Join-Irreducible Cross Product Varieties of Groups

Abstract

Let $\mathfrak {U},\mathfrak {B}$ be varieties of groups which have finite coprime exponents, let $\mathfrak {U}$ be metabelian and nilpotent with “small” nilpotency class, and let $\mathfrak {B}$ be abelian. The product variety $\mathfrak {U}\mathfrak {B}$ is shown to be join-irreducible if and only if $\mathfrak {U}$ is join-irreducible. This is done by obtaining a simple description for the critical groups generating $\mathfrak {U}\mathfrak {B}$ when $\mathfrak {U}$ is join-irreducible and finding a word which is not a law in $\mathfrak {U}\mathfrak {B}$ but is a law in every proper subvariety of $\mathfrak {U}\mathfrak {B}$