A Robinson characterization of finite $Psigma T$-groups

Let $sigma ={sigma_{i} | iin I}$ 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 $G$.