On the \(\mathcal F\)-abnormal maximal subgroups of finite groups.
type: Item
instance of: scholarly article
title: On the \(\mathcal F\)-abnormal maximal subgroups of finite groups.
zbMATH Open document ID: 1110.20016
author: Ni Du
author: Shirong Li
published in: Journal of Pure and Applied Algebra
publication date: 2006-12-07T00:00:00Z
review text: For any saturated formation \(\mathcal F\) of finite groups containing all supersolvable groups, the groups in \(\mathcal F\) are characterized in the paper by the \(\mathcal F\)-abnormal maximal subgroups. Let \(\mathcal F\) be a saturated formation of finite groups and let \(\mathcal U\) be the class of all finite supersolvable groups. Let \(G\) be a finite group. A maximal subgroup \(M\) of \(G\) is called \(\mathcal F\)-normal in \(G\) if \(G/M_G\in{\mathcal F}\), where \(M_G\) denotes the core of \(M\) in \(G\); otherwise \(M\) is said to be \(\mathcal F\)-abnormal in \(G\). Let \({\mathcal M}_c (G)\) denote the set of \(\mathcal F\)-abnormal maximal subgroups \(M\) of \(G\) such that \(|G:M|\) is composite. Let \(\mathcal F\) be a saturated formation containing \(\mathcal U\). In the paper, the authors give the definition of \(s\)-\(\theta\)-completion for the first time and prove the following main results: (i) \(G\in{\mathcal F}\) if and only if every \(\mathcal F\)-abnormal maximal subgroup of \(G\) has index a prime; (ii) \(G\in{\mathcal F}\) if and only if, for every \(M\in{\mathcal M}_c(G)\), \(M\) has an \(s\)-\(\theta\)-completion \(C\) such that \(G=CM\) and \(C/M_G\) has square free order; (iii) Let \(G\) be a finite group which is \(S_4\)-free. Then \(G\in{\mathcal F}\) if and only if every \(M\in{\mathcal M}_c(G)\) has an \(s\)-\(\theta\)-completion \(C\) such that \(C/M_G\) is cyclic with \(|C/M_G|\geq|G:M|\).
Mathematics Subject Classification ID: 20D10
Mathematics Subject Classification ID: 20D40
Mathematics Subject Classification ID: 20D30
Mathematics Subject Classification ID: 20E28
Mathematics Subject Classification ID: 20D35
zbMATH DE Number: 5078575
zbMATH Keywords: saturated formations
zbMATH Keywords: supersolvable groups
zbMATH Keywords: abnormal maximal subgroups
reviewed by: Rodica Covaci
MaRDI profile type: Publication
full work available at URL: j.jpaa.2006.01.019
OpenAlex ID: W2133090906
cites work: A note on minimal subgroups of finite groups
cites work: On minimal subgroups of finite groups (III)
cites work: ON c-NORMAL MAXIMAL AND MINIMAL SUBGROUPS OF SYLOW SUBGROUPS OF FINITE GROUPS
cites work: Finite soluble groups
cites work: Q2762626
cites work: On the desking completions, theta completions and theta pairs for maximal subgroups*
cites work: The Deskins index complex and the supersolvability of finite groups
cites work: On the intersection of a class of maximal subgroups of a finite group. II
cites work: Q5577154
cites work: On finite groups with dihedral Sylow 2-subgroups
DOI: 10.1016/J.JPAA.2006.01.019
description: scientific article; zbMATH DE number 5078575