The Criterion of Shmel'kin and Varieties Generated by Wreath Products of Finite Groups

TL;DR

This paper establishes a general criterion for when the variety generated by the wreath product of two finite groups equals the product of their individual varieties, extending previous work on abelian groups and utilizing Shmel'kin's criterion.

Contribution

It introduces a new, general criterion for the equality of varieties generated by wreath products of finite groups, broadening the understanding of their structure and classification.

Findings

Provides a criterion for var(A Wr B) = var(A) var(B) for finite groups A and B

Extends previous results from abelian groups to more general finite groups

Utilizes Shmel'kin's criterion and techniques on critical groups in nilpotent-by-abelian varieties

Abstract

We present a general criterion under which the equality var(A Wr B) = var(A) var(B) holds for finite groups A and B. This continues our previous research on varieties, generated by wreath products of abelian groups, and generalizes some existing results in this direction in literature. The classification is based on criterion of A.L. Shmel'kin for product varieties of groups and on technics developed by R. Burns et al. on critical groups in nilpotent-by-abelian varieties.