On properties of formations of quasinilpotent groups
type: Item
instance of: scholarly article
title: On properties of formations of quasinilpotent groups
zbMATH Open document ID: 0810.20015
DOI: 10.1007/BF01208320
published in: Mathematical Notes
publication date: 1994-07-13T00:00:00Z
review text: A finite group \(G\) is said to be quasinilpotent if \(C_ G(H/K)H = G\), for any chief factor \(H/K\) of \(G\). Let \({\mathcal N}^*\) be the class of all quasinilpotent finite groups and \(\mathcal N\) the class of all nilpotent finite groups. Both \(\mathcal N\) and \({\mathcal N}^*\) are radical formations. The paper confirms the dualism between \(\mathcal N\) and \({\mathcal N}^*\). The author proves that: i) for any two subnormal subgroups \(A\) and \(B\) of any arbitrary finite group \(G\), we have \(\langle A, B\rangle^{{\mathcal N}^*} \equiv \langle A^{{\mathcal N}^*}, B^{{\mathcal N}^*}\rangle\); ii) the quasinilpotent radical of a subnormal subgroup of an arbitrary finite group \(G\) commutes with any of its other subnormal subgroups.
Mathematics Subject Classification ID: 20D10
Mathematics Subject Classification ID: 20D15
Mathematics Subject Classification ID: 20D35
Mathematics Subject Classification ID: 20F17
Mathematics Subject Classification ID: 20D40
zbMATH DE Number: 567874
zbMATH Keywords: chief factor
zbMATH Keywords: quasinilpotent finite groups
zbMATH Keywords: nilpotent finite groups
zbMATH Keywords: radical formations
zbMATH Keywords: subnormal subgroups
zbMATH Keywords: quasinilpotent radical
author: S. F. Kamornikov
reviewed by: Rodica Covaci
MaRDI profile type: Publication
cites work: On groups with abelian Sylow 2-subgroups
cites work: Q3659962
cites work: Fittingklassen endlicher Gruppen, in denen gewisse Hauptfaktoren einfach sind
cites work: Nilpotent injectors in finite groups
cites work: Vertauschbare nachinvariante Untergruppen
cites work: Q3033956
cites work: Q3961010
cites work: Q3994689
cites work: Über den Normalisator der subnormalen Untergruppen
full work available at URL: https://doi.org/10.1007/bf01208320
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description: scientific article; zbMATH DE number 567874