Some classes of finite groups and mutually permutable products.
type: Item
instance of: scholarly article
title: Some classes of finite groups and mutually permutable products.
zbMATH Open document ID: 1148.20013
author: Adolfo Ballester-Bolinches
author: M. Asaad
author: J. C. Beidleman
author: R. Esteban-Romero
published in: Journal of Algebra
publication date: 2008-07-11T00:00:00Z
full work available at URL: http://hdl.handle.net/10251/19092
review text: A group \(G=AB\) is said to be a `mutually permutable product' of its subgroups \(A\) and \(B\) if \(A\) permutes with every subgroup of \(B\) and \(B\) permutes with every subgroup of \(A\). This paper is devoted to the study of mutually permutable products of finite groups in relation with certain classes of groups. The authors consider the class \(\mathcal Y\) of finite groups satisfying a converse of Lagrange's theorem in the following sense: \(G\) is a `\(\mathcal Y\)-group' if for every subgroup \(H\) and all primes \(q\) dividing the index \(|G:H|\) there exists a subgroup \(K\) of \(G\) such that \(H\) is contained in \(K\) and \(|K:H|=q\). \(\mathcal Y\)-groups have been studied in [\textit{M. Weinstein} (ed.), Between nilpotent and solvable. Passaic, New Jersey: Polygonal Publishing House (1982; Zbl 0488.20001)] and it is proved, in particular, that they are supersoluble groups. Moreover, in the paper under review, the authors also consider finite groups whose chief factors are simple, the so called `SC-groups'. It is proved that the mutually permutable product of a \(\mathcal Y\)-group and an SC-group is an SC-group, by means of local versions, depending on prime numbers, of both classes of groups. A local characterization of \(\mathcal Y\)-groups was obtained by the last three authors [in J. Algebra 312, No. 1, 445-454 (2007; Zbl 1119.20019)]. -- Another result in the paper states that the product of pairwise mutually permutable \(\mathcal Y\)-groups is supersoluble. \textit{H. Heineken} and the third author proved [in J. Algebra 297, No. 2, 409-416 (2006; Zbl 1104.20024)] the following: If a finite group \(G\) is the mutually permutable product of subgroups \(A\) and \(B\) and \(G\) is a PST-group (that is, a group in which every subnormal subgroup permutes with all Sylow subgroups), then \(A\) and \(B\) are PST-groups. In the paper under review the authors present a local version of this result.
reviewed by: Ana Martínez-Pastor
Mathematics Subject Classification ID: 20D40
Mathematics Subject Classification ID: 20D10
Mathematics Subject Classification ID: 20D35
Mathematics Subject Classification ID: 20D20
zbMATH DE Number: 5300812
zbMATH Keywords: finite groups
zbMATH Keywords: products of groups
zbMATH Keywords: mutually permutable products
zbMATH Keywords: permutability
zbMATH Keywords: \(\mathcal Y\)-groups
zbMATH Keywords: SC-groups
zbMATH Keywords: PST-groups
zbMATH Keywords: supersoluble groups
MaRDI profile type: Publication
OpenAlex ID: W1971255965
cites work: Finite Groups whose Subnormal Subgroups Permute with all Sylow Subgroups
cites work: Permutable products of supersoluble groups.
cites work: Finite soluble groups with permutable subnormal subgroups
cites work: Characterization of finite groups with some \(S\)-quasinormal subgroups.
cites work: On the supersolvability of finite groups
cites work: On some classes of supersoluble groups.
cites work: Groups in which Sylow subgroups and subnormal subgroups permute.
cites work: A Local Approach to Certain Classes of Finite Groups
cites work: Totally permutable products of finite groups satisfying SC or PST.
cites work: On totally permutable products of finite groups.
cites work: On mutually permutable products of finite groups.
cites work: Sylow permutable subnormal subgroups of finite groups. II
cites work: Sylow permutable subnormal subgroups of finite groups
cites work: On finiteJ-groups
cites work: Totally permutable products of certain classes of finite groups.
cites work: Totally permutable torsion subgroups
cites work: Criteria for permutability to be transitive in finite groups
cites work: On the fitting core of a formation
cites work: Mutually permutable subgroups and group classes.
cites work: Group classes and mutually permutable products.
cites work: The Wielandt Subgroup of a Finite Soluble Group
cites work: Finite soluble groups
cites work: Gruppen, in denen das Normalteilersein transitiv ist.
cites work: Endliche Gruppen I
cites work: Produkte nilpotenter Gruppen
cites work: Sylow-Gruppen und Subnormalteiler endlicher Gruppen
cites work: Q3744347
cites work: A Note on Finite Groups in Which Normality is Transitive
cites work: The structure of finite groups in which permutability is a transitive relation
cites work: Q5518443
cites work: Q3949183
cites work: Über das Produkt paarweise vertauschbarer nilpotenter Gruppen
cites work: Q5510848
DOI: 10.1016/J.JALGEBRA.2007.12.001
Recommended article: Q5869362
Recommended article: Q4913124
Recommended article: Totally permutable products of finite groups satisfying SC or PST.
Recommended article: On mutually permutable products of finite groups.
Recommended article: Group classes and mutually permutable products.
Recommended article: Mutually permutable subgroups and group classes.
Recommended article: Q3442599
Recommended article: Some results on products of finite groups
Recommended article: On pairwise mutually permutable products
Recommended article: Q4609059
description: scientific article; zbMATH DE number 5300812