On the nilpotency index of the radical of a group algebra. VII (Q1073880)

[For part V of this series cf. J. Algebra 90, 251-258 (1984; Zbl 0543.16005); VIII cf. Proc. Am. Math. Soc. 92, 327-328 (1984; Zbl 0548.16006).] Let p be a prime, let K be a field of odd prime characteristic p and let G be a finite p-solvable group with a p-Sylow subgroup P of order \(p^ m\). Let t(G) be the nilpotency index of the radical J(KG) of the group algebra KG of G over K. It is shown that if \(t(G)=p^{m-1}\) then \(p=3\) and \(P\cong M(3)\). The proof depends heavily on previous papers of the author.