On the lattice of the $σ$-permutable subgroups of a finite group (1703.01773v2)

Abstract: 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.