Finite groups with permutable Hall subgroups

TL;DR

This paper investigates the structure of finite groups that have a complete Hall σ-set and where certain subgroups permute with all members of this set, revealing new structural insights.

Contribution

It introduces the concept of permutable subgroups within complete Hall σ-sets and analyzes their impact on the structure of finite groups.

Findings

Characterization of groups with permutable Hall σ-subgroups

Conditions under which subgroup permutability influences group structure

New structural theorems for finite groups with specific subgroup properties

Abstract

Let $\sigma =\{\sigma_{i} | i\in I\}$ be a partition of the set of all primes $\Bbb{P}$ and $G$ a finite group. A set ${\cal H}$ of subgroups of $G$ is said to be a \emph{complete Hall $\sigma $-set} of $G$ if every member $\ne 1$ of ${\cal H}$ is a Hall $\sigma _{i}$-subgroup of $G$ for some $i\in I$ and $\cal H$ contains exactly one Hall $\sigma _{i}$-subgroup of $G$ for every $i$ such that $\sigma _{i}\cap \pi (G)\ne \emptyset$. In this paper, we study the structure of $G$ assuming that some subgroups of $G$ permutes with all members of ${\cal H}$.