An Analog for the Frattini Factorization of Finite Groups

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 $G$ such that $G=HN_G(S)$. 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.