\(p\)-nilpotence of finite groups and minimal subgroups.
type: Item
instance of: scholarly article
title: \(p\)-nilpotence of finite groups and minimal subgroups.
zbMATH Open document ID: 1072.20020
published in: Journal of Algebra
publication date: 2004-02-03T00:00:00Z
review text: Let \(G\) be a finite group. A group \(G\) is called \(p\)-nilpotent if \(G_{p'}\) is normal in \(G\). It is proved that a finite group \(G\) is \(p\)-nilpotent if every minimal subgroup of \(P\cap O^p(G)\) is quasinormal in \(P\) and \(N_G(P)\) is \(p\)-nilpotent, and when \(p= 2\) either \([\Omega_2(P\cap O^p(G)),P]\leq\Omega_1(P\cap O^p(G))\) or \(P\) is quaternion free, where \(p\) is a prime dividing the order of \(G\) and \(P\) is a Sylow \(p\)-subgroup of \(G\). By using this result, the authors also obtain a series of corollaries for \(p\)-nilpotence and some results about solvability and supersolvability of finite groups.
reviewed by: N. M. Kurnosenko
Mathematics Subject Classification ID: 20D15
Mathematics Subject Classification ID: 20D20
Mathematics Subject Classification ID: 20D10
Mathematics Subject Classification ID: 20D40
zbMATH DE Number: 2037074
zbMATH Keywords: \(p\)-nilpotence
zbMATH Keywords: solvability
zbMATH Keywords: supersolvability
zbMATH Keywords: minimal subgroups
zbMATH Keywords: Sylow subgroups
zbMATH Keywords: permutable subgroups
author: Xiuyun Guo
author: K. P. Shum
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2003.05.004
OpenAlex ID: W2032548447
cites work: Finite groups with nilpotent maximal A-invariant subgroups
cites work: Some results on \(p\)-nilpotence and solubility of finite groups
cites work: Finite groups whose minimal subgroups are normal
cites work: M-groups and 2-groups
cites work: Solvability of groups of odd order
cites work: Q5577154
cites work: The influence of minimal subgroups of focal subgroups on the structure of finite groups
cites work: Endliche Gruppen I
cites work: Q4161846
cites work: Q3961021
cites work: Zum Satz von Sylow
DOI: 10.1016/J.JALGEBRA.2003.05.004
description: scientific article; zbMATH DE number 2037074