# A characterization of finite $σ$-soluble $PσT$-groups

### Chenchen Cao

Ningbo University, China### Dein Wong

China University of Mining and Technology, Xuzhou, China### Chi Zhang

China University of Mining and Technology, Xuzhou, China

## Abstract

Let $F$ be a non-empty class of groups, $G$ a finite group and $L(G)$ be the lattice of all subgroups of $G$. A chief factor $H/K$ of $G$ is $F$-central in $G$ if $(H/K)⋊(G/C_{G}(H/K))∈F$. Let $L_{cF}(G)$ be the set of subgroups $A$ of $G$ such that every chief factor of $G$ between $A_{G}$ and $A_{G}$ is $F$-central in $G$; let $L_{F}(G)$ be the set of subgroups $A$ of $G$ such that $A_{G}/A_{G}∈F$. In this paper, we study the influence of $L_{F}(G)$ and $L_{cF}(G)$ on the structure of $G$, where $F$ is a normally hereditary saturated formation containing all $σ$-nilpotent groups and $D=G_{F}$ is $σ$-soluble. Moreover, we give a new characterization of a finite $σ$-soluble group to be a $PσT$-group.

## Cite this article

Chenchen Cao, Dein Wong, Chi Zhang, A characterization of finite $σ$-soluble $PσT$-groups. Rend. Sem. Mat. Univ. Padova 148 (2022), pp. 203–212

DOI 10.4171/RSMUP/108