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 ${ \cal H}$.
Categories: math.GR