  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 $G$.
