On p-radical groups
Let G be a finite group and let K be a field of characteristic \(p>0\). Let JK(G) be the Jacobson radical of the group ring K(G). Let P be a p-Sylow subgroup of G. G is said to be p-radical if \(\cap_{x\in G}K(G)JK(P^ x)=JK(G)\). [Cf. \textit{W. Feit}, The Representation Theory of Finite Groups (1982; Zbl 0493.20007), pp. 265-268.] If G is p-radical and H is a normal subgroup of G then H and G/H are p- radical groups. If H is a normal p-subgroup of G and G/H is p-radical then G is p-radical. \textit{T. Okuyama} has shown that a p-radical group is necessarily p-solvable (to appear). The author shows that if G has an abelian p-complement then G is p-radical. The main result, established using block and character theory (not all notation is explained), is that if \(G=PH\) is p-nilpotent where \(H=O_{p'}(G)\) then G is p-radical if and only if \([K,D]\cap C_ K(D)=1\) for any p-subgroup of G.
scientific article; zbMATH DE number 3961873