  Let $\sigma =\{\sigma_{i} | i\in I\}$ be a partition of the set of all primes
$\Bbb{P}$ and $G$ a finite group. A set ${\cal H}$ of subgroups of $G$ is said
to be a \emph{complete Hall $\sigma $-set} of $G$ if every member $\ne 1$ of
${\cal H}$ is a Hall $\sigma _{i}$-subgroup of $G$ for some $i\in I$ and $\cal
H$ contains exactly one Hall $\sigma _{i}$-subgroup of $G$ for every $i$ such
that $\sigma _{i}\cap \pi (G)\ne \emptyset$. In this paper, we study the
structure of $G$ assuming that some subgroups of $G$ permutes with all members
of ${\cal H}$.
