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The Quasisimple Case of the k(GV)-Conjecture
Journal of Algebra 235, 45᎐65 Ž2001. doi:10.1006rjabr.2000.8461, available online at http:rrwww.idealibrary.com on
The Quasisimple Case of the k Ž GV . -Conjecture Udo Riese1 Uni¨ ersitat Mathematisches Institut, Auf der Morgenstelle 10, D-72076 ¨ Tubingen, ¨ Tubingen, Germany ¨ E-mail: [email protected] Communicated by Walter Feit Received May 5, 1999
1. INTRODUCTION Let G be a finite group of order < G <. The k Ž GV .-conjecture states that, for a given prime p not dividing < G < and every elementary abelian p-group V on which G acts faithfully, the number of conjugacy classes of the semidirect product GV is bounded from above by the order of V, that is, k Ž GV . F < V < . For p-solvable groups, this assertion is equivalent to R. Brauer’s celebrated k Ž B .-conjecture, claiming in general that the number of irreducible characters in a p-block B is bounded by the order of a defect group of B. In view of the Clifford reduction in wRTx and wR2x, it will follow from our main theorem, the result for p s 2 by Gluck wGx, and from recent work of Gluck and Magaard wGMx that the k Ž GV .-conjecture holds for all primes p different from 3, 5, 7, 11, 13, 19, 31. In attacking the k Ž GV .-problem, using a result of Gallagher wGax, one can assume irreducibility of V as an ⺖p G-module. By the combined work of Knorr ¨ wKx, Gow wGox, Robinson wR1, R2x, and Robinson and Thompson wRTx, the k Ž GV .-conjecture holds whenever, for some ¨ g V, the restriction of V to CG Ž ¨ . contains a self-dual faithful submodule. Such a vector will be called weakly real. In case VC G Ž ¨ . itself is a self-dual module, the vector is called real. The notion is motivated from the fact that in this semisimple Žco-prime. situation, a module is self-dual precisely when its 1 The author was supported by the German academic exchange service DAAD, Grant 332400510.
45 0021-8693r01 $35.00 Copyright 䊚 2001 by Academic Press All rights of reproduction in any form reserved.
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Brauer character is real valued. More generally, if G acts F-linear for some extension field F of the prime field, a vector ¨ g V is called F-real if VC G Ž ¨ . is self-dual with respect to F. The search for Žweakly. real vectors allows an inductive approach to the k Ž GV .-conjecture via Clifford theory. Suppose V is a faithful FG-module for some finite extension field F of ⺖p . Carrying out the Clifford reduction as in wRTx or wR2x, one can assume a situation where there is a minimal nonabelian normal subgroup E of G which acts irreducibly on V such that End F G Ž V . s End F E Ž V .. Furthermore, we may assume the module V is tensor indecomposable. This reduces the problem either to the quasisimple case, that is, the generalized Fitting subgroup F*Ž G . is the central product of the center Z s ZŽ G . and a quasisimple nonabelian normal subgroup E, or to the extraspecial case, where F*Ž G . s F Ž G . is the central product of the center Z and E, where E is an extraspecial q-group for some prime q / p. A real vector for G is real for all subgroups of G. We embed E into GLŽ V . through the faithful representation. If the normalizer NGL ŽV .Ž E . is a p⬘-group Žwhich is always true in our cases; see Section 3., we may assume that G s NGL ŽV .Ž E ., so G is the largest possible group. In this paper we deal exclusively with the quasisimple case. The first definitive result was obtained by Liebeck wLx. The starting point of our investigation is the main result of D. Goodwin’s Ph.D. thesis wGdx Žsee also wGd1x and wGd2x.. In his work ‘‘Regular Orbits of Linear Groups,’’ he gives a list of triples Ž E, n, < F <., where E is a nonabelian quasisimple group, n is the dimension of an irreducible E-module V over the field F, where char Ž F . s p is a prime not dividing < G <, where G is isomorphic to a subgroup of NGL ŽV .Ž E . containing E. This list contains all possibilities of such triples Ž E, n, < F <. where G s NGL ŽV .Ž E . has no regular orbit in its permutation action on V. We call these triples Ž E, n, < F <. Goodwin triples. We remark that, up to a few exceptions, a Goodwin triple Ž E, n, < F <. does determine the isomorphism type of the module V. We will restate Goodwin’s result in Section 2. Note that if ¨ G is a regular G-orbit in V, then ¨ is a real vector for the group G. We prove: THEOREM. Let G be a finite group. Suppose V is a faithful FG-module of F-dimension n, where F is a finite field of characteristic co-prime to < G <. Assume there is a quasisimple normal subgroup E of G which is irreducible on V. Then there exists an F-real ¨ ector in V for G except when F is the prime field and one of the following holds: ŽA.
E ( 2. A 5 , n s 2, < F < g 11, 19, 314 ,
ŽB.
E ( 2. A 6 , n s 4, < F < s 7,
ŽC.
E ( 2.U4 Ž2., n s 4, < F < g 7, 13, 19, 314 .
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In ŽC. for < F < s 31, there is a weakly real ¨ ector in V for G. In all other exceptional cases, the group NGL ŽV .Ž E . is a p⬘-group and has no weakly real ¨ ector in V. We mention that the assumptions in the theorem imply F*Ž G . s E) ZŽ G . and, of course, the irreducibility of V as a G-module. Combined with Gluck’s and Magaard’s work on the extraspecial case wGMx and wRSx for p s 2, this proves: COROLLARY 1. Let G be a finite group. Suppose V is a faithful irreducible FG-module where char Ž F . s p does not di¨ ide < G <. Assume there is no real ¨ ector in V for G. Then F s ⺖p , G is absolutely irreducible on V, and p is one of the primes 3, 5, 7, 11, 13, 19, or 31. COROLLARY 2. The k Ž GV .-conjecture holds true for all primes p / 3, 5, 7, 11, 13, 19, 31. So in order to solve the k Ž GV .-problem, only these special situations have to be dealt with. If no vector is weakly real, we can usually find ¨ g V such that CG Ž ¨ . is abelian. This leads to Žsee Section 6. COROLLARY 3. The k Ž GV .-conjecture holds true whene¨ er G has an irreducible normal quasisimple subgroup. Excluding the cases where E is an alternating group Žof degree less than char Ž F .. and V is its ‘‘deleted’’ permutation module over F Žsee Section 2., where always real vectors exist, the remaining triples in Goodwin’s list are finite in number. The occurrence of all the larger groups in this finite set is related to the action of these groups as automorphisms of well-known lattices which give rise to extraordinarily small character degrees. We show by a case-by-case analysis that a real vector exists in all but the three exceptional cases ŽA., ŽB., and ŽC. in the theorem. Furthermore, all these groups are fortunately Atlas-groups wAx, and our work is completely based on the character tables given in the Atlas, except for the group 2.U4Ž2. and its four-dimensional modules, where we used the computer system GAP wSx. 2. GOODWIN’S RESULT In this section we give the exact result of Goodwin. Let c G 5 and let F be a finite extension of ⺖p . The F-vector space V s Ž ¨ 1 , . . . , ¨ c .: ¨ i g F, ݨ i s 04 is a module for the alternating group A c via the Žnatural . permutation action on the coordinates. We call this module the deleted permutation module over F for A c . This is a self-dual module, so every vector is real under the action of A c Ževen for Sc .. In his Ph.D. thesis, Goodwin always assumes F s ⺖p is the prime field. We work over arbitrary finite fields F.
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The FG-module V viewed as an ⺖p G-module is the multiple of an irreducible ⺖p G-module, say V0 . Let F0 s ⺖p Ž V . : F be the field generated by the character to V Žvia embedding roots of unity into an algebraic closure of ⺖p .. Then w F : F0 x is the multiplicity of V0 in V wI, 9.18x, and we see that E is still irreducible on V0 . Assume there is a regular orbit on V0 . Then there is a regular orbit on V. Hence Goodwin’s result gives possible groups E. His proof for the size of the prime field for the remaining E’s relies on the fact that in case that no regular orbit exists, we have < V < F Ý1 / g g G < C¨ Ž g .<. Investigating this inequality, he gives indeed upper n bounds for '< V < s < F <. Now we restate Goodwin’s main result for arbitrary F, which can be deduced from the proof given in the prime field case in wGd1x and wGd2x. THEOREM Žsee wGdx.. Suppose V is a faithful FG-module of F-dimension n, where F is a finite field of char Ž F . s p not di¨ iding < G <. Assume there is a quasisimple normal subgroup E of G which is irreducible on V. Then G has a regular orbit on V or one of the following holds: ŽA. E is an alternating group A c , c - char Ž F ., and V is the deleted permutation module o¨ er F for E. ŽB. The triple Ž E, n, < F <. belongs to the following list: ŽI. Alternating groups: A 5 , n s 3, < F < s 11, A 5 , n s 4, < F < g 7, 11, 134 , Žc. 2. A 5 , n s 2, < F < F 61, Žd. 2. A 5 , n s 4, < F < g 7, 11, 134 , Ža. Žb.
Že. A 6 , n s 5, < F < g 7, 11, 13, 17, 194 , Žf. 2. A 6 , n s 4, < F < g 7, 11, 134 , Žg. 3. A 6 , n s 3, < F < F 49, Žh. 3. A 6 , n s 6, < F < g 7, 11, 134 , Ži. 2. A 7 , n s 4, < F < s 11. ŽII.
Linear groups: Ža. L2 Ž7., n s 3, < F < g 11, 23, 254 , Žb. L2 Ž7., n s 6, < F < s 5, Žc. 6. L3 Ž4., n s 6, < F < s 13.
ŽIII. Unitary groups: Ža. U3 Ž3., n s 6, < F < g 5, 114 , Žb. U3 Ž3., n s 7, < F < s 5,
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Žc. U4 Ž2., n s 5, < F < F 49, Žd. U4Ž2., n s 6, < F < F 43, Že. U4 Ž2., n s 10, < F < s 7, Žf. 2.U4Ž2., n s 4, < F < F 49, Žg. 6 1.U4Ž3., n s 6, < F < F 151, Žh. 6 1.U4Ž3., n s 12, < F < s 11, Ži. U5 Ž2., n s 10, < F < g 7, 134 , Žj. U5 Ž2., n s 11, < F < s 7. ŽIV. Other quasisimple groups: Ža. 2. J 2 , n s 6, < F < g 11, 194 , Žb. 6.Suz, n s 12, < F < s 19, Žc. S6 Ž2., n s 7, < F < F 79, Žd. 2.S6 Ž2., n s 8, < F < s 11, Že. 2.⍀q Ž . < < 8 2 .2, n s 8, F F 271. It is well known how to settle case ŽA. of Goodwin’s result Žsee Proposition 1 below.. Thus the main part of this work deals with the triples occurring in case ŽB..
3. THE GROUP G s NGL ŽV .Ž E . We describe the groups G s NGL ŽV .Ž E . for the various groups E occurring in Goodwin’s list. Let FG s End F G Ž V . and FE s End F E Ž V .. Then Z s ZŽ G . ( FGU . The Brauer character afforded by the irreducible FG-module V of dimension n is always denoted by . Let g Irr Ž E . be a constituent of the Žcomplex. character E . We can view as a Brauer character and get FE equals the field of character values F Ž . Žembedding the Žcomplex. < G <-th roots of unity into ⺖p. wI, 9.14x. We have n s Ž 1 . s < FE : F < Ž 1 . s < FE : F < ⭈ n 0 . If there are different characters in question, we use the numbering of the Atlas for . As E is irreducible on V, we see that the central product G 0 s E)CG Ž E . is normal in G and CG Ž E . ( FEU . The structure of G can now be determined as follows.
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With the help of the Atlas wAx, it is straightforward to check the following: Ži. FE s F and V is an absolutely irreducible FE-module except possibly in the cases IŽd. < F < s 7, 13; IŽh. < F < s 7, 11, 13; IIŽb. < F < s 5; and IIIŽh. < F < s 11. In these cases there exists a module V of dimension n over F such that < FE : F < s 2 Žhence Ž1. s n 0 s nr2.. We call these cases where FE / F the exceptional cases of induced type. Žii. FG s F, so Z s ZŽ G . ( F*. We always have F*Ž G . s E) Z and GrCG Ž E . is isomorphic to a subgroup of AutŽ E .. In all cases there is a largest group Žup to isoclinism. H s E.r ŽAtlas notation., with ZŽ H . s ZŽ E ., such that viewed as a complex character is extendible to H. Inspection of Goodwin’s list shows r s 1 or r s 2 in all cases except IIIŽh., where r s 2 ⭈ 2. If H Žin some isoclinic version. acts on V, then H F G and G ( H ) Z, or we are in an exceptional case of induced type where always FEU ( CG Ž E . and G 0 s E) FEU is a normal subgroup of index < r < / 1 in G and G is not a central product. If H g G, then it may occur that GrCG Ž E . ( HrZŽ H .. Clearly, in this case, < H : E < s 2 and GrE must be cyclic of order divisible by 4. We conclude that G is isomorphic to a subgroup of index 2 in the central ˆ s H )C2 < Z < , where a cyclic group of order s is denoted by C s . product G To see that G acts on V, one has to calculate the character values of for ˆ be a generator of C2 < Z < . Then for x g H : G, ˆ elements G _ G 0 . Let s g G we have xs g G and Ž xs . s Ž x . Ž s . for some g Irr Ž C2 < Z < .. Again the Atlas gives: Žiii. H F G except in the following cases: IŽh. p s 7, 13; IIŽb. p s 5; IIŽc. p s 13; IIIŽa. p s 5, 11; and IIIŽi. p s 7, 13, which we call the exceptional cases of invariant type. Now it is obvious that, in all cases, the group NGL ŽV .Ž E . is indeed a p⬘-group as E and FE are p⬘-groups, and < GrG 0 < F 2 Žexcept in case IIIŽh... We summarize the properties of G in the following way: COROLLARY. Let Ž E, n, < F <. be a Goodwin triple. Then G ( NGL ŽV .Ž E . is a p⬘-group. Except in case IIIŽh., the central product G 0 ( E) FE has index at most 2 in G, and one of the following holds: ŽI. The group G is nonexceptional; that is, G s H ) Z is a central product o¨ er ZŽ E . for some group H, uniquely determined up to isoclinism, with Z ( FE s F. ŽII. G is not a central product and either FE s F and G is exceptional of in¨ ariant type, or else FE / F and G is exceptional of induced type.
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How we work with the character tables in wAx to determine the various groups is explained in the following: EXAMPLE. Suppose we are in case IŽh. Ž3. A 6 , 6, < F <., < F < g 7, 11, 134 . If the restriction of to E ( 3. A 6 is absolutely irreducible, then s 16 or 16* and affords a primitive third root of unity on ZŽ E .. Hence < F < / 11 in this case. Furthermore, Goodwin’s proof shows that in this case for < F < s 13, a regular orbit exists. So < F < s 7. The automorphisms of A 6 leading to A 6 .2 1 and A 6 .2 2 do not centralize ZŽ E .. Therefore 16 and 16* are not invariant under these automorphisms, but both characters extend to H ( 3. A 6 .2 3 . involving 'y 2 on H _ E. As < F < s 7, we see that H is not a subgroup of GLŽ V . and G is of invariant type. Calculating the character values as indicated above shows that indeed < GrG 0 < s 2. Now we assume that E is not absolutely irreducible Žbut irreducible.. Then G is of induced type and E equals the sum of two irreducible characters belonging to the set 14 , 14* , 15 , 15*4 on which Out Ž A 6 . acts regularly. All four characters in question involve '5 and a primitive third root of unity . If < F < s 7 or 13, then '5 f F, but g F and E extends either to H1 ( 3. A 6 .2 1 or H2 ( 3. A 6 .2 2 Ždepending on E ., e.g., if E s 14 q 15* , then E extends to H2 by the action of Out Ž A 6 . on the four characters in question. If < F < s 11, then '5 g F and f F. Hence either H2 or H3 ( 3. A 6 .2 3 are subgroups of G Ždepending on E .. 4. PRELIMINARIES In this section we describe our method for the nonexceptional cases, where G is a central product, and for the exceptional ones. We write in the nonexceptional cases G ( H ) Z, and in this situation for given 0 / ¨ g V, let C s CG Ž ¨ ., CH s CH Ž ¨ ., and NH s NH Ž ¨ . s NH Ž² ¨ :.. Then C l Z s 1 and C is isomorphic to a subgroup of NH containing CH . Inspecting the character tables of E, we will often apply the following: LEMMA 1. Let ¨ g V and assume G ( H ) Z. Suppose that the restriction of to CH takes only real ¨ alues. Then each of the following conditions implies that ¨ is a real ¨ ector for G in V: Ža.
NH rCH is an elementary abelian 2-group and takes only real
¨ alues on NH ,
Žb.
for all x g NH _ CH , Ž x . s 0 holds or x is an in¨ olution,
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Žc. < Z : ZŽ H .< is relati¨ ely prime to < NH Ž CH . : CH <, Žd. NH s CH = ZŽ H .. Proof. We have either NG Ž ¨ . s NH ) Z or NG Ž ¨ . s NH = Z. To any x g NH there exists a unique z x g Z such that xz x g C. Conversely, every c g C can be written c s xz s zx with x g H and z g Z; then x g NH and z s z x . The assignment : x ¬ z x is a homomorphism of NH into Z with kernel CH . Taking character values given the claim in Ža. and Žb.. In Žc. and Žd. we have C s CH . Lemma 1 similar to wGM, Lemma 1.11x. Now we deal with case ŽA.. Our argument follows wRT, Corollary 6x. PROPOSITION 1.
In case ŽA. of Goodwin’s theorem, a real ¨ ector exists.
Proof. We have G ( Sc = Z for every c G 5 Žfor c s 6, inspect the character table in wAx.. Let H s Sc . Let V be the deleted permutation module over F and w F : ⺖p x s n. Then V s nV0 as an ⺖p S c-module wI, 9.18x, where V0 is the deleted permutation module over ⺖p . Write p s k Ž c y 1. q l. As p ) c, we have k / l, and for ¨ 0 s Ž k, . . . , k, l . g V0 : V, we have CH Ž ¨ 0 . s Scy1 , which is maximal and self-normalizing in H. Now apply Lemma 1. Now we briefly discuss our method in the exceptional cases. First assume we are in an exceptional case of induced type. Then the n-dimensional FG-module V corresponds to the complex character s q , where generates the group GalŽ FE rF . of order 2. We remark that the permutation representation of G 0 on the sets V and W are equal where V is the irreducible FG 0-module with character s q and W0 is the irreducible FE G 0-module with character . The following lemma enables us to extend our methods of the nonexceptional type ŽLemma 1. to the induced type, as in this case the central product G 0 s E) FEU is of index 2 in G. LEMMA 2. Suppose FE is not central in G Ž so that w FE : F x s 2.. Then there is an absolutely irreducible FE G 0-module W which is permutation isomorphic to V as a G 0-module. If there is a real ¨ ector for G 0 in W Ž self-duality with regard to FE ., there is a real ¨ ector in V for G. Proof. We may identify W s V as sets. Let w g W be the real vector for G 0 s ECG Ž E .. As Ž x . s 0 for all x g G _ G 0 and C G Ž ¨ . is real 0 valued, the claim follows as in Lemma 1. Placing this lemma in a more general context shows that after applying the Clifford reduction, one can assume that the restriction of to every normal subgroup is absolutely irreducible; see the proof of wRT, Theorem
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12x. But here we work with Goodwin’s result, so we do not assume absolute irreducibility of restricted to E; hence some groups of induced type occur. If the group G is exceptional of invariant type, we have H g G, E s G⬘, and G 0 s E) Z s F*Ž G . is of index 2 in G. As G 0 is a central product, we may apply Lemma 1 to G 0 and then the following lemma to find real vectors in V. LEMMA 3. Let G be a group of in¨ ariant type. Suppose ¨ is a real ¨ ector for G 0 . If CE Ž ¨ . s CG 0Ž ¨ ., then CG Ž ¨ . s CE Ž ¨ . and ¨ is a real ¨ ector for G in V. Proof. We must have CG Ž ¨ . s CE Ž ¨ ., because otherwise G would be isomorphic to the central product of ECG Ž ¨ . and Z, contradicting our assumption on G. We finally investigate the existence of regular orbits in terms of the size of the field of scalars. Our arguments are similar to those in wPPx. We call an F-subspace U of V a fixed space provided U s C V Ž X . for some subgroup X of G. In this case we call Xˆ s CG Ž U . s CG Ž C V Ž X . . the closure of X with regard to V, and X is a closed subgroup Žwith ˆ Observe that C V Ž X . s C V Ž Xˆ.. If Xˆ / 1, then respect to V . if X s X. ˆ Ž . C V X / V by faithfulness. A regular G-orbit on V corresponds to a vector of V which is not contained in C V Ž X . for any Žminimal. closed subgroup X / 1 of G. We may embed any group of p⬘-roots of unity into an algebraic closure ⺖p of ⺖p . This induces an embedding of complex character values of any p⬘-group into ⺖p. LEMMA 4. There is an integral polynomial f s f , G s Ý0 F i F n a i t i of degree n and highest coefficient a n s 1 with the following properties: It takes nonnegati¨ e ¨ alues at each prime power which is co-prime to < G < for which the ¨ alues of may be embedded into the field of corresponding order. If L is such a field, there is an LG-module W affording , and f Ž< L <. G 0 equals the number of ¨ ectors contained in regular G-orbits on W. In particular, f Ž< L <. s 0 if and only if there exists no regular G-orbit on W. Proof. Let f nŽ t . s t n y 1. Then f nŽ< F <. is the number of nontrivial vectors of V. Let L be as in the lemma. Then there is an LG-module W affording the Brauer character . For any subgroup X of G, dim F C V Ž X . s w X , 1 X x s dim L CW Ž X . . Let Xˆ s CG Ž C V Ž X ... Then X : Xˆ and C V Ž X . s C V Ž Xˆ .. It follows that CW Ž X . = CW Ž Xˆ .. But dim F C V Ž Xˆ . s dim L CW Ž Xˆ . by the above argu-
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ment Žreplacing X by Xˆ .. Consequently, CW Ž X . s CW Ž Xˆ . and Xˆ s CG Ž CW Ž X ... Hence the closed subgroups of G with regard to V and W are the same. These form a lattice with respect to intersection and the closure of the usual join. Similarly, the fixed subspaces of V, W form lattices with regard to intersection and taking the smallest fixed subspace containing the usual join. The assignments Xˆ ¬ C V Ž Xˆ . and Xˆ ¬ CW Ž Xˆ . are lattice antiisomorphisms from the closed subgroups of G onto the fixed subspaces of V and W, respectively, having pairwise the same dimensions. Thus we have a lattice isomorphism between the fixed subspaces of V and W preserving dimensions. We count the number of nonzero vectors which belong to some Žproper. maximal fixed subspace in V, resp. W. This leads to an integral polynomial of degree at most n y 1 which only depends on . The polynomial f of the lemma is just the difference of f n and this polynomial. Thus f is as asserted. It takes nonnegative values on < F < and < L <. There is a regular G-orbit on W if and only if there is a one-dimensional subspace of W which is not contained in any maximal fixed subspace of W. This is equivalent to saying that f Ž< L <. ) 0. 5. PROOF OF THE THEOREM First we introduce some more notation which will be used in the proofs. Elements of a given group are always denoted by the number of their conjugacy class in the character table in wAx; e.g., X s ²3 A: denotes a cyclic group of order 3 generated by an element belonging to class 3 A, a Frobenius group of order 21 containing elements in class 3 A, and 7A is denoted by Y s ²3 A, 7A: and similar notations are used. Furthermore, regŽ X . is always the regular character of the group X, and X - Y means that X is a proper subgroup of Y. If isoclinic groups exist, we always refer to the group whose character table is given in the Atlas as Gq and to the other one by Gy Žin our cases there are always at most two isoclinic groups.. If our arguments apply for both groups, we also write just G for both groups. We recall that in the nonexceptional cases, we can write G ( H ) Z, where H s E.2 or H s E. Furthermore, n s Ž1. s Ž1. s n 0 and FG s FE s F. Now we deal with the various Goodwin triples Ž E, n, < F <.. 5.1. Alternating Groups Ža. (A 5 , 3, 11). Here G ( A 5 = Z is of nonexceptional type, Ž1. s 3, and H ( A 5 . Let X s ²3 A: F H. We calculate X s regŽ X ., and if X - Y - H, then Y is isomorphic either to S3 or to A 4 . The restriction of
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to an A 4 subgroup of H is irreducible and we have w Y , 1x s 0 if Y is an S3 subgroup of H. Hence there is ¨ g V such that CH s CH Ž ¨ . s X. Since NH Ž CH . ( S3 and the restriction of to the latter group is real, Lemma 1 shows that ¨ is real. Žb. (A 5 , 4,
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Že. (A 6 , 5,
< F < g 7, 134 , s 8 , and H s Hy, or s 9 and H s Hq. < F < s 11, s 8 , and H s Hq, or s 9 and H s Hy.
We deal first with the case < F < s 7, s 9 , and H s Hq. The case 8 , Hy will follow by interchanging 3 A and 3 B, 6 A and 6 B. Let H0 F H such that H0 ( 2. A 6 . We calculate the orbits of G on nonzero vectors. In H0 only elements in class 3 A centralize any nonzero vector. Let X s ²3 A:. Then w X , 1x s 2. As < H0 : NH 0Ž X .< s 20, < NH Ž X . : X < s 12, and 7 2 y 1 s 48, we see that for 0 / u g U s C V Ž X ., 0 u H 0 l U s 12. We conclude that there are four orbits of length 240 and two regular orbits of length 720 under the action of H0 . The 12 vectors of u H 0 l U of each orbit are distributed on the eight one-dimensional subspaces of U as Ž6, 6, 0, 0, 0, 0, 0, 0., resp. three times Ž0, 0, 2, 2, 2, 2, 2, 2.. This can be seen from the action of a Sylow 3-subgroup of H0 and the fact that H0 contains a central element of order 2. We call these orbits 6 and 2i for i g 1, 2, 34 in the obvious way. Let G 0 s H0 ) Z. Let u g 6 . Then < CG 0Ž u.< s 3 < X < s 9. The orbits 2i are fused under the action of Z, so for u g 2i we have < CG Ž u.< s 3. So we get orbits of length 240, 720, 720, 720 under the action of G 0 . We remark that < G : G 0 < s 2. An element in class 2 B centralizes a two-dimensional subspace of V, and any nonzero vector ¨ which is centralized by some element x g G _ G 0 must be centralized by an element in class 2 B. This shows that two orbits of length 720 must fuse in G, so the orbit lengths are 240, 720, and 1440. By looking at the centralizers of order 3, 6, and 18, it is straightforward to see that no vector is real. For < F < s 13, we can deduce from the case < F < s 7 that there is a regular orbit in V under a subgroup X ( 2.S6 )C3 of G. Hence there is ¨ g V such that < CG Ž ¨ .< F 2. If < F < s 11, we conclude that there is an orbit ¨ H of length 720 under the action of H and CG Ž ¨ . s CH Ž ¨ . s ²2 B : by Lemma 1Žc.. Žg. (3.A 6 , 3,
QUASISIMPLE CASE OF THE
k Ž GV .-CONJECTURE
57
NH Ž X . is a dihedral group of order 8. Applying Lemma 1 gives the existence of a real vector. Žh. (3.A 6 , 6,
58
UDO RIESE
A 4-subgroup of H is irreducible. Furthermore, if X ( S3 : H, then w X , 1x s 0 and w ² 7 A: , 1x s 0. This shows that there is ¨ g V such that CH Ž ¨ . is cyclic of order 3. As NH Ž CH Ž ¨ .. s S3 and is real on the latter group, we apply Lemma 1. Žb. (L 2(7), 6, 5). Suppose first that E is absolutely irreducible. Then Ž1. s Ž1. and the complex character can be extended to H ( L2 Ž7..2, affording '2 . As < F < s 5, we see that G is of invariant type. Let H0 ( L2 Ž7. F G and X ( S4 : H0 . We calculate w X , 1x s 1 and X is maximal in H0 . We apply Lemma 1 and Lemma 3 to get a real ¨ g V. Now suppose G is of induced type. Then H ( L2 Ž7..2 and the restriction of to H0 s E ( L2 Ž7. equals 2 q 3 . The assertion follows from the results in IIŽa. and Lemma 2 applied to G 0 s E) FEU . Žc. (6.L 3(4), 6, 13). The complex character of degree 6 of H0 s E ( 6. L3 Ž4. is extendible to H ( 6. L3 Ž4..2 1 , affording '2 , resp. 'y 2 Ždepending on the isoclinism type of H .. As < F < s 13, the group G is of invariant type, and both isoclinic groups Hq and Hy give rise to the same group G. If we view H0 as a maximal subgroup of X s 6 1.U4Ž3..2 2 , this representation is just the restriction of the six-dimensional representation of X. It follows that C0 s CH 0Ž ¨ . s 2 4 : A 5 for some ¨ g V and NH 0Ž ¨ . s ZŽ H0 . = C0 Žsee also IIIŽg... Now apply Lemma 1 and Lemma 3 to obtain a real ¨ g V. 5.3. Unitary Groups Ža. (U3(3), 6,
QUASISIMPLE CASE OF THE
k Ž GV .-CONJECTURE
59
Žii. Here G ( U3 Ž3. = Z is also nonexceptional, but s 4 or s 5 . The restriction of to a maximal subgroup isomorphic to L2 Ž7. contains the 1-character and the irreducible character 4 of L2 Ž7., which is real. Žc. (U4(2), 5,
. .
y 6480 g X ⭈ Ž t y 1 . r Ž NH Ž X H . rX H y 17280 g Y ⭈ Ž t y 1 . r Ž NH Ž YH . rYH
. .,
60
UDO RIESE
where g X Žresp. g Y . is a normalized linear polynomial which counts the number u X Žresp. u Y . of one-dimensional subspaces UX Žresp. UY . of C V Ž X H . Žresp. C V Ž YH .. for which CH ŽUX . s X H Žresp. CH ŽUY . s YH .. But g X and g Y have their root at t s 7, so g X s g Y s t y 7 and f , H Ž t . s t 4 y 330t 2 q 2240t y 1911. This enables us to determine the lengths of H-orbits on V as 2 = 240, 2 = 2160, 6480, and 17280 for < F < s 13. For < F < s 19, we get 3 = 240, 3 = 2160, 3 = 6480, 3 = 17280, and 51840. This shows that for < F < s 13, 19, no vector is real. Now let G 1 s H )C6 . We note that the given lengths of H-orbits for H and < F < s 7, 13, 19 are also those for G 1. The nontrivial closed subgroups of G 1 are C, D, X, Y, B, where C l H s CH , D l H s DH , X l H s X H , Y l H s YH , and B l H s 1 and BZŽ G 1 . s ²3 A: ZŽ G 1 .. We have dim C V Ž B . s 3 and f , G 1Ž t . s f , H Ž t . y g B ⭈ Ž Ž t y 1 . r1296. ⭈ 51840, where g B is a normalized quadratic polynomial defined analogously to g X . But g B has roots at 7 and 13, hence g B Ž t . s Ž t y 7.Ž t y 13. and f , G 1Ž t . s t 4 y 40 t 3 q 510t 2 y 2200t q 1729. Let < F < s 31. We have f , G 1Ž31. s 155,520. Hence the group G 1 has exactly one regular orbit on V for < F < s 31. Let be a primitive fifth root of unity. We conclude that for z g Z of order 5, the element 5 A ⭈ z s diagŽ 2 , 3 , 4 , 1. centralizes some vector 0 / ¨ g V. Let < F < s 37. We have f , G 1Ž37. s 3 ⭈ 155,520. Let G 2 s H )C12 . As there is no element of order 36 in H, for all ¨ g V, we have CG 2Ž ¨ . s CG Ž ¨ ., so always three G 2-orbits of the same length are fused under G. We know that G 1 has exactly three regular orbits on V, hence there is ¨ g V such that < CG 2Ž ¨ .< F 2 and we are done. Let < F < s 43. We have f , G 1Ž43. s 7 ⭈ 155,520. Now as < G : G 1 < s 7, it follows that these seven regular orbits under G 1 are fused to one regular orbit of G. So there is a real ¨ g V. Let < F < s 49. We have f , G 1Ž49. s 14 ⭈ 155,520, and there are exactly 14 regular orbits in V under G 1. If a regular orbit ¨ G 1 does not give rise to a real vector, then < CG Ž ¨ .< is divisible by 4. As Ž4 B . s Ž2 B . s 0 and an element x belonging to class 4 A in E normalizes only two different subspaces of dimension 1 outside C V Ž x ., the result follows with Lemma 1. Žg. (6 1 . U4(3), 6,
QUASISIMPLE CASE OF THE
k Ž GV .-CONJECTURE
61
of H plays no role in our argument. H has a maximal subgroup M ( Ž2 6 = 3. A 6 , and M l H0 s Ž2 5 = 3. A 6 . The elementary abelian normal subgroup W of H0 of order 2 5 is an indecomposable A 6-module with composition series a four-dimensional module atop of the trivial one. W is completely reducible as a A 5-module with the same composition factors. This shows that there is ¨ g V such that C0 s CH 0Ž ¨ . s 2 4 : A 5 and NH Ž ¨ . s ZŽ H . = C2 = C0 s ZŽ H . = C, where C s CH Ž ¨ . s C0 = C2 . We apply Lemma 1Žd.. Žh. (6 1 .U4(3), 12, 11). Here G ( 6 1.U4Ž3..Ž2 1 2 2 . is exceptional of induced type, Ž1. s 12, and < F < s 11. We can apply our results from IIIŽg. and Lemma 2 to the group G 0 ( 6 1.U4Ž3..2 1 ) FEU and get a real vector. We remark that, similar to IŽh., this case is only relevant in view of Goodwin’s theorem, as ZŽ E . is not central in G. Ži. (U5 (2), 10,
1 2
žÝ
nŽ x . q
xgS 1
Ý ygS 2
CV Ž y .
/
,
where nŽ x . s < ¨ g V N x s Ž x . ¨ , is a linear faithful character of X 4< Ž X s ² x :.. Now we use the character table of E and calculate 7 10 ) 176 ⭈ 2 ⭈ 7 5 q 3520 Ž 7 4 q 2 ⭈ 7 3 . q 3520 ⭈ 2 ⭈ 7 2 q 21120 ⭈ 2 ⭈ 7 4 . Hence a real vector exists for < F < s 7.
62
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Now let < F < s 13. Then GrG⬘ is cyclic of order 24, E s G ( U5 Ž2., and E is real valued. If ¨ g V is not a real vector, we infer that < NE Ž ¨ .rCE Ž ¨ .< must be divisible by 3 or by 4. Now let S1 and S2 be as before, and let S3 s x g E
1 2
žÝ
xgS 1
nŽ x . q
Ý xgS 2
CV Ž x . q
Ý nŽ x . q Ý xgS 3
/
mŽ x . ,
xgS 4
where mŽ x . s < ¨ g V < ¨ x s y¨ 4<. This yields to 1310 ) 352 ⭈ 13 5 q 3520 Ž 13 4 q 2 ⭈ 13 3 . q 3520 ⭈ 2 ⭈ 13 2 q 21120 ⭈ 2 ⭈ 13 4 q 11880 ⭈ 13 4 q 35640 Ž 13 4 q 13 2r2 . q 142560 ⭈ 13 2 . Hence a real vector exists for < F < s 13. Žj. (U5 (2), 11, 7). Here G ( U5 Ž2. = Z is nonexceptional, Ž1. s 11, and H ( U5 Ž2.. There is a maximal subgroup X ( L2 Ž11. in H which centralizes a real vector ¨ g V. 5.4. Other Quasisimple Groups Ža. (2.J2 , 6,
QUASISIMPLE CASE OF THE
k Ž GV .-CONJECTURE
63
Žc. (S 6 (2), 7,
64
UDO RIESE
the k Ž GV .-conjecture holds in case CG Ž ¨ . is abelian for some ¨ g V wK, Theorem 3.7x. This settles ŽA., ŽB., and ŽC. for < F < s 13, 19. In the remaining case < F < s 7, we can count the number of conjugacy classes as k Ž GV . s
Ý
g⍀
CG Ž ¨ . ,
where ¨ Ž g ⍀ . are representatives of the different G-orbits in V. Now apply the description of the orbits given in Section 4 IIIŽf. and get k Ž GV . F 3Ž34 q 24 q 216. s 822 - 2401 s < V <.
ACKNOWLEDGMENTS The author did this work during a visit at Yale University. He thanks the Department of Mathematics for its hospitality and especially Walter Feit for many discussions on this subject. Furthermore, the author is indebted to Herbert Pahlings for pointing out an error in the treatment of IŽf. Ž E ( 2. A 6 and n s 4. in an earlier version of this work.
REFERENCES wAx wGx wGax wGMx wGdx wGd1x wGd2x wGox wIx wKx wLx wPPx wRSx
J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson, ‘‘Atlas of Finite Groups,’’ Clarendon, Oxford, 1985. D. Gluck, On the k Ž GV . problem, J. Algebra 89 Ž1984., 46᎐55. P. X. Gallagher, The number of conjugacy classes in a finite group, Math. Z. 118 Ž1970., 175᎐179. D. Gluck and K. Magaard, The extraspecial case of the k Ž GV . problem, Trans. Amer. Math. Soc., submitted. D. Goodwin, ‘‘Regular Orbits of Linear Groups,’’ Ph.D. thesis, Imperial College, London, 1998. D. Goodwin, Regular orbits of linear groups with an application to the k Ž GV .-problem Žpart 1., J. Algebra 227 Ž2000., 395᎐432. D. Goodwin, Regular orbits of linear groups with an application to the k Ž GV .-problem Žpart 2., J. Algebra 227 Ž2000., 433᎐473. R. Gow, On the number of characters in a block and the k Ž GV . problem for selfdual V, J. London Math. Soc. 48 Ž1993., 441᎐451. I. M. Isaacs, ‘‘The Character Theory of Finite Groups,’’ Academic Press, New York, 1976. R. Knorr, ¨ Oan the number of characters in a p-block of a p-solvable group, Illinois J. Math. 28 Ž1984., 181᎐210. M. W. Liebeck, Regular orbits of linear groups, J. Algebra 184 Ž1996., 1136᎐1142. H. Pahlings and W. Plesken, Group actions on Cartesian powers with applications to representation theory, J. Reine Angew. Math. 380 Ž1987., 178᎐195. U. Riese and P. Schmid, Self-dual modules and real vectors for solvable groups, J. Algebra 227 Ž2000., 159᎐171.
QUASISIMPLE CASE OF THE
wR1x wR2x wRTx wSx wWx
k Ž GV .-CONJECTURE
65
G. R. Robinson, Some remarks on the k Ž GV . problem, J. Algebra 172 Ž1995., 159᎐166. G. R. Robinson, Further reductions for the k Ž GV .-problem, J. Algebra 195 Ž1997., 141᎐150. G. R. Robinson and J. G. Thompson, On Brauer’s k Ž B . problem, J. Algebra 184 Ž1996., 1143᎐1160. M. Schonert, et al., ‘‘GAPᎏGroups, Algorithms, and Programming,’’ 4th ed., 1994. ¨ R. A. Wilson, The complex Leech lattice and maximal subgroups of the Suzuki group, J. Algebra 84 Ž1983., 151᎐188.
The Quasisimple Case of the k Ž GV . -Conjecture Udo Riese1 Uni¨ ersitat Mathematisches Institut, Auf der Morgenstelle 10, D-72076 ¨ Tubingen, ¨ Tubingen, Germany ¨ E-mail: [email protected] Communicated by Walter Feit Received May 5, 1999
1. INTRODUCTION Let G be a finite group of order < G <. The k Ž GV .-conjecture states that, for a given prime p not dividing < G < and every elementary abelian p-group V on which G acts faithfully, the number of conjugacy classes of the semidirect product GV is bounded from above by the order of V, that is, k Ž GV . F < V < . For p-solvable groups, this assertion is equivalent to R. Brauer’s celebrated k Ž B .-conjecture, claiming in general that the number of irreducible characters in a p-block B is bounded by the order of a defect group of B. In view of the Clifford reduction in wRTx and wR2x, it will follow from our main theorem, the result for p s 2 by Gluck wGx, and from recent work of Gluck and Magaard wGMx that the k Ž GV .-conjecture holds for all primes p different from 3, 5, 7, 11, 13, 19, 31. In attacking the k Ž GV .-problem, using a result of Gallagher wGax, one can assume irreducibility of V as an ⺖p G-module. By the combined work of Knorr ¨ wKx, Gow wGox, Robinson wR1, R2x, and Robinson and Thompson wRTx, the k Ž GV .-conjecture holds whenever, for some ¨ g V, the restriction of V to CG Ž ¨ . contains a self-dual faithful submodule. Such a vector will be called weakly real. In case VC G Ž ¨ . itself is a self-dual module, the vector is called real. The notion is motivated from the fact that in this semisimple Žco-prime. situation, a module is self-dual precisely when its 1 The author was supported by the German academic exchange service DAAD, Grant 332400510.
45 0021-8693r01 $35.00 Copyright 䊚 2001 by Academic Press All rights of reproduction in any form reserved.
46
UDO RIESE
Brauer character is real valued. More generally, if G acts F-linear for some extension field F of the prime field, a vector ¨ g V is called F-real if VC G Ž ¨ . is self-dual with respect to F. The search for Žweakly. real vectors allows an inductive approach to the k Ž GV .-conjecture via Clifford theory. Suppose V is a faithful FG-module for some finite extension field F of ⺖p . Carrying out the Clifford reduction as in wRTx or wR2x, one can assume a situation where there is a minimal nonabelian normal subgroup E of G which acts irreducibly on V such that End F G Ž V . s End F E Ž V .. Furthermore, we may assume the module V is tensor indecomposable. This reduces the problem either to the quasisimple case, that is, the generalized Fitting subgroup F*Ž G . is the central product of the center Z s ZŽ G . and a quasisimple nonabelian normal subgroup E, or to the extraspecial case, where F*Ž G . s F Ž G . is the central product of the center Z and E, where E is an extraspecial q-group for some prime q / p. A real vector for G is real for all subgroups of G. We embed E into GLŽ V . through the faithful representation. If the normalizer NGL ŽV .Ž E . is a p⬘-group Žwhich is always true in our cases; see Section 3., we may assume that G s NGL ŽV .Ž E ., so G is the largest possible group. In this paper we deal exclusively with the quasisimple case. The first definitive result was obtained by Liebeck wLx. The starting point of our investigation is the main result of D. Goodwin’s Ph.D. thesis wGdx Žsee also wGd1x and wGd2x.. In his work ‘‘Regular Orbits of Linear Groups,’’ he gives a list of triples Ž E, n, < F <., where E is a nonabelian quasisimple group, n is the dimension of an irreducible E-module V over the field F, where char Ž F . s p is a prime not dividing < G <, where G is isomorphic to a subgroup of NGL ŽV .Ž E . containing E. This list contains all possibilities of such triples Ž E, n, < F <. where G s NGL ŽV .Ž E . has no regular orbit in its permutation action on V. We call these triples Ž E, n, < F <. Goodwin triples. We remark that, up to a few exceptions, a Goodwin triple Ž E, n, < F <. does determine the isomorphism type of the module V. We will restate Goodwin’s result in Section 2. Note that if ¨ G is a regular G-orbit in V, then ¨ is a real vector for the group G. We prove: THEOREM. Let G be a finite group. Suppose V is a faithful FG-module of F-dimension n, where F is a finite field of characteristic co-prime to < G <. Assume there is a quasisimple normal subgroup E of G which is irreducible on V. Then there exists an F-real ¨ ector in V for G except when F is the prime field and one of the following holds: ŽA.
E ( 2. A 5 , n s 2, < F < g 11, 19, 314 ,
ŽB.
E ( 2. A 6 , n s 4, < F < s 7,
ŽC.
E ( 2.U4 Ž2., n s 4, < F < g 7, 13, 19, 314 .
QUASISIMPLE CASE OF THE
k Ž GV .-CONJECTURE
47
In ŽC. for < F < s 31, there is a weakly real ¨ ector in V for G. In all other exceptional cases, the group NGL ŽV .Ž E . is a p⬘-group and has no weakly real ¨ ector in V. We mention that the assumptions in the theorem imply F*Ž G . s E) ZŽ G . and, of course, the irreducibility of V as a G-module. Combined with Gluck’s and Magaard’s work on the extraspecial case wGMx and wRSx for p s 2, this proves: COROLLARY 1. Let G be a finite group. Suppose V is a faithful irreducible FG-module where char Ž F . s p does not di¨ ide < G <. Assume there is no real ¨ ector in V for G. Then F s ⺖p , G is absolutely irreducible on V, and p is one of the primes 3, 5, 7, 11, 13, 19, or 31. COROLLARY 2. The k Ž GV .-conjecture holds true for all primes p / 3, 5, 7, 11, 13, 19, 31. So in order to solve the k Ž GV .-problem, only these special situations have to be dealt with. If no vector is weakly real, we can usually find ¨ g V such that CG Ž ¨ . is abelian. This leads to Žsee Section 6. COROLLARY 3. The k Ž GV .-conjecture holds true whene¨ er G has an irreducible normal quasisimple subgroup. Excluding the cases where E is an alternating group Žof degree less than char Ž F .. and V is its ‘‘deleted’’ permutation module over F Žsee Section 2., where always real vectors exist, the remaining triples in Goodwin’s list are finite in number. The occurrence of all the larger groups in this finite set is related to the action of these groups as automorphisms of well-known lattices which give rise to extraordinarily small character degrees. We show by a case-by-case analysis that a real vector exists in all but the three exceptional cases ŽA., ŽB., and ŽC. in the theorem. Furthermore, all these groups are fortunately Atlas-groups wAx, and our work is completely based on the character tables given in the Atlas, except for the group 2.U4Ž2. and its four-dimensional modules, where we used the computer system GAP wSx. 2. GOODWIN’S RESULT In this section we give the exact result of Goodwin. Let c G 5 and let F be a finite extension of ⺖p . The F-vector space V s Ž ¨ 1 , . . . , ¨ c .: ¨ i g F, ݨ i s 04 is a module for the alternating group A c via the Žnatural . permutation action on the coordinates. We call this module the deleted permutation module over F for A c . This is a self-dual module, so every vector is real under the action of A c Ževen for Sc .. In his Ph.D. thesis, Goodwin always assumes F s ⺖p is the prime field. We work over arbitrary finite fields F.
48
UDO RIESE
The FG-module V viewed as an ⺖p G-module is the multiple of an irreducible ⺖p G-module, say V0 . Let F0 s ⺖p Ž V . : F be the field generated by the character to V Žvia embedding roots of unity into an algebraic closure of ⺖p .. Then w F : F0 x is the multiplicity of V0 in V wI, 9.18x, and we see that E is still irreducible on V0 . Assume there is a regular orbit on V0 . Then there is a regular orbit on V. Hence Goodwin’s result gives possible groups E. His proof for the size of the prime field for the remaining E’s relies on the fact that in case that no regular orbit exists, we have < V < F Ý1 / g g G < C¨ Ž g .<. Investigating this inequality, he gives indeed upper n bounds for '< V < s < F <. Now we restate Goodwin’s main result for arbitrary F, which can be deduced from the proof given in the prime field case in wGd1x and wGd2x. THEOREM Žsee wGdx.. Suppose V is a faithful FG-module of F-dimension n, where F is a finite field of char Ž F . s p not di¨ iding < G <. Assume there is a quasisimple normal subgroup E of G which is irreducible on V. Then G has a regular orbit on V or one of the following holds: ŽA. E is an alternating group A c , c - char Ž F ., and V is the deleted permutation module o¨ er F for E. ŽB. The triple Ž E, n, < F <. belongs to the following list: ŽI. Alternating groups: A 5 , n s 3, < F < s 11, A 5 , n s 4, < F < g 7, 11, 134 , Žc. 2. A 5 , n s 2, < F < F 61, Žd. 2. A 5 , n s 4, < F < g 7, 11, 134 , Ža. Žb.
Že. A 6 , n s 5, < F < g 7, 11, 13, 17, 194 , Žf. 2. A 6 , n s 4, < F < g 7, 11, 134 , Žg. 3. A 6 , n s 3, < F < F 49, Žh. 3. A 6 , n s 6, < F < g 7, 11, 134 , Ži. 2. A 7 , n s 4, < F < s 11. ŽII.
Linear groups: Ža. L2 Ž7., n s 3, < F < g 11, 23, 254 , Žb. L2 Ž7., n s 6, < F < s 5, Žc. 6. L3 Ž4., n s 6, < F < s 13.
ŽIII. Unitary groups: Ža. U3 Ž3., n s 6, < F < g 5, 114 , Žb. U3 Ž3., n s 7, < F < s 5,
QUASISIMPLE CASE OF THE
k Ž GV .-CONJECTURE
49
Žc. U4 Ž2., n s 5, < F < F 49, Žd. U4Ž2., n s 6, < F < F 43, Že. U4 Ž2., n s 10, < F < s 7, Žf. 2.U4Ž2., n s 4, < F < F 49, Žg. 6 1.U4Ž3., n s 6, < F < F 151, Žh. 6 1.U4Ž3., n s 12, < F < s 11, Ži. U5 Ž2., n s 10, < F < g 7, 134 , Žj. U5 Ž2., n s 11, < F < s 7. ŽIV. Other quasisimple groups: Ža. 2. J 2 , n s 6, < F < g 11, 194 , Žb. 6.Suz, n s 12, < F < s 19, Žc. S6 Ž2., n s 7, < F < F 79, Žd. 2.S6 Ž2., n s 8, < F < s 11, Že. 2.⍀q Ž . < < 8 2 .2, n s 8, F F 271. It is well known how to settle case ŽA. of Goodwin’s result Žsee Proposition 1 below.. Thus the main part of this work deals with the triples occurring in case ŽB..
3. THE GROUP G s NGL ŽV .Ž E . We describe the groups G s NGL ŽV .Ž E . for the various groups E occurring in Goodwin’s list. Let FG s End F G Ž V . and FE s End F E Ž V .. Then Z s ZŽ G . ( FGU . The Brauer character afforded by the irreducible FG-module V of dimension n is always denoted by . Let g Irr Ž E . be a constituent of the Žcomplex. character E . We can view as a Brauer character and get FE equals the field of character values F Ž . Žembedding the Žcomplex. < G <-th roots of unity into ⺖p. wI, 9.14x. We have n s Ž 1 . s < FE : F < Ž 1 . s < FE : F < ⭈ n 0 . If there are different characters in question, we use the numbering of the Atlas for . As E is irreducible on V, we see that the central product G 0 s E)CG Ž E . is normal in G and CG Ž E . ( FEU . The structure of G can now be determined as follows.
50
UDO RIESE
With the help of the Atlas wAx, it is straightforward to check the following: Ži. FE s F and V is an absolutely irreducible FE-module except possibly in the cases IŽd. < F < s 7, 13; IŽh. < F < s 7, 11, 13; IIŽb. < F < s 5; and IIIŽh. < F < s 11. In these cases there exists a module V of dimension n over F such that < FE : F < s 2 Žhence Ž1. s n 0 s nr2.. We call these cases where FE / F the exceptional cases of induced type. Žii. FG s F, so Z s ZŽ G . ( F*. We always have F*Ž G . s E) Z and GrCG Ž E . is isomorphic to a subgroup of AutŽ E .. In all cases there is a largest group Žup to isoclinism. H s E.r ŽAtlas notation., with ZŽ H . s ZŽ E ., such that viewed as a complex character is extendible to H. Inspection of Goodwin’s list shows r s 1 or r s 2 in all cases except IIIŽh., where r s 2 ⭈ 2. If H Žin some isoclinic version. acts on V, then H F G and G ( H ) Z, or we are in an exceptional case of induced type where always FEU ( CG Ž E . and G 0 s E) FEU is a normal subgroup of index < r < / 1 in G and G is not a central product. If H g G, then it may occur that GrCG Ž E . ( HrZŽ H .. Clearly, in this case, < H : E < s 2 and GrE must be cyclic of order divisible by 4. We conclude that G is isomorphic to a subgroup of index 2 in the central ˆ s H )C2 < Z < , where a cyclic group of order s is denoted by C s . product G To see that G acts on V, one has to calculate the character values of for ˆ be a generator of C2 < Z < . Then for x g H : G, ˆ elements G _ G 0 . Let s g G we have xs g G and Ž xs . s Ž x . Ž s . for some g Irr Ž C2 < Z < .. Again the Atlas gives: Žiii. H F G except in the following cases: IŽh. p s 7, 13; IIŽb. p s 5; IIŽc. p s 13; IIIŽa. p s 5, 11; and IIIŽi. p s 7, 13, which we call the exceptional cases of invariant type. Now it is obvious that, in all cases, the group NGL ŽV .Ž E . is indeed a p⬘-group as E and FE are p⬘-groups, and < GrG 0 < F 2 Žexcept in case IIIŽh... We summarize the properties of G in the following way: COROLLARY. Let Ž E, n, < F <. be a Goodwin triple. Then G ( NGL ŽV .Ž E . is a p⬘-group. Except in case IIIŽh., the central product G 0 ( E) FE has index at most 2 in G, and one of the following holds: ŽI. The group G is nonexceptional; that is, G s H ) Z is a central product o¨ er ZŽ E . for some group H, uniquely determined up to isoclinism, with Z ( FE s F. ŽII. G is not a central product and either FE s F and G is exceptional of in¨ ariant type, or else FE / F and G is exceptional of induced type.
QUASISIMPLE CASE OF THE
k Ž GV .-CONJECTURE
51
How we work with the character tables in wAx to determine the various groups is explained in the following: EXAMPLE. Suppose we are in case IŽh. Ž3. A 6 , 6, < F <., < F < g 7, 11, 134 . If the restriction of to E ( 3. A 6 is absolutely irreducible, then s 16 or 16* and affords a primitive third root of unity on ZŽ E .. Hence < F < / 11 in this case. Furthermore, Goodwin’s proof shows that in this case for < F < s 13, a regular orbit exists. So < F < s 7. The automorphisms of A 6 leading to A 6 .2 1 and A 6 .2 2 do not centralize ZŽ E .. Therefore 16 and 16* are not invariant under these automorphisms, but both characters extend to H ( 3. A 6 .2 3 . involving 'y 2 on H _ E. As < F < s 7, we see that H is not a subgroup of GLŽ V . and G is of invariant type. Calculating the character values as indicated above shows that indeed < GrG 0 < s 2. Now we assume that E is not absolutely irreducible Žbut irreducible.. Then G is of induced type and E equals the sum of two irreducible characters belonging to the set 14 , 14* , 15 , 15*4 on which Out Ž A 6 . acts regularly. All four characters in question involve '5 and a primitive third root of unity . If < F < s 7 or 13, then '5 f F, but g F and E extends either to H1 ( 3. A 6 .2 1 or H2 ( 3. A 6 .2 2 Ždepending on E ., e.g., if E s 14 q 15* , then E extends to H2 by the action of Out Ž A 6 . on the four characters in question. If < F < s 11, then '5 g F and f F. Hence either H2 or H3 ( 3. A 6 .2 3 are subgroups of G Ždepending on E .. 4. PRELIMINARIES In this section we describe our method for the nonexceptional cases, where G is a central product, and for the exceptional ones. We write in the nonexceptional cases G ( H ) Z, and in this situation for given 0 / ¨ g V, let C s CG Ž ¨ ., CH s CH Ž ¨ ., and NH s NH Ž ¨ . s NH Ž² ¨ :.. Then C l Z s 1 and C is isomorphic to a subgroup of NH containing CH . Inspecting the character tables of E, we will often apply the following: LEMMA 1. Let ¨ g V and assume G ( H ) Z. Suppose that the restriction of to CH takes only real ¨ alues. Then each of the following conditions implies that ¨ is a real ¨ ector for G in V: Ža.
NH rCH is an elementary abelian 2-group and takes only real
¨ alues on NH ,
Žb.
for all x g NH _ CH , Ž x . s 0 holds or x is an in¨ olution,
52
UDO RIESE
Žc. < Z : ZŽ H .< is relati¨ ely prime to < NH Ž CH . : CH <, Žd. NH s CH = ZŽ H .. Proof. We have either NG Ž ¨ . s NH ) Z or NG Ž ¨ . s NH = Z. To any x g NH there exists a unique z x g Z such that xz x g C. Conversely, every c g C can be written c s xz s zx with x g H and z g Z; then x g NH and z s z x . The assignment : x ¬ z x is a homomorphism of NH into Z with kernel CH . Taking character values given the claim in Ža. and Žb.. In Žc. and Žd. we have C s CH . Lemma 1 similar to wGM, Lemma 1.11x. Now we deal with case ŽA.. Our argument follows wRT, Corollary 6x. PROPOSITION 1.
In case ŽA. of Goodwin’s theorem, a real ¨ ector exists.
Proof. We have G ( Sc = Z for every c G 5 Žfor c s 6, inspect the character table in wAx.. Let H s Sc . Let V be the deleted permutation module over F and w F : ⺖p x s n. Then V s nV0 as an ⺖p S c-module wI, 9.18x, where V0 is the deleted permutation module over ⺖p . Write p s k Ž c y 1. q l. As p ) c, we have k / l, and for ¨ 0 s Ž k, . . . , k, l . g V0 : V, we have CH Ž ¨ 0 . s Scy1 , which is maximal and self-normalizing in H. Now apply Lemma 1. Now we briefly discuss our method in the exceptional cases. First assume we are in an exceptional case of induced type. Then the n-dimensional FG-module V corresponds to the complex character s q , where generates the group GalŽ FE rF . of order 2. We remark that the permutation representation of G 0 on the sets V and W are equal where V is the irreducible FG 0-module with character s q and W0 is the irreducible FE G 0-module with character . The following lemma enables us to extend our methods of the nonexceptional type ŽLemma 1. to the induced type, as in this case the central product G 0 s E) FEU is of index 2 in G. LEMMA 2. Suppose FE is not central in G Ž so that w FE : F x s 2.. Then there is an absolutely irreducible FE G 0-module W which is permutation isomorphic to V as a G 0-module. If there is a real ¨ ector for G 0 in W Ž self-duality with regard to FE ., there is a real ¨ ector in V for G. Proof. We may identify W s V as sets. Let w g W be the real vector for G 0 s ECG Ž E .. As Ž x . s 0 for all x g G _ G 0 and C G Ž ¨ . is real 0 valued, the claim follows as in Lemma 1. Placing this lemma in a more general context shows that after applying the Clifford reduction, one can assume that the restriction of to every normal subgroup is absolutely irreducible; see the proof of wRT, Theorem
QUASISIMPLE CASE OF THE
k Ž GV .-CONJECTURE
53
12x. But here we work with Goodwin’s result, so we do not assume absolute irreducibility of restricted to E; hence some groups of induced type occur. If the group G is exceptional of invariant type, we have H g G, E s G⬘, and G 0 s E) Z s F*Ž G . is of index 2 in G. As G 0 is a central product, we may apply Lemma 1 to G 0 and then the following lemma to find real vectors in V. LEMMA 3. Let G be a group of in¨ ariant type. Suppose ¨ is a real ¨ ector for G 0 . If CE Ž ¨ . s CG 0Ž ¨ ., then CG Ž ¨ . s CE Ž ¨ . and ¨ is a real ¨ ector for G in V. Proof. We must have CG Ž ¨ . s CE Ž ¨ ., because otherwise G would be isomorphic to the central product of ECG Ž ¨ . and Z, contradicting our assumption on G. We finally investigate the existence of regular orbits in terms of the size of the field of scalars. Our arguments are similar to those in wPPx. We call an F-subspace U of V a fixed space provided U s C V Ž X . for some subgroup X of G. In this case we call Xˆ s CG Ž U . s CG Ž C V Ž X . . the closure of X with regard to V, and X is a closed subgroup Žwith ˆ Observe that C V Ž X . s C V Ž Xˆ.. If Xˆ / 1, then respect to V . if X s X. ˆ Ž . C V X / V by faithfulness. A regular G-orbit on V corresponds to a vector of V which is not contained in C V Ž X . for any Žminimal. closed subgroup X / 1 of G. We may embed any group of p⬘-roots of unity into an algebraic closure ⺖p of ⺖p . This induces an embedding of complex character values of any p⬘-group into ⺖p. LEMMA 4. There is an integral polynomial f s f , G s Ý0 F i F n a i t i of degree n and highest coefficient a n s 1 with the following properties: It takes nonnegati¨ e ¨ alues at each prime power which is co-prime to < G < for which the ¨ alues of may be embedded into the field of corresponding order. If L is such a field, there is an LG-module W affording , and f Ž< L <. G 0 equals the number of ¨ ectors contained in regular G-orbits on W. In particular, f Ž< L <. s 0 if and only if there exists no regular G-orbit on W. Proof. Let f nŽ t . s t n y 1. Then f nŽ< F <. is the number of nontrivial vectors of V. Let L be as in the lemma. Then there is an LG-module W affording the Brauer character . For any subgroup X of G, dim F C V Ž X . s w X , 1 X x s dim L CW Ž X . . Let Xˆ s CG Ž C V Ž X ... Then X : Xˆ and C V Ž X . s C V Ž Xˆ .. It follows that CW Ž X . = CW Ž Xˆ .. But dim F C V Ž Xˆ . s dim L CW Ž Xˆ . by the above argu-
54
UDO RIESE
ment Žreplacing X by Xˆ .. Consequently, CW Ž X . s CW Ž Xˆ . and Xˆ s CG Ž CW Ž X ... Hence the closed subgroups of G with regard to V and W are the same. These form a lattice with respect to intersection and the closure of the usual join. Similarly, the fixed subspaces of V, W form lattices with regard to intersection and taking the smallest fixed subspace containing the usual join. The assignments Xˆ ¬ C V Ž Xˆ . and Xˆ ¬ CW Ž Xˆ . are lattice antiisomorphisms from the closed subgroups of G onto the fixed subspaces of V and W, respectively, having pairwise the same dimensions. Thus we have a lattice isomorphism between the fixed subspaces of V and W preserving dimensions. We count the number of nonzero vectors which belong to some Žproper. maximal fixed subspace in V, resp. W. This leads to an integral polynomial of degree at most n y 1 which only depends on . The polynomial f of the lemma is just the difference of f n and this polynomial. Thus f is as asserted. It takes nonnegative values on < F < and < L <. There is a regular G-orbit on W if and only if there is a one-dimensional subspace of W which is not contained in any maximal fixed subspace of W. This is equivalent to saying that f Ž< L <. ) 0. 5. PROOF OF THE THEOREM First we introduce some more notation which will be used in the proofs. Elements of a given group are always denoted by the number of their conjugacy class in the character table in wAx; e.g., X s ²3 A: denotes a cyclic group of order 3 generated by an element belonging to class 3 A, a Frobenius group of order 21 containing elements in class 3 A, and 7A is denoted by Y s ²3 A, 7A: and similar notations are used. Furthermore, regŽ X . is always the regular character of the group X, and X - Y means that X is a proper subgroup of Y. If isoclinic groups exist, we always refer to the group whose character table is given in the Atlas as Gq and to the other one by Gy Žin our cases there are always at most two isoclinic groups.. If our arguments apply for both groups, we also write just G for both groups. We recall that in the nonexceptional cases, we can write G ( H ) Z, where H s E.2 or H s E. Furthermore, n s Ž1. s Ž1. s n 0 and FG s FE s F. Now we deal with the various Goodwin triples Ž E, n, < F <.. 5.1. Alternating Groups Ža. (A 5 , 3, 11). Here G ( A 5 = Z is of nonexceptional type, Ž1. s 3, and H ( A 5 . Let X s ²3 A: F H. We calculate X s regŽ X ., and if X - Y - H, then Y is isomorphic either to S3 or to A 4 . The restriction of
QUASISIMPLE CASE OF THE
k Ž GV .-CONJECTURE
55
to an A 4 subgroup of H is irreducible and we have w Y , 1x s 0 if Y is an S3 subgroup of H. Hence there is ¨ g V such that CH s CH Ž ¨ . s X. Since NH Ž CH . ( S3 and the restriction of to the latter group is real, Lemma 1 shows that ¨ is real. Žb. (A 5 , 4,
56
UDO RIESE
Že. (A 6 , 5,
< F < g 7, 134 , s 8 , and H s Hy, or s 9 and H s Hq. < F < s 11, s 8 , and H s Hq, or s 9 and H s Hy.
We deal first with the case < F < s 7, s 9 , and H s Hq. The case 8 , Hy will follow by interchanging 3 A and 3 B, 6 A and 6 B. Let H0 F H such that H0 ( 2. A 6 . We calculate the orbits of G on nonzero vectors. In H0 only elements in class 3 A centralize any nonzero vector. Let X s ²3 A:. Then w X , 1x s 2. As < H0 : NH 0Ž X .< s 20, < NH Ž X . : X < s 12, and 7 2 y 1 s 48, we see that for 0 / u g U s C V Ž X ., 0 u H 0 l U s 12. We conclude that there are four orbits of length 240 and two regular orbits of length 720 under the action of H0 . The 12 vectors of u H 0 l U of each orbit are distributed on the eight one-dimensional subspaces of U as Ž6, 6, 0, 0, 0, 0, 0, 0., resp. three times Ž0, 0, 2, 2, 2, 2, 2, 2.. This can be seen from the action of a Sylow 3-subgroup of H0 and the fact that H0 contains a central element of order 2. We call these orbits 6 and 2i for i g 1, 2, 34 in the obvious way. Let G 0 s H0 ) Z. Let u g 6 . Then < CG 0Ž u.< s 3 < X < s 9. The orbits 2i are fused under the action of Z, so for u g 2i we have < CG Ž u.< s 3. So we get orbits of length 240, 720, 720, 720 under the action of G 0 . We remark that < G : G 0 < s 2. An element in class 2 B centralizes a two-dimensional subspace of V, and any nonzero vector ¨ which is centralized by some element x g G _ G 0 must be centralized by an element in class 2 B. This shows that two orbits of length 720 must fuse in G, so the orbit lengths are 240, 720, and 1440. By looking at the centralizers of order 3, 6, and 18, it is straightforward to see that no vector is real. For < F < s 13, we can deduce from the case < F < s 7 that there is a regular orbit in V under a subgroup X ( 2.S6 )C3 of G. Hence there is ¨ g V such that < CG Ž ¨ .< F 2. If < F < s 11, we conclude that there is an orbit ¨ H of length 720 under the action of H and CG Ž ¨ . s CH Ž ¨ . s ²2 B : by Lemma 1Žc.. Žg. (3.A 6 , 3,
QUASISIMPLE CASE OF THE
k Ž GV .-CONJECTURE
57
NH Ž X . is a dihedral group of order 8. Applying Lemma 1 gives the existence of a real vector. Žh. (3.A 6 , 6,
58
UDO RIESE
A 4-subgroup of H is irreducible. Furthermore, if X ( S3 : H, then w X , 1x s 0 and w ² 7 A: , 1x s 0. This shows that there is ¨ g V such that CH Ž ¨ . is cyclic of order 3. As NH Ž CH Ž ¨ .. s S3 and is real on the latter group, we apply Lemma 1. Žb. (L 2(7), 6, 5). Suppose first that E is absolutely irreducible. Then Ž1. s Ž1. and the complex character can be extended to H ( L2 Ž7..2, affording '2 . As < F < s 5, we see that G is of invariant type. Let H0 ( L2 Ž7. F G and X ( S4 : H0 . We calculate w X , 1x s 1 and X is maximal in H0 . We apply Lemma 1 and Lemma 3 to get a real ¨ g V. Now suppose G is of induced type. Then H ( L2 Ž7..2 and the restriction of to H0 s E ( L2 Ž7. equals 2 q 3 . The assertion follows from the results in IIŽa. and Lemma 2 applied to G 0 s E) FEU . Žc. (6.L 3(4), 6, 13). The complex character of degree 6 of H0 s E ( 6. L3 Ž4. is extendible to H ( 6. L3 Ž4..2 1 , affording '2 , resp. 'y 2 Ždepending on the isoclinism type of H .. As < F < s 13, the group G is of invariant type, and both isoclinic groups Hq and Hy give rise to the same group G. If we view H0 as a maximal subgroup of X s 6 1.U4Ž3..2 2 , this representation is just the restriction of the six-dimensional representation of X. It follows that C0 s CH 0Ž ¨ . s 2 4 : A 5 for some ¨ g V and NH 0Ž ¨ . s ZŽ H0 . = C0 Žsee also IIIŽg... Now apply Lemma 1 and Lemma 3 to obtain a real ¨ g V. 5.3. Unitary Groups Ža. (U3(3), 6,
QUASISIMPLE CASE OF THE
k Ž GV .-CONJECTURE
59
Žii. Here G ( U3 Ž3. = Z is also nonexceptional, but s 4 or s 5 . The restriction of to a maximal subgroup isomorphic to L2 Ž7. contains the 1-character and the irreducible character 4 of L2 Ž7., which is real. Žc. (U4(2), 5,
. .
y 6480 g X ⭈ Ž t y 1 . r Ž NH Ž X H . rX H y 17280 g Y ⭈ Ž t y 1 . r Ž NH Ž YH . rYH
. .,
60
UDO RIESE
where g X Žresp. g Y . is a normalized linear polynomial which counts the number u X Žresp. u Y . of one-dimensional subspaces UX Žresp. UY . of C V Ž X H . Žresp. C V Ž YH .. for which CH ŽUX . s X H Žresp. CH ŽUY . s YH .. But g X and g Y have their root at t s 7, so g X s g Y s t y 7 and f , H Ž t . s t 4 y 330t 2 q 2240t y 1911. This enables us to determine the lengths of H-orbits on V as 2 = 240, 2 = 2160, 6480, and 17280 for < F < s 13. For < F < s 19, we get 3 = 240, 3 = 2160, 3 = 6480, 3 = 17280, and 51840. This shows that for < F < s 13, 19, no vector is real. Now let G 1 s H )C6 . We note that the given lengths of H-orbits for H and < F < s 7, 13, 19 are also those for G 1. The nontrivial closed subgroups of G 1 are C, D, X, Y, B, where C l H s CH , D l H s DH , X l H s X H , Y l H s YH , and B l H s 1 and BZŽ G 1 . s ²3 A: ZŽ G 1 .. We have dim C V Ž B . s 3 and f , G 1Ž t . s f , H Ž t . y g B ⭈ Ž Ž t y 1 . r1296. ⭈ 51840, where g B is a normalized quadratic polynomial defined analogously to g X . But g B has roots at 7 and 13, hence g B Ž t . s Ž t y 7.Ž t y 13. and f , G 1Ž t . s t 4 y 40 t 3 q 510t 2 y 2200t q 1729. Let < F < s 31. We have f , G 1Ž31. s 155,520. Hence the group G 1 has exactly one regular orbit on V for < F < s 31. Let be a primitive fifth root of unity. We conclude that for z g Z of order 5, the element 5 A ⭈ z s diagŽ 2 , 3 , 4 , 1. centralizes some vector 0 / ¨ g V. Let < F < s 37. We have f , G 1Ž37. s 3 ⭈ 155,520. Let G 2 s H )C12 . As there is no element of order 36 in H, for all ¨ g V, we have CG 2Ž ¨ . s CG Ž ¨ ., so always three G 2-orbits of the same length are fused under G. We know that G 1 has exactly three regular orbits on V, hence there is ¨ g V such that < CG 2Ž ¨ .< F 2 and we are done. Let < F < s 43. We have f , G 1Ž43. s 7 ⭈ 155,520. Now as < G : G 1 < s 7, it follows that these seven regular orbits under G 1 are fused to one regular orbit of G. So there is a real ¨ g V. Let < F < s 49. We have f , G 1Ž49. s 14 ⭈ 155,520, and there are exactly 14 regular orbits in V under G 1. If a regular orbit ¨ G 1 does not give rise to a real vector, then < CG Ž ¨ .< is divisible by 4. As Ž4 B . s Ž2 B . s 0 and an element x belonging to class 4 A in E normalizes only two different subspaces of dimension 1 outside C V Ž x ., the result follows with Lemma 1. Žg. (6 1 . U4(3), 6,
QUASISIMPLE CASE OF THE
k Ž GV .-CONJECTURE
61
of H plays no role in our argument. H has a maximal subgroup M ( Ž2 6 = 3. A 6 , and M l H0 s Ž2 5 = 3. A 6 . The elementary abelian normal subgroup W of H0 of order 2 5 is an indecomposable A 6-module with composition series a four-dimensional module atop of the trivial one. W is completely reducible as a A 5-module with the same composition factors. This shows that there is ¨ g V such that C0 s CH 0Ž ¨ . s 2 4 : A 5 and NH Ž ¨ . s ZŽ H . = C2 = C0 s ZŽ H . = C, where C s CH Ž ¨ . s C0 = C2 . We apply Lemma 1Žd.. Žh. (6 1 .U4(3), 12, 11). Here G ( 6 1.U4Ž3..Ž2 1 2 2 . is exceptional of induced type, Ž1. s 12, and < F < s 11. We can apply our results from IIIŽg. and Lemma 2 to the group G 0 ( 6 1.U4Ž3..2 1 ) FEU and get a real vector. We remark that, similar to IŽh., this case is only relevant in view of Goodwin’s theorem, as ZŽ E . is not central in G. Ži. (U5 (2), 10,
1 2
žÝ
nŽ x . q
xgS 1
Ý ygS 2
CV Ž y .
/
,
where nŽ x . s < ¨ g V N x s Ž x . ¨ , is a linear faithful character of X 4< Ž X s ² x :.. Now we use the character table of E and calculate 7 10 ) 176 ⭈ 2 ⭈ 7 5 q 3520 Ž 7 4 q 2 ⭈ 7 3 . q 3520 ⭈ 2 ⭈ 7 2 q 21120 ⭈ 2 ⭈ 7 4 . Hence a real vector exists for < F < s 7.
62
UDO RIESE
Now let < F < s 13. Then GrG⬘ is cyclic of order 24, E s G ( U5 Ž2., and E is real valued. If ¨ g V is not a real vector, we infer that < NE Ž ¨ .rCE Ž ¨ .< must be divisible by 3 or by 4. Now let S1 and S2 be as before, and let S3 s x g E
1 2
žÝ
xgS 1
nŽ x . q
Ý xgS 2
CV Ž x . q
Ý nŽ x . q Ý xgS 3
/
mŽ x . ,
xgS 4
where mŽ x . s < ¨ g V < ¨ x s y¨ 4<. This yields to 1310 ) 352 ⭈ 13 5 q 3520 Ž 13 4 q 2 ⭈ 13 3 . q 3520 ⭈ 2 ⭈ 13 2 q 21120 ⭈ 2 ⭈ 13 4 q 11880 ⭈ 13 4 q 35640 Ž 13 4 q 13 2r2 . q 142560 ⭈ 13 2 . Hence a real vector exists for < F < s 13. Žj. (U5 (2), 11, 7). Here G ( U5 Ž2. = Z is nonexceptional, Ž1. s 11, and H ( U5 Ž2.. There is a maximal subgroup X ( L2 Ž11. in H which centralizes a real vector ¨ g V. 5.4. Other Quasisimple Groups Ža. (2.J2 , 6,
QUASISIMPLE CASE OF THE
k Ž GV .-CONJECTURE
63
Žc. (S 6 (2), 7,
64
UDO RIESE
the k Ž GV .-conjecture holds in case CG Ž ¨ . is abelian for some ¨ g V wK, Theorem 3.7x. This settles ŽA., ŽB., and ŽC. for < F < s 13, 19. In the remaining case < F < s 7, we can count the number of conjugacy classes as k Ž GV . s
Ý
g⍀
CG Ž ¨ . ,
where ¨ Ž g ⍀ . are representatives of the different G-orbits in V. Now apply the description of the orbits given in Section 4 IIIŽf. and get k Ž GV . F 3Ž34 q 24 q 216. s 822 - 2401 s < V <.
ACKNOWLEDGMENTS The author did this work during a visit at Yale University. He thanks the Department of Mathematics for its hospitality and especially Walter Feit for many discussions on this subject. Furthermore, the author is indebted to Herbert Pahlings for pointing out an error in the treatment of IŽf. Ž E ( 2. A 6 and n s 4. in an earlier version of this work.
REFERENCES wAx wGx wGax wGMx wGdx wGd1x wGd2x wGox wIx wKx wLx wPPx wRSx
J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson, ‘‘Atlas of Finite Groups,’’ Clarendon, Oxford, 1985. D. Gluck, On the k Ž GV . problem, J. Algebra 89 Ž1984., 46᎐55. P. X. Gallagher, The number of conjugacy classes in a finite group, Math. Z. 118 Ž1970., 175᎐179. D. Gluck and K. Magaard, The extraspecial case of the k Ž GV . problem, Trans. Amer. Math. Soc., submitted. D. Goodwin, ‘‘Regular Orbits of Linear Groups,’’ Ph.D. thesis, Imperial College, London, 1998. D. Goodwin, Regular orbits of linear groups with an application to the k Ž GV .-problem Žpart 1., J. Algebra 227 Ž2000., 395᎐432. D. Goodwin, Regular orbits of linear groups with an application to the k Ž GV .-problem Žpart 2., J. Algebra 227 Ž2000., 433᎐473. R. Gow, On the number of characters in a block and the k Ž GV . problem for selfdual V, J. London Math. Soc. 48 Ž1993., 441᎐451. I. M. Isaacs, ‘‘The Character Theory of Finite Groups,’’ Academic Press, New York, 1976. R. Knorr, ¨ Oan the number of characters in a p-block of a p-solvable group, Illinois J. Math. 28 Ž1984., 181᎐210. M. W. Liebeck, Regular orbits of linear groups, J. Algebra 184 Ž1996., 1136᎐1142. H. Pahlings and W. Plesken, Group actions on Cartesian powers with applications to representation theory, J. Reine Angew. Math. 380 Ž1987., 178᎐195. U. Riese and P. Schmid, Self-dual modules and real vectors for solvable groups, J. Algebra 227 Ž2000., 159᎐171.
QUASISIMPLE CASE OF THE
wR1x wR2x wRTx wSx wWx
k Ž GV .-CONJECTURE
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