Brauer's \(k(B)\)-conjecture for solvable \(3'\)-groups (Q1325066)

R. Brauer asked whether the number of ordinary irreducible characters in a \(p\)-block of a finite group is bounded by the order of its defect group. In the case of \(p\)-solvable groups it suffices to prove a statement about the number of characters in certain group extensions. Here very significant contributions have been given by R. Knörr. The author proves: Main theorem. Let \(G\) be a solvable \(3'\)-group and let \(V\) be a faithful \(KG\)-module, where \(\text{char}(K) \neq 3\) and \(\text{char}(K) \nmid | G|\). Then there exists \(v \in V\) such that \(V_{C_ G(v)}\) is a permutation module. By a recent result Knörr this verifies the conjecture for solvable \(3'\)-groups.