Mutually permutable products of finite groups.
type: Item
instance of: scholarly article
title: Mutually permutable products of finite groups.
zbMATH Open document ID: 1227.20018
DOI: 10.5402/2011/867082
author: Rola A. Hijazi
published in: ISRN Algebra
publication date: 2011-10-28T00:00:00Z
review text: Summary: Let \(G\) be a finite group and \(G_1,G_2\) are two subgroups of \(G\). We say that \(G_1\) and \(G_2\) are mutually permutable if \(G_1\) is permutable with every subgroup of \(G_2\) and \(G_2\) is permutable with every subgroup of \(G_1\). We prove that if \(G=G_1G_2=G_1G_3=G_2G_3\) is the product of three supersolvable subgroups \(G_1\), \(G_2\), and \(G_3\), where \(G_i\) and \(G_j\) are mutually permutable for all \(i\) and \(j\) with \(i\neq j\) and the Sylow subgroups of \(G\) are Abelian, then \(G\) is supersolvable. As a corollary of this result, we also prove that if \(G\) possesses three supersolvable subgroups \(G_i\) (\(i=1,2,3\)) whose indices are pairwise relatively prime, and \(G_i\) and \(G_j\) are mutually permutable for all \(i\) and \(j\) with \(i\neq j\), then \(G\) is supersolvable.
Mathematics Subject Classification ID: 20D40
Mathematics Subject Classification ID: 20D20
Mathematics Subject Classification ID: 20D10
zbMATH DE Number: 5964872
zbMATH Keywords: finite groups
zbMATH Keywords: mutually permutable subgroups
zbMATH Keywords: triple factorisations
zbMATH Keywords: products of supersolvable subgroups
zbMATH Keywords: Abelian Sylow subgroups
zbMATH Keywords: supersolvability
Wikidata QID: Q58690060
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.5402/2011/867082
OpenAlex ID: W2041057878
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description: scientific article; zbMATH DE number 5964872