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

## TL;DR

This paper characterizes finite groups whose lattice of ${\sigma}$-permutable subgroups is distributive, based on the structure of Hall ${\sigma}$-subgroups and their permutability properties.

## Contribution

It provides a new characterization of finite groups with a distributive lattice of ${\sigma}$-permutable subgroups, extending previous subgroup lattice theories.

## Key findings

- Identifies conditions for the distributivity of the subgroup lattice
- Links subgroup permutability with lattice properties
- Provides structural insights into ${\sigma}$-permutable subgroups

## 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.