Two problems on varieties of groups generated by wreath products
type: Item
instance of: scholarly article
title: Two problems on varieties of groups generated by wreath products
zbMATH Open document ID: 1011.20029
DOI: 10.1155/S0161171202012528
author: Vahagn H. Mikaelian
published in: International Journal of Mathematics and Mathematical Sciences
publication date: 2002-08-13T00:00:00Z
full work available at URL: https://eudml.org/doc/49971
full work available at URL: S0161171202012528.html
review text: This article poses two problems concerning varieties of groups generated by Cartesian or wreath products of groups. We write \(N\text{ Wr }H\) for the wreath product of groups \(N\) and \(H\), and for sets of groups \(X\) and \(Y\) their wreath product is \(X\text{ Wr }Y=\{N\text{ Wr }H\mid N\in X,\;H\in Y\}\). Denoting the variety generated by \(N\) as \(\text{var}(N)\), the two problems are: (1) Let \(N\) and \(H\) be arbitrary groups. Find a criterion by which \(\text{var}(N\text{ Wr }H)=\text{var}(N)\cdot\text{var}(H)\) holds. (2) Let \(X\) and \(Y\) be arbitrary sets of groups. Find a criterion by which \(\text{var}(X\text{ Wr }Y)=\text{var}(X)\cdot\text{var}(Y)\) holds. Since the Cartesian wreath product and the direct wreath product of two groups or sets of groups generates the same variety of groups, there is no need to specify in the problems or notation which type of wreath product is being considered. The two problems have recently (2001) been solved by the author in the Abelian group case. Here, he outlines the method used for Abelian groups and provides examples showing that the same criteria do not work for non-Abelian groups.
Mathematics Subject Classification ID: 20E10
Mathematics Subject Classification ID: 20E22
Mathematics Subject Classification ID: 20K25
zbMATH DE Number: 1780402
zbMATH Keywords: varieties of groups
zbMATH Keywords: Abelian groups
zbMATH Keywords: wreath products
zbMATH Keywords: products of varieties
reviewed by: Ross P. Abraham
MaRDI profile type: Publication
OpenAlex ID: W2091973074
description: scientific article; zbMATH DE number 1780402