On the nilpotency index of the radical of a group algebra. VI
[For Part V of this series cf. ibid. 90, 251-258 (1984; Zbl 0543.16005); VII, ibid. 101, 299-302 (1986; Zbl 0589.16012).] Let F be a field of characteristic \(p>0\). Let G be a finite p-solvable group with a p-Sylow subgroup P of order \(p^ m\). Let \(M=O_{p'}(G)\), let \(H=O_{p',p}(G)\) and let F/M be the Frattini subgroup of H/M. The main result is that if \(t(G)>p^{m-1}\) and if G is not of p-length 1 then \(p=2\), F/M is cyclic and G/F is the symmetric group of order 24. A very similar result was proved and crucially used by \textit{S. Koshitani} [J. Algebra 94, 106-112 (1985; reviewed above)] but the present proof is less involved.
scientific article; zbMATH DE number 3959688