Primitive and measure-preserving systems of elements on the varieties of metabelian and metabelian profinite groups

Abstract

Given a free metabelian group S of finite rank r, r ≥ 2, we prove that a system of elements g1, ..., gn ∈ S for n = 1 or n = r preserves measure on the variety of all metabelian groups if and only if the system is primitive. Similar results hold for a free profinite group \(\hat S\) and the variety of finite metabelian groups for each n, 1 ≤ n ≤ r. Some corollaries to these theorems are derived.