A Classification Theorem for Varieties Generated by Wreath Products of Groups

TL;DR

This paper establishes a precise criterion determining when the variety generated by the wreath product of certain nilpotent and abelian groups equals the product of their individual varieties, extending previous results and offering new applications.

Contribution

It provides a new classification criterion for varieties generated by wreath products of nilpotent and abelian groups, generalizing earlier restricted cases.

Findings

The criterion characterizes when var(A Wr B) equals var(A) var(B).

It extends previous work on wreath products of abelian and finite groups.

Applications demonstrate the criterion's usefulness in group theory.

Abstract

We suggest a criterion under which for a nilpotent group of finite exponent $A$ and for an abelian group $B$ the variety $var(A \,Wr\, B)$ generated by their wreath product $A \,Wr\, B$ is equal to the product of varieties $var(A)$ and $var(B)$ generated by $A$ and $B$. Namely the equality holds if and only if either the group $B$ is not of some non-zero exponent; or if $B$ is of a non-zero exponent $n$, and $B$ contains a subgroup isomorphic to $C_{d}^c \times C_{n/d}^\infty$, where $c$ is the nilpotency class of $A$, $d$ is the largest divisor of $n$ coprime with $m$, $C_{d}^c$ is the direct power of $c$ copies of the cycle $C_d$ of order $d$, $C_{n/d}^\infty$ is the direct power of countably many copies of the cycle $C_{n/d}$ of order $n/d$. This criterion continues our previous work on cases when the similar criterions were given for wreath products of abelian groups or of finite groups. Also, this is a generalization of known results in literature, which solve the same problem for much more restricted cases. Some applications of the criterion are considered at the end of paper.