A remark on p-radical groups
Let FG be the group algebra of a finite group G over an algebraically closed field F of characteristic \(p>0\). Let P be a p-Sylow subgroup of G. Following \textit{K. Motose} and \textit{Y. Ninomiya} [Math. J. Okayama Univ. 17, 171-176 (1975; Zbl 0316.20004)], G is said to be p-radical if the induced module \((F_ P)^ G\) from the trivial FP-module \(F_ P\) is a direct sum of simple FG-modules. \textit{T. Okuyama} proved that a p-radical group is p-soluble [Hokkaido Math. J. 10, 299-318 (1981; Zbl 0488.20013)]. The author derives a useful sufficient condition that G should be p-radical, namely that for any simple FG-module S whose vertex is Q the restriction to \(F[N_ G(Q)]\) should be a simple \(F[N_ G(Q)]\)- module.
scientific article; zbMATH DE number 4173217