Nilpotency indices of the radicals of finite \(p\)-solvable group algebras. IV (Q2783648)

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 \(J\). With \(p\) odd, in a series of papers the author determines the groups \(G\) for which \(p^{m-2}<t(G)<p^{m-1}\). In the first one [J. Aust. Math. Soc. 71, No. 1, 117-133 (2001; Zbl 0997.20005)] the case \(p>3\) is completed. In the present one (and in the second and third one; see the preceding reviews Zbl 1005.20002 and Zbl 1005.20003) the extremely difficult case \(p=3\) is dealt with by means of complicated calculations.