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

TL;DR

This paper characterizes finite groups where the property of --permutability is transitive, extending the understanding of subgroup permutability within the framework of --groups.

Contribution

It provides a Robinson-type characterization of finite --groups, focusing on the transitivity of --permutability.

Findings

Characterization of --permutability transitivity in finite groups

Extension of Robinson's characterization to --groups

Conditions under which --permutability is transitive

Abstract

Let $\sigma =\{\sigma_{i} | i\in I\}$ be some partition of the set of all primes $\Bbb{P}$ and let $G$ be a finite group. Then $G$ is said to be $\sigma $-full if $G$ has a Hall $\sigma _{i}$-subgroup for all $i$. A subgroup $A$ of $G$ is said to be $\sigma$-permutable in $G$ provided $G$ is $\sigma $-full and $A$ permutes with all Hall $\sigma _{i}$-subgroups $H$ of $G$ (that is, $AH=HA$) for all $i$. We obtain a characterization of finite groups $G$ in which $\sigma$-permutability is a transitive relation in $G$, that is, if $K$ is a ${\sigma}$-permutable subgroup of $H$ and $H$ is a ${\sigma}$-permutable subgroup of $G$, then $K$ is a ${\sigma}$-permutable subgroup of $G$.