On the lattice of the $\sigma$-permutable subgroups of a finite group

Abstract

Let $\sigma =\{{\sigma }_i|i\in I\}$ be some partition of the set of all primes $\mathbb{P}$, $G$ a finite group and $\sigma (G)=\{{\sigma }_i|{\sigma }_i\cap \pi (G)\ne \varnothing \}$. A set $\mathcal{H}$ of subgroups of $G$ is said to be a complete Hall $\sigma$-set of $G$ if every member $\ne 1$ of $\mathcal{H}$ is a Hall ${\sigma }_i$-subgroup of $G$ for some ${\sigma }_i\in \sigma$ and $\mathcal{H}$ 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.

arXiv comment13 pages