The influence of SS-quasinormality of some subgroups on the structure of finite groups.
type: Item
instance of: scholarly article
title: The influence of SS-quasinormality of some subgroups on the structure of finite groups.
zbMATH Open document ID: 1152.20019
author: Zhencai Shen
author: Xiaochun Liu
author: Shirong Li
author: Jian-Jun Liu
published in: Journal of Algebra
publication date: 2008-07-29T00:00:00Z
review text: The authors take up the concept of S-quasinormality occurring in \textit{O. H. Kegel} [Math. Z. 78, 205-221 (1962; Zbl 0102.26802)] and now generalise it to SS-quasinormality: a subgroup \(H\) of a finite group \(G\) is defined as being an SS-quasinormal subgroup of \(G\) if \(H\) is supplemented by, \(B\) say, in \(G\) such that \(H\) permutes with every Sylow subgroup of \(B\) (Definition 1.1). That new concept is being used to derive a number of sufficient conditions for groups being \(p\)-nilpotent (Theorem 1.1, 1.2, 1.7), \(p\)-supersoluble (Theorem 1.3, 1.4), belonging to saturated formations containing the class of supersoluble groups (Theorem 1.5), and belonging to classes of groups with the Sylow tower property of supersoluble type (Theorem 1.8). Throughout the paper, SS-quasinormality is used together with the concept of \(\mathcal M(G)\), the family of all maximal subgroups of all Sylow subgroups of \(G\); moreover, if \(P\) is a \(p\)-group and \(d\) the minimal number of generators of \(P\), that of \(\mathcal M_d(P)=\{P_1,\dots,P_d\}\), a set of maximal subgroups of \(P\) such that \(\bigcap^d_{i=1}P_i\) equals the Frattini subgroup of \(P\).
reviewed by: Hans Lausch
Mathematics Subject Classification ID: 20D35
Mathematics Subject Classification ID: 20D40
Mathematics Subject Classification ID: 20D20
Mathematics Subject Classification ID: 20D10
Mathematics Subject Classification ID: 20D15
Mathematics Subject Classification ID: 20E28
zbMATH DE Number: 5304648
zbMATH Keywords: \(p\)-nilpotency
zbMATH Keywords: \(p\)-supersolubility
zbMATH Keywords: quasinormal subgroups
zbMATH Keywords: maximal subgroups
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2008.01.030
OpenAlex ID: W2044174215
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DOI: 10.1016/J.JALGEBRA.2008.01.030
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description: scientific article; zbMATH DE number 5304648