Some new characterizations of $PST$-groups

TL;DR

This paper investigates the influence of quasipermutable subgroups on finite group structures and provides new characterizations of soluble PST-groups based on these properties.

Contribution

It introduces new characterizations of soluble PST-groups using the concepts of quasipermutable and S-quasipermutable subgroups.

Findings

Characterizations of soluble PST-groups derived from subgroup properties

Analysis of the influence of quasipermutable subgroups on group structure

New criteria for group solubility based on subgroup permutability

Abstract

Let $H$ and $B$ be subgroups of a finite group $G$ such that $G=N_{G}(H)B$. Then we say that $H$ is \emph{quasipermutable} (respectively \emph{$S$-quasipermutable}) in $G$ provided $H$ permutes with $B$ and with every subgroup (respectively with every Sylow subgroup) $A$ of $B$ such that $(|H|, |A|)=1$. In this paper we analyze the influence of $S$-quasipermutable and quasipermutable subgroups on the structure of $G$. As an application, we give new characterizations of soluble $PST$-groups.