Varieties generated by wreath products of abelian and nilpotent groups

TL;DR

This paper surveys recent research on varieties generated by wreath products of groups, providing classifications for abelian groups and exploring new cases involving non-abelian finite groups.

Contribution

It offers a complete classification for wreath products of abelian groups and extends the analysis to non-abelian finite groups, including new unpublished results.

Findings

Classification of wreath products of abelian groups

Results on varieties generated by non-abelian finite groups

Extensions to sets of abelian groups

Abstract

We present a survey of our recent research on varieties, generated by wreath products of groups. In particular, the full classification of all cases, when the (cartesian or direct) wreath product of any abelian groups $A$ and $B$ generates the product variety $\var{A} \var{B}$, is given. The analog of this is given for sets of abelian groups. We also present new, unpublished research on cases when the similar question is considered for non-abelian, finite groups.