Further reductions for the \(k(GV)\)-problem
Let \(p\) be a prime, let \(G\) be a finite \(p'\)-group, and let \(V\) be a faithful (irreducible) \(\mathbb{F}_pG\)-module. Then \(k(GV)\) denotes the class number of the semidirect product \(GV\). The ``\(k(GV)\)-problem'' is to show that \(k(GV)\leq|V|\). A breakthrough on this problem was achieved in a joint paper of the author and \textit{J. G. Thompson} [J. Algebra 184, No. 3, 1143-1160 (1996)] where it was shown that the \(k(GV)\)-problem has a positive answer for primes \(p>5^{30}\). In the paper under review the methods of that paper are refined. In combination with results of D. Goodwin this yields a positive solution to the \(k(GV)\)-problem for all primes \(p>211\). As a consequence, for every prime \(p>211\), every \(p\)-block of every finite \(p\)-solvable group with defect group \(D\) contains at most \(|D|\) irreducible characters.
scientific article; zbMATH DE number 1076269