The McKay Conjecture Is True for the Sporadic Simple Groups

The McKay Conjecture Is True for the Sporadic Simple Groups

207, 294᎐305 Ž1998. JA987450 JOURNAL OF ALGEBRA ARTICLE NO. The McKay Conjecture Is True for the Sporadic Simple Groups Robert A. Wilson School of M...

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207, 294᎐305 Ž1998. JA987450

JOURNAL OF ALGEBRA ARTICLE NO.

The McKay Conjecture Is True for the Sporadic Simple Groups Robert A. Wilson School of Mathematics and Statistics, The Uni¨ ersity of Birmingham, Edgbaston, Birmingham B15 2TT, England Communicated by Jan Saxl Received November 12, 1997

The McKay conjecture states that the number of irreducible complex characters of a group G that have degree prime to p is equal to the same number for the Sylow p-normalizer in G. We verify this conjecture for the 26 sporadic simple groups. 䊚 1998 Academic Press

1. INTRODUCTION McKay’s conjecture in its simplest form states that if G is a finite group, p a prime, and P a Sylow p-subgroup of G, then the number of irreducible complex characters of G whose degree is not divisible by p is equal to the number of irreducible complex characters of NG Ž P . with the same property. This conjecture is related to the Alperin and Dade conjectures Žsee, for example, w5x., and all three seem to be true for very deep reasons, although a proof may still be a long way off. It is known to be true in the case in which P is cyclicᎏthis follows from the extensive theory of Brauer treesᎏsee w7x. The work described in this paper was mostly done well over 10 years ago, but was not written up at the time, partly because of difficulties in completing one or two tricky cases. Some of the results were obtained earlier by Ostermann w10x, as corollaries of his calculations of the character tales of many of the Sylow p-normalizers of sporadic groups. Our proofs here, however, use far less information about the Sylow p-normalizers than this. Again, some of the results have since been obtained independently by Jianbei An and others, as part of the much more general verification of the Dade conjectures for the smaller sporadic simple groups 294 0021-8693r98 $25.00 Copyright 䊚 1998 by Academic Press All rights of reproduction in any form reserved.

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Žsee, for example, w1x, w2x, w8x, w9x, etc... Nevertheless, we considered it worthwhile to present all of these results again, for completeness and because our proofs are often simpler. In what follows, the word ‘‘character’’ will always mean an irreducible complex character. In general, we note that the degrees of the characters of P are powers of p, so they are either divisible by p or equal to 1. In the latter case, they are lifts of characters of PrPX . It follows immediately, by Clifford’s theorem, that all characters of NG Ž P . that have degree coprime to p arise in the same way as lifts of characters of NG Ž P .rP X . On the other hand, a closer inspection of the character table of NG Ž P .rP X , using Clifford theory again, shows that every character has degree dividing < NG Ž P .rP <, which is prime to p. Thus our task is reduced to counting the number of characters Žor, equivalently, conjugacy classes . of NG Ž P .rP X , for noncyclic Sylow p-subgroups P. Now NG Ž P .rP X is a split extension of an abelian p-group by a pX-group, isomorphic to NG Ž P .rP. Clifford theory gives us a simple parameterization of the characters of a group of this shape: consider the orbits of NG Ž P .rP on the characters of PrPX , and for each orbit pick the stabilizer I␹ i of a representative point ␹ i Žcalled the inertial group.. Then the characters of NG Ž P .rP X are in one-to-one correspondence with the pairs Ž ␹ i , ␰ j . as i runs over orbits, and ␰ j runs over the characters of I␹ . i Thus the total number of characters of NG Ž P .rP X is equal to the sum over the orbits of NG Ž P . on characters ␹ of PrPX , of the number of characters of the inertial subgroup I␹ in NG Ž P .rP. It turns out that in all cases in the sporadic groups, PrPX is elementary abelian, although this is not always immediately obvious. This slightly simplifies some of the calculations. We first present a table of results Žsee Table I., showing the structure of NG Ž P .rP X , and the number of its conjugacy classes, in each case. In later sections we present some examples of the proofs.

2. PROOFS FOR LARGE PRIMES, p G 7 For p s 13, the only case that arises is the Monster, M, in which the Sylow 13-normalizer is a maximal subgroup of the shape 131q 2 :Ž3 = 4S4 .. Thus NG Ž P .rP X ( 13 2 :Ž3 = 4S4 ., in which the action of 3 = 4S4 is given by the Žunique. embedding into GL2 Ž13. ( 3 = 2.Ž2 = L2 Ž13...2. A straightforward calculation Žusing 2 = 2 matrices over GF Ž13., if necessary. shows that 3 = 4S4 has just two orbits, of lengths 72 and 96, on the 168 nontrivial characters of PrPX ( 13 2 , and so the inertial groups of the

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TABLE I Results G M11 M12 J1 M22 J2

M23 HS

J3 M24 McL

He

Ru

Suz

O’N

Co 3

Co 2

Fi 22

HN

p 2 3 2 3 2 2 3 2 3 5 2 3 2 3 5 2 3 2 3 2 3 5 2 3 5 7 2 3 5 2 3 5 2 3 7 2 3 5 2 3 5 2 3 5 2 3 5

NG Ž P .rP 2

X

2 3 2 :SD 16 23 S3 = S3 2 3 :7:3 23 3 2 :Q8 2 = A4 3 2 :8 5 2 : D 12 23 3 2 :SD 16 23 2 = 3 2 :SD 16 5 2 :QD16 2 = A4 3 2 :8 24 32 : D8 23 Ž3 2 :4 = 3.⭈2 5 2 :3:8 24 32 : D8 5 2 :4 A 4 7 2 :Ž S3 = 3. 23 3 2 :SD 16 5 2 :Ž4 X 2. A4 = 2 2 Ž3 = 3 2 : D 8 .:2 5 2 :Ž4 = S3 . 23 3 4 :2 1q4 D 10 7 2 :Ž3 = D 8 . 24 S3 = 3 2 :SD 16 5 2 :24:2 25 S3 = 3 2 :SD 16 5 2 :4S4 24 S3 = 3 2 :2 5 2 :4S4 2 2 = A4 S3 = 3 2 :4 D 10 = 5:4

No. of characters 4 9 8 9 8 8 6 8 9 14 8 9 8 18 13 8 9 16 9 8 12 13 16 9 16 20 8 9 20 16 18 16 8 18 20 16 27 20 32 27 20 16 18 20 16 18 20

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TABLE IᎏContinued G Ly

Th

Fi 23

Co1

J4 X

Fi 24

B

M

p 2 3 5 2 3 5 7 2 3 5 2 3 5 7 2 3 11 2 3 5 7 2 3 5 7 2 3 5 7 11 13

NG Ž P .rP 3

X

2 S3 = 3 2 :SD 16 5:4 = 5:4 24 1 Ž S = S3 = S3 . 3 2 5 2 :4S4 7 2 :Ž3 = 2 S4 . 24 S3 = S3 = S3 5 2 :4S4 25 S3 = 3 2 :SD 16 5:4 = 5:4 7 2 :Ž3 = 2 A 4 . 25 Ž2 = 3 2 :8.:2 112 :Ž5 = 2 S4 . 25 S3 = S3 = 3 2 :2 Ž A 4 = 5 2 :4 A 4 .:2 7 2 :Ž6 = S3 . 26 S3 = S3 = S3 5:4 = 5:4 Ž2 2 = 7 2 :Ž3 = 2 A 4 ..⭈2 26 S3 = 3 2 :SD 16 = S3 5:4 = 5 2 :Ž4 = S3 . 7:6 = 7:6 112 :Ž5 = 2 A 5 . 13 2 :Ž3 = 4S4 .

No. of characters 8 27 25 16 15 20 27 16 27 20 32 27 25 27 32 18 42 32 54 56 25 64 27 25 81 64 81 80 49 50 55

nontrivial characters have orders 4 and 3, respectively. Moreover, the group 4S4 is well understood, and has 16 characters. Thus NG Ž P .rP X has exactly 3 = 16 q 4 q 3 s 55 characters. For p s 11, two cases arise: the Monster, in which P X s 1 and NG Ž P . ( 112 :Ž5 = 2 A 5 ., and J4 , in which NG Ž P . ( 111q 2 :Ž5 = 2 S4 ., so NG Ž P .rP X ( 112 :Ž5 = 2 S4 .. In both cases, the calculations are very similar to the above. In M, the group 5 = 2 A 5 has just 5 = 9 s 45 characters, and acts transitively on the 120 nontrivial characters of P. The inertial subgroup therefore has order 5, and the total number of characters of NG Ž P . is 45 q 5 s 50.

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Similarly, in J4 , the group 5 = 2 S4 acts transitively on the 120 nontrivial characters of PrPX ; the inertial subgroup therefore has order 2, and the total number of characters of NG Ž P .rP X is 5 = 8 q 2 s 42. In the case p s 7, again it turns out that in every case PrPX ( 7 2 , and similar calculations produce the results without too much difficulty. The cases are listed in Table II, together with a subgroup H, in which the structure of NG Ž P . can easily be seen. In each case there is, up to conjugacy, only one subgroup of GL2 Ž7. which acts as NG Ž P .rP does on PrPX , and therefore it is again a simple calculation to obtain the results given in Table I. For G ( He, there are five orbits of NG Ž P .rP on nontrivial characters of PrPX , with lengths 6, 6, 9, 9, 18, and inertial groups of orders 3, 3, 2, 2, 1. Since S3 = 3 itself has 3 = 3 s 9 characters, we obtain a total of 9 q 3 q 3 q 2 q 2 q 1 s 20 characters of NG Ž P .rP X . For G ( O’N, NG Ž P .rP X ( 3 = D 8 has 3 = 5 s 15 characters, and the nontrivial orbit lengths are 12, 12, 24, corresponding to inertial groups of orders 2, 2, 1. Thus NG Ž P .rP X has 15 q 2 q 2 q 1 s 20 characters. For G ( Th, NG Ž P .rP X ( 3 = 2 S4 has 3 = 8 s 24 characters, and is transitive on the nontrivial characters of PrPX , with inertial group of order 3. Thus NG Ž P .rP X has 24 q 3 s 27 characters. For G ( Co1 , NG Ž P .rP X ( 3 = 2 A 4 has 3 = 7 s 21 characters, and has two orbits of size 24 on the nontrivial characters of PrPX , each with inertial group of order 3. Thus NG Ž P .rP X has 21 q 3 q 3 s 27 characters. For G ( FiX24 , we calculate similarly that there are 18 q 3 q 2 q 2 s 25 characters of NG Ž P .rP X , and for G ( M we have 7 = 7 s 49 characters. Finally, we consider the case G ( B. It is easy to see that there are just two inertial groups, namely Ž2 3 = Ž3 = 2 A 4 ...2 and 2 2 = 3. The latter group clearly has 12 characters. To see how many characters the former group has, we need to do a little more work. First note that the normal subgroup of order 4 has two conjugacy classes fixed by the outer automor-

TABLE II Sylow 7-Normalizers X

G

P

H

NG Ž P .rP

He O’N Th Co1 X Fi 24 B M

7 1q2 7 1q2 72 72 7 1q2 72 1q4 7 :7

7 2 :SL 2 Ž7. L3 Ž7.:2 ᎏ ᎏ He:2 Ž2 2 = F4 Ž2..:2 7 1q4 :Ž3 = 2 S7 .

7 2 :Ž S3 = 3. 7 2 :Ž3 = D 8 . 7 2 :Ž3 = 2 S4 . 7 2 :Ž3 = 2 A 4 . 7 2 :Ž S3 = 6. Ž2 2 = 7 2 :Ž3 = 2 A 4 ..⭈2 7:6 = 7:6

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phism, and one pair of classes swapped by this automorphism. Similarly, the group 3 = 2 A 4 has a total of 21 conjugacy classes, of which nine are fixed, and six pairs swapped, by the automorphism. Thus the number of conjugacy classes in the inner half of the inertial group is 2 = 9 q 2 = 6 q 1 = 9 q 2 = 1 = 6 s 51. The number of classes in the outer half of 2 2 :2 ( D 8 is 2, while the number in the outer half of Ž3 = 2 A 4 ..2 ( 3 = 2 S4 is 3 = 3 s 9, giving a total count of 2 = 9 s 18 outer classes, and a total number of characters 51 q 18 q 12 s 81. 3. THE CASE p s 5 In the cases G ( J 2 , He, Suz, Fi 22 , Fi 23 , and FiX24 , the Sylow 5-subgroup is an elementary abelian group 5 2 , and the structure of its normalizer is clear. In the cases G ( HS, McL, Ru, Co 3 , Co 2 , and Th, the Sylow group is an extraspecial group 51q 2 , so again PrPX ( 5 2 , and the calculations we are required to do are very similar. For G ( J 2 , we have D 12 ( 2 = S3 acting, with four orbits of size 6, giving 6 q 2 q 2 q 2 q 2 s 14 characters. For G ( HS, we have QD16 s ² a, b < a8 s b 2 s 1, a b s a5 : acting, with nontrivial orbit sizes 8 and 16, and with 10 conjugacy classes of elements. Thus there are 10 q 2 q 1 s 13 characters of 5 2 :QD16 in total. For G ( McL, we have 3:8 s ² a, b < a3 s b 8 s 1, a b s ay1 : acting, with a single orbit of size 24. The group 3:8 itself has eight linear characters and four of degree 2, giving a total of 12 q 1 s 13 characters of 5 2 :3:8. For G ( He, we have 4 A 4 acting, with a single orbit of size 24. The inertial groups are therefore 4 A 4 and 2, giving a total of 14 q 2 s 16 characters. A similar argument works for the cases G ( Co 2 , Fi 22 , Th, and Fi 23 , where NG Ž P .rP X ( 5 2 :4S4 , and shows that there are 16 q 4 s 20 characters. For G ( Ru, we have 4 X 2 acting, with nontrivial orbit sizes 8 and 16, giving a total of 14 q 4 q 2 s 20 characters. The case G ( Suz is discussed together with the Monster below. The case G ( Co 3 has a transitive group 24:2 ( ² x, y < x 24 s y 2 s 1, y x s x 5 : acting, with an inertial group of order 2. Thus there are 18 q 2 s 20 characters in total. The groups Co 2 , Fi 22 , Th, and Fi 23 all have a group 5 2 :4S4 for NG Ž P .rP X . There is just one proper inertial subgroup, of order 4, and we have already seen that 4S4 has 16 conjugacy classes, so the total number of characters of NG Ž P .rP X is 4 q 16 s 20. Finally, for G ( FiX24 , the inertial groups are Ž A 4 = 4 A 4 .:2 and S4 = 2. The latter has 10 characters, while the former can be shown to have 46, giving a total of 56. To see this, note that A 4 has two classes fixed by the

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outer automorphism, and one pair of swapped classes. Similarly, 4.A 4 has six fixed classes, and four pairs of swapped classes. Thus the number of inner classes is 2 = 6 q 1 = 6 q 2 = 4 q 2 = 1 = 4 s 34. Furthermore, S4 has two outer classes, while 4S4 has six outer classes, giving 2 = 6 s 12 outer classes in all. The remaining cases Žwhere the Sylow group has order bigger than 5 3 . are listed in Table III, along with a maximal subgroup H in which the structure of the Sylow 5-normalizer can be easily elucidated. In the case of Co1 , the shape of NG Ž P .rP X is obvious. The three cases G ( HN, Ly, and B all have isomorphic Sylow 5-subgroups, and we see, for example, from w11x that PrPX ( 5 2 , from which the shape of NG Ž P .rP X follows immediately. ŽSpecifically, the group 4S6 has two orbits, of lengths 36 and 120, on the 156 one-dimensional subspaces of 51q 4r5, so a simple counting argument shows that an element of order 5 in 4S6 fixes just one of these one-dimensional subspaces. Similarly, the 5-element fixes just one hyperplane, namely, the orthogonal complement of the fixed 1-space.. A similar calculation in 51q 6 :4. J 2 .2 is carried out in w13x, with the same result. It is now a triviality to calculate the number of characters of NG Ž P .rP X in all cases except the Monster, where we see that 4 = S3 has orbits 1 q 12 q 12 on the characters of 5 2 , so that 5 2 :Ž4 = S3 . has 4 = 3 q 2 q 2 s 16 characters. ŽThis calculation also occurs for the Suzuki group.. Therefore 5:4 = 5 2 :Ž4 = S3 . has 5 = 16 s 80 characters. 4. THE CASE p s 3 In many ways this is the hardest case, because the structure of NG Ž P .rP X is sometimes quite hard to determine precisely. As in the previous cases, the key is to find a suitable subgroup H in which the structure of the Sylow p-normalizer can be seen. The smaller cases, up to and including M24 , are all quite straightforward, as are the cases He and Ru. We treat the other 15 cases individually.

TABLE III The Larger Cases for p s 5 X

G

H

NG Ž P .

NG Ž P .rP

HN Ly Co1 B M

51q4 .2 1q4 .5.4 51q 4 .4S6 1q2 5 .GL2 Ž5. 51q4 .2 1q4 . A 5 .4 51q6 .4 J 2 .2

51q4 :Ž2 = 5:4. 51q4 :Ž4 = 5:4. 51q2 :Ž4 = 5:4. 51q4 :Ž4 = 5:4. 1q6 Ž 5 : 4 = 5 2 :Ž4 = S3 ..

5:2 = 5:4 5:4 = 5:4 5:4 = 5:4 5:4 = 5:4 5:4 = 5 2 :Ž4 = S3 .

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Taking the McLaughlin group first, we see the Sylow 3-normalizer as a subgroup of shape 31q 4 :Ž4.S3 . inside the maximal subgroup 31q 4 :2.S5 , from which we deduce that NG Ž P .rP ( Q8 and NG Ž P .rP X ( Ž3 2 :4 = 3..2. This enables us to calculate the orbits of Q8 on the characters of the 3 3 as 1 q 2 q 8 q 8 q 8, with inertial groups Q8 , 4, 1, 1, 1, respectively. Next we look at the Suzuki group, and use the two subgroups 3 5 : M11 and 3 2q 4 :2.Ž A 4 = 2 2 ..2 to show that NG Ž P .rP ( SD 16 , and PrPX ( 3 3 , on which the action is given by NG Ž P .rP X ( Ž3 = 3 2 :Q8 ..2. The inertial groups are now SD 16 , Q8 , and three copies of the trivial group. The case Co 3 is very similar, as the Sylow 3-normalizer is contained in a group 3 5 :Ž2 = M11 ., and we obtain NG Ž P .rP X ( S3 = 3 2 :SD 16 . Now the second factor has inertial groups SD 16 and 2, and so has 7 q 2 s 9 characters, making 3 = 9 s 27 for the whole group. In Conway’s second group, the Sylow normalizer is contained in the subgroup 3 4 :Ž2 = A 6 .2 2 . inside U4 Ž3.. D 8 , so again we obtain NG Ž P .rP X ( S3 = 3 2 :SD 16 . The argument for the Lyons group is also very similar, since it also contains a group of the shape 3 5 :Ž2 = M11 ., and we obtain NG Ž P .rP X ( S3 = 3 2 :SD 16 again. In the big Conway group Co1 we use the subgroup 3 6 :2.M12 to come to the same conclusion again in this group. The Sylow 3-normalizer in the O’Nan group is the maximal subgroup of shape 3 4 :2 1q 4 : D 10 , which is transitive on the 80 nontrivial characters of the 3 4 , with inertial group therefore of order 4. Now the group 2 1q 4 D 10 itself has 14 irreducible characters, made up of the four characters of D 10 , six more characters of 2 4 : D 10 , and four faithful characters. Thus the total number of characters of the Sylow 3-normalizer is 4 q 14 s 18. For the smallest Fischer group Fi 22 , we use the subgroup 31q 6 .2 3q 4 .3 2 .2, and perform explicit calculation with the generators given in w12x. This shows that the 3 2 fixes a unique hyperplane in its action on the 3 6 , and therefore NG Ž P .rP X ( S3 = 3 2 :2. The second factor has just six characters, so the whole group has 18. For the group Fi 23 , we can use the analogous subgroup 31q 8.2 1q 6 .31q 2 . 2 S4 , or alternatively, note that the Sylow 3-normalizer actually lies in the 1q8 .Ž Ž . part of this group that is in Oq :2 A 4 = A 4 = A 4 ..2.S3 . 8 3 :S3 , namely, 3 Since the top S3 permutes the three copies of A 4 , and since the action of 2.Ž A 4 = A 4 = A 4 . is given by the tensor product action SL 2 Ž3. m SL 2 Ž3. m SL 2 Ž3., we pick up a factor of 3 toward PrPX from each ‘‘layer,’’ and therefore NG Ž P .rP X ( S3 = S3 = S3 . In the Harada᎐Norton group we use 3 4 :2Ž A 4 = A 4 ..4, in which the action is given by some version of the orthogonal group O4qŽ 3.. In particular, the action of 2Ž A 4 = A 4 . is given by the natural action of SL 2 Ž3. m SL 2 Ž3.. This means that, as usual, there is a unique fixed hyper-

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plane in the action of the Sylow subgroup on the 3 4 , whence NG Ž P .rP X ( S3 = 3 2 :4, so the total number of characters is 3 = Ž4 q 2. s 18. The case of J4 is actually not hard, but is worth giving in some detail, as the isomorphism type of the Sylow 3-normalizer has been wrongly stated in otherwise authoritative papers. We have NG Ž P . ( Ž2 = 31q 2 :8.:2, and NG Ž P . rP ( Ž 2 = 8 . :2 ( ² x, y, z < x 2 s y 8 s z 2 s w x, y x s w x, z x s 1, y z s xy 5 : . The latter group has just two orbits, of lengths 1 and 8, on the 9 characters of PrPX ( 3 2 , and, moreover, has 14 conjugacy classes, so the total number of characters of NG Ž P .rP X is 14 q 4 s 18. In the largest Fischer group FiX24 we use the subgroup 3 2q4q8 Ž A 5 = 2.A 4 ..2. Now it is clear that we can factor out the normal 3 6 , and we are left with the group A 5 = 2.A 4 acting on 3 8 as the tensor product of the deleted permutation representation of A 5 with the natural representation of SL 2 Ž3.. Thus we pick up a contribution of 3 2 to PrPX from the 3 8 , and calculate NG Ž P .rP X ( S3 = S3 = 3 2 :2. In the Baby Monster, a similar calculation in the subgroup 3 2q 3q6 Ž S4 = . 2 S4 ., in which the action on the 3 6 is given by O 3 Ž3. m GL2 Ž3., shows that NG Ž P .rP X ( S3 = S3 = S3 . Similarly, in the Monster we use 3 2q 5q10 Ž M11 = 2 S4 ., in which the action of M11 = 2 S4 on 310 is the tensor product of a 5-dimensional representation of M11 and the natural representation of GL2 Ž3.. Thus there is a unique fixed hyperplane, and the usual argument shows that NG Ž P .rP X ( S3 = S3 = 3 2 :SD 16 , and there are 3 = 3 = 9 s 81 characters in total. Finally, we are left with the Thompson group. Here we use Aschbacher’s description of the structure of the 3-local subgroups, in Chapter 14 of w3x. He states explicitly that PrPX ( 3 3, and we deduce from his other results that the action of NG Ž P .rP ( 2 2 on it is the sum of the three nontrivial irreducibles. This implies that the orbits of 2 2 on the 27 characters of 3 3 are one of size 1 Žthe trivial character ., three of size 2 Žthe fixed points of the three involutions., and five of size 4 Žthe rest.. Thus we obtain a total of 4 q 3 = 2 q 5 = 1 s 15 characters. 5. THE CASE p s 2 In this case, the groups NG Ž P .rP X all have very straightforward structure, so the number of characters can very easily be determined. Moreover, it is easy to see that in all but five cases, P is self-normalizing. The

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exceptions are J1 , where NG Ž P .rP ( 7:3, and J 2 , J 3 , Suz, and HN, where NG Ž P .rP ( 3. The only real problem, therefore, is, in certain cases, to work out the order of PrPX . In M11 , the Sylow 2-subgroup is SD 16 s ² a, b < a8 s b 2 s 1, a b s a3 : and PrPX ( 2 2 . In J1 , the Sylow 2-subgroup is elementary abelian, and NG Ž P .rP X ( 2 7:7:3. In M22 , M23 , and McL, the Sylow 2-normalizer is a subgroup 2 4 : D 8 of 2 4 : A 6 , so we have NG Ž P .rP X ( 2 3. Now using 2 10 : M22 in Fi 22 , or its double cover in Fi 23 , we see that the Sylow 2-subgroup fixes a unique hyperplane in the 2 10 , and therefore PrPX has order 2 4 . In M24 and the Held group, NG Ž P . ( 2 6 :Ž2 = D 8 . - 2 6 :3S6 , in which it is easily seen using the hexacode that there is a unique hyperplane in the 2 6 fixed by the Sylow 2-subgroup, so that NG Ž P .rP X ( 2 4 . Now using the subgroups of shape 2 11 M24 in Co1 , J4 , and FiX24 , we see that the Sylow 2-subgroup of M24 fixes a unique hyperplane in the 2 11 , so that NG Ž P .rP X has order 2 5. Similarly, using 2 1q24 Co1 in the Monster, we get NG Ž P .rP X of order 2 6 . In J 2 and J 3 , the subgroup 2 1q 4 : A 5 shows us that NG Ž P .rP X ( 2 = A 4 . In M12 , the Sylow normalizer 4 2 :2 2 is seen inside 4 2 : D 12 , from which it is clear that NG Ž P .rP X ( 2 3. In the Higman᎐Sims group HS, the Sylow normalizer 4 3 : D 8 is seen inside 4 3 : L3 Ž2., which shows that NG Ž P .rP X is isomorphic to that in the affine group AGL3 Ž2. ( 2 3 : L3 Ž2.. This is well known to be 2 3. A similar argument produces the same answer in the O’Nan group, using the subgroup 4 3.L3 Ž2.. In the Rudvalis group Ru we use the subgroup 2 3q 8 L3 Ž2., and the normal 2 3 is obviously contained in P X , so we can quotient it out. Thus we look at 2 8 : D 8 inside 2 8 : L3 Ž2., where the action of L3 Ž2. on 2 8 is the adjoint representation Ži.e., the Steinberg module.. This is well known to restrict to the regular representation of D 8 , and so NG Ž P .rP X ( 2 3. In the Suzuki group, we use 2 4q 6 :3.A 6 , or rather its quotient group 2 6 :3.A 6 . The 2 6 is really a three-dimensional GF Ž4.-module for 3.A 6 , in which the Sylow 2-normalizer 3 = D 8 fixes a unique hyperplane, so NG Ž P .rP X ( A 4 = 2 2 . In Co 3 we can use the involution centralizer 2.S6 Ž2.. In the quotient S6 Ž2., the Sylow 2-normalizer is contained in 2 5 :S6 , the action of S6 on 2 5 being that on the space of even subsets of six letters. Direct calculation shows that NG Ž P .rP X ( 2 4 . In the Lyons group, the Sylow 2-normalizer is in 2.A11 , and so in 2.S8 . Now in S8 , we have Sylow 2-normalizer 2 4 : D 8 - S4 X 2, and by calculation, NG Ž P .rP X ( 2 3. In Co 2 we use the subgroup 2 10 : M22 :2, and first determine the image of PrPX in M22 :2, using the subgroup 2 4 :S6 ( 2 4 :S4Ž2.. Here we have the Sylow 2-subgroup 2 = D 8 in S4Ž2., giving a contribution of 2 3 to PrPX , and

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we pick up another factor of 2 from the normal subgroup 2 4 . Similarly, in 2 10 : M22 :2 the Sylow 2-subgroup fixes a unique hyperplane in the 2 10 , so we pick up another factor of 2, giving PrPX ( 2 5 in Co 2 . Similarly for the Baby Monster, we can use the subgroup 2 2q 10q20 :Ž M22 :2 = S3 ., and as usual work just in the quotient group 2 20 :Ž M22 :2 = S3 .. Again we have a contribution of 2 5 to PrPX from the quotient M22 :2 = S3 , and the representation on 2 20 is the tensor product of a 10-dimensional representation of M22 :2 and the two-dimensional representation of S3 . Thus there is again a unique fixed hyperplane for the Sylow 2-subgroup, and so in B we have PrPX of order 2 6 . Next we consider the Thompson group. Here the argument is a little trickier, as both the useful 2-local subgroups 2 4 .L5 Ž2. and 2 1q 8.A 9 are nonsplit extensions. First we use the involution centralizer, and work first in the quotient A 9 . Here the Sylow 2-normalizer is isomorphic to that in A 8 ( L4Ž2., so PrPX ( 2 3. Now in the action on 2 8 there is a unique fixed hyperplane, so in Th we have PrPX of order at most 2 4 . On the other hand, in L5 Ž2. we see PrPX of order at least 2 4 . Finally, we consider the Harada᎐Norton group HN. Again we need to use two different 2-local subgroups to obtain the required information without going into too much detail in one particular group. We use the involution centralizer 2 1q 8 ⭈ Ž A 5 = A 5 .:2, in which the A 5 = A 5 acts on the 2 8 as O4q Ž4., i.e., as L2 Ž4. m L2 Ž4.. Hence P⬘ contains at least a 2 5 out of the 2 1q 8 , and moreover, we know that the normalizer of this 2 5 in HN has the shape 2 5 Ž2 6 :Ž L3 Ž2. = 3... In the latter group, the L3 Ž2. acts on 2 6 as two copies of the natural module, so we have NG Ž P .rP X ( 2 2 = A 4 .

ACKNOWLEDGMENTS I gratefully acknowledge the help of Klaus Lux, who introduced me to this problem many years ago, and translated it from character theory into group theory for me. I also acknowledge the stimulus provided by J. Alperin’s 10 lectures on representation theory, held in Birmingham under the auspices of the London Mathematical Society, 14᎐18 April 1997, which encouraged me to complete the three remaining cases and write up the results.

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