On Levi quasivarieties generated by nilpotent groups
type: Item
instance of: scholarly article
title: On Levi quasivarieties generated by nilpotent groups
zbMATH Open document ID: 0956.20015
DOI: 10.1007/BF02674590
author: L. V. Taranina
author: Alexander Ivanovich Budkin
published in: Siberian Mathematical Journal
publication date: 2000-11-21T00:00:00Z
full work available at URL: https://eudml.org/doc/120818
review text: Let \(\mathcal M\) be a class of groups. By \(L({\mathcal M})\) we denote the class of groups \(G\) such that, for every \(x\in G\), the normal closure \((x)^G\) belongs to \(\mathcal M\). This yields the notion of the Levi class generated by a class of groups. By \(q{\mathcal M}\) the authors denote the quasivariety generated by \(\mathcal M\). The authors prove the following theorem: Let \(\mathcal K\) be an arbitrary set of nilpotent groups of class \(2\) without elements of order \(2\). Suppose that, in every group in \(\mathcal K\), the centralizer of each element outside the center of the group is an Abelian group. Then, for \({\mathcal M}=q{\mathcal K}\), we have \(L({\mathcal M})\subseteq{\mathcal N}_3\).
Mathematics Subject Classification ID: 20E10
Mathematics Subject Classification ID: 20F18
Mathematics Subject Classification ID: 20F45
Mathematics Subject Classification ID: 08C15
zbMATH DE Number: 1534537
zbMATH Keywords: quasivarieties of groups
zbMATH Keywords: Levi classes
zbMATH Keywords: centralizers
zbMATH Keywords: nilpotent groups
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1534537