A Robinson characterization of finite $PσT$-groups

Abstract

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