Varieties of groups and normal-subgroup lattices -- a survey
type: Item
instance of: scholarly article
title: Varieties of groups and normal-subgroup lattices -- a survey
zbMATH Open document ID: 0815.20016
DOI: 10.1007/BF01190820
published in: Algebra Universalis
publication date: 1994-11-17T00:00:00Z
review text: This is mainly a survey paper dealing with the problem of distinguishing group varieties by means of their congruence varieties (i.e., lattice varieties generated by the (congruence=) normal subgroup lattices of all groups belonging to each of the given group varieties). It follows from a result of \textit{D. W. Barnes} [J. Aust. Math. Soc. 2, 17-34 (1961; Zbl 0115.253)] that if \(G\) is a finite group such that for each prime \(p\) the Sylow \(p\)-subgroups of \(G\) have nilpotency class \(< p\), then the congruence variety of the group variety generated by \(G\) is the same as the congruence variety of abelian groups of exponent dividing the exponent of \(G\). On the other hand, \textit{Cs. Szabó} and the reviewer [Algebra Univers. 33, No. 2, 191-195 (1995)] found a lattice identity distinguishing the congruence variety of groups of exponent 4 and nilpotency class 2 from the congruence variety of abelian groups of exponent 4. As the third main result in this area, the authors formulate a corollary to a theorem of \textit{Ch. Herrmann} and \textit{A. P. Huhn} [Math. Z. 144, 185-194 (1975; Zbl 0316.06006)]: group varieties of distinct finite exponents have different congruence varieties. Here they give a new, elegant proof, using a sequence of new, very simple lattice identities, which might deserve further study.
reviewed by: Péter P. Pálfy
Mathematics Subject Classification ID: 20E10
Mathematics Subject Classification ID: 08B10
Mathematics Subject Classification ID: 20E15
Mathematics Subject Classification ID: 06B20
zbMATH DE Number: 679613
zbMATH Keywords: verbally abelian groups
zbMATH Keywords: survey
zbMATH Keywords: group varieties
zbMATH Keywords: congruence varieties
zbMATH Keywords: normal subgroup lattices
zbMATH Keywords: variety of groups
zbMATH Keywords: nilpotency class
zbMATH Keywords: groups of exponent 4
zbMATH Keywords: finite exponents
zbMATH Keywords: lattice identities
author: Robert G. Burns
author: Sheila Oates-Williams
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01190820
OpenAlex ID: W2005950850
description: scientific article; zbMATH DE number 679613