  Let $\sigma =\{\sigma_{i} | i\in I\}$ be some partition of the set of all
primes $\Bbb{P}$ and $G$ a finite group. $G$ is said to be $\sigma$-soluble if
every chief factor $H/K$ of $G$ is a $\sigma _{i}$-group for some $i=i(H/K)$.
  A set ${\cal H}$ of subgroups of $G$ is said to be a complete Hall $\sigma
$-set of $G$ if every member $\ne 1$ of ${\cal H}$ is a Hall $\sigma
_{i}$-subgroup of $G$ for some $\sigma _{i}\in \sigma $ and ${\cal H}$ contains
exactly one Hall $\sigma _{i}$-subgroup of $G$ for every $i \in I$ such that
$\sigma _{i}\cap \pi (G)\ne \emptyset$. A subgroup $A$ of $G$ is said to be
${\sigma}$-permutable in $G$ if $G$ has a complete Hall $\sigma$-set $\cal H$
such that $AH^{x}=H^{x}A$ for all $x\in G$ and all $H\in \cal H$.
  We obtain characterizations of finite $\sigma$-soluble groups $G$ in which
$\sigma$-permutability is a transitive relation in $G$.
