The \(k(GV)\) conjecture for modules in characteristic 31
This is a further contribution by the authors to the solution of Brauer's \(k(B)\)-problem for a finite \(p\)-solvable group. As is usual in this kind of work, we consider the following configuration. Let \(p\) be an odd prime and let \(G\) be a finite \(p'\)-group. Let \(V\) be a faithful irreducible \(FG\)-module, where \(F\) is the field of order \(p\). The problem is to show that the number of conjugacy classes \(k(GV)\) in the semidirect product \(GV\) is at most \(|V|\). Robinson and Thompson have given a sufficient condition for the inequality to hold in terms of a self-duality criterion for a special type of submodule of \(V\). The combined work of various authors has shown that we do have \(k(GV)\leq|V|\) unless possibly \(p\) is one of \(3\), \(5\), \(7\), \(11\), \(13\), \(19\) or \(31\). The paper under review eliminates the prime \(31\) from this list. The present authors also remark that their techniques will probably extend to eliminate the primes \(11\) and \(19\). The main problem that arises in the case when \(p=31\) is associated with various actions of the group \(\text{SL}(2,5)\times Z_{15}\) on a two-dimensional module over the field of order \(31\). The proof is subdivided into an analysis of primitive and imprimitive modules in an attempt to show that the Robinson-Thompson condition generally applies. When the condition does not apply -- and this can occur in the imprimitive case -- the authors make use of a theorem of Liebeck and Pyber that estimates the number of conjugacy classes in a permutation group of degree \(n\). Eventually, they are able to obtain the desired inequality in all circumstances.
scientific article; zbMATH DE number 1762098