On Brauer's \(k(B)\)-problem (Q1815021)

This paper gives (in the case of \(p\)-solvable groups) a significant contribution to R. Brauer's famous conjecture, that the number of characters of a \(p\)-block in a finite group is bounded by the order of the defect group of the block. This has for \(p\)-solvable groups been reduced to the so-called \(k(GV)\)-problem, which is treated in the paper. The main result is that the conjecture is true for sufficiently large prime \(p\) (\(p>5^{30}\)). The bound comes from a theorem of \textit{M. W. Liebeck} [J. Algebra 184, No. 3, 1136-1142 (1996; Zbl 0901.20006)] and the proof relies on the classification of the finite simple groups. It should be remarked that after the publication of this paper the bound on the prime in Liebeck's theorem and consequently in the present paper has been lowered dramatically.