On \(p\)-radical groups \(G\) and the nilpotency indices of \(J(kG)\). II (Q1356387)

[For part I cf. ibid. 31, No. 4, 773-785 (1994; Zbl 0824.20005).] Let \(F\) be an algebraically closed field of characteristic \(p>0\). A finite group \(G\) with Sylow \(p\)-subgroup \(P\) of order \(p^a\) is called \(p\)-radical if the Jacobson radical \(J(FG)\) of the group algebra \(FG\) satisfies \(J(FG)=J(FP)FG\). \textit{T. Okuyama} has proved [Osaka J. Math. 23, 467-469 (1986; Zbl 0611.20006)] that a \(p\)-radical group \(G\) is always \(p\)-solvable. Thus the Loewy length of its group algebra \(FG\) is at least \(a(p-1)+1\). The author shows that, for a \(p\)-radical group \(G\), the following assertions are equivalent: (1) \(FG\) has Loewy length \(a(p-1)+1\); (2) the projective cover of the trivial \(FG\)-module has Loewy length \(a(p-1)+1\); (3) \(O_{p'}(G)=N\rtimes H\) where \(N\) is an elementary abelian \(p\)-group, \(H=M\rtimes Q\) where \(M\) is a \(p'\)-group and \(Q\) is an elementary abelian \(p\)-group, and every element in \(N\) has full \(p\)-defect.