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Two problems on varieties of groups generated by wreath products. (English) Zbl 1011.20029
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.

20E10 Quasivarieties and varieties of groups
20E22 Extensions, wreath products, and other compositions of groups
20K25 Direct sums, direct products, etc. for abelian groups
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