Ž1., N4 .Q8Ž1. x:. First we calculate the order of the generator j9 of N4 .Q8Ž1.. For this purpose, we must determine which conjugacy class, 4A or 4B, of elements of order 4 in M22 contains j. There are three conjugacy classes of elements of order 4 in SLŽ3, 4., represented by the matrices 1 T 9Ž b. s 1 0

0 1 b

ž

0 0 , 1

/

for b s 1, v , v 2 . We let T Ž b . be the image of T 9Ž b . in PSLŽ3, 4.. Then T Ž b . has a centralizer of order 16 in PSLŽ3, 4.. The element T Ž1. has a centralizer of order 32 in M22 , and thus lies in the conjugacy class 4A of that group. However, the two classes of T Ž v . and T Ž v 2 . fuse in M22 to form part of the class 4B, whose elements all have centralizers of order 16. It is straightforward to calculate that the matrix 0 As v 0



1 v 0

1 v v2

0

lies in SLŽ3, 4. and satisfies Aj9 s T 9Ž v 2 . A. Thus j lies in the conjugacy class of T Ž v 2 . in PSLŽ3, 4., and in the class 4B in M22 . We know from the Atlas that the element j9 of 12. M22 with image j9 in 3. M22 has order 8. Hence Ž j9. 2 g Z9 has order 4. It follows that Ž j9. 2 does not lie in the elementary subgroup ² a2 , w N4 .Q8Ž1., N4 .Q8Ž1. x:, and hence that the quotient Q modulo that subgroup does not have exponent 2. Of course, Q is abelian of order 8 and has a factor group isomorphic to Z2 = Z2 . Therefore, Q

32

JANICE F. HUANG

must be isomorphic to Z4 = Z2 . All of this tells us that Ž j9. 2 must be an element of order 4 in Z9, which forces Ž j9. 4 to equal a2 . In particular, j9 has order 8 modulo w N4 .Q8Ž1., N4 .Q8Ž1. x. The entire group N4 .Q8Ž1. has exponent dividing 8, since N4 .Q8Ž1.rZ9 has exponent 2 and Z9 has exponent 4. Hence both N4 .Q8Ž1. and N4 .Q8Ž1.rw N4 .Q8Ž1., N4 .Q8Ž1. x have exponent 8. This implies that Y s N4 .Q8Ž1.r N4 .Q8Ž1. , N4 .Q8Ž1. , Z8 = Z2 . As generators of this group Y we can take the image j Y s j9w N4 .Q8Ž1., N4 .Q8Ž1. x of j9, and some element h Y of order 2 which is not in ² j Y : , Z8 . Then the image of N4 s ² a: in Y is ²Ž j Y . 2 :, and that of ² a2 : is ²Ž j Y . 4 :. Evidently, conjugation by sX4 must leave invariant the unique maximum elementary subgroup ²Ž j Y . 4 , h Y : of Y, as well as the subgroup ²Ž j Y . 4 : of all fourth powers of elements of Y. It follows that w h Y , sX4 x lies in ²Ž j Y . 4 :. From the action of sX4 on i9 and j9, it follows that w j Y , sX4 x lies in ²Ž j Y . 2 : = ² h Y :, but not in ²Ž j Y . 2 :. We conclude that w N4 .Q8Ž1., sX4 x²Ž j Y . 4 : s ²w j Y , sX4 x, Ž j Y . 4 : has order 4 and intersects ² j Y : in ²Ž j Y . 4 :. This implies that any linear character r of ² a:, trivial on ² a2 :, has two extensions to linear characters of N4 .Q8Ž1., trivial on w N4 .Q8Ž1., sX4 x, and two extensions to linear characters of N4 .Q8Ž1., nontrivial on w N4 .Q8Ž1., sX4 x. Obviously, the former two extensions are fixed by sX4 , and the latter two are interchanged by sX4 . The remaining character in IrrŽ N4 .Q8Ž1. N r . is the only one of degree 2, and so is fixed by sX4 . This completes the proof of the lemma. We are now ready to calculate the values in the remaining column of Table 1. As in Sects. 3.4 and 3.5, we begin our determination of these values by letting n s 1. We recall from Ž3.1.6. that the normalizer NM 22Ž C2, 3 . s NM 22Ž H2 . is isomorphic to H2 i Q8 . The centralizer CM 22Ž H2 . must have a normal 3-complement K. Using Ž3.4.7., we determine that there must be exactly one 3-block, 1.bY0 , of NM 22Ž C2, 3 ., corresponding to the single conjugacy class of characters in IrrŽ K .. The block 1.bY0 has six characters, each with 3-defect d s 2. In addition, 1.bY0 has defect group H2 and induces the block 1. B0 of M22 . We determine that the normalizer NM 22 .2 Ž C2, 3 . must fuse exactly two characters in 1.bY0 and fix every other character. We let n 1 s 1 and have the first two entries in the fourth column of Table 1. Further, k Ž NM 22Ž C2, 3 ., B, d, O N 1. s 0 for any other combination of B, d, and O satisfying the hypotheses of Theorem 1. Therefore, the third entry in this column of Table 1 is 0. Next we let n s 2 and let n 2 be a faithful linear character of the center N2 of 2. M22 . The normalizer N2. M 22 .2 Ž C2, 3 . has the structure Z2 = NM 22 .2 Ž C2, 3 .. Then N2. M 22 .2 Ž C2, 3 . has exactly one 3-block, 2.bY0 s B Ž n 2 . = 1.bY0 lying over B Ž n 2 .. This block has defect group H2 and induces 2. B0 .

COUNTING CHARACTERS IN BLOCKS

33

Furthermore, IrrŽ1.bY0 . is isomorphic to IrrŽ2.bY0 N n 2 . as an N2. M 22 .2 Ž C2, 3 .set. The isomorphism sends any f g IrrŽ1.bY0 . to n 2 = f g IrrŽ2.bY0 N n 2 .. Hence it preserves both degrees and 3-defects. Once again the normalizer N2. M 22 .2 Ž C2, 3 . must fuse exactly two characters in 2.bY0 and fix every other character. This gives us the fourth and fifth entries in the fourth column of Table 1. Further, k Ž N2. M 22Ž C2, 3 ., B, d, O N n 2 . s 0 for any other combination of B, d, and O satisfying the hypotheses of Theorem 1. Therefore, the sixth entry in this column of Table 1 is 0. We now let n s 4 and let n4 be a faithful linear character of the center N4 of 4. M22 . The centralizer C4. M 22 .2 Ž H2 . must have a normal 3-complement, telling us that N4. M 22 .2 Ž C2, 3 . has a single block 4.bY0 of characters corresponding to the single class of characters of this complement. This block has defect group H2 and induces 4. B0 . We again have a defectpreserving isomorphism between IrrŽ4.bY0 N n4 . and IrrŽ1.bY0 .. However, in this case there is no fusion of characters. We choose O s Z1 and have the seventh entry in the fourth column of Table 1. Further, k Ž N4. M 22Ž C2, 3 ., B, d, O N n4 . s 0 for any other combination of B, d, and O satisfying the hypotheses of Theorem 1. We next let n s 3 and let n 3 be a faithful linear character of the center N3 of 3. M22 . Using Ž3.4.7., we find there is one 3-block, 3.bY0 , of N3. M 22Ž C2, 3 ., corresponding to the single conjugacy class of characters of the normal 3-complement of the centralizer. The set IrrŽ3.bY0 N n 3 . consists of five characters, each with 3-defect d s 2. It has defect group H2X and induces the block 3. B0 of 3. M22 . We choose O s Z1 and let n 3 be a faithful linear character of N3 . We then have the eighth entry in the fourth column of Table 1. Further, k Ž N3. M 22Ž C2, 3 ., B, d, O N n 3 . s 0 for any other combination of B, d, and O satisfying the hypotheses of Theorem 1. Therefore, the ninth entry in this column is 0. Now we let n s 6 and let n 6 be a faithful linear character of the center N6 of 6. M22 . As in the case when n s 2, the normalizer N6. M 22Ž C2, 3 . has the form Z2 = N3. M 22Ž C2, 3 .. Then N6. M 22 .2 Ž C2, 3 . has exactly one 3-block, 6.bY0 s B Ž n 6 . = 3.bY0 lying over B Ž n 6 .. This block has defect group H2X and induces 6. B0 . Furthermore, IrrŽ3.bY0 N n 3 . is isomorphic to IrrŽ6.bY0 N n 6 . as an N6. M 22Ž C2, 3 .-set. The isomorphism preserves both degrees and 3-defects. We choose O s Z1. This gives us the 10th entry in the fourth column of Table 1 and tells us that k Ž N6. M 22Ž C2, 3 ., B, d, O N n 6 . s 0 for any other combination of B, d, and O satisfying the hypotheses of Theorem 1. Therefore, the 11th entry in this column is 0. Finally, we let n s 12 and let n 12 be a faithful linear character of the center N12 of 12. M22 . The normalizer N12. M 22Ž C2, 3 . has the structure Z4 = N3. M 22Ž C2, 3 .. Thus N12. M 22Ž C2, 3 . has exactly one 3-block, 12.bY0 lying over B Ž n 12 .. This block has defect group H2X and induces 12. B0 . Furthermore, IrrŽ3.bY0 N n 3 . is isomorphic to IrrŽ12.bY0 N n 12 . as an N12. M 22Ž C2, 3 .-set.

34

JANICE F. HUANG

The isomorphism preserves both degrees and 3-defects. We choose O s Z1. This gives us the last entry in the fourth column of Table 1. Further, k Ž N12. M 22Ž C2, 3 ., B, d, O N n 12 . s 0 for any other combination of B, d, and O satisfying the hypotheses of Theorem 1. This completes the verification of all entries in Table 1 and hence completes the proof of Theorem 1. We shall turn our attention next to the prime 2. 4. THE PRIME p s 2 4.1. Radical 2-Subgroups of M22 and Their Normalizers We shall make a systematic search for the radical 2-subgroups of M22 . The observation in Ž1.1.2., that P is a radical p-subgroup of G if and only if it is one of NG Ž V Ž ZŽ P ..., will simplify our calculations. In the process of identifying the radical 2-subgroups of M22 , Ž4.1.1. we shall let A run through the M22 -conjugacy classes of elementary abelian 2-subgroups. For each such A we shall find all groups P such that P is a radical 2-subgroup of N Ž A. and such that A s V Ž ZŽ P ... This procedure will yield all radical 2-subgroups of M22 . We shall first identify the maximal elementary abelian 2-subgroups of M22 and their normalizers. We shall refer to the work of Adem and Milgram w1x extensively in what follows. We begin our task by considering a Sylow 2-subgroup Ts

~¡ 1a

¢b

¦¥ §

0 1 c

0 0 ; a, b, c g F4 1

Ž 4.1.2.

of PSLŽ3, 4.. We point out that T has order 2 6 and, from Ž2.1.6., the involution s2 normalizes T. We then define a Sylow 2-subgroup S s ²T , s2 :

Ž 4.1.3.

of M22 , which has order 2 7. As Adem and Milgram w1, p. 389x point out, there are three conjugacy classes of maximal elementary abelian 2-subgroups of M22 . As representatives of two of these classes we shall use the subgroups of T given by Es

~¡ 10

¢b

0 1 c

¦¥ §

0 0 ; b, c g F4 1

Ž 4.1.4.

35

COUNTING CHARACTERS IN BLOCKS

and Fs

~¡ 1a

¢b

0 1 0

¦¥ §

0 0 ; a, b g F4 , 1

Ž 4.1.5.

where E s EX , the pointwise stabilizer of the hexad X of Ž2.3.1.. The normalizer of EX was constructed in Sect. 2.3 as NM 22Ž E . , E i AŽ2. 6 , Ž . Ž . Ž . where AŽ2. was defined in 2.3.13 as the intersection M l M 6 22 X 22 Z of the stabilizers of two disjoint hexads. In addition, we pointed out there that N Ž E . acts as S4Ž2.9 , A 6 on the elements of E while preserving the symplectic form Ž2.3.7.. We note that the Sylow 2-subgroup S of M22 identified in Ž4.1.3. is also a Sylow 2-subgroup of the normalizer NM 22Ž E . of the pointwise stabilizer E s EX of the hexad X, since the elements of S all stabilize X. Further, all elements of N Ž E . , E i AŽ2. 6 are in one of the forms a1 a3 a5

a2 a4 a6

0 0 1

or

a1 a3 a5

a2 a4 a6

0 aX1 0 s2 aX3 1 aX5

aX2 aX4 aX6

0 0 , 1

Ž 4.1.6.

for some a i , aXj g F4 , where the determinants are all 1. We turn next to the group F, which is our representative of the other conjugacy class of maximal elementary 2-subgroups of M22 with order 2 4 , described in Ž4.1.5.. It has only two fixed points in P, namely u and w1, 0, 0x. It follows that the normalizer NP S LŽ3, 4.Ž F . is the stabilizer PSLŽ3, 4.w1, 0, 0x s Ž M22 . u, w1, 0, 0x whose elements are described in Ž2.1.5.. It Ž2. can also be identified as the semidirect product F i AŽ2. 5 , where A 5 is the complement to F consisting of all elements of the form 1 v b2 c2 v e2 f 2

0 b e

0 c , f

Ž 4.1.7.

for some b, c, e, f g F4 , where bf y ce s 1. From Ž2.1.6. it is clear that the Ž . element s2 normalizes both F and AŽ2. 5 , L 2 4 , while acting on NP S LŽ3, 4.Ž F .rF as the Galois automorphism of L2 Ž4.. We set S5Ž2. equal to ² s2 :. Then AŽ2. 5 i NM 22 Ž F . s F i S5Ž2. .

Ž 4.1.8.

ŽSee w1, p. 389x.. It is clear that the Sylow 2-subgroup S of M22 is a subgroup of NM 22Ž F ., since it stabilizes the subset  u, w1, 0, 0x4 of P. We

36

JANICE F. HUANG

note that NM 22Ž F . consists precisely of all elements in one of the following forms: 1 b1 b4

0 b2 b5

0 b3 b6

or

1 s2 b1 b4

0 b2 b5

0 b3 , b6

Ž 4.1.9.

for some bi , bj g F4 , where the determinants are all 1. We point out that E l F is the center ZŽT . of the Sylow 2-subgroup T of PSLŽ3, 4., and that T s EF. The normalizer NP S LŽ3, 4.ŽT . is clearly the subgroup of PSLŽ3, 4. of all elements of the form 1 a c

0 b d

0 0 e

with b ? e s 1. Thus NP S LŽ3, 4.ŽT . s ²T, g :, where 1 gs 0 0

0 v 0

0 0 v2

Ž 4.1.10.

was identified in Sect. 3.1. Since s2 normalizes T, it follows that NM 22 Ž T . s ² S, g : .

Ž 4.1.11.

If we set S3Ž2. s ² s2 , g :, then NM 22ŽT . is isomorphic to a semidirect product of T by S3Ž2.. As the representative of the third and last conjugacy class of maximal elementary abelian 2-subgroups of M22 we shall use the subgroup of order 23 D s ² d , z , s2 : ,

Ž 4.1.12.

where 1 ds 0 1

0 1 0

0 0 gElF 1

Ž 4.1.13.

is the central involution of the Sylow 2-subgroup S of Ž4.1.3. and where 1 zs 1 0

0 1 0

0 0 g F. 1

Ž 4.1.14.

37

COUNTING CHARACTERS IN BLOCKS

Then the normalizer NM 22Ž D . of D is isomorphic to a semidirect product D i AutŽ D . , D i L3 Ž2.. ŽSee w1, p. 389x.. We point out that we have so far determined four radical 2-subgroups of M22 , namely S, since it is a Sylow 2-subgroup of M22 , as well as E, F, and D, since their normalizers have the form of a semidirect product of a simple Žor almost simple in the case of F . group with each of them. We note that S is equal to the product EFD and calculate that it is self-normalizing in M22 . In the notation of Ž4.1.1., we now let A s E and point out that the only radical 2-subgroup P of N Ž A. , E i A 6 which satisfies A s E s V Ž ZŽ P .. is P s E. Similarly, if A s F then P s F, and if A s D then P s D. We turn next to the remaining conjugacy classes of elementary abelian 2-subgroups. We recall Lemma 2.3.9, which states that if A F E s EX and A / 1 then N Ž A. F N Ž E .. The following lemma will also be useful for our calculations. LEMMA 4.1.15.

If K F F and < K < ) 2 then N Ž K . F N Ž F ..

Proof. We may assume that K F F is a group which contains distinct involutions 1 ss a b

0 1 0

0 0 1

and

1 t s a9 b9

0 1 0

0 0 . 1

Then, if an element w x, y, z x g PGŽ2, 4. with w x, y, z x / w1, 0, 0x is fixed by both s and t, we must have ay q bz s 0 and a9 y q b9z s 0. Since y and z are not both zero, this happens iff the determinant of Ž a9a b9b . is 0. Hence if the determinant of Ž a9a b9b . is not zero, the only fixed points of K are u and w1, 0, 0x. In this case, N Ž K . F Ž M22 .u, w1, 0, 0x4 s N Ž F .. On the other hand, if this determinant is 0, then K is N Ž F .-conjugate to the subgroup ZŽT . s E l F. Adem and Milgram w1, p. 389x confirm our calculations that C Ž E l F . s T while N Ž E l F .rC Ž E l F . , S3 . It follows that N Ž E l F . s T i S3Ž2., where S3Ž2. s ² s2 , g : was identified earlier. The group N Ž E l F . is then precisely the normalizer N ŽT . s N Ž EF . s ² S, g :, identified in Ž4.1.11.. Since this is a subgroup of N Ž F ., this completes the proof of Lemma 4.1.15. We proceed by considering the remaining conjugacy classes of elementary abelian 2-subgroups of M22 . The Atlas w8x identifies a single conjugacy class of involutions in M22 . Parrott w25x determined, and Adem and Milgram w1, p. 389x confirmed, that the normalizer of an involution of M22 is of the form E i S4 . If we let A s ² d :, where d was identified in Ž4.1.13. as the central involution of the Sylow 2-subgroup S, then the normalizer N Ž² d :. can be calculated as E i S4Ž1., where S4Ž1. s ² S3Ž1., V2Ž1. :,

38

JANICE F. HUANG

with V2Ž1. s O 2 Ž S4Ž1. . s ² z , s2 : F D and S3Ž1. s ² a , k :. Here 1 as 0 0

1 1 0

0 v 0 s2 1 1 0

1 0 0

0 0 g AŽ2. 6 1

Ž 4.1.16.

and 1 k s v2 1

0 1 0

0 0 g S. 1

Ž 4.1.17.

We then have LEMMA 4.1.18. The only groups P which are radical 2-subgroups of N Ž² d :. and which satisfy ² d : s V Ž ZŽ P .. are the Sylow 2-subgroup S and the group ED s E i V2Ž1. s O 2 Ž N Ž² d :... Proof. The radical 2-subgroups of N Ž² d :. s E i S4Ž1. are the inverse images of the radical 2-subgroups of S4Ž1., which are the Sylow 2-subgroup Ž1. Ž² :. of D Ž1. D Ž1. 8 and the normal 4-group V2 . The inverse image in N d 8 is the Sylow 2-subgroup S of M22 . Clearly ² d : s V Ž ZŽ S ... The group V2Ž1. s O 2 Ž S4Ž1. . has the inverse image ED in N Ž² d :.. Thus ED s O 2 Ž N Ž² d :... Since E is a regular F 2 V2Ž1.-module, it follows that CE Ž V2Ž1. . s ² d :, which is the center of ED. Hence N Ž ED . s N Ž² d :., so that ED is a radical 2-subgroup of N Ž² d :.. Then ² d : s V Ž ZŽ ED ... This completes the proof of Lemma 4.1.18. We note that N Ž E . l N Ž D . / N Ž E l D . s N Ž ² d :. .

Ž 4.1.19.

In fact, Adem and Milgram w1, p. 393x point out that N Ž E . l N Ž D . , D i S4 , where S4 represents one of the two conjugacy classes of S4 in L3 Ž2.. We confirm that elements in S of the form 1 0 0

0 1 c

0 0 , 1

with c / 0, 1,

centralize the element d but do not normalize D. In particular, we note that S is not a subgroup of N Ž D .. We conclude that N Ž E l D . is a subgroup of N Ž E ., by Lemma 2.3.9, but not of N Ž D .. As stated in Adem and Milgram w1, p. 393x, there are two conjugacy classes of elementary abelian 2-subgroups of order 2 3 in M22 , in addition to the maximal class represented by D. One of these classes contains every subgroup of order 2 3 in E, and the other contains every subgroup of order

COUNTING CHARACTERS IN BLOCKS

39

Ž2. 2 3 in F. The actions of AŽ2. 6 and S5 are both transitive on these classes in E and F, respectively. As a representative of the AŽ2. 6 -conjugacy class of subgroups of order 2 3 in E we may take the group

~¡ 01

¢

0 1 c

¦¥ §

0 0 ; b g F4 and c g F 2 , b 1 which is the space perpendicular to ² d : with respect to the symplectic form Ž2.3.7.. We can then prove Bs

LEMMA 4.1.20. There are no radical 2-subgroups P of M22 such that B s V Ž ZŽ P ... Proof. We note that N Ž B . must be a subgroup of N Ž E . by Lemma 2.3.9. Since N Ž² d :. normalizes the three-dimensional subspace B perpendicular to ² d :, it follows that N Ž B . is the same group as N Ž² d :.. In fact, 1 F ² d : F B F E is the only N Ž² d :. composition series of E. The proof of Lemma 4.1.20 is then a direct result of the proof of Lemma 4.1.18. As a representative of the N Ž F .-conjugacy class of elementary abelian subgroups of order 2 3 in F, we choose the group Vs

~¡ 1a

¢b

0 1 0

¦¥ §

0 0 ; a g F 2 and b g F4 . 1

We then prove LEMMA 4.1.21. There are no radical 2-subgroups P of N Ž V . which satisfy V s V Ž ZŽ P ... Proof. We find that N Ž² d :. l N Ž F . s S. Here the group S acting on its subgroup F has the form S s F i D 8 . We quickly determine that NP S LŽ3, 4.Ž V . is precisely T and that NNŽ F .Ž V . s S s T² s2 :. We note that N Ž V . is a subgroup of N Ž F . by Lemma 4.1.15. Hence N Ž V . s S. The only radical 2-subgroup of N Ž V . is then S. However, S clearly does not satisfy V s V Ž ZŽ S .., since ZŽ S . s ² d :. This completes the proof of Lemma 4.1.21. Finally, we turn our attention to the conjugacy classes of elementary abelian 2-subgroups of M22 with order 2 2 . Adem and Milgram w1, p. 393x confirm our calculations, which indicate that there are four such classes. One of these can be represented by E l F, which lies in both E and F. There are three additional classes, one having representatives contained in E and two having representatives contained in F. We prove first LEMMA 4.1.22. The only group P which is a radical 2-subgroup of N Ž E l F . and which satisfies E l F s V Ž ZŽ P .. is the Sylow 2-subgroup T s EF of PSLŽ3, 4..

40

JANICE F. HUANG

Proof. The group E l F is the center ZŽT . of the Sylow 2-subgroup T s EF of PSLŽ3, 4. defined in Ž4.1.2.. We pointed out in Ž4.1.11. and in the following sentence that N ŽT . s ² S, g : s T i S3Ž2.. Clearly N Ž E l F .rC Ž E l F . is isomorphic to a subgroup of S3 , AutŽ E l F .. Since C Ž E l F . s T and since the elements of S3Ž2. all normalize E l F, we conclude that N Ž E l F . s N ŽT . s T i S3Ž2.. We shall, in fact, prove more here than is stated in the lemma, and use this opportunity to develop an alternative structure of N ŽT . as E i S4 . We calculate that the element fs

1 0 v2

0 v 1

0 0 g g E, v2

Ž 4.1.23.

which has order 3, normalizes the 4-group V2Ž2. s ² z , k : of a copy of D 8 given by D 8Ž2. s ² z , k :. The element z was identified in Ž4.1.14. as a generator of D, and 1 k s s2 1 1

0 v2 1

0 0 , v

Ž 4.1.24.

while k 2 s k was identified in Ž4.1.17.. We point out that z , k , and k are all elements of N ŽT . , T i S3Ž2.. In fact, z and k lie in F. Setting S4Ž2. s ² D 8Ž2., f : gives us a copy of S4 complementary to E and also contained in NM 22ŽT .. We conclude that NM 22 Ž E l F . s NM 22Ž T . s E i S4Ž2.

Ž 4.1.25.

Ž2. and notice that S4Ž2. F AŽ2. 6 , since the elements of S4 stabilize both hexads X and Z. We know that N Ž E . l N Ž F . F N Ž E l F .. But Lemmas 2.3.9 and 4.1.15 verify the reverse inclusion. It follows that these two groups are identical, that is, NŽ E. l NŽ F . s NŽ E l F . . Ž 4.1.26.

Adem and Milgram w1, p. 393x confirm that N Ž E . l N Ž F . s E i S4Ž2., where S4Ž2. represents a second conjugacy class of S4 in A 6 . ŽThe first such class is represented by S4Ž1. in N Ž² d :. s E i S4Ž1... The radical 2-subgroups of N Ž E l F . s E i S4Ž2. are then the inverse images of the radical 2-subgroups of S4Ž2., which are the Sylow 2-subgroup Ž2. Ž2. Ž . D Ž2. 8 and the normal 4-group V2 . The inverse image in N E l F of D 8 Ž2. is the Sylow 2-subgroup S of M22 , while the inverse image of V2 in N Ž E l F . is T s EF s E i V2Ž2.. The group S does not satisfy E l F s V Ž ZŽ S .. since ZŽ S . s ² d :. However, the group T clearly satisfies E l F s V Ž ZŽT ... This completes the proof of Lemma 4.1.22.

41

COUNTING CHARACTERS IN BLOCKS

Next we turn our attention to the remaining conjugacy class of elementary abelian 2-subgroups of order 2 2 in E. We choose as a representative of this class the group U s ² d , c :, where

cs

1 0 v2

0 1 v

0 0 . 1

The group U is nonsingular with respect to the symplectic form Ž2.3.7.. Adem and Milgram w1, p. 393x find that the centralizer of the group U is of the form E i Z3 , while the normalizer is Ž E i Z3 . i S3 . In our calculations we find that an involution in S3 can be chosen as the element k g F of Ž4.1.17., and that k inverts the subgroup Z3 = Z3 of the normalizer. It follows that N ŽU . , E i wŽ Z3 = Z3 . i Z2 x. The radical 2-subgroup P s E i ² k : of this normalizer clearly does not satisfy U s V Ž ZŽ P .. since ZŽ P . s ZŽT . s E l F. This gives us the following result. LEMMA 4.1.27. There is no group P which is a radical 2-subgroup of N ŽU . and which satisfies U s V Ž ZŽ P ... We are ready now to consider the final two conjugacy classes of elementary abelian 2-subgroups of M22 with order 2 2 . Both of these classes contain representatives lying in the group F. We choose as a representative of one of these classes the group M s ² d , z :, where d was identified in Ž4.1.13. as the central element of the Sylow 2-subgroup S of M22 , and z was identified in Ž4.1.14.. Then M F D. We recall that N Ž F . , F i S5Ž2., where S5Ž2. acts on F as L2 Ž4. together with its Galois automorphism. The group M is fixed elementwise by the third generator of D, the element s2 , acting as the Galois automorphism of L3 Ž2.. We prove LEMMA 4.1.28. The one group P which is a radical 2-subgroup of N Ž M . and which satisfies M s V Ž ZŽ P .. is the group FD. Proof. We note that M s F l D. We have that N Ž M . F N Ž F . by Lemma 4.1.15. Adem and Milgram w1, p. 393x confirm our calculations that C Ž M . s F i Z2Ž3., where Z2Ž3. s ² s2 :. That is, C Ž M . s FD. As before, we have that N Ž M .rC Ž M . , S3 . We calculate that N Ž M . , F i Ž Z2Ž3. = S3Ž3. . by finding that s2 commutes with the generators 1 ms 0 v

0 0 1

0 1 1

and

1 ûs 0 v

0 1 1

0 0 1

42

JANICE F. HUANG

of a copy S3Ž3. of S3 . Notice that m and û have orders 3 and 2, respectively. ŽWe note that S5 has a conjugacy class of maximal subgroups in the form Z2 = S3 .. The radical 2-subgroups of N Ž M . are then F i Z2Ž3. s FD s O 2 Ž N Ž M .., whose normalizer is F i Ž Z2Ž3. = S3Ž3. . and the conjugates of the Sylow 2-subgroup P s F i Ž Z2 = Z2 .. The group P does not satisfy M s V Ž ZŽ P .. since ZŽ P . s ² d :. However, FD clearly satisfies M s V Ž ZŽ FD ... This completes the proof of Lemma 4.1.28. We note that NŽ F l D. s NŽ F . l NŽ D. .

Ž 4.1.29.

Adem and Milgram w1, p. 393x calculate that N Ž F . l N Ž D . , D i S4 , where S4 represents the second conjugacy class of S4 in L3 Ž2.. ŽThe first such class is represented by a subgroup in N Ž E . l N Ž D . , D i S4 .. We find that the element m of order 3 in N Ž F l D . normalizes the 4-group ² : of D 8 . Here k was identified in V2Ž3. s ² k , b 2 : of a copy D Ž3. 8 s k, b Ž4.1.17. and 1 b s v2 1

0 1 1

0 0 . 1

: gives a copy of S4 which lies in N Ž F l D . and Setting S4Ž3. s ² D Ž3. 8 ,m satisfies S4Ž3. l D s 1. We conclude that N Ž F l D . can be constructed both as F i Ž Z2Ž3. = S3Ž3. . and as D i S4Ž3., and that N Ž F . l N Ž D . s N Ž F l D .. However, we point out that S is not a subgroup of this normalizer, since it does not lie in N Ž D .. As a representative of the final class of elementary abelian 2-subgroups of M22 to be considered, we choose L s ² d , k :, where d is again the central element of the Sylow 2-subgroup S of M22 and k was identified in Ž4.1.17.. This subgroup represents the conjugacy class made up of subgroups none of which is fixed by the Galois automorphism s2 of L2 Ž4.. Adem and Milgram w1, p. 393x confirm our calculations that C Ž L. s F, while the normalizer N Ž L. , F i S3 . The radical 2-subgroup of N Ž L. is then P , F i Z2 . However, P does not satisfy L s V Ž ZŽ P .. since ZŽ P . s F. This completes the proof of LEMMA 4.1.30. There are not radical 2-subgroups P of N Ž L. which satisfy L s V Ž ZŽ P ... This concludes our search for the radical 2-subgroups of M22 . We combine our results in the form of the following proposition. PROPOSITION 4.1.31. The following is a complete list of representati¨ es of the conjugacy classes of radical 2-subgroups of M22 : 1, E, F, D, T s EF, ED, FD, and S s EFD.

43

COUNTING CHARACTERS IN BLOCKS

4.2. Radical 2-Chains of M22 In this section we shall prove For the prime p s 2,

PROPOSITION 4.2.1. < <

Ý Cg RŽ M22 .rM 22

Ž y1. C k Ž Nn . M 22 Ž C . , n. B, d, O N nn .

s k Ž Nn . M 22 Ž 1 . , n. B, d, O N nn . y k Ž Nn . M 22 Ž 1 - E . , n. B, d, O N nn . y k Ž Nn . M 22 Ž 1 - F . , n. B, d, O N nn . q k Ž Nn . M 22 Ž 1 - EF . , n. B, d, O N nn . y

< <

Ý Cg RŽ N Ž D ..rN Ž D .

Ž y1. C k Ž Nn . M 22 Ž C . , n. B, d, O N nn . . Ž 4.2.2.

Proof. We begin by setting k Ž C . s k Ž Nn. M 22Ž C ., n. B, d, O N nn . for any 2-chain C of M22 . Then k Ž C . depends only on the M22-conjugacy class of C. Furthermore, as we noted in Ž1.4.2., k Ž C . is equal to k Ž C9. whenever C and C9 are 2-chains of M22 with the same normalizer NM 22 .2 Ž C . s NM 22 .2 Ž C9. in M22 .2. As usual, we write R s RŽ M22 . for the family of all radical 2-chains of M22 . We know from Definition 1.1.4 that any chain C: P0 - P1 - ??? - Pn in R starts with P0 s O 2 Ž M22 . s 1. If < C < s n ) 0, then the second subgroup P1 in C is a nontrivial radical 2-subgroup of M22 . For any nontrivial radical 2-subgroup Q of M22 , we define RQ to be the subfamily of all chains C g R with < C < ) 0 such that the second subgroup P1 in C is M22 -conjugate to Q. Evidently RQ is closed under M22-conjugation, as is the complementary subfamily RQX s R y RQ . We apply the above definitions when Q is the nontrivial radical 2-subgroup D of M22 . Since R is the disjoint union of the M22-invariant subfamilies RD and RDX , any subfamily RrM22 of representatives for the M22 -orbits in R is the disjoint union of similar subfamilies RD rM22 and RDX rM22 of RD and RDX , respectively. It follows that the sum on the left-hand side of Ž4.2.2. is

Ý

Cg RrM 22

< <

Ž y1. C k Ž C. s

Ý

Cg RD rM 22

< <

Ž y1. C k Ž C . q

Ý

X D rM 22

< <

Ž y1. C k Ž C . .

Cg R

Ž 4.2.2A.

44

JANICE F. HUANG

Let Rw D x be the subfamily of all chains C g RD having D as their second subgroup P1. By the definition of RD , any chain in RD is M22-conjugate to some chain in Rw D x. Clearly Rw D x is closed under conjugation by elements of N Ž D . s NM 22Ž D .. Furthermore, any element g g M22 sending some chain C g Rw D x to another chain C g g Rw D x must normalize the common second subgroup D in both these chains, and hence lie in N Ž D .. It follows that any family Rw D xrN Ž D . of representatives for the N Ž D .-orbits in Rw D x is also a family of RD rM22 of representatives for the M22-orbits in RD . Let C: P0 s 1 - P1 s D - P2 - ??? - Pn be a 2-chain in M22 with length n ) 0 and the indicated first and second subgroups. Since D is a radical 2-subgroup of M22 , it follows from Definition 1.1.4 that C is a radical 2-chain of M22 if and only if C 1 : P1 - P2 - ??? - Pn is a radical 2-chain of N Ž D .. Hence the map C ¬ C 1 is a bijection of Rw D x onto RŽ N Ž D ... Clearly this bijection preserves N Ž D .-conjugacy, and hence sends RD rN Ž D . onto a family RŽ N Ž D ..rN Ž D . of reprsentatives for the N Ž D .-orbits in RŽ N Ž D ... Since any chain C g Rw D x starts with P0 s 1, it has the same normalizer NM 22 .2 Ž C . s NM 22 .2 Ž C 1 . in M22 .2 as its image C 1. Hence k Ž C . s k Ž C 1 .. Because < C < s < C 1 < q 1, we conclude that the first sum on the right-hand side of Ž4.2.2A. is

Ý

< <

Ž y1. C k Ž C . s

Cg RD rM 22

< <

Ž y1. C k Ž C .

Ý Cg Rw D xrN Ž D .

sy

Ý

< <

Ž y1. C k Ž C . .

Cg RŽ N Ž D ..rN Ž D .

Substituting this last expression for that sum in Ž4.2.2A., we see that Ž4.2.2. will hold, and the proposition will be true, if and only if

Ý

X D rM 22

< <

Ž y1. C k Ž C . s k Ž 1 . y k Ž 1 - E . y k Ž 1 - F . q k Ž 1 - EF . .

Cg R

Ž 4.2.2 B . A subgroup Q of S s EFD is said to be weakly closed in S with respect to M22 .2 if it is the only M22 .2-conjugate Q g of itself contained in S. Because E and F are the only elementary abelian subgroups of S with order 16, but are not M22 -conjugate to each other, they are both weakly closed in S with respect to M22 .2. Hence so is their product EF. If Q is any of these three subgroups, then RG Q will denote the subfamily of all chains C g R with < C < ) 0 such that the second subgroup P1 in C contains some M22 -conjugate Q g of Q. Evidently RG Q is closed under M22-conjugation. The unique radical 2-chain of M22 with length 0 is the chain 1 consisting only of the trivial subgroup of M22 . This chain lies in RDX . Any other chain

45

COUNTING CHARACTERS IN BLOCKS

C g RDX has length < C < ) 0, and a second subgroup P1 which is a nontrivial radical 2-subgroup of M22 not conjugate to D. In view of Proposition 4.1.31, such a P1 must be M22-conjugate to exactly one of the groups E, F, EF, ED, FD, or EFD. Since each of these groups contains either E or F, we conclude that C lies in RG E j RG F . On the other hand, D, which has order 8, cannot contain any conjugate of E or F, which have order 16. So no chain in RD can lie in either RG E or RG F . Therefore, RDX is the disjoint union of  14 and RG E j RG F . It follows that RDX rM22 is the disjoint union of  14 and a family Ž RG E j RG F .rM22 of reprsentatives for the M22 -orbits in RG E j RG F . Evidently any chain in RG EF lies in RG E l RG F . Conversely, if C g RG E l RG F , then the second subgroup P1 in C contains both an M22-conjugate E g of E and M22-conjugate F h of F. Because P1 is a 2-subgroup of M22 , it is contained in some M22-cony1 jugate S k of the Sylow 2-subgroup S. So the M22 -conjugate E g k of E is y1 contained in S. Since E is weakly closed in S, we must have E g k s E. Hence E g s E k . Similarly F h s F k . So P1 contains the M22-conjugate E k F k s Ž EF . k of EF. Therefore, C belongs to RG EF . This completes the proof that RG EF s RG E l RG F . We may choose a family RG EF rM22 of representatives for the M22-orbits in RG EF , and then extend it to two families RG ErM22 and RG F rM22 of representatives for the M22 -orbits in RG E and RG F , respectively. We may choose Ž RG E j RG F .rM22 to be the union Ž RG ErM22 . j Ž RG F rM22 . of the resulting families. Then RDX rM22 is the union of the three subfamilies  14 , RG ErM22 , and RG F rM22 . The first of these subfamilies is disjoint from the other two, which intersect in RG EF rM22 . It follows that the sum on the left-hand side of Ž4.2.2B. is

Ý X

< <

Ž y1. C k Ž C . s k Ž 1 . q

Cg RD rM 22

Ý

< <

Ž y1. C k Ž C .

Cg RG E rM 22

q

Ý

< <

Ž y1. C k Ž C .

Cg RG F rM 22

y

Ý

< <

Ž y1. C k Ž C . .

Cg RG EF rM 22

Hence Ž4.2.2B. will hold, and the proposition will be true, if

Ý

< <

Ž y1. C k Ž C . s yk Ž 1 - Q . ,

Ž 4.2.2C .

Cg RG Q rM 22

for each Q s E, F, EF. The group Q is now one of the nontrivial radical 2-subgroups E, F, or EF of M22 . Furthermore, Q is weakly closed in S with respect to M22 .2. Suppose that C: P0 - ??? - Pn is any 2-chain of M22 starting with P0 s 1,

46

JANICE F. HUANG

such that n ) 0 and P1 ) Q g for some g g M22 . We claim that Q g is the only M22 .2-conjugate of Q contained in P1. Indeed, the 2-subgroup P1 of M22 must be contained in some M22-conjugate S k of the Sylow 2-subgroup y1 S. Then Q g k is an M22 .2-conjugate of Q contained in S. Since Q is y1 weakly closed in S, we must have Q g k s Q. So Q g s Q k . It follows similarly that Q h s Q k whenever Q h is an M22 .2-conjugate of Q contained in P1. Thus any such Q h must equal Q g . The fact that Q g is unique allows us to define a 2-chain f Ž C .: P0 s 1 - Q g - P1 - ??? - Pn of length < C < q 1 depending only on C. If h g M22 .2 normalizes C, then Q g h Ž P1 . h s P1. So the uniqueness of Q g implies that Q g h s Q g , that is, that h normalizes f Ž C .. Since any h g M22 .2 normalizing f Ž C . must normalize C, we conclude that C and f Ž C . have the same normalizer in M22 .2. Hence k Ž C . s k Ž f Ž C .. and NM 22Ž C . s NM 22Ž f Ž C ... For each i s 1, 2, . . . , n we may apply the function f to the initial subchain Ci : P0 - P1 - ??? - Pi of C with length i. It gives us the initial subchain f Ž Ci . s f Ž C . iq1: P0 Q g - P1 - ??? - Pi of f Ž C . with length i q 1. Applying the arguments in the preceding paragraph with Ci in place of C, we obtain NM 22Ž Ci . s NM 22Ž f Ž Ci .. s NM 22Ž f Ž C . iq1 .. Hence O 2 Ž NM 22Ž Ci .. is equal to the ith subgroup Pi in C if and only if O 2 Ž NM 22Ž f Ž C . iq1 .. is equal to the Ž i q 1.st subgroup Pi in f Ž C .. Because the first subgroup P0 in both C and f Ž C . is 1, and the second subgroup Q g in f Ž C . is a radical 2-subgroup of M22 , we conclude that C is a radical 2-chain of M22 if and only if f Ž C . is a radical 2-chain of M22 . We now define R) Q to be the subfamily of RG Q consisting of all chains C g R with < C < ) 0 such that the second group P1 in C satisfies Q g - P1 for some g g M22 . Then the above arguments imply that f is an injection of R) Q into RQ , and that the image f Ž R) Q . consists of all chains C g RQ such that C is not M22-conjugate to 1 - Q. Evidently f preserves M22 -conjugation. Since RG Q is the disjoint union of RQ and R) Q , we conclude that it is the disjoint union of three M22-invariant subfamilies, R) Q , f Ž R) Q ., and the M22-conjugacy class of 1 - Q. Let R) QrM22 be any family of representatives for the M22-orbits in R) Q . Then f Ž R) QrM22 . is a family of representatives for the M22-orbits in f Ž RQ .. So we may choose our family RG QrM22 to be the disjoint union of the three subfamilies R) QrM22 , f Ž R) QrM22 ., and  1 - Q4 . Then the sum on the left-hand side of Ž4.2.2C. is

Ý

Cg RG Q rM 22

< <

Ž y1. C k Ž C . s

< <

Ž y1. C k Ž C .

Ý

Cg R) Q rM 22

q

Ý Cgf Ž R) Q rM 22 .

< <

Ž y1. C k Ž C . y k Ž 1 - Q .

COUNTING CHARACTERS IN BLOCKS

s

Ý

47

< <

Ž y1. C k Ž C .

Cg R) Q rM 22

q Ž y1 .

< f Ž C .<

k Ž f Ž C . . y k Ž1 - Q. .

But, k Ž C . s k Ž f Ž C .. and < f Ž C .< s < C < q 1 for any C g R) Q rM22 . So Žy1.
If the abo¨ e assumption holds, then

Ý Cg RŽ N Ž D ..rN Ž D .

< <

Ž y1. C k Ž Nn . M 22 Ž C . , n. B, d, O N n . s 0 Ž 4.3.2.

for any di¨ isor n of 12, any 2-block n. B of n. M22 , any integer d, any faithful linear character n of N s Nn s ZŽ n. M22 ., and any subgroup O of OutŽ n. M22 N N .n . Proof. Let n. N Ž D . s Nn. M 22Ž D . be the inverse image of N Ž D . in n. M22 . We know that < D < s 2 3 and that N Ž D . s D i L, where L , L3 Ž2. , L2 Ž7. acts faithfully on D as AutŽ D .. So D is a normal 2-subgroup of N Ž D . with CNŽ D .Ž D . s D. The inverse image n. D of D in n. N Ž D . is a nilpotent normal subgroup with Cn. NŽ D .Ž n. D . s ZŽ n. D .. In fact, n. D is the direct product Ž n. D . 2 = N2 9 of its Sylow 2-subgroup Ž n. D . 2 and the Hall 29-subgroup N2 9 of N. Since Ž n. D . 2 has image D s CNŽ D .Ž D ., we have Cn. NŽ D .ŽŽ n. D . 2 . s ZŽ n. D . s ZŽŽ n. D . 2 . = N2 9. Thus Cn. NŽ D .ŽŽ n. D . 2 . has exactly one 2-block b9 lying over B Ž n .. In fact 1 b9 s 1n X2 s 1 BŽ n . , where n X2 is the restriction of n to N2 9. It follows from w14, 5.3.11x that Ž4.3.3. n. N Ž D . has exactly one 2-block n.b lying over B Ž n .. Indeed, 1 n.b s 1 b9 s 1 BŽ n . . We shall prove Proposition 4.3.1 in a series of lemmas. We denote by IrrŽ Nn. NŽ D .Ž C ., d, O N n . the set of all characters x g IrrŽ Nn. NŽ D .Ž C . N n . such that dŽ x . s d and such that Nn. NŽ D ..2 Ž C .x has image O in OutŽ n. M22

48

JANICE F. HUANG

N N .n , and by k Ž Nn. NŽ D .Ž C ., d, O N n . the order of IrrŽ Nn. NŽ D .Ž C ., d, O N n .. Here n. N Ž D ..2 s Nn. M 22 .2 Ž D .. We first prove LEMMA 4.3.4. equi¨ alent to

Equation Ž4.3.2. holds tri¨ ially if n. B / Ž n.b . n. M 22 , and is

Ý

< <

Ž y1. C k Ž Nn . NŽ D . Ž C . , d, O N n . s 0

Ž 4.3.5.

Cg RŽ N Ž D ..rN Ž D .

if n. B s Ž n.b . n. M 22 . Proof. Any C g RŽ N Ž D .. begins with P0 s O 2 Ž N Ž D .. s D. Hence Nn. M 22Ž C . F Nn. M 22Ž D . s n. N Ž D .. So Nn. M 22Ž C . s Nn. NŽ D .Ž C .. Any character x g IrrŽ Nn. M 22Ž C . N n . defines a 2-block B Ž x . of Nn. M 22Ž C . s Nn. NŽ D .Ž C . inducing 2-blocks of both n. M22 and n. N Ž D .. In view of Ž4.3.3., this implies that B Ž x . n. NŽ D . s n.b and B Ž x . n. M 22 s Ž n.b . n. M 22 . This tells us that IrrŽ Nn. M 22Ž C ., n. B, d, O N n . is empty unless n. B is the fixed 2-block Ž n.b . n. M 22 , in which case this set is equal to IrrŽ Nn. NŽ D .Ž C ., d, O N n .. The conclusion of the lemma follows. We look next at the structure of n. N Ž D ..2. Because n. N Ž D . is a central extension of a perfect group N Ž D . by N, it is the central product n. N Ž D . s HN of its derived group H s w n. N Ž D ., n. N Ž D .x and N. Furthermore, H is perfect, and both factors H and N are normal in n. N Ž D ..2. Since N Ž D . is perfect, it must be the derived group of N Ž D ..2 s NM 22 .2 Ž D .. The Atlas tells us that NM 22 .2 Ž D . is isomorphic to N Ž D . = Z2 , which has N Ž D . = 1 as its derived group. It follows that N Ž D ..2 s N Ž D . = ² b : for some involution b g N Ž D ..2 y N Ž D .. Let b9 be any element of n. N Ž D ..2 having b as its image in N Ž D ..2, and let N9 s H l N. Then b9 acts as an automorphism of the perfect group H centralizing HrN9 , N Ž D .. Since N9 is central in H, this implies that b9 centralizes H. Hence Ž4.3.6. n. N Ž D ..2 is the central product n. N Ž D ..2 s HCn. NŽ D ..2 Ž H ., where Cn. NŽ D ..2 Ž H . s ² b9, N : covers OutŽ n. M22 . , Z2 . Since we know that b9 inverts N and centralizes H, it must both invert and centralize the cyclic group N9 s H l N. Hence < N9 < s < H l N < F 2. It follows that the natural epimorphism g: H ª N Ž D . with kernel N9 induces a bijection of RŽ H . onto RŽ N Ž D .., sending any radical 2-chain C: P0 s O 2 Ž H . - P1 - ??? - Pn of H to the radical 2-chain g Ž C .: g Ž P0 . s O 2 Ž N Ž D .. s D - g Ž P1 . - ??? - g Ž Pn . of N Ž D .. In that case NH Ž C . is the inverse image gy1 Ž NNŽ D .Ž g Ž C ..., and Nn. NŽ D .Ž g Ž C .. is the central product Nn. NŽ D .Ž g Ž C .. s NH Ž C . N, with NH Ž C . l N s N9. Let D9 be the inverse image gy1 Ž D . of D in H. Then D9 is a central extension of the elementary abelian group D of order 2 3 by the cyclic group N9 of order 1 or 2. Since H acts irreducibly on D9rN9 , D, this

COUNTING CHARACTERS IN BLOCKS

49

implies that D9 s O 2 Ž H . is elementary abelian of order 2 3 or 2 4 . For arbitrary subgroup K of H, where K G D9, we denote by IrrŽ K, d N n . the set of all x g IrrŽ K N n . with dŽ x . s d. Let n 9 be the restriction of n to a faithful linear character of N9 s NH Ž C . l N s H l N. Furthermore, let d9 s d y aŽw N : N9x., where aŽ n. is the function of n defined in Sect. 1.2. We prove LEMMA 4.3.7. Equation Ž4.3.5. holds tri¨ ially if O / OutŽ n. M22 N N .n , and is equi¨ alent to

Ý

< <

Ž y1. C k Ž NH Ž C . , d9 N n 9 . s 0

Ž 4.3.8.

Cg RŽ H .rH

if O s OutŽ n. M22 N N .n . Proof. There is a bijection of IrrŽ NH Ž C . N n 9. onto IrrŽ Nn. NŽ D .Ž z Ž C .. N n ., sending any character x in the former set to its central product xn with n . Since x and xn have the same degree, while w Nn. NŽ D .Ž g Ž C .. : NH Ž C .x s w N : N9x, we have d Ž xn . s a Ž < Nn . NŽ D . Ž g Ž C . . < . y a Ž xn Ž 1 . . s a Ž w N : N9 x . q a Ž < NH Ž C . < . y a Ž x Ž 1 . . s a Ž w N : N9 x . q d Ž x . . In particular, dŽ xn . s d if and only if dŽ x . s d9. It follows from Ž4.3.6. that Nn. N Ž D ..2 Ž g Ž C .. is the central product Nn. N Ž D ..2 Ž g Ž C .. s NH Ž C .² b9, N :, where ² b9, N : l NH Ž C . s N9. Since b9 g Nn. NŽ D ..2 Ž g Ž C .. centralizes NH Ž C ., it fixes x g IrrŽ NH Ž C . N n 9.. It follows that b9 fixes xn if and only if it fixes n . We conclude that Nn. NŽ D ..2 Ž g Ž C ..xn has image OutŽ n. M22 N N .n for every x g IrrŽ NH Ž C . N n 9.. This tells us that the set IrrŽ Nn. NŽ D .Ž g Ž C .., d, O N n . is empty if O / OutŽ n. M22 N N .n and is in one-to-one correspondence with the set IrrŽ NH Ž C ., d9 N n 9. when O s OutŽ n. M22 N N .n . The conclusion of the lemma follows. Suppose that K is a subgroup of H such that D9 F K. Recall from w12, 17.1x that a weight of K with respect to D9 is any character x g IrrŽ K . such that dŽ x . s dŽ f . for any f g IrrŽ D9. lying under x . Since D9 is abelian, f Ž1. s 1 and dŽ f . s aŽ< D9 <. y aŽ f Ž1.. s aŽ< D9 <.. Hence the set Wt Ž K N D9, n 9. of all weights of K with respect to D9 lying over n 9 is the set IrrŽ K, aŽ< D9 <. N n 9., and its order w Ž K N D9, n 9. is w Ž K N D9, n 9. s k Ž K, aŽ< D9 <. N n 9.. In particular, the number w Ž K, d9 N D9, n 9. of x g Wt Ž K N D9, n 9. with dŽ x . s d9 is 0 if d9 / aŽ< D9 <. and is w Ž K N D9, n 9. s k Ž K, aŽ< D9 <. N n 9. if d s aŽ< D9 <.. The hypothesis of Theorem 17.6 of w12x includes the statement that the M-A-D conjecture must hold when G is any subgroup of HrD9 , L3 Ž8.. But any such subgroup G is either

50

JANICE F. HUANG

solvable or isomorphic to L3 Ž8.. So our assumption made at the beginning of this section implies that the conjecture holds for G. Applying w12, 17.6x with H and D9 s O 2 Ž H . substituted appropriately, we see that < <

Ž y1. C k Ž NH Ž C . , d9 N n 9 .

Ý Cg RŽ H .rH

s

< <

Ž y1. C w Ž NH Ž C . , d9 N D9, n 9 . .

Ý Cg RŽ H .rH

This and the above arguments give us the following lemma. LEMMA 4.3.9. equi¨ alent to

Equation Ž4.3.8. holds tri¨ ially if d9 / aŽ< D9 <., and is

Ý

< <

Ž y1. C w Ž NH Ž C . N D9, n 9 . s 0

Ž 4.3.10.

Cg RŽ H .rH

if d9 s aŽ< D9 <.. We have already remarked that N Ž D . , HrN9 is a semidirect product D i L, where L , L3 Ž2. acts on D as AutŽ D .. It follows that H is the semicentral product H s D9L9 of D9 with the derived group L9 s w gy1 Ž L., gy1 Ž L.x of the inverse image of L. Hence N0 s D9 l L9 s ZŽ L9. F N9 and L9 is isomorphic to either L3 Ž2. or its double cover 2. L3 Ž2. , SL 2 Ž7. Žsee the Atlas w8x.. We fix a subgroup K of H containing D9. Then K has the form K s D9K 9, where K 9 s K l L9 is a subgroup of L9 containing N0 s ZŽ L9.. Conjugation by both K and K 9 has the same effect on the eight characters in IrrŽ D9 N n 9., permuting them in various orbits. The stabilizer Kf in K for fixed f g IrrŽ D9 N n 9. is the semicentral product Kf s D9KfX , where KfX s Kf l K 9 is the stabilizer of f in K 9. Let n 0 be the restriction of n 9 to N0. Then the map sending x g IrrŽ KfX N n 0 . to the well-defined function fx : s k ¬ f Ž s . x Ž k . for s g D9 and k g KfX is a bijection of IrrŽ KfX N n 0 . onto IrrŽ Kf N f .. Combining this with the inductive bijection of IrrŽ Kf N f . onto IrrŽ K N f . we obtain a bijection x ¬ Ž fx . K of IrrŽ KfX N n 0 . onto IrrŽ K N f .. Fix x g IrrŽ KfX N n 0 .. By definition, the character Ž fx . K g IrrŽ K N f . is a weight with respect to D9 if and only if dŽŽ fx . K . s aŽ< D9 <.. Since Ž fx . K Ž1. s w K : Kf xfx Ž1. s w K : Kf x x Ž1., this holds if and only if aŽ< D9 <. s aŽ< K <. y aŽŽ fx . K Ž1.. s aŽ< Kf <. y aŽ x Ž1... But aŽ< Kf <. y aŽ< D9 <. s aŽ< KfrD9 <. s aŽ< KfX rN0 <.. Hence Ž4.3.11. the number w Ž K N D9, f . of weights of K with respect to D9 lying over a fixed character f g IrrŽ D9 N n 9. is equal to the number k Ž KfX , aŽ< N0 <. N n 0 . of characters x g IrrŽ KfX N n 0 . satisfying aŽ x Ž1.. s aŽ< KfX rN0 <..

COUNTING CHARACTERS IN BLOCKS

51

It follows that w Ž K N D9, n 9 . s

Ý

w Ž K N D9, f .

f gIrr Ž D 9Nn 9.rK

s

Ý f gIrr Ž D 9Nn 9.rK

k Ž KfX , a Ž < N0 < . N n 0 . .

Ž 4.3.12.

The following lemma will be useful in what follows. LEMMA 4.3.13.

If O 2 Ž KfX . ) N0 then k Ž KfX , aŽ< N0 <. N n 0 . s 0.

Proof. Any character x g IrrŽ KfX N n 0 . lies over some KfX -conjugacy class of characters c g IrrŽ O 2 Ž KfX N n 0 .. Fix any such c . It follows from Clifford theory that x Ž1.rc Ž1. divides w KfX : O 2 Ž KfX .x. Since N0 is central, the degree c Ž1. divides
NLŽ C .

1 1 - V1 1 - V2 1 - V1V2 s D 8 1 - V1 - V1V2 1 - V2 - V1V2

L NLŽ V1 . , S4 NLŽ V2 . , S4 D8 D8 D8

52

JANICE F. HUANG

We have also given the normalizers of each of these chains. Since the radical 2-chains of H are the inverse images of those of L , HrO 2 Ž H ., we conclude that LEMMA 4.3.14.

Equation Ž4.3.10. is equi¨ alent to

w Ž D9L9 N D9, n 9 . y w Ž D9NL9 Ž V1 . N D9, n 9 . y w Ž D9NL9 Ž V2 . N D9, n 9 . q w Ž D9DX8 N D9, n 9 . s 0,

Ž 4.3.15.

where DX8 is the in¨ erse image of D 8 in L9. The Atlas w8x tells us that the maximal subgroups of L s L3 Ž2. have indices 7 or 8. Since
In the case where L fixes some f g IrrŽ D9 N n 9., eq.

Proof. Here D9 is the direct product D9 s D0 = N9, where D0 s kerŽ f . is normal in H and isomorphic to D as an L-group. Since N0 s D9 l L9 F N9 and H s DL9 is perfect, this implies that N0 s N9 and H s D0 i L9 , D i L9. We treat the four terms in eq. Ž4.3.15. individually. Since L9 acts on D0 as AutŽ D0 ., it permutes transitively the seven nontrivial characters in IrrŽ D0 .. It follows that L9 has two orbits in IrrŽ D9 N n 9., one represented by f with stabilizer LXf s L9, and one represented by a character f 9 with stabilizer LXf s NL9Ž V2 .. The character tables in the Atlas tell us that IrrŽ L9 N n 0 . has exactly one character x with aŽ< x <. s aŽ< L9rN0 <. s 3, whether L9 is L3 Ž2. or 2. L3 Ž2.. Because O 2 Ž NL9Ž V2 .. is the inverse image V2X of V2 , and hence contains N0, by Lemma 4.3.13 there can be no characters x g IrrŽ NL9Ž V2 . N n 0 . with aŽ< x <. s aŽ< NL9Ž V2 .rN0 <. s 3. We conclude from this and Ž4.3.12. that w Ž D9L9 N D9, n 9 . s 1 q 0 s 1.

Ž 4.3.17.

The subgroup V1 of L fixes all four characters of D9 s D0 = N0 of the form f 9 s l0 = n 0, where l0 g IrrŽ D0 . is trivial on the subgroup w D0, V1 x of order 2. It follows that, if V1X is the inverse image of V1 in L9, then N0 - V1X F O 2 Ž NL9Ž V1 .f 9 . for any such f 9. Hence w Ž NL9Ž V1 . N D9, f 9. s 0 for these f 9 by Lemma 4.3.13. However, V1 permutes regularly and transitively the four characters f 0 g IrrŽ D9 N n 9. such that f 0 is nontrivial on w D0, V1 x. It follows that the stabilizer NLŽ V1 .f 0 of any such f 0 is a complement to V1 in NLŽ V1 . , S4 , and hence must be isomorphic to S3 . Since the Schur multiplier of S3 is trivial Ž S3 has only cyclic Sylow subgroups!. this means that NL9Ž V1 .f 0 , N9 = S3 . So IrrŽ NL9Ž V1 .f 0 N n 9.

COUNTING CHARACTERS IN BLOCKS

53

contains exactly one character x with aŽ x Ž1.. s aŽw NL9Ž V1 .f 0 : N0 x. s 1, corresponding to the character of degree 2 in IrrŽ S3 .. From this and Ž4.3.12. we conclude that w Ž D9NL9 Ž V1 . N D9, n 9 . s 1.

Ž 4.3.18.

If f 9 g IrrŽ D9 N n 9., then kerŽ f 9. l D0 has order at least 2 2 . So kerŽ f 9. l w D0, V2 x has order at least 2 Žbecause
Ž 4.3.19.

Similarly 1 - Ž V2 .f 9 F O 2 ŽŽ D 8 .f 9 . s Ž D 8 .f 9 for any f 9 g IrrŽ D9 N n 9.. Hence w Ž D9DX8 N D9, n 9 . s 0.

Ž 4.3.20.

In view of Ž4.3.17., Ž4.3.18., Ž4.3.19., and Ž4.3.20., eq. Ž4.3.15. is 1 y 1 y 0 q 0 s 0, which is correct. Thus the equation holds when L fixes some f g IrrŽ D9 N n 9.. This concludes the proof of Lemma 4.3.16. The final step in the proof of Proposition 4.3.1 is the following lemma. LEMMA 4.3.21. N n 9. transiti¨ ely.

Equation Ž4.3.15. holds when L permutes all f g IrrŽ D9

Proof. In this case Lf , Z 7 i Z3 is a Frobenius group of order 21. Therefore, the Sylow 2-subgroup D 8 is a complement to Lf in L, since D 8 l Lf s 1 and < D 8 < ? < Lf < s 8 ? 7 ? 3 s < L <. Since D 8 is contained in each of L, NLŽ V1 ., NLŽ V2 ., and D 8 , we conclude that each of these groups has a single orbit in IrrŽ D9 N n 9., with stabilizers Lf , Z 7 i Z3 , NLŽ V1 .f , Z3 , NLŽ V2 .f , Z3 , and Ž D 8 .f s 1, for any f in that orbit. Each of these stabilizers has odd order, and hence splits over N0 , Z2 or Z1. Furthermore, each irreducible character of these stabilizers has odd degree, and hence is a weight with respect to N0. It follows from this and Ž4.3.12. that w Ž D9L9 N D9, n 9. s
54

JANICE F. HUANG

4.4. The M-A-D Conjecture and the Prime p s 2 In verifying the M-A-D conjecture for the prime p s 2 and the group M22 we shall be concerned only with the following four radical 2-chains, listed together with their nomalizers in M22 : C0, 2 : 1,

NM 22 Ž C0, 2 . s M22 ,

C1, 2 : 1 - E,

NM 22 Ž C1, 2 . s E i AŽ2. 6 , E i A6 ,

C2, 2 : 1 - EF ,

NM 22 Ž C2, 2 . s E i S4Ž2. , E i S4 ,

C3, 2 : 1 - F ,

NM 22 Ž C3, 2 . s F i S5Ž2. , F i S5 .

Notice that the second subscript j in Ci, j represents the prime p s 2, to distinguish these chains from those found in Sect. 3. In view of Propositions 4.2.1 and 4.3.1, the conjecture will be correct if and only if the equation k Ž Nn . M 22 Ž C0, 2 . , n. B, d, O N nn . y k Ž Nn . M 22Ž C1, 2 . , n. B, d, O N nn . q k Ž Nn . M 22 Ž C2, 2 . , n. B, d, O N nn . y k Ž Nn . M 22 Ž C3, 2 . , n. B, d, O N nn . s 0

Ž 4.4.1.

holds for each n s 1, 2, 4, 3, 6, 12, each faithful linear character nn of the center Nn of n. M22 , each integer d G 0, each subgroup O of OutŽ n. M22 N Nn ., and each block n. B satisfying Ž4.4.2. n. B is a 2-block of n. M22 lying over the 2-block B Ž nn . of Nn containing nn , but not having the Sylow 2-subgroup of Nn as a defect group. ŽSee Ž1.4.1. and Ž1.4.2. as well as the statement of the M-A-D conjecture in Sect. 1.4.. The correctness of Ž4.4.1. will follow from Table 2. The index for each column in the table is one of the radical 2-chains Ci, j listed above. Underneath that index we write the sign Žq or y. of the corresponding term in Ž4.4.1.. Each row of Table 2 has three indices, a block n. B, an integer d, and a subgroup Zk of OutŽ M22 .. For a fixed divisor n of 12, the n. B appearing in Table 2 is the only 2-block of n. M22 having a defect group strictly larger than the Sylow 2-subgroup of Nn . These 2-blocks will be described explicitly in the next section. The table entry for the row with indices n. B, d, Zk and column with index Ci, j is the value of k Ž Nn. M 22Ž Ci, j ., n. B, d, Zk N nn ., where nn is a faithful linear character of Nn . These values will be computed in the next four sections. We

55

COUNTING CHARACTERS IN BLOCKS

TABLE 2

1. B 1. B 1. B 1. B 2. B 2. B 2. B 2. B 4. B 4. B 4. B 3. B 3. B 3. B 6. B 6. B 6. B 12. B 12. B

7 6 5 4 7 6 5 7 6 5 4 7 6 5 7 6 5 6 5

Z2 Z2 Z2 Z1 Z2 Z2 Z2 Z1 Z1 Z1 Z1 Z1 Z1 Z1 Z1 Z1 Z1 Z1 Z1

C0, 2 q

C1, 2 y

C2, 2 q

C3, 2 y

8 2 0 2 4 0 3 4 2 4 2 8 2 0 8 0 1 2 4

8 2 0 2 4 0 3 4 2 2 2 8 2 0 8 0 1 2 2

8 2 2 0 4 2 1 4 2 1 0 8 2 2 8 2 1 2 1

8 2 2 0 4 2 1 4 2 3 0 8 2 2 8 2 1 2 3

shall also show that k Ž Nn. M 22Ž Ci, j ., n. B, d, Zk N nn . s 0 for any combination of n. B, d, Zk and Ci, j not appearing in Table 2. Therefore, eq. Ž4.4.1. will hold if and only if the alternating sum of the values in any row of Table 2 is 0. Since this is obviously true, the verification that Table 2 is correct will complete the proof of THEOREM 2. The M-A-D conjecture holds when the group G is any co¨ ering group n. M22 of M22 for some n s 1, 2, 3, 4, 6, or 12, the cyclic central subgroup N of G is the center Nn of n. M22 , the p-block B of G is any 2-block n. B of n. M22 lying o¨ er the block B Ž nn . containing a faithful linear character nn g IrrŽ Nn . and not ha¨ ing the Sylow 2-subgroup of Nn as a defect group, the defect d is any nonnegati¨ e integer, and the group O is any subgroup of OutŽ n. M22 N Nn .n n. 4.5. The Chain C0, 2 As in Sect. 3.3, we shall summarize information gained from the character table in the Atlas for the groups Nn. M 22 .2 Ž C0, 2 . s n. M22 .2. Using Ž3.3.1., we determine that, for each divisor n of 12, there is only one 2-block for n. M22 having a defect group strictly larger than the Sylow

56

JANICE F. HUANG

2-subgroups of Nn . We begin with n s 1 and let n 1 s 1 since ZŽ M22 . s N1 is trivial. Then there is only one 2-block 1. B of M22 where Irr Ž 1. B N n 1 . s Irr Ž 1. B . s  x 1 , x 2 , x 3 , x4 , x 5 , x6 , x 7 , x 8 , x 9 , x 10 , x 11 , x 12 4 . Ž 4.5.1. These characters have 2-defects 7, 7, 7, 7, 7, 7, 6, 6, 7, 4, 4, and 7, respectively. The Atlas w8x indicates that Ž4.5.2. the two characters x 10 and x 11 of M22 , with 2-defect 4, fuse in M22 .2. No other characters of M22 fuse under this action. These results give us the 1. B entries in the first column of Table 2 and tell us that k Ž NM 22Ž C0, 2 ., 1. B, d, O N 1. is zero for all other combinations of d and O satisfying the hypotheses of Theorem 2. Next we let n s 2, let n 2 be a faithful linear character of the center N2 of 2. M22 , and determine that there is exactly one 2-block 2. B of 2. M22 satisfying Ž4.4.2. for n s 2. In the notation of the Atlas w8x, we have Irr Ž 2. B N n 2 . s  x 13 , . . . , x 23 4 .

Ž 4.5.3.

These characters have 2-defects 7, 7, 5, 5, 7, 7, 7, 7, 7, 7, and 5, respectively. In addition, the Atlas confirms that Ž4.5.4. the pair of characters x 17 and x 18 of 2. M22 , with 2-defect 7, and the pair of characters x 19 and x 20 also of 2. M22 , with 2-defect 7, fuse in 2. M22 .2. No other characters of 2. M22 fuse under this action. These values give us the 2. B entries in the first column of Table 2. Also, k Ž N2. M 22Ž C0, 2 ., 2. B, d, O N n 2 . is zero for all other combinations of d and O satisfying the hypotheses of Theorem 2. We then let n s 4, let n4 be a faithful linear character of the center N4 of 4. M22 , and determine that there is only one 2-block 4. B of 4. M22 where Irr Ž 4. B N n4 . s  x 24 , . . . , x 31 4 .

Ž 4.5.5.

The 2-defects of these characters are 6, 6, 5, 5, 4, 4, 5, and 5, respectively. We recall that the central elements of 4. M22 and 3. M22 are inverted by the outer automorphism group of M22 Žsee Atlas w8, p. 40x.. Thus the only possible value of the subgroup O of OutŽ n. M22 N Nn . s 1 is O s Z1 whenever n ) 2. These results give us the 4. B entries in the first column of Table 2, and tell us that k Ž N4. M 22Ž C0, 2 ., 4. B, d, O N n4 . is zero for all other combinations of d and O satisfying the hypotheses of Theorem 2. Now let n s 3, let n 3 be a faithful linear character of the center N3 of 3. M22 , and determine that there are two 2-blocks, 3. B and 3. B9, of 3. M22 where Irr Ž 3. B N n 3 . s  x 32 , . . . , x41 4 Ž 4.5.6.

COUNTING CHARACTERS IN BLOCKS

57

and Irr Ž 3. B9 N n 3 . s  x42 4 . These characters have 2-defects 7, 7, 7, 7, 7, 7, 6, 7, 7, 6, and 0, respectively. Note that the block 3. B9 does not satisfy Ž4.4.2.. The values here give us the 3. B entries in the first column of Table 2 and tell us that k Ž N3. M 22Ž C0, 2 ., B, d, O N n 3 . is zero for all other combinations of B, d, and O satisfying the hypotheses of Theorem 2. We next let n s 6, let n 6 be a faithful linear character of the center N6 of 6. M22 , and determine that there are two 2-blocks, 6. B and 6. B9, of 6. M22 where Irr Ž 6. B N n 6 . s  x43 , . . . , x 51 4

Ž 4.5.7.

and Irr Ž 6. B9 N n 6 . s  x 52 4 . These characters have 2-defects 7, 7, 5, 7, 7, 7, 7, 7, 7, and 1, respectively. We note that the block 6. B9 does not satisfy Ž4.4.2.. These values give us the 6. B entries in the first column of Table 2 and tell us that k Ž N6. M 22Ž C0, 2 ., B, d, O N n 6 . is zero for all other combinations of B, d, and O satisfying the hypotheses of Theorem 2. Finally, we let n s 12, let n 12 be a faithful linear character of the center N12 of 12. M22 , and determine that there are two 2-blocks, 12. B and 12. B9, of 12. M22 where Irr Ž 12. B N n 12 . s  x 53 , . . . , x 58 4

Ž 4.5.8.

and Irr Ž 12. B9 N n 12 . s  x 59 4 . These characters have 2-defects 6, 6, 5, 5, 5, 5, and 2, respectively. The block 12. B9 does not satisfy the hypothesis of Theorem 2. The values here give us the 12. B entries in the first column of Table 2 and tell us that k Ž N12. M 22Ž C0, 2 ., B, d, O N n 12 . is zero for all other combinations of B, d, and O satisfying the hypotheses of Theorem 2. 4.6. The Chain C1, 2 We consider next the radical 2-chain C1, 2 , whose normalizer is NM 22Ž C1, 2 . s E i AŽ2. 6 . The character table for this group can be found in Janko’s paper w21, p. 11x. It is clear that the first seven characters must lie over the principal character of E, while the next five lie over a linear

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character of E which has stabilizer S4 . In this and in the following sections, we shall make use of the result from w14, 5.3.11x which states that Ž4.6.1. if P is a p-group and CG Ž P . F P, then there is exactly one p-block of NG Ž P .. Then the 2-defects of the 12 characters in the only 2-block of E i AŽ2. 6 can be seen to be 7, 7, 6, 4, 4, 7, 7, 7, 7, 6, 7, and 7, respectively. This block clearly induces the block 1. B of M22 . We consider the normalizer NM 22 .2 Ž E . of E in M22 .2. We know that this group is E i S6Ž2. by Ž2.3.15., where S6Ž2. s AŽ2. 6 .2. The character table for A 6 in the Atlas w8, p. 5x tells us that this means that NM 22 .2 Ž E . fuses two Ž2. conjugacy classes of elements of AŽ2. 6 of order 5 and two characters of A 6 with degree 8 and 2-defect 4. These are clearly x4 and x 5 in Janko’s Ž2. w x character table. Since AŽ2. 6 .2 s S6 , we know from 8, p. 5 that S4 .2 , S4 = Z2 . Therefore, no other characters are fused by this action. Since the center ZŽ M22 . is trivial, we set n 1 equal to 1. The results above give us the 1. B entries in the second column of Table 2 and tell us that k Ž NM 22Ž C1, 2 ., 1. B, d, O N 1. is zero for all other combinations of d and O satisfying the hypotheses of Theorem 2. We turn now to the factor group 2. M22 of the covering group. We choose a faithful character n 2 of the center N2 , N4r² a2 : of 2. M22 . We recall that a2 s y1. We prove LEMMA 4.6.2. The 2. B entries in the second column of Table 2 are as listed. Proof. It is clear that the elementary abelian 2-subgroup E splits over N2 . The 16 extensions of n 2 to linear characters of 2. E are completely determined by their kernels, which are 16 complements to N2 in 2. E. One of these complements is E9 s ² ˜ ezi ˜ e z j ; 1 F i - j F 6, i , j / 4: , where ˜ e z i is the image in the factor group 2. E s Gr "14 of the element Ž2. e z i of G. We determine that E9 is stabilized by a subgroup AŽ3. 5 of A 6 Ž3. where A 5 s A z 1 , z 2 , z 3 , z 5, z 6 4. We recall from the construction of NM 22Ž E . in Ž2. Sect. 2.3 that AŽ2. 6 splits over N4 . Further, A 6 does not stabilize E9, since if it did that would imply that a Sylow 2-subgroup of M22 splits over N2 , which would mean that 2. M22 is a split extension. Since the index of A 5 in A 6 is six, there are six copies of E in the orbit containing E9. We find a second complement to N2 in 2. E, which we label E0. The complement E0 is the direct product of the two 4-groups V1 s ² ˜ ezi ˜ e z j ; 1 F i - j F 3:

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and V2 s ² ˜ ezi ˜ e z j ; 4 F i - j F 6: . Then E0 is stabilized by the normalizer in A Z of the partition of Z into the two sets  z1 , z 2 , z 34 and  z 4 , z 5 , z 6 4 , which is a Frobenius group of the form Ž Z3 = Z3 . i Z4 . Again we note that E0 is not stabilized by AŽ2. 6 since that would imply that a Sylow 2-subgroup of M22 splits over N2 , which would mean that 2. M22 is a split extension. Since Ž Z3 = Z3 . i Z4 is of index 10 in A 6 , there are 10 copies of E in this orbit containing E0. Thus the orbits of E9 and E0 include all 16 s 6 q 10 complements to N2 in E. For details of the above calculations, see w17x. Ž3. Ž2. Since AŽ2. 6 splits over N4 , we can conclude that A 5 F A 6 splits over Ž . N2 . The degrees of the characters of N2. M 22 C1 which lie over n 2 are then simply the degrees of the characters of A 5 , whcih are 1, 3, 3, 4, and 5, multiplied by 6, and the degrees of the characters of Ž Z3 = Z3 . i Z4 , which are 1, 1, 1, 1, 4, and 4, multiplied by 10. This yields characters with 2-defects 7, 7, 7, 5, 7, 7, 7, 7, 7, 5, and 5, respectively. As usual these characters lie in a single 2-block which must induce the only 2-block 2. B of 2. M22 . Next we consider the action of NM 22 .2 Ž E . on the characters of 2.Ž E i Ž2. . A 6 . From the Atlas w8, p. 39x we determine that NM 22 .2 Ž E . s E i S6 and that w8, p. 4x if A 6 .2 s S6 then A 5 .2 s S5 for A 5 F A 6 . We conclude that there exists an element in M22 .2, which together with AŽ3. 5 , the stabilizer of E9, generates a copy of S5 acting on E. This element of M22 .2 must combine the conjugacy classes of AŽ3. 5 containing elements of order 5. In addition, it must fuse the two characters of N2. M 22Ž C1 . which have degree 18 s 3 = 6 and 2-defect 7. In determining the action of M22 .2 on the group Ž Z3 = Z3 . i Z4 , which is the stabilizer of E0, we first note w8, p. 4x that the Atlas indicates that if A 6 .2 s S6 then wŽ Z3 = Z3 . i Z4 x.2 s Ž Z3 = Z3 . i D 8 for Ž Z3 = Z3 . i Z4 F A 6 . The two orbits of complements of N2 in 2. E must be stable under the action of M22 .2 since they are of different lengths, namely, 10 and 6. Therefore, there exists an element in NM 22 .2 Ž E . which leaves invariant the complement E0. This element generates from Ž Z3 = Z3 . i Z4 , the normalizer in A 6 of a Sylow 3-subgroup, a copy of Ž Z3 = Z3 . i D 8 s NS 6Ž Z3 = Z3 .. We note that the two characters of Ž Z3 = Z3 . i Z4 with degree 4 cannot fuse under this action since the classes of elements of order 3 in A 6 are stabilized by S6 . However, the two characters of Ž Z3 = Z3 . i Z4 found by multiplying the faithful characters of Z4 by the principal character of Z3 = Z3 do fuse under the action of M22 .2, which generates D 8 from Z4 . These characters have degree 10 s 1 = 10 and 2-defect 7.

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We conclude that two pairs of characters in the single 2-block of N2. M 22Ž C1, 2 ., all with 2-defect 7, fuse in N2. M 22 .2 Ž C1, 2 .. The above considerations yield the 2. B entries in the second column of Table 2 and tell us that k Ž N2. M 22Ž C1, 2 ., 2. B, d, O N n 2 . is zero for all other combinations of d and O satisfying the hypotheses of Theorem 2. This concludes the proof of Lemma 4.6.2. We next consider the normalizer of C1, 2 in the factor group 4. M22 of the covering group. We choose a faithful character n4 of the center N4 of 4. M22 and prove LEMMA 4.6.3. The 4. B entries in the second column of Table 2 are as gi¨ en. Proof. We recall from Sect. 2.3 the structure of N4. M 22Ž E . as G i AŽ2. 6 . We note that the group E does not split over N4 , and the group G, which is the inverse image of E in 4. M22 , is the central product of an extra special group of order 2 5 and N4 . Then IrrŽ G N n4 . contains just one character C, which has degree 4 and which must be invariant under SZ . ŽWe recall that the set IrrŽ G i A Z N n4 . s IrrŽ G i A Z N C . is in one-to-one correspondence with the set IrrŽ A Z N j ., where A Z is a central extension of A Z by a cyclic central subgroup L and j is a faithful linear character of L.. We know from w8, p. 5x that the Schur multiplier of A 6 is cyclic of order 6. Therefore, Ž4.6.4. the Clifford extension of AŽ2. 6 ? G for the character C must correspond to 2. A 6 . It follows that to find the degrees of the characters of N4. M 22Ž C1, 2 ., we must use the degrees of the characters of 2. A 6 , which are 4, 4, 8, 8, 10, and 10, and multiply them by 4 Žsee the Atlas w8x.. This procedure yields characters with 2-defects 5, 5, 4, 4, 6, and 6, respectively. As usual, these characters lie in a single block, which must induce the only 2-block 4. B of 4. M22 . Since the outer automorphisms of M22 invert the generator of N4 , the above calculations yield the 4. B entries of Table 2 and tell us that k Ž N4. M 22Ž C1, 2 ., 4. B, d, O N n4 . is zero for all other combinations of d and O satisfying the hypotheses of Theorem 2. This completes the proof of Lemma 4.6.3. We note for future reference the following corollary to Lemma 4.6.3. LEMMA 4.6.5. The degrees of the two characters of the Sylow 2-subgroup 4.S of 4. M22 and of 4. NM 22Ž E . which lie o¨ er n4 are 8 and 8, the degrees of the characters of the nonsplit extension D 16 of D 8 multiplied by 4.

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Proof. We know from cohomology theory w7, 10.1x that restriction from a finite group G to a Sylow q-subgroup Gq induces a monomorphism of the Sylow q-subgroup of H n Ž G, A. into H n Ž Gq , A., for any prime q and any integer n. Thus we must look at the inverse images of the Sylow q-subgroups of the factor group AŽ2. 6 . The only Sylow subgroups of A 6 which are not cyclic occur when q s 3 or q s 2. The Sylow 3-subgroup is Z3 = Z3 . Its inverse image is G i Ž Z3 = Z3 .. Using a result of Gallagher w20, 8.15x, since the order 9 s < Z3 = Z3 < is relatively prime to the degree 4 of C, the character C can be extended in this case. We conclude from the above remarks that C cannot be extended to a character of the inverse image of the Sylow 2-subgroup D 8 of A 6 , since if it did that would imply that the complement AŽ2. 6 splits over 4. E s G in the Clifford extension. This gives us the result of the corollary. We look next at the factor group 3. M22 with center N3 , Z3 . We choose a faithful character n 3 of N3 . We prove LEMMA 4.6.6. The 3. B entries in the second column of Table 2 are as gi¨ en. Proof. It is immediately clear that the elementary abelian 2-subgroup E of M22 splits over N3 . In considering the group AŽ2. 6 , which lies in the normalizer NM 22Ž C1, 2 ., we note that A 6 has an nonsplit extension by Z3 and that the characters of 3. A 6 are listed in the Atlas w8x. The construction of the extension 3. M22 was described in Sect. 2.2, using the central extension SLŽ3, 4. of PSLŽ3, 4. by the central subgroup N3 , Z3 . We recall that any Sylow 3-subgroup of AŽ2. 6 is also a Sylow 3-subgroup of M22 and hence is M22 -conjugate to a Sylow 3-subgroup of PSLŽ3, 4.. Then it is clear that in the extension 3. M22 the group AŽ2. 6 does not split because a 3-Sylow subgroup of AŽ2. 6 , which has form Z 3 = Z 3 , does not split. In fact, the inverse image of the Sylow 3-subgroup is an extra special group of order 3 3 and exponent 3. Therefore, the group N3. M 22Ž C1, 2 . is isomorphic to the semidirect product E i Ž3. A Z . and has five characters which lie over the principal character of E with degrees the same as the degrees of the characters for the extension 3. A Z , namely, 3, 3, 6, 9, and 15 Žsee the Atlas w8x.. There are also five characters which lie over an irreducible character l / 1 of E which is stabilized by S4 s Ž A 6 .l . Then 3.S4 s 3.Ž A 6 .l must be isoclinic to a split extension in 3. M22 since the cyclic Sylow 3-subgroup of S4 splits over N3 . Hence the degrees of these five characters will be equal to the degrees of the characters of S4 , which are 1, 1, 2, 3, and 3, multiplied by 15 s w A 6 : S4 x. This means the 2-defects of the characters are 7, 7, 6, 7, 7, 7, 7, 6, 7, and 7, respectively. As usual, the characters lie in a single block which must induce the only 2-block 3. B of 3. M22 which has defect group larger that N3 . Since the outer automor-

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phisms of M22 invert a generator of N3 , these calculations yield the 3. B entries in the second column of Table 2 and tell us that k Ž N3. M 22Ž C1, 2 ., B, d, O N n 3 . is zero for all other combinations of B, d, and O satisfying the hypotheses of Theorem 2. This completes the proof of Lemma 4.6.6. We next consider the normalizer of the radical 2-chain C1, 2 in the extension 6. M22 which has center N6 s N2 = N3 . We choose a faithful character n 6 of N6 . We prove LEMMA 4.6.7. The 6. B entries in the second column of Table 2 are as listed. Proof. We determined above that the group E splits in both 2. M22 and 3. M22 and thus it must split in 6. M22 . In the proof of Lemma 4.6.2 we found a complement E9 to N2 in 2. E which was stabilized by AŽ3. 5 , a subgroup of AŽ2. . We also found that a second complement E0 to N 6 2 in 2. E was stabilized by a subgroup Ž Z3 = Z3 . i Z4 F AŽ2. . In the extension 6 3. M22 the group AŽ2. does not split over N3 , and it is clear that the 6 subgroup Ž Z3 = Z3 . i Z4 does not split over N3 either, since Z3 = Z3 is Ž3. Ž2. a Sylow 3-subgroup of AŽ2. 6 . However, the subgroup A 5 of A 6 must split over Z3 since its Sylow 3-subgroup is cyclic. Therefore, since ZŽ6. M22 . s N6 s N2 = N3 , the degrees of the characters of N6. M 22Ž C1, 2 . which lie over n 6 are the degrees of the characters of A 5 , which are 1, 3, 3, 4, and 5, multiplied by 6 and those of the nonsplit extension 3.ŽŽ Z3 = Z3 . i Z4 ., which are 3, 3, 3, and 3, multiplied by 10. This means that the 2-defects of the characters are 7, 7, 7, 5, 7, 7, 7, 7, and 7, respectively. As usual, the characters lie in a single block which must induce the only 2-block 6. B of 6. M22 which has defect group larger than O 2 Ž N6 .. Since the outer automorphisms of M22 invert a generator of N6 , these calculations yield the 6. B entries in the second column of Table 2 and tell us that k Ž N6. M 22Ž C1, 2 ., B, d, O N n 6 . is zero for all other combinations of B, d, and O satisfying the hypotheses of Theorem 2. This completes the proof of Lemma 4.6.7. Finally, we consider the maximal covering group 12. M22 . We choose a faithful character n 12 of the center N12 s N3 = N4 of 12. M22 . We prove LEMMA 4.6.8. The 12. B entries in the second column of Table 2 are as gi¨ en. Proof. The elementary abelian group E does not split over N4 and hence it does not split over N12 . We found in the proof of Lemma 4.6.6 that the extension for n 3 in 3. AŽ2. 6 does not split. Since n 3 is a faithful linear character of N3 , its Clifford extension is equivalent to that extension itself. The Clifford extension for a direct product of two invariant charac-

COUNTING CHARACTERS IN BLOCKS

63

ters is the product of the Clifford extensions for these characters. Therefore, we conclude that, since the inverse image of E in 12. M22 is G = N3 s 12. E, the Clifford extension for the product of C and n 3 in 12.Ž E . is 6. A 6 . It follows that in determining the degrees of the characi AŽ2. 6 ters in IrrŽ N12. M 22Ž C1, 2 . N n 12 . we must use those of the nonsplit extension 6. A 6 of A 6 , which are 6, 6, 12, and 12, multiplying them by the degree 4 of the only character in IrrŽ G = N3 N n 12 .. This means that the 2-defects of the characters are 6, 6, 5, and 5, respectively. As usual, the characters lie in a single block which must induce the only 2-block 12. B of 12. M22 which has defect group larger than N4 . Since the outer automorphisms of M22 invert a generator of N12 , these calculations yield the 12. B entries in the second column of Table 2 and tell us that k Ž N12. M 22Ž C1, 2 ., B, d, O N n 12 . is zero for all other combinations of B, d, and O satisfying the hypotheses of Theorem 2. This completes the proof of Lemma 4.6.7. This completes our consideration of the entries in column 2 of Table 2. We turn our attention to the chain C2, 2 , which yields entries in column 3. 4.7. The Chain C2, 2 The next radical 2-chain to consider, C2, 2 , has normalizer NM 22Ž C2, 2 . , Ž . E i S4Ž2., where S4Ž2. F AŽ2. 6 . This normalizer is a subgroup of NM 22 E s Ž2. Ž . NM 22 C1, 2 s E i A 6 . We can then quickly determine the degrees of the characters of Nn. M 22Ž C2, 2 ., as well as the action of the outer automorphism group, in each of the covering groups n. M22 , from the work done in determining the results in the previous section. We know from Ž2.3.15. that NM 22 .2 Ž E . s E i S6Ž2., that the inverse image of NM 22Ž E . in 4. M22 is the Ž . semidirect product G i AŽ2. 6 , and, from 2.3.17 , that the inverse image of Ž . NM 22 .2 E in 4. M22 .2 can be identified with G i S6Ž2.. It follows, first, from information about the maximal subgroups of A 6 contained in the Atlas w8, p. 5x, that NM 22 .2 Ž C2, 2 . , E i Ž S4Ž2. = Z2Ž2. . ,

Ž 4.7.1.

where we can choose Z2Ž2. to be the group ² s : generated by s of Ž2.3.14.. We recall the construction of S4Ž2. in Sect. 4.1. This group is generated by the 4-group V2Ž2. s  1, k , z , kz 4 , and by a copy SX3 of S3 , where SX3 s ² f, k 3z :. Here k , z , f, and k were identified in Ž4.1.17., Ž4.1.14., Ž4.1.23., and Ž4.1.24., respectively. The groups V2Ž2. and SX3 can be identified with corresponding groups acting on the hexad Z as subgroups of SZ . In particular, we calculate that V2Ž2. can be identified with V s  1, Ž z1 , z 2 .Ž z 3 , z 4 ., Ž z 3 , z 4 .Ž z 5 , z 6 ., Ž z1 , z 2 .Ž z 5 , z 6 .4 , and the group SX3 can be identified with SX3 which is ²Ž z1 , z 5 , z 4 .Ž z 2 , z6 , z 3 ., Ž z1 , z 4 .Ž z 2 , z 3 .:, where the z i were identified in Ž2.3.12. as the six elements of the hexad Z in P.

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The element s can be identified with the involution s s Ž z1 , z 2 .Ž z 3 , z 4 .Ž z 5 , z6 .. We calculate that s centralizes V and SX3 , and hence centralizes S4Ž2. s ² V, SX3 :. Therefore, we can identify Ž4.7.2. the inverse image of NM Ž C2, 2 . in 4. M22 as the semidirect 22 product G i S4Ž2., and Ž4.7.3. the inverse image of NM .2 Ž C2, 2 . in 4. M22 .2 as the semidirect 22 product G i Ž S4Ž2. = Z2Ž2. .. When n s 1, we find immediately that NM 22Ž C2, 2 . has 12 irreducible characters, 5 of which, with degrees 1, 1, 2, 3, and 3, lie over the trivial character l1 of E, 5 of which, with degrees 3, 3, 3, 3, and 6, lie over a Ž2. linear character l2 of E which has stabilizer D Ž2. 8 in S4 , and 2 of which, with degrees 12 and 12, lie over a linear character l3 of E with stabilizer Z2 in S4Ž2.. This yields characters with 2-defects 7, 7, 6, 7, 7, 7, 7, 7, 7, 6, 5, and 5, respectively. As usual, the characters lie in a single block which must induce the only 2-block 1. B of M22 . The structure of NM 22 .2 Ž C2, 2 . implies that no characters of NM 22Ž C2, 2 . are fused, since the stabilizers of l1 , l2 , and l3 in S4Ž2. = Z2Ž2. are S4Ž2. = Z2Ž2., D 8Ž2. = Z2Ž2., and Z2 = Z2Ž2., respectively. We let n 1 s 1. This gives us the 1. B entries in the third column of Table 2 and tells us that k Ž NM 22Ž C2, 2 ., 1. B, d, O N 1. is zero for all other combinations of d and O satisfying the hypotheses of Theorem 2. We next let n s 2 and let n 2 be a faithful linear character of the center N2 of 2. M22 . We let H s S4Ž2. and K s S4Ž2. = Z2Ž2., where Z2Ž2. is generated by s . We prove LEMMA 4.7.4. The 2. B entries in the third column of Table 2 are as gi¨ en. Proof. The elements of the elementary abelian group GrG9 of order 2 5 are the images ˜ eU of the eU , where U runs over the 2 5 subsets of even order in Z. We seek the H-orbits and the K-orbits in the set IrrŽ GrG9 N n 2 . of all 16 linear characters of GrG9 lying over the nontrivial linear character n 2 of N2 s N4rG9. The permutation f s Ž z1 , z5 , z 4 .Ž z 2 , z6 , z 3 . has orbits  ˜ e z1 ˜ e z5 , ˜ e z5 ˜ e z4, ˜ e z4 ˜ e z 14 and  ˜ ez2 ˜ e z6 , ˜ e z6 ˜ e z3 , ˜ e z3 ˜ e z 2 4 consisting of the nontrivial elements in two different 4-groups V3 and V4 of GrG9. We have GrG9 s V3 = V4 = Ž N4rG9.. Since s exchanges V3 and V4 , it leaves invari˜1 g ant V3 = V4 s w GrG9, f x. It follows that the unique linear character l ˜ IrrŽ GrG9 N n 2 ., with kerŽ l1 . s V3 = V4 , is fixed by s . Thus the stabilizer of ˜1 in K is l Kl˜1 s NK Ž ² f : . s ² s : = NH Ž ² f : . , ² s : = SX3 .

Ž 4.7.5.

˜1 belongs to an H-orbit L 1 of four characters, which is also a Hence l K-orbit. Note that Kl˜1 leaves invariant each of the three characters in ˜1 .. Thus K leaves invariant the corresponding characIrrŽŽ GrG9. i Hl˜1 N l ters of degrees 4, 4, and 8 induced by these in IrrŽŽ GrG9. i H N n 2 ..

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Clearly none of the remaining 12 characters in IrrŽ GrG9 N n 2 . can be fixed by a Sylow 3-subgroup of H. If x, y, z, w are four distinct elements of Z, then we compute that GrG9, Ž x, y . Ž z, w . s ² ˜ ex ˜ ey , ˜ ez ˜ ew : ,

Ž 4.7.6.

which is a 4-subgroup not containing N2 . We apply Ž4.7.6. to each of the three nontrivial elements Ž z1 , z 3 .Ž z 2 , z 4 ., Ž z1 , z 2 .Ž z 3 , z 4 ., and Ž z1 , z 4 .Ž z 3 , z 2 . of a 4-group H2 of H. It follows that

w GrG9, H2 x s ² ˜e z1 ˜e z 3 , ˜e z 2 ˜e z 4 , ˜e z1 ˜e z 2 , ˜e z 3 ˜e z 4 , ˜e z1 ˜e z 4 , ˜e z 2 ˜e z 3 : is the subgroup of order 2 3 consisting of the ˜ eU , where U runs over all the subsets of even order in  z1 , z 3 , z 2 , z 4 4 . Clearly w GrG9, H2 x l ² a ˜: s 1. ˜2 and l˜X2 in IrrŽ GrG9 N n 2 . having Thus there are two linear characters l w GrG9, H2 x in their kernel. Neither of these characters belongs to the orbit L 1 since Hl˜1 intersects the normal 4-subgroup V of H trivially, while H2 F Hl˜2 does not. Using Ž4.7.6., we see that w GrG9, V x contains a. ˜ Therefore, V cannot fix any character in IrrŽ GrG9 N n 2 .. It follows that Hl˜2 is a 2-subgroup of H containing H2 but not V. The only such 2-subgroup is H2 itself. Thus ˜2 and l˜X2 , since Hl˜2 s HlX2 s H2 , Z2 = Z2 . Further, V must interchange l these are the only characters in IrrŽ GrG9 N n 2 . fixed by H2 , and V normalizes H2 . Therefore, there is a single H-orbit L 2 of length w H : H2 x s 6 ˜2 and l˜X2 . The element s centralizes H2 F H. Since containing both l

w GrG9, s x s 1, ˜e z1 ˜e z 2 ˜e z 3 ˜e z 4 , ˜e z 3 ˜e z 4 ˜e z 5 ˜e z 6 , ˜e z 5 ˜e z 6 ˜e z1 ˜e z 2

½

5

Ž 4.7.7.

we have a ˜ g w GrG9, s x ? w GrG9, H2 x. So s cannot fix l˜2 , that is, s ˜2 and l˜X2 . Since Ž z1 , z 2 .Ž z5 , z6 . also exchanges l˜2 and l˜X2 , we exchanges l ˜2 and l˜X2 . Hence conclude that Ž z 3 , z 4 . s s Ž z1 , z 2 .Ž z5 , z 6 . fixes both l Kl˜2 s ²Ž z 3 , z 4 ., H2 : s H2 i ²Ž z 3 , z 4 .: , D 8 . In particular, Kl˜2 exchanges two of the four linear characters in IrrŽŽ GrG9. i Hl˜2 N l 2 . and fixes the other two. Thus K exchanges two of the four characters of degree 6 in IrrŽŽ GrG9. i Ž H N n 2 . induced by these four linear characters and fixes the other two. There must be 16 y 4 y 6 s 6 other characters in IrrŽ GrG9 N n 2 .. We calculate that the element k s Ž z1 , z 4 , z 2 , z 3 . Ž z5 , z6 .

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Ž2. of H, corresponds to the element of order four k in D Ž2. 8 F S4 defined in 2 Ž4.1.21.. Its square is k s Ž z1 , z 2 .Ž z 3 , z 4 .. We compute that w GrG9, k x s ²˜ e z1 ˜ ez2 , ˜ e z3 ˜ e z4, ˜ e z1 ˜ e z4 ˜ e z5 ˜ e z 6 : has order 2 3 and does not contain a. ˜ It follows ˜3 and l˜X3 in that there are precisely two ² k :-invariant characters l IrrŽ GrG9 N n 2 .. The group V normalizes ² k : and fixes no character in ˜3 and l˜X3 . Since no Sylow IrrŽ GrG9 N n 2 .. Therefore, V interchanges l X ˜ ˜ 3-subgroup of H fixes either l3 or l 3 , we conclude that Hl˜3 s ² k :. Thus ˜3 and l˜X3 belong to the same orbit L 3 of length w H : Hl 3 x s 6 in l IrrŽ GrG9 N n 2 .. It follows from Ž4.7.7. that a ˜ g w GrG9, k x ? w GrG9, s x. Therefore, s ˜3 and l˜X3 . Now Ž z1 , z 2 .Ž z5 , z6 . does not fix l˜3 since must also interchange l ˜3 , but Ž z1 , z 2 .Ž z 3 , z 4 . g V l ² k : does fix l3 . Hence V does not fix l ˜3 and l˜X3 . Thus Kl˜3 s Ž z 3 , z 4 . s s Ž z 1 , z 2 .Ž z 5 , z 6 . fixes both l ²Ž z 3 , z 4 ., ² k :: s ² k : i ²Ž z 3 , z 4 .: , D 8 . In particular, Kl˜ exchanges two 3 ˜3 . and fixes the other of the four linear characters in IrrŽŽ GrG9. i Hl˜3 N l two. Thus K exchanges two of the four characters of degree 6 in IrrŽŽ GrG9. i H N n 2 . induced by these linear characters and fixes the other two. In summary, there are three H-orbits in IrrŽ GrG9 N n 2 ., the orbits L 1 , ˜1 , l˜2 , and l˜3 , respectively. They have orders 4, 6, and 6, L 2 , and L 3 of l ˜1 . contains three characters with respectively. The set IrrŽŽ GrG9. i H N l ˜2 . contains degrees 4, 4, and 8, all fixed by K. The set IrrŽŽ GrG9. i H N l four characters with degrees 6, 6, 6, and 6, two of which are fixed by K and ˜3 . also two of which are exchanged by K. The set IrrŽŽ GrG9. i H N l contains four characters with degrees 6, 6, 6, and 6, two of which are fixed by K and two of which are exchanged by K. These characters must lie in a single block, which must induce the only 2-block 2. B of 2. M22 . Their 2-defects are 6, 6, 5, 7, 7, 7, 7, 7, 7, 7, and 7, respectively. This gives us the 2. B entries in the third column of Table 2 and tells us that k Ž N2. M 22Ž C2, 2 ., 2. B, d, O N n 2 . is zero for all other combinations of d and O satisfying the hypotheses of Theorem 2. This completes the proof of Lemma 4.7.4.

For n s 4 we let n4 be a faithful linear character of the center N4 of 4. M22 and recall that we found in Ž4.6.4. that the Clifford extension of Ž2. AŽ2. has the same Sylow 6 .G for C does not split. Since the group S4 Ž2. 2-subgroup as A 6 , we conclude that the Clifford extension of S4Ž2..G for C does not split either. Thus the degrees of the characters of N4. M 22Ž C2, 2 . which lie over n4 are the degrees of the characters of a nonsplit extension 2.S4 of S4 , which are 2, 2, and 4, multiplied by 4. This yields characters with 2-defects 6, 6, and 5, respectively. As usual, the characters lie in a single block, which must induce the only 2-block 4. B of 4. M22 . This gives

COUNTING CHARACTERS IN BLOCKS

67

us the 4. B entries in the third column of Table 2 and tells us that k Ž N4. M 22Ž C2, 2 ., 4. B, d, O N n4 . is zero for all other combinations of d and O satisfying the hypotheses of Theorem 2. We next let n s 3 and let n 3 be a faithful linear character of the center N3 of 3. M22 . In the extension 3. M22 , we found, in Lemma 4.6.6, that E splits. It is also true that 3.S4 must be a split extension of N3 , since S4 has no central extension by Z3 . Therefore, the degrees of the characters of N3. M 22Ž C2, 2 . which lie over n 3 are the same as those of NM 22Ž C2, 2 .. As usual, the characters lie in a single 2-block b. We recall from Sect. 4.5 that there are two 2-blocks of 3. M22 . The block 3. B9 has a Sylow 2-subgroup Ž N3 . 2 of N3 as a defect group. However, the defect group of b must be at least as big as O 2 Ž N3. M 22Ž C2 .. which is certainly larger than Ž N3 . 2 . The block b 3. M 22 induced by b must have a defect group containing the defect group of b. Hence b 3. M 22 must be the other 2-block 3. B of 3. M22 . This gives us the 3. B entries in the third column of Table 2 and tells us that k Ž N3. M 22Ž C2, 2 ., B, d, O N n 3 . is zero for all other combinations of B, d, and O satisfying the hypotheses of Theorem 2. Now we let n s 6, let n 6 be a faithful linear character of the center N6 of 6. M22 , and recall that 3.S4Ž2. is a split extension of N3 . It follows that the normalizer 6.S4Ž2. must also be a split extension of N6 in the extension 6. M22 of 2. M22 by N3 . Therefore the degrees of the characters of N6. M 22Ž C2, 2 . which lie over n 6 are the same as the degrees of the characters of N2. M 22Ž C2, 2 . which lie over n 2 . As usual, the characters lie in a single block and, using an argument similar to the one above in the case where n s 3, we conclude that this block must induce the block 6. B of 6. M22 . This gives us the 6. B entries in the third column of Table 2 and tells us that k Ž N6. M 22Ž C2, 2 ., B, d, O N n 6 . is zero for all other combinations of B, d, and O satisfying the hypotheses of Theorem 2. Finally, we let n s 12 and let n 12 be a faithful linear character of the center N12 of 12. M22 . From the above remarks it is clear that the normalizer 12.S4Ž2. must also be a split extension of N12 in 12. M22 , which is an extension of 4. M22 by N3 . Thus the degrees of the characters of N12. M 22Ž C2, 2 . which lie over n 12 must be the same as the degrees of the characters of N4. M 22Ž C2, 2 . which lie over n4 . As usual, the characters lie in a single block which must induce the block 12. B of 12. M22 , using an argument similar to the one in the case where n s 3. This gives us the 12. B entries in the third column of Table 2 and tells us that k Ž N12. M 22Ž C2, 2 ., B, d, O N n 12 . is zero for all other combinations of B, d, and O satisfying the hypotheses of Theorem 2. We have yet to consider the chain C3, 2 and the entries in the last column of Table 2. We shall do so in the next and final section.

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4.8. The Chain C3, 2 We turn our attention to the chain C3, 2 , whose normalizer NM 22Ž C3, 2 . is equal to F i S5Ž2., where S5Ž2. acts on F as S L2 Ž4.. We begin with n s 1. Since the center N1 of M22 is trivial, we set n 1 s 1. We prove LEMMA 4.8.1. The 1. B entries in the last column of Table 2 are as gi¨ en. Proof. We notice that the 15 nontrivial linear characters of F form a single S5-orbit. We then calculate that the group F i S5Ž2. has 12 irreducible characters, 7 of which, x 1 , . . . , x 7 with degrees 1, 1, 4, 4, 5, 5, and 6, lie over the principal character of F, and 5 of which, x 8 , . . . , x 12 with degrees 15, 15, 15, 15, and 30, lie over a linear character of F which has stabilizer D 8 in S5 . These characters then have 2-defects 7, 7, 5, 5, 7, 7, 6, 7, 7, 7, 7, and 6, respectively. They lie in a single 2-block which must therefore induce the block 1. B of M22 . We consider next the normalizer NM 22 .2 Ž F . of F in M22 .2. Suppose that l is a nontrivial linear character of F. In particular, we may take l to be the character which maps 1 us a b

0 1 0

0 0 gF 1

to lŽ u . s Žy1. TrŽ a. where TrŽ a. s Tr FF24 Ž a.. Then clearly l is fixed by s4 , whose image generates OutŽ M22 ., as well as by s2 . ŽSee Ž2.1.10. and Ž2. Ž2.1.6... Further, the stabilizer Ž AŽ2. . 5 l of l in A 5 , the complement of F in NP S LŽ3, 4.Ž F . described in Ž4.1.7., is the set of all elements of the form 1 0 v e2

0 1 e

0 0 , 1

since

l



1 a b

0 1 0

0 0 1

0

sl



1 a b

0 1 0

0 0 1

1 v e2d 2 v e2 f 2

0

0

c

d

e

f

0

if and only if TrŽ fa q db. s TrŽ a. for all a, b g F4 , which implies that d s 0, f s 1, and, therefore, c s 1. The kernel of l clearly contains all

69

COUNTING CHARACTERS IN BLOCKS

elements of the form u with a s 0. We see that 1 0 v e2

0 1 e

0 0 1

s2

s

1 0 ve

0 1 e2

0 0 , 1

and 1 0 v e2

0 1 e

0 0 1

s4

s

1 0 v 2e

0 1 e2

0 1 0 ' 0 1 ve

0 1 e2

0 0 1

mod ker Ž l . .

Therefore, conjugation by s4 s2 g M22 .2 y M22 centralizes N Ž F .l modulo the kernel of l. ŽClearly s4 s2 centralizes F.. The same result holds if l is the trivial character of F. Thus there is no fusion of characters of NM 22Ž F . in NM 22 .2 Ž F .. These considerations give us the 1. B entries in the last column of Table 2 and tell us that k Ž NM 22Ž C3, 2 ., 1. B, d, O N 1. is zero for all other combinations of d and O satisfying the hypotheses of Theorem 2. This completes the proof of Lemma 4.8.1. We turn next to the factor group 2. M22 of the covering group. The center of 2. M22 is N2 , N4r² y 1:. We choose a faithful character n 2 of N2 . LEMMA 4.8.2. The 2. B entries in the last column of Table 2 are as gi¨ en. Proof. We recall the construction of 4. N Ž E . as G i AŽ2. 6 in Sect. 2.3. It is clear that the elementary abelian group F of order 2 4 is a subgroup of NM 22Ž E . , E i AŽ2. 6 , since the latter consists of all elements of either form Ž4.1.6., where the a i and aXj lie in F4 and all determinants are 1. It is straightforward to calculate that F splits over N2 . We let F˜ denote the inverse image of F in 2. M22 . A Sylow 5-subgroup of S5 has form Z5 . It ˜ 2 . Since 5 and 2 are acts faithfully and irreducibly on the factor group FrN w x relatively prime, this implies 15, 5.3.5 that the group F˜ is a direct product ˜ Z5 x = N2 . Thus there is one complement of F to N2 in F˜ which is w F, ˜ the unique linear character of F, ˜ lying over stabilized by Z5 . Denote by l ˜ Z5 x. Then clearly NS5Ž Z5 . , Z5 i Z4 F Ž S5 .l˜. We note n 2 , with kernel w F, ˜ because this would imply that a Sylow that S5 does not stabilize l 2-subgroup of M22 splits over N2 , which it does not. Therefore, a Frobenius group of the form Z5 i Z4 , which is maximal in S5 and has order 20, ˜ in S5 . The resulting S5-orbit in IrrŽ F˜ N n 2 . has length is the stabilizer of l 6 s w S5 : Z4 h Z5 x. A Sylow 3-subgroup of S5 has form Z3 . Using the same argument as above, as well as the fact that w F, Z3 x s F, we conclude that the normalizer of Z3 in S5 , which is of the form Z2 = S3 and is ˜1 in IrrŽ F˜ N n 2 .. The resulting maximal in S5 , stabilizes a unique character l

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S5-orbit in IrrŽ F˜ N n 2 . has length 10 s w S5 : Z2 = S3 x. Since 10 q 6 s 16, this and the preceding orbit account for all the characters of IrrŽ F˜ N n 2 .. We claim that the inverse image N2 .S5 of S5 in 2. M22 is isoclinic to a split extension. The derived group of the inverse image N4 .S5 of S5 in 4. M22 is isomorphic to either 2. A 5 or A 5 . So this derived group intersects N4 in at most V Ž N4 .. We factor this last group out to obtain N2 . This means that the derived group of N2 .S5 is A 5 , which implies that N2 .S5 is isoclinic to a split extension. Since N2 .Ž Z2 = S3 . F N2 .S5 and N2 .Ž Z4 h Z5 . F N2 .S5 , it follows that N2 .Ž Z2 = S3 . and N2 .Ž Z4 h Z5 . are also isoclinic to split extensions. Therefore, the degrees of the characters of N2. M 22Ž C3 . which lie over n 2 are the degrees for the group Z4 h Z5 , which are 1, 1, 1, 1, and 4, multiplied by 6, together with the degrees for the group Z2 = S3 , which are 1, 1, 2, 1, 1, and 2, multiplied by 10. The 2-defects of these 11 characters in IrrŽ N2. M 22Ž C2, 3 . N n 2 . are 7, 7, 7, 7, 5, 7, 7, 6, 7, 7, and 6, respectively. The characters lie in a single block, which must induce the only 2-block 2. B of 2. M22 . We turn our attention to N2. M 22 .2 Ž F .. We compute first the normalizer in 2. N Ž F ..2 s N2. M 22 .2 Ž F . of a Sylow 3-subgroup Z3 of 2. N Ž F . s N2. M 22Ž F .. We make use of the results of Lemma 3.4.2, where we found the normalizer N4. M 22 .2 Ž H1 . of H1 , which is conjugate to Z3 , to be isomorphic to Ž N4 ( Q8 . i Ž² e 3 , r 2 : = ² s 3 , i 2 :.. Let s 3 , e 3 , i 2 , r 2 , and Q8 be the elements and subgroup used to construct N4. M 22 .2 Ž H1 .. Then  "s 3 4 ,  "e 34 ,  "i 2 4 ,  "r 2 4 , and Q8r "14 are their respective images in N2. M .2 Ž H1 .. 22 There is some ˜ t g 2. M22 such that s˜3 s  "s 34t˜ generates Z3 g Syl 3 Ž2. N Ž F ... We set ˜ e 3 s  "e 34t˜, ˜i 2 s  "i 2 4t˜, r˜2 s  "r 2 4t˜, and V˜4 s t ˜ Ž Q8r "14. in 2. M22 .2. Then s˜3 , ˜ e 3 , ˜i 2 , and r˜2 have orders 3, 3, 2, and 2, respectively, while V˜4 is a 4-group. The subgroup ² s˜3 , e˜3 , ˜i 2 , r˜2 : is the direct product ² s˜3 , ˜i 2 : = ² ˜ e 3 , r˜2 : of ² s˜3 , ˜i 2 : , S3 and ² e˜3 , r˜2 : , S3 . The factor ² s˜3 , ˜i 2 : centralizes N2 = V˜4 , while the other factor ² ˜ e3 , r˜2 : centralizes N2 and acts faithfully on V˜4 as the full automorphism group AutŽ V4 . , S3 . In view of Lemma 3.4.1 we have N2. M 22 .2 Ž² s˜3 :. s N2 = Ž V˜4 i ² ˜ e3 , r˜2 :. = ² s˜3 , ˜i 2 : , Z2 = S4 = S3 . In particular, Ž4.8.3.

N2. M 22 .2 Ž² s˜3 :. splits over N2 .

The normalizer 2. N Ž F . is the semidirect product Ž2.F . i S5 of the inverse image 2.F of F in 2. M22 with an S5 acting faithfully on F as S L2 Ž4. acts on F42 . Since OutŽ S5 . s 1, the factor group 2. N Ž F ..2r2.F , S5 .2 is a direct product 2. N Ž F ..2r2.F s S5 = Z2 of the image S5 of S5 with some cyclic group Z2 of order 2. We may assume that ² s˜3 : F S5 . Then the image s˜3X of s˜3 in 2. N Ž F ..2r2.F generates a Sylow 3-subgroup ² s˜3X : of S5 . The normalizer NS =Z Ž² s˜3X :. s NS Ž² s˜3X :. = Z2 is the image 5 2 5 of N2. NŽ F ..2 Ž² s˜3 :. in 2. N Ž F ..2r2.F. Since s˜3 fixes no nontrivial element

COUNTING CHARACTERS IN BLOCKS

71

of F, we have that N2. F Ž² s˜3 :. s C2. F Ž s˜3 . s N2 . Hence we have an exact sequence F

1 ª N2 ª N2. NŽ F ..2 Ž ² s˜3 : . ª NS 5Ž ² s˜3X : . = Z2 ª 1. Since N2. M 22 .2 Ž² s˜3 :. splits over N2 by Ž4.8.3., so does it subgroup N2. NŽ F ..2 Ž² s˜3 :.. Therefore, the group N2. NŽ F ..2 Ž² s˜3 :. is isomorphic to N2 = NS 5Ž² s˜3X :. = Z2 in such a way that N2. NŽ F .Ž² s˜3 :. is isomorphic to N2 = NS 5Ž² s˜3X :.. It follows that N2. NŽ F ..2 Ž ² s˜3 : . s N2. NŽ F . Ž ² s˜3 : . = Z2 ,

Ž 4.8.4.

for some Z2 , Z2 . Thus every character in IrrŽ N2. NŽ F .Ž² s˜3 :.. is fixed by N2. NŽ F ..2 Ž² s˜3 :.. Therefore, the characters in IrrŽ N2. M 22Ž C3, 2 . N n 2 . with degrees 10, 10, 20, 10, 10, and 20 are all fixed by 2. N Ž F ..2. We recall that a Sylow 5-subgroup of S5 has form Z5 , with normalizer Z5 i Z4 . We consider next the normalizer in 2. N Ž F ..2 of this Z5 . The inverse image of Z5 in 2. M22 is N2 = Z5 . By the character table for M22 in the Atlas, the inverse image of Z4 is a split extension of the form N2 = Z4 . The group Z5 fixes exactly two points in P, namely the two Ž . points w1, 0, 0x and u fixed by PSLŽ3, 4.w1, 0, 0x s F i AŽ2. 5 . Therefore, N Z5 is contained in Ž M22 .w1, 0, 0x, u4 s N Ž F .. So 2. N Ž Z5 . s N2. M 22Ž Z5 . s N2 = Ž Z5 i Z4 .. Evidently 2. N Ž Z5 ..2 s N2. M .2 Ž Z5 . s 2. N Ž Z5 . ? ² b˜:, where 22 b˜ 2 g 2. N Ž Z5 . and b˜ f 2. N Ž Z5 .. Conjugation by b˜ must leave invariant the normal Sylow 5-subgroup Z5 of 2. N Ž Z5 .. Since Z4 acts by conjugation as AutŽ Z5 ., we may multiply b˜ by an element of Z4 and assume that b˜ centralizes Z5 . This forces b˜ 2 to lie in N2 s C2. NŽ Z 5 .Ž Z5 .. We may choose b˜ so that ² b˜, 2.Z4 : is a 2-group. Then the commutator w b˜, D x, where D is a generator of Z4 , must lie in the two normal subgroups C2. NŽ Z 5 ..2 Ž Z5 . and 2. N Ž Z5 . of 2. N Ž Z5 ..2. Hence it lies in their intersection N2 = Z5 where N2 s ² a ˜:. It follows that either b˜ centralizes 2. N Ž Z5 ., or b˜ that D s D a. ˜ We shall show that the latter case holds. We shall use the fact that, as Dade has proved in w13x, the M-A-D conjecture holds for 2. M22 .2 and the prime 5, since the Sylow 5-subgroup of 2. M22 is cyclic. This tells us that the outer automorphism acts on the five characters in IrrŽ2. N Ž Z5 . N n 2 . Žof degrees 1, 1, 1, 1, and 4. exactly as it acts on the five characters lying in the unique 5-block 2. B5 of 2. M22 lying over n 2 and having Z5 as its defect group. From the Atlas, the characters in IrrŽ2. B5 . s IrrŽ2. B5 N n 2 . are x 15 , x 17 , x 18 , x 19 , and x 20 , whose orbits under the outer automorphism are  x 15 4 ,  x 17 , x 18 4 , and  x 19 , x 20 4 . So the outer automorphism must permute the characters in IrrŽ2. N Ž Z5 . N n 2 . in three orbits of lengths 1, 2, and 2. This tells us that b˜ cannot centralize 2. N Ž Z5 .. Hence we must have that D b˜ s D a. ˜

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JANICE F. HUANG

All of this implies that two pairs of conjugacy classes in N2. M 22Ž F . are fused in N2. M 22 .2 Ž F .. It follows that two pairs of characters are also fused under this action. Now b˜ centralizes N Ž Z5 .. Hence the characters of N2. M 22Ž Z5 . lying over the trivial character of N2 cannot be fused. Therefore, the four characters in IrrŽ N2. M 22Ž Z5 .. moved by b˜ must all lie over n 2 . These characters have degrees 1, 1, 1, 1, and 4, which means that the characters moved by b˜ must be the four characters of degree 1. We conclude that N2. M 22 .2 Ž F . moves the corresponding characters of degree 6 and 2-defect 7 in IrrŽ2. M22 N n 2 .. This gives us the 2. B entries of the last column of Table 2 and tells us that k Ž N2. M 22Ž C3, 2 ., 2. B, d, O N n 2 . is zero for all other combinations of d and O satisfying the hypotheses of Theorem 2. This completes the proof of Lemma 4.8.2. We look next at the factor group 4. M22 of the covering group. We set n s n4 , a faithful linear character of the center N4 of 4. M22 , and prove LEMMA 4.8.5. The 4. B entries in the last column of Table 2 are as gi¨ en. Proof. The factor group 4. M22 has center N4 . We recall that S5Ž2. acts on F as the semidirect product of L2 Ž4. with its Galois automorphism. The Sylow 2-subgroup S of N Ž C3, 2 . , F i S5Ž2. is also a Sylow 2-subgroup of N Ž C1, 2 . , E i AŽ2. 6 . It has structure F i D 8 , as well as E i D 8 . We determined in Ž4.6.5. that the degrees of the irreducible characters of 4.S which lie over n4 are 8 and 8, the degrees of the characters of the nonsplit extension 2. D 8 of D 8 multiplied by 4. We calculate in 4. N Ž E . that the derived group w4.F, 4.F x is nontrivial. For example, we take elements 1 ss 1 0

0 1 0

0 0 e z 5 e z 6 g 4.F , 1

which maps to 1 1 v

0 1 0

0 0 g F, 1

and 1 ts v 1

0 1 0

0 0 e z 3 e z 4 g 4.F , 1

COUNTING CHARACTERS IN BLOCKS

73

which maps to 1 v v

0 1 0

0 0 g F, 1

and find that w s, t x s y1. Therefore, the group 4.F has the same structure as 4. E s G. Then 4.F has a unique character C9 with degree 4 lying over n4 . The Sylow p-subgroups of S5 are all cyclic except when p s 2. Therefore, the character C9 can be extended to the inverse image of Sylow p-subgroups of S5 in the cases when p / 2. If the character C9 could be extended for p s 2 we would get a character of 4.S with degree 4 lying over n4 , which we do not. Thus the Clifford extension of S5 for c cannot be isoclinic to a split extension. Hence it must be isoclinic to 2.S5 s Ž2. A 5 ..2. It follows that the degrees of the characters of N4. M Ž C3 ., which 22 lie over n4 , are found by taking the degrees of the characters of 2.S5 , which are 4, 4, 4, 6, and 6, and multiplying them by 4. ŽSee the Atlas w8x.. This procedure yields the 2-defects 5, 5, 5, 6, and 6, respectively. These characters lie in a single block, which must induce the only 2-block 4. B of 4. M22 . This gives us the 4. B entries of the last column of Table 2 and tells us that k Ž N4. M 22Ž C3, 2 ., 4. B, d, O N n4 . is zero for all other combinations of d and O satisfying the hypotheses of Theorem 2. This completes the proof of Lemma 4.8.5. We look next at the factor group 3. M22 with center of the form N3 , Z3 . We choose a faithful character n 3 of N3 and prove LEMMA 4.8.6. The 3. B entries in the last column of Table 2 are as gi¨ en. Proof. It is immediately clear that the elementary abelian 2-subgroup F of M22 splits over the subgroup N3 . Similarly, S5 must split in 3. M22 , since S5 has no central extension over Z3 . Therefore, the degrees of the characters of N3. M 22Ž C3, 2 . which lie over n 3 are the same as those of the normalizer NM 22Ž C3, 2 . which were identified in Lemma 4.8.1 above. These characters, therefore, have 2-defects 7, 7, 5, 5, 7, 7, 6, 7, 7, 7, 7, and 6, respectively. They lie in a single block, which must induce the only 2-block 3. B of 3. M22 which has defect group larger than N3 . This gives us the 3. B entries of the last column of Table 2 and tells us that k Ž N3. M 22Ž C3, 2 ., B, d, O N n 3 . is zero for all other combinations of B, d, and O satisfying the hypotheses of Theorem 2. This completes the proof of Lemma 4.8.6. Next we consider the normalizer of chain C3, 2 in the factor group 6. M22 . The center of 6. M22 is N6 , N3 = N2 . We choose a faithful character n 6 of N6 and prove LEMMA 4.8.7. The 6. B entries in the last column of Table 2 are as gi¨ en.

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JANICE F. HUANG

Proof. Since the normalizer N3. M 22Ž C3, 2 . is a split extension of N3 in 3. M22 , it follows that in the extension 6. M22 of 2. M22 by N3 the group N6. M 22Ž C3, 2 . must also be a split extension of N3 . Therefore, the degrees of the characters of N6. M 22Ž C3, 2 . which lie over n 6 are the same as the degrees of the characters of N2. M 22Ž C3, 2 . which lie over n 2 . These were calculated in Lemma 4.8.2 above. They have 2-defects 7, 7, 7, 7, 5, 7, 7, 6, 7, 7, and 6, respectively. These characters lie in a single block, which must induce the only 2-block 6. B of 6. M22 which has defect group larger than O 2 Ž N6 .. This gives us the 6. B entries of the last column of Table 2 and tells us that k Ž N6. M 22Ž C3, 2 ., B, d, O N n 6 . is zero for all other combinations of B, d, and O satisfying the hypotheses of Theorem 2. This completes the proof of Lemma 4.8.7. Finally we consider the covering group 12. M22 , which has center N12 , N3 = N4 . We choose a faithful character n 12 of N12 and prove LEMMA 4.8.8. The last two entries in the last column of Table 2 are as gi¨ en. Proof. Since the normalizer N3. M 22Ž C3, 2 . is a split extension of N3 in 3. M22 , it follows that in the extension 12. M22 of 4. M22 by N3 the group N12. M 22Ž C3, 2 . must also be a split extension of N3 . Therefore, the degrees of the characters of N12. M 22Ž C3, 2 . which lie over n 12 are the same as the degrees of the characters of N4. M 22Ž C3, 2 . which lie over n4 . These were calculated in Lemma 4.8.5 above. These characters have 2-defects 5, 5, 5, 6, and 6, respectively. They lie in a single block, which must induce the only 2-block 12. B of 12. M22 which has defect groups larger than N4 . This gives us the 12. B entries of the last column of Table 2 and tells us that k Ž N12. M 22Ž C3, 2 ., B, d, O N n 12 . is zero for all other combinations of B, d, and O satisfying the hypotheses of Theorem 2. This completes the proof of Lemma 4.8.8. This completes the verification of all entries in Table 2 and, hence, completes the proof of Theorem 2. Therefore, we have verified that the M-A-D conjecture holds for any proper covering group of the Mathieu group M22 and any prime p.

ACKNOWLEDGMENTS This paper had its origins in my Ph.D. thesis w17x. I thank my advisor, E. C. Dade, who introduced me to the subject and gave numerous suggestions.

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