Let $\sigma =\{ \sigma_{ i} | i\in I\}$ be some partition of the set of all primes $\Bbb{ P}$, $G$ a finite group and $\sigma (G) =\{ \sigma_{ i} |\sigma_{ i}\cap \pi (G)\ne \emptyset \}$. 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 $\sigma_{ i}\in \sigma (G)$. A subgroup $A$ of $G$ is said to be ${ \sigma}$-permutable in $G$ if $G$ possesses a complete Hall $\sigma $-set and $A$ permutes with each Hall $\sigma_{ i}$-subgroup $H$ of $G$, that is, $AH=HA$ for all $i \in I$. We characterize finite groups with distributive lattice of the ${ \sigma}$-permutable subgroups.
Categories: math.GR