On classes of finite groups with simple non-abelian chief factors

Abstract

Let $\mathfrak{J}$ be a class of non-abelian simple groups and $\mathfrak{X}$ be a class of groups. A chief factor $H/K$ of a group $G$ is called $\mathfrak{X}$-central in $G$ provided $(H/K)⋊G/C_G(H/K)\in \mathfrak{X}$. We say that $G$ is a $\mathfrak{J}cs$-$\mathfrak{X}$-\emph{group} if every chief $\mathfrak{X}$-factor of $G$ is $\mathfrak{X}$-central and other chief factors of $G$ are simple $\mathfrak{J}$-groups. We use ${\mathfrak{X}}_{\mathfrak{J}cs}$ to denote the class of all $\mathfrak{J}cs$-$\mathfrak{X}$-groups. A subgroup $U$ of a group $G$ is called $\mathfrak{X}$-\emph{maximal} in $G$ provided that $(a)$ $U\in \mathfrak{X}$, and $(b)$ if $U\le V\le G$ and $V\in \mathfrak{X}$, then $U=V$. In this paper we described the structure of $\mathfrak{J}cs$-$\mathfrak{H}$-groups for a solubly saturated formation $\mathfrak{H}$ and all hereditary saturated formations $\mathfrak{F}$ containing all nilpotent groups such that the ${\mathfrak{F}}_{\mathfrak{J}cs}$-hypercenter of $G$ coincides with the intersection of all ${\mathfrak{F}}_{\mathfrak{J}cs}$-maximal subgroups of $G$ for every group $G$.