On some products of finite groups
A. Ballester-Bolinches, S. Y. Madanha, M. C. Pedraza-Aguilera, X . Wu

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 which is the product of two normal supersoluble subgroups is supersoluble if and only if is nilpotent. In this article we show that if is the product of supersoluble (respectively, -supersoluble) subgroups and , is normal in , permutes with every maximal subgroup of each Sylow subgroup of , then is supersoluble (respectively, -supersoluble) provided that is nilpotent. We also investigate products of subgroups defined above when and obtain more general results.
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Taxonomy
TopicsFinite Group Theory Research · graph theory and CDMA systems
