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

Abstract

Let $\sigma =\{{\sigma }_i|i\in I\}$ be some partition of the set of all primes $\mathbb{P}$ 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 $G$.

arXiv comment15 pages. arXiv admin note: text overlap with arXiv:1704.02509