Permutable products of supersoluble groups.
type: Item
instance of: scholarly article
title: Permutable products of supersoluble groups.
zbMATH Open document ID: 1083.20019
author: Manuel J. Alejandre
author: Adolfo Ballester-Bolinches
author: John Cossey
published in: Journal of Algebra
publication date: 2004-11-19T00:00:00Z
review text: Let the finite group \(G=AB\) be the product of two supersoluble subgroups \(A\) and \(B\). It is well-known that \(G\) need not to be supersoluble, even if \(A\) and \(B\) are normal in \(G\). The authors consider the case that the subgroup \(A\) commutes with every subgroup of \(B\) and \(B\) commutes with every subgroup of \(A\), i.e. that \(G\) is the permutable product of \(A\) and \(B\). They show that in this case the Fitting factor group \(G/F(G)\) must be supersoluble and metabelian; also \(G/\text{Core}_G(A\cap B)\) is always supersoluble. This also provides a new proof of a theorem of Baer and a result of Asaad and Shaalan by which a mutually product of two supersoluble subgroups is supersoluble, provided that its commutator subgroup is nilpotent.
Mathematics Subject Classification ID: 20D40
Mathematics Subject Classification ID: 20D10
zbMATH DE Number: 2116868
zbMATH Keywords: finite groups
zbMATH Keywords: factorized groups
zbMATH Keywords: permutable products
zbMATH Keywords: supersoluble groups
reviewed by: Bernhard Amberg
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2003.01.002
OpenAlex ID: W2070000637
cites work: Q3134546
cites work: On the supersolvability of finite groups
cites work: Classes of finite groups and their properties
cites work: ON PRODUCTS OF FINITE SUPERSOLUBLE GROUPS
cites work: Q4240510
cites work: \(p\)-supersolvability of factorized finite groups
cites work: Q4240516
cites work: Finite soluble groups
cites work: Über das Produkt von zwei abelschen Gruppen
DOI: 10.1016/J.JALGEBRA.2003.01.002
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description: scientific article; zbMATH DE number 2116868