An Analog for the Frattini Factorization of Finite Groups

Abstract

Using the classification of finite simple groups, we prove that if H is an insoluble normal subgroup of a finite group G, then H contains a maximal soluble subgroup S such that G=HNG(S). Thereby Problem 14.62 in the “Kourovka Notebook” is given a positive solution. As a consequence, it is proved that in every finite group, there exists a subgroup that is simultaneously a \({\mathfrak{S}}\)-projector and a \({\mathfrak{S}}\)-injector in the class, \({\mathfrak{S}}\), of all soluble groups.