Regular orbits and the \(k(GV)\)-problem (Q2759631)

Brauer's \(k(B)\)-problem is to show that the number \(k(B)\) of irreducible complex characters in a \(p\)-block \(B\) satisfies \(k(B)\leq|D|\) where \(D\) is a defect group of \(B\). Nagao has proved that Brauer's \(k(B)\)-problem, for \(p\)-blocks of \(p\)-solvable groups, is equivalent to what is now known as the \(k(GV)\)-problem. The \(k(GV)\)-problem is to show that, for a finite \(p'\)-group \(G\) and a faithful finite \(\mathbb{F}_pG\)-module \(V\), the number \(k(GV)\) of conjugacy classes of the semidirect product \(GV\) satisfies \(k(GV)\leq|V|\).NEWLINENEWLINENEWLINEThe work of several authors, including R.~Knörr, G.~R.~Robinson and J.~Thompson, had lead to a positive solution of the \(k(GV)\)-problem for all but finitely many primes \(p\). The main result of the present paper shows that the only exceptions to the \(k(GV)\)-problem can (possibly) occur for the primes 3, 5, 7, 11, 13, 19, 31. This fact has been obtained independently by Gluck, Magaard and Riese. Since the present paper appeared in print, Gluck, Magaard, P.~Schmid and U. Riese have made further progress so that, at present, 5 is the only prime for which the \(k(GV)\)-problem is not yet known to have a positive solution.NEWLINENEWLINEFor the entire collection see [Zbl 0959.00030].