THE EXTRASPECIAL CASE OF THE k(GV ) PROBLEM

Let E be an extraspecial-type group and V a faithful, absolutely irreducible k[E]-module, where k is a finite field. Let G be the normalizer in GL(V ) of E. We show that, with few exceptions, there exists a v ∈ V such that the restriction of V to CH(v) is self-dual whenever H ≤ G and (|H|, |V |) = 1. Let G be a finite group and let V be a faithful irreducible finite G-module such that (|G| , |V |) = 1. The k(GV ) problem is to show that k(GV ), the number of conjugacy classes in the semidirect product GV, is at most |V | . This is a special but very difficult case of Brauer’s famous conjecture that the number of ordinary irreducible characters in an r-block of a finite group is at most the order of its defect group. Indeed, if r is the prime divisor of |V |, then GV has a unique r-block, and its defect group is |V | . On the other hand, an affirmative solution to the k(GV ) problem implies Brauer’s conjecture for r-solvable groups. The naive approach of directly estimating k(GV ) has not led to any substantial result. Recent progress on the k(GV ) problem is based instead on the remarkable idea of “centralizer criteria.” These criteria assert that k(GV ) ≤ |V | if there exists v ∈ V such that CG(v) satisfies some condition. The first such criteria were proved by Knörr [K] who showed, for example, that k(GV ) ≤ |V | if there exists v ∈ V such that CG(v) is abelian. Building on work of Knörr [K], Gow [Gw], and Robinson [R1], Robinson and Thompson [RT] proved the most powerful centralizer criterion to date; k(GV ) ≤ |V | provided that there exists v ∈ V such that VCG(v) has a faithful self-dual summand. For convenience, such vectors v will be called RT vectors. Unfortunately, not all modules V contain RT -vectors. Indeed, it is not hard to see that V may contain no RT vector when |V | = 5, 7, or 13 and G/Z(G) ∼= S4. For these modules, it is of course easy to show that k(GV ) ≤ |V |. The point, however, is that imprimitive modules which are induced from these two-dimensional modules, also, in general, do not contain RT -vectors. Such imprimitive modules are probably the greatest obstacle to the complete solution of the k(GV ) problem. Clifford-theoretic reductions as in [RT, Sect. 4] lead to two major cases of the k(GV ) problem. In the quasisimple case, F ∗(G) = QZ(G), where Q is quasisimple and Q acts absolutely irreducibly on V. In the extraspecial case, there is a prime p such that Op(G) acts absolutely irreducibly on V and every characteristic abelian subgroup of Op(G) is central. Since, in the extraspecial case, one may also assume that V is a primitive module, a theorem of P. Hall [Go, 5.4.9] implies that Op(G) = E0Z, where E0 is extraspecial of odd prime exponent or exponent 4 and Z consists Received by the editors August 12, 1999 and, in revised form, January 2, 2001. 2000 Mathematics Subject Classification. Primary 20C20; Secondary 20C33, 20D06, 20E28. Research of both authors partially supported by NSA grants. c ©2001 American Mathematical Society