Influence of permutizers of subgroups on the structure of finite groups.
type: Item
instance of: scholarly article
title: Influence of permutizers of subgroups on the structure of finite groups.
zbMATH Open document ID: 1156.20014
author: Guohua Qian
author: Shouhong Qiao
author: Yanming Wang
published in: Journal of Pure and Applied Algebra
publication date: 2008-07-10T00:00:00Z
review text: The `permutiser' of a subgroup \(H\) in a group \(G\) is the subgroup generated by all cyclic subgroups of \(G\) that permute with \(H\). A group \(G\) satisfies the `permutiser condition' if a proper subgroup of \(G\) is a proper subgroup of its permutiser, and satisfies the `maximal permutiser condition' if the permutiser of every maximal subgroup of \(G\) coincides with \(G\). In this paper the authors study conditions under which a finite group satisfying the (maximal) permutiser condition is supersoluble. For instance, they characterise the supersoluble groups as the groups in which every normal subgroup with index at most~\(2\) satisfies the permutiser condition (Theorem~3.1.1). They also characterise the supersoluble groups as the groups \(G\) satisfying the maximal permutizer condition and contained in a quotient-closed class of groups containing \(G\) but not the symmetric group \(S_4\) (Theorem~3.1.3). This theorem is used to prove the equivalence of several group properties for groups satisfying the maximal permutizer condition. In the last part of the paper, the authors study some conditions under which a group \(G\) with a normal subgroup \(H\) such that \(G/H\) is supersoluble turns out to be supersoluble. This happens when \(G\) satisfies the maximal permutiser condition and \(H\) is \(V_4\)-free, where \(V_4\) is the Klein four-group (Theorem~3.2.1). When \(H\cap M\) is either \(H\) or a maximal subgroup of \(H\) for every maximal subgroup \(M\) of \(G\) and \(H\) satisfies the maximal permutiser condition, then \(G\) is \(p\)-supersoluble for all odd primes \(p\), \(G\) is supersoluble if \(H\) has odd order, and \(G\) is supersoluble if \(H\) is \(V_4\)-free (Theorem~3.2.2). Moreover, if every almost maximal subgroup \(L\) of \(H\) is supplemented in \(H\) by a cyclic subgroup, then \(G\) is supersoluble (Theorem~3.2.3). As a consequence, if \(H\) has all Sylow subgroups cyclic, then \(G\) is supersoluble (Corollary~3.2.4), and the classical result…
reviewed by: Adolfo Ballester-Bolinches
Mathematics Subject Classification ID: 20D10
Mathematics Subject Classification ID: 20D40
Mathematics Subject Classification ID: 20D15
Mathematics Subject Classification ID: 20D20
Mathematics Subject Classification ID: 20E28
zbMATH DE Number: 5299446
zbMATH Keywords: finite groups
zbMATH Keywords: supersolubility
zbMATH Keywords: permutability
zbMATH Keywords: permutizer condition
zbMATH Keywords: permutizers
zbMATH Keywords: maximal subgroups
MaRDI profile type: Publication
full work available at URL: j.jpaa.2008.03.004
OpenAlex ID: W2022214627
cites work: A characterization of the finite soluble groups
cites work: On maximal subgroups of finite groups
cites work: On finite groups satisfying the permutizer condition
cites work: Endliche Gruppen I
cites work: Implications of Permutizers of Some Subgroups in Finite Groups
cites work: A first course in group theory
cites work: Q3739339
DOI: 10.1016/J.JPAA.2008.03.004
description: scientific article; zbMATH DE number 5299446