On the lattice of the $σ$-permutable subgroups of a finite group

Abstract

Let $σ=\{σ_{i} | i\in I\}$ be some partition of the set of all primes $\Bbb{P}$, $G$ a finite group and $σ(G) =\{σ_{i} |σ_{i}\cap π(G)\ne \emptyset \}$. A set ${\cal H}$ of subgroups of $G$ is said to be a complete Hall $σ$-set of $G$ if every member $\ne 1$ of ${\cal H}$ is a Hall $σ_{i}$-subgroup of $G$ for some $σ_{i}\in σ$ and ${\cal H}$ contains exactly one Hall $σ_{i}$-subgroup of $G$ for every $σ_{i}\in σ(G)$. A subgroup $A$ of $G$ is said to be $σ$-permutable in $G$ if $G$ possesses a complete Hall $σ$-set and $A$ permutes with each Hall $σ_{i}$-subgroup $H$ of $G$, that is, $AH=HA$ for all $i \in I$. We characterize finite groups with distributive lattice of the $σ$-permutable subgroups.