Some results on \(p\)-nilpotence and solubility of finite groups
type: Item
instance of: scholarly article
title: Some results on \(p\)-nilpotence and solubility of finite groups
zbMATH Open document ID: 0961.20016
author: Adolfo Ballester-Bolinches
author: Xiuyun Guo
published in: Journal of Algebra
publication date: 2001-05-29T00:00:00Z
review text: Let \(G\) be a finite group, \(P\) a Sylow \(p\)-subgroup of \(G\), and let \(A\) be a \(p\)-group. The subgroup of \(A\) generated by its elements of order \(p^i\) is denoted by \(\Omega_i(A)\). For \(p=2\), \(\Omega(A)\) stands for the subgroup of \(A\) generated by \(\Omega_1(A)\) and \(\Omega_2(A)\), while otherwise \(\Omega(A)=\Omega_1(A)\). If \(\Omega(P\cap G')\subseteq Z(N_G(P))\), then \(G\) is \(p\)-nilpotent (Theorem 1). The toughest proof of the paper is that of Theorem 2: If \(p=2\), \(P\) is quaternion-free, \(N_G(P)\) is 2-nilpotent, and \(\Omega_1(P\cap G')\subseteq Z(P)\), then \(G\) is 2-nilpotent. Finally, if \(M\) is a nilpotent maximal subgroup of \(G\), \(Q\) a Sylow 2-subgroup of \(M\) and \(\Omega(Q\cap G')\subseteq Z(Q)\), then \(G\) is soluble (Theorem 3).
reviewed by: Hans Lausch
Mathematics Subject Classification ID: 20D20
Mathematics Subject Classification ID: 20D10
Mathematics Subject Classification ID: 20D15
zbMATH DE Number: 1471824
zbMATH Keywords: finite groups
zbMATH Keywords: Sylow subgroups
zbMATH Keywords: \(p\)-nilpotent groups
zbMATH Keywords: nilpotent maximal subgroups
zbMATH Keywords: soluble groups
MaRDI profile type: Publication
full work available at URL: jabr.1999.8274
OpenAlex ID: W2027427531
cites work: M-groups and 2-groups
cites work: Q3961021
cites work: On finite insoluble groups with nilpotent maximal subgroups
cites work: On Two Theorems of Thompson
DOI: 10.1006/JABR.1999.8274
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description: scientific article; zbMATH DE number 1471824