A characterization of finite $\sigma$-soluble $P\sigma T$-groups

Abstract

Let $\mathfrak{F}$ be a non-empty class of groups, $G$ a finite group and $\mathcal{L}(G)$ be the lattice of all subgroups of $G$. A chief factor $H/K$ of $G$ is $\mathfrak{F}$-central in $G$ if $(H/K)⋊(G/C_G(H/K))\in \mathfrak{F}$. Let ${\mathcal{L}}_{c\mathfrak{F}}(G)$ be the set of subgroups $A$ of $G$ such that every chief factor of $G$ between $A^G$ and $A_G$ is $\mathfrak{F}$-central in $G$; let ${\mathcal{L}}_{\mathfrak{F}}(G)$ be the set of subgroups $A$ of $G$ such that $A^G/A_G\in \mathfrak{F}$. In this paper, we study the influence of ${\mathcal{L}}_{\mathfrak{F}}(G)$ and ${\mathcal{L}}_{c\mathfrak{F}}(G)$ on the structure of $G$, where $\mathfrak{F}$ is a normally hereditary saturated formation containing all $\sigma$-nilpotent groups and $D=G^{\mathfrak{F}}$ is $\sigma$-soluble. Moreover, we give a new characterization of a finite $\sigma$-soluble group to be a $P\sigma T$-group.