On \(G\)-covering subgroup systems of finite groups.
type: Item
instance of: scholarly article
title: On \(G\)-covering subgroup systems of finite groups.
zbMATH Open document ID: 1249.20015
author: A. N. Skiba
author: Wenbin Guo
published in: Acta Mathematica Hungarica
publication date: 2012-02-13T00:00:00Z
review text: Let \(\mathcal F\) be a class of groups and \(G\) be a finite group. A set \(\Sigma\) of subgroups of \(G\) is called a \(G\)-covering subgroup system for the class \(\mathcal F\) if \(\Sigma\subseteq\mathcal F\) implies \(G\in\mathcal F\). Let \(p\) be an odd prime dividing the size of a finite group \(G\). For a \(p\)-Sylow subgroup \(P\) of \(G\), Thompson's \(p\)-nilpotence criterion states that the set \(\{N_G(J(P)),C_G(Z(G))\}\) is a \(G\)-covering subgroup system for the class of all \(p\)-nilpotent groups. By the Glauberman-Thompson theorem the set \(\{N_G(Z(J(P)))\}\) is a \(G\)-covering subset for the class of all \(p\)-nilpotent groups. Let \(\Sigma\) be the set of all normalizers of all Sylow subgroups of \(G\). Then \(\Sigma\) is a \(G\)-covering subgroup system for the class of all nilpotent groups by \textit{M.~Bianchi, A.~Gillio Berta Mauri} and \textit{P.~Hauck} [Arch. Math. 47, 193-197 (1986; Zbl 0605.20017)], but not for the class of all supersoluble groups (see \textit{V.~Fedri} and \textit{L.~Serena} [Arch. Math. 50, No. 1, 11-18 (1988; Zbl 0638.20013)]). The paper continues this approach by considering the following sets of subgroups. Let \(p,q\) be distinct primes dividing the size of a finite group \(G\) and let \(P\) be a \(p\)-Sylow subgroup of \(G\). A set of subgroups of \(G\) containing at least one supplement in \(G\) to each maximal subgroup of \(P\) will be denoted by \(\Sigma_p\). The notation \(\Sigma_q\) is defined similarly. The main result is the following. Theorem A. The set \(\Sigma_p\cup\Sigma_q\) is a \(G\)-covering subgroup system for the classes of all metanilpotent, all supersoluble, all \(p\)-supersoluble, all \(p\)-nilpotent and all Abelian groups.
Mathematics Subject Classification ID: 20D10
Mathematics Subject Classification ID: 20D15
Mathematics Subject Classification ID: 20D20
Mathematics Subject Classification ID: 20D30
zbMATH DE Number: 6006193
zbMATH Keywords: supplements of subgroups
zbMATH Keywords: covering subgroup systems
zbMATH Keywords: Sylow subgroups
zbMATH Keywords: maximal subgroups
zbMATH Keywords: \(p\)-supersoluble groups
zbMATH Keywords: \(p\)-nilpotent groups
zbMATH Keywords: \(p\)-decomposable groups
zbMATH Keywords: metanilpotent groups
MaRDI profile type: Publication
full work available at URL: s10474-011-0088-0
OpenAlex ID: W2018867203
cites work: Classes of Finite Groups
cites work: On finite groups with nilpotent Sylow-normalizers
cites work: Finite soluble groups
cites work: Finite soluble groups with supersoluble Sylow normalizers
cites work: \(G\)-covering subgroup systems for the classes of supersoluble and nilpotent groups.
cites work: Q5715057
cites work: Q2762626
cites work: Q5478072
cites work: On weakly \(s\)-permutable subgroups of finite groups.
cites work: Endliche Gruppen I
cites work: On the \(\mathcal X\Phi\)-hypercentre of finite groups.
cites work: Über die Normalstruktur von mehrfach faktorisierten Gruppen
cites work: Q3949183
cites work: Q5577154
DOI: 10.1007/S10474-011-0088-0
Recommended article: \(G\)-covering subgroup systems for the classes of supersoluble and nilpotent groups.
Recommended article: Q5715057
Recommended article: A covering subgroup system for the class of \(p\)-nilpotent groups.
Recommended article: \(G\)-covering subgroup systems for the class of all \(\sigma \)-nilpotent finite groups
Recommended article: On one criterion for the \(\sigma \)-nilpotency of a finite group
Recommended article: Q5294749
Recommended article: Q3369457
Recommended article: Some necessary and sufficient conditions for p-nilpotence of finite groups
Recommended article: A contribution to the theory of finite supersolvable groups
Recommended article: The influence of ℋ-subgroups on the structure of finite groups
description: scientific article; zbMATH DE number 6006193