# Subvariety structures in certain product varieties of groups

## TL;DR

This paper classifies when wreath products of different groups generate the same variety, enabling the analysis of subvarieties in nilpotent-by-abelian product varieties and revealing distinct subvarieties with identical algebraic properties.

## Contribution

It provides a classification of wreath product-generated varieties and explores subvarieties with identical properties but different structures in nilpotent-by-abelian contexts.

## Key findings

- Identified conditions for wreath products to generate the same variety.
- Found subvarieties with identical nilpotency, solubility length, and exponent that are distinct.
- Strengthened previous classifications of varieties generated by wreath products.

## Abstract

We classify certain cases when the wreath products of distinct pairs of groups generate the same variety. This allows us to investigate the subvarieties of some nilpotent-by-abelian product varieties ${\mathfrak U}{\mathfrak V}$ with the help of wreath products of groups. In particular, using wreath products we find such subvarieties in nilpotent-by-abelian ${\mathfrak U}{\mathfrak V}$, which have the same nilpotency class, the same length of solubility, and the same exponent, but which still are distinct subvarieties. Obtained classification strengthens our recent work on varieties generated by wreath products.