A contribution to the theory of finite supersoluble groups
type: Item
instance of: scholarly article
title: A contribution to the theory of finite supersoluble groups
zbMATH Open document ID: 0809.20009
published in: Rendiconti del Seminario Matematico della Università di Padova
publication date: 1995-03-29T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/108283
review text: Criteria for \(p\)-supersolubility of \(p\)-soluble groups and for supersolubility of groups are provided in terms of subgroups possessing the cover-avoidance property (CAP). A subgroup \(M\) of a group \(G\) is said to be a CAP-subgroup of \(G\), if each chief factor of \(G\) is either covered or avoided by \(M\). Let \(G\) be a \(p\)-soluble group. Then the following statements are equivalent: (i) \(G\) is \(p\)-supersoluble; (ii) all \(p\)-subgroups of \(G\) are CAP-subgroups of \(G\), (iii) all maximal subgroups of the Sylow \(p\)-subgroups of \(G\) are CAP-subgroups of \(G\); (iv) all maximal subgroups of \(O_{p'p}(G)\) containing \(O_{p'}(G)\) are CAP-subgroups of \(G\); (v) there exists a normal subgroup \(H\) of \(G\) such that \(G/H\) is \(p\)-supersoluble and all maximal subgroups of any Sylow \(p\)-subgroup of \(H\) are CAP-subgroups of \(G\); (vi) there exists a normal subgroup \(H\) of \(G\) such that \(G/H\) is \(p\)-supersoluble and all maximal subgroups of \(O_{p'p} (H)\) containing \(O_{p'}(H)\) are CAP-subgroups of \(G\) (Corollary 1). The criterion for supersolubility is of similar nature. Let \(G\) be a group and let \(p\) denote the largest prime dividing the order of \(G\). Assume that for all primes \(q\) different from \(p\) every maximal subgroup of the Sylow \(q\)-subgroups of \(G\) is a CAP-subgroup of \(G\). Then (i) \(G\) possesses a Sylow tower and (ii) \(G/O_ p (G)\) is supersoluble.
reviewed by: Hans Lausch
Mathematics Subject Classification ID: 20D10
Mathematics Subject Classification ID: 20D30
Mathematics Subject Classification ID: 20D20
zbMATH DE Number: 492571
zbMATH Keywords: \(p\)-supersolubility of \(p\)-soluble groups
zbMATH Keywords: supersolubility of groups
zbMATH Keywords: cover-avoidance property
zbMATH Keywords: chief factor
zbMATH Keywords: \(p\)-subgroups
zbMATH Keywords: CAP-subgroups
zbMATH Keywords: maximal subgroups
zbMATH Keywords: Sylow \(p\)-subgroups
zbMATH Keywords: Sylow tower
author: Luis M. Ezquerro
MaRDI profile type: Publication
cites work: Influence of \(\pi\)-quasinormality on maximal subgroups of Sylow subgroups of Fitting subgroup of a finite group
cites work: On maximal subgroups of finite groups
cites work: Endliche Gruppen I
cites work: Influence of normality on maximal subgroups of Sylow subgroups of a finite group
cites work: Two sufficient conditions for supersolvability of finite groups
description: scientific article; zbMATH DE number 492571