On the number of characters in a \(p\)-block of a \(p\)-solvable group (Q791630)

R. Brauer conjectured that the number of irreducible characters in a \(p\)-block of a finite group \(G\) is bounded by the order of the defect group. In this important paper, the author gives a first significant contribution to the proof of the conjecture when \(G\) is \(p\)-solvable. By Fong and Nagao one has ``only'' to consider the following question, which itself is very intriguing: Let \(V\) be an elementary abelian \(p\)-group and \(H\subseteq\Aut(V)\) a \(p'\)-group. Is the number \(k(HV)\) of conjugacy classes of \(HV\) bounded by \(|V|\)? In the first part of the paper an original and powerful method is developed to tackle this question. Then the question is answered in the affirmative, when \(H\) is supersolvable. Thus Brauer's conjecture is true, when \(G\) is a \(p\)-solvable group with a supersolvable \(p\)-complement. A more general case has been done by \textit{D. Gluck} [J. Algebra 89, 46-55 (1984; reviewed below)] and by the author (unpublished). In the method mentioned above the generalized character \(\delta\) of \(H\) defined by \(\delta(h)=|V:C_V(h)|\) plays a key role because of the following result: ``If for some \(v\in V\) we have \((\gamma\delta,\gamma)_C\geq k(C)\) for each generalized character \(\delta\) of \(C=C_H(v)\) with \(p\nmid\gamma(1)\), then \(k(HV)\leq|V|\)''. A lengthy analysis shows the existence of such an element \(v\), if \(H\) is supersolvable.