On a lattice characterization of finite soluble $PST$-groups

TL;DR

This paper characterizes finite soluble PST-groups using lattice-theoretic properties of subgroups related to a class of groups, providing a new criterion based on centrality conditions of chief factors.

Contribution

It introduces a lattice-based characterization of finite soluble PST-groups, linking subgroup properties to the structure of chief factors and centrality conditions.

Findings

Finite soluble PST-groups are characterized by subgroup centrality conditions.

A subgroup lattice criterion involving nilpotent groups determines PST-group status.

The structure of chief factors is key to understanding PST-group properties.

Abstract

Let $\mathfrak{F}$ be a class of finite groups and $G$ a finite group. Let ${\cal L}_{\mathfrak{F}}(G)$ be the set of all subgroups $A$ of $G$ with $A^{G}/A_{G}\in \mathfrak{F}$. A chief factor $H/K$ of $G$ is $\mathfrak{F}$-central in $G$ if $(H/K)\rtimes (G/C_{G}(H/K)) \in\mathfrak{F}$. We study the structure of $G$ under the hypothesis that every chief factor of $G$ between $A_{G}$ and $A^{G}$ is $\mathfrak{F}$-central in $G$ for every subgroup $A\in {\cal L}_{\mathfrak{F}}(G)$. As an application, we prove that a finite soluble group $G$ is a $PST$-group if and only if $A^{G}/A_{G}\leq Z_{\infty}(G/A_{G})$ for every subgroup $A\in {\cal L}_{\mathfrak{N}}(G)$, where $\mathfrak{N}$ is the class of all nilpotent groups.