On some products of finite groups

TL;DR

This paper extends classical results on the supersolubility of finite groups formed as products of subgroups, showing new conditions under which the product remains supersoluble or w-supersoluble, especially involving subgroup normality and permutation properties.

Contribution

It introduces new criteria involving subgroup normality and permutation conditions that ensure the product of supersoluble subgroups is supersoluble, generalizing previous results.

Findings

If G=AB with A normal and B permuting with maximal subgroups of A, then G is supersoluble if G' is nilpotent.

Similar results hold for w-supersoluble groups under the same conditions.

The paper also explores cases where A and B intersect trivially, leading to broader applicability.

Abstract

A classical result of Baer states that a finite group $ G $ which is the product of two normal supersoluble subgroups is supersoluble if and only if $ G' $ is nilpotent. In this article we show that if $ G=AB $ is the product of supersoluble (respectively, $ w $-supersoluble) subgroups $ A $ and $ B $, $ A $ is normal in $ G $, $ B $ permutes with every maximal subgroup of each Sylow subgroup of $ A $, then $ G $ is supersoluble (respectively, $ w $-supersoluble) provided that $ G' $ is nilpotent. We also investigate products of subgroups defined above when $ A\cap B=1 $ and obtain more general results.