Finite Groups with $\mathfrak F$-Subnormal Subgroups

Abstract

Let $G$ be a finite group, let $M$ be a maximal subgroup of $G$, and let $\mathfrak F$ be a hereditary formation consisting of solvable groups. The metanilpotency of the $\mathfrak F$-residual $G^\mathfrak F$ is established under the assumption that all subgroups maximal in $M$ are $\mathfrak F$-subnormal in $G$, and the nilpotency of $G^\mathfrak F$ is established in the case where $\mathfrak F$ is saturated. Properties of the group $G$ are indicated in more detail for the formation of all solvable groups with Abelian Sylow subgroups, for the formation of all supersolvable groups, and for the formation of all groups with nilpotent commutator subgroup.