A note on \(p\)-nilpotence and solvability of finite groups.
type: Item
instance of: scholarly article
title: A note on \(p\)-nilpotence and solvability of finite groups.
zbMATH Open document ID: 1169.20012
author: Jiangtao Shi
author: Shi Wujie
author: Cui Zhang
published in: Journal of Algebra
publication date: 2009-04-29T00:00:00Z
review text: Let \(G\) be a finite group, \(p\) a prime number, \(P\) a \(p\)-Sylow subgroup. In recent papers, for instance, in \textit{A. Ballester-Bolinches} and \textit{X. Guo} [J. Algebra 228, No. 2, 491-496 (2000; Zbl 0961.20016)], characterisations were given for \(G\) to be \(p\)-nilpotent, solvable, supersolvable under certain hypotheses. The authors provide examples that show the necessity of these hypotheses. Further, certain results in these papers are extended by the authors, such as: if \(\Omega_1(P\cap O^p(G))\) is central in \(O^p(G)\) and \(P\) is quaternion-free in case \(p=2\) then the group \(G\) is nilpotent; moreover, under the same hypotheses for \(p=2\) and in case there is a nilpotent maximal subgroup then the group is solvable.
reviewed by: János Kurdics
Mathematics Subject Classification ID: 20D20
Mathematics Subject Classification ID: 20D15
Mathematics Subject Classification ID: 20D10
zbMATH DE Number: 5549378
zbMATH Keywords: \(p\)-nilpotence
zbMATH Keywords: solvability
zbMATH Keywords: supersolvability
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2008.12.004
OpenAlex ID: W1989317104
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DOI: 10.1016/J.JALGEBRA.2008.12.004
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description: scientific article; zbMATH DE number 5549378