Nilpotency indices of the radicals of finite \(p\)-solvable group algebras. I (Q2748395)

Let \(kG\) be the group algebra with Jacobson radical \(J\) of the finite \(p\)-solvable group \(G\) over the field \(k\) of characteristic \(p\). It is well-known that \(J\) is a nilpotent ideal and \(t(G)\leq p^m\), where \(p^m\) is the highest power of \(p\) dividing \(|G|\) and \(t(G)\) is the nilpotency index of \(J\). There are many known results connecting \(t(G)\) and the structure of \(G\), see \textit{G. Karpilovsky} [The Jacobson radical of group algebras, North-Holland, Amsterdam (1987; Zbl 0618.16001)] or the author [Proc. Edinb. Math. Soc., II. Ser. 37, No. 3, 509-517 (1994; Zbl 0818.20004)]. With \(p\) odd, the author determines the groups \(G\) for which \(p^{m-2}<t(G)<p^{m-1}\). The present paper is the first of a forthcoming series.