On varieties of groups generated by wreath products of abelian groups

TL;DR

This paper characterizes when wreath products of abelian groups generate the product variety, extending classical results by providing necessary and sufficient conditions based on group exponents and prime divisors.

Contribution

It generalizes Higman and Houghton's results to arbitrary abelian groups, establishing precise criteria for the generated variety by wreath products.

Findings

Wreath product generates the product variety if one group is not of finite exponent.

If both groups have finite exponents, specific divisibility and infiniteness conditions must hold.

Provides a complete characterization for varieties generated by wreath products of abelian groups.

Abstract

Generalizing results of Higman and Houghton on varieties generated by wreath products of finite cycles, we prove that the (direct or cartesian) wreath product of arbitrary abelian groups $A$ and $B$ generates the product variety $var (A) \cdot var (B)$ if and only if one of the groups $A$ and $B$ is not of finite exponent, or if $A$ and $B$ are of finite exponents $m$ and $n$ respectively and for all primes $p$ dividing both $m$ and $n$, the factors $B[p^k]/B[p^{k-1}]$ are infinite, where $B[s]=\langle b\in B|\,b^{s}=1 \rangle$ and where $p^k$ is the highest power of $p$ dividing $n$.