Cover-avoidance properties and the structure of finite groups
type: Item
instance of: scholarly article
title: Cover-avoidance properties and the structure of finite groups
zbMATH Open document ID: 1028.20014
DOI: 10.1016/S0022-4049(02)00327-4
published in: Journal of Pure and Applied Algebra
publication date: 2003-06-18T00:00:00Z
review text: Let \(G\) be a finite group. The authors call a subgroup \(A\) of the group \(G\) a CAP-subgroup of \(G\) if for any chief factor \(H/K\) of \(G\) \(H\cap A=K\cap A\) or \(HA=KA\). Some characterizations for a finite group to be solvable are obtained when some of its maximal subgroups or \(2\)-maximal subgroups are CAP-subgroups. In particular, it is proved that a group \(G\) is solvable if and only if there exists a solvable \(2\)-maximal subgroup \(L\) of \(G\) such that \(L\) is a CAP-subgroup of \(G\). It is also proved that \(G\) is \(p\)-solvable if and only if a Sylow \(p\)-subgroup \(P\) of \(G\) is a CAP-subgroup of \(G\).
Mathematics Subject Classification ID: 20D10
Mathematics Subject Classification ID: 20D20
Mathematics Subject Classification ID: 20D30
Mathematics Subject Classification ID: 20E28
zbMATH DE Number: 1930186
zbMATH Keywords: \(p\)-solvability
zbMATH Keywords: \(p\)-nilpotency
zbMATH Keywords: finite groups
zbMATH Keywords: maximal subgroups
zbMATH Keywords: CAP-subgroups
zbMATH Keywords: Sylow subgroups
author: Xiuyun Guo
author: K. P. Shum
reviewed by: N. M. Kurnosenko
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1930186