On \(p\)-radical groups \(G\) and the nilpotency indices of \(J(kG)\) (Q1804688)

Let \(kG\) be the group algebra of a finite group over an algebraically closed field of characteristic \(p > 0\) and let \(P\) be a Sylow \(p\)- subgroup of \(G\). \(G\) is said to be \(p\)-radical if the induced module \((k_ p)^ G\) of the trivial \(kP\) module \(k_ p\) is completely reducible as a right \(kG\)-module. \textit{T. Okuyama} has shown that a \(p\)- radical group is \(p\)-solvable [Osaka J. Math. 23, 467-469 (1986; Zbl 0611.20006)]. \textit{Y. Tsushima} has given group-theoretic characterisations of a \(p\)-radical \(p\)-nilpotent group [J. Algebra 103, 80-86 (1986; Zbl 0597.16011)]. This paper examines the structure of a certain \(p\)-radical group of \(p\)- length 2, the actual conditions being somewhat complicated. It is also shown that a \(p\)-radical group of minimal nilpotency index has \(p\)-length at most 2.