Primitivity in the free groups of the variety UmUn

Abstract

For each m,n >= 0, let Gm,n denote the free group of rank r in the variety UmUn. The main results in this paper are: (i) a necessary and sufficient condition for a system of r elements {v1,…, vr} to form a basis of Gm,ni (ii) necessary and sufficient conditions for a system of l elements {v1,…, vl}, l <= r, to be included in a basis of Gm,0. In particular, (i), (ii) yield corresponding results for the free metabelian group of rank r.