Torsion-free nilpotent groups of small Hirsch length with isomorphic finite quotients

Abstract

Let T denote the class of finitely generated torsion-free nilpotent groups. For a group G let F(G) be the set of isomorphism classes of finite quotients of G. Pickel proved that if G∈T, then the set g(G) of isomorphism classes of groups H∈T with F(G)=F(H) is finite. We give an explicit description of the sets g(G) for the T-groups G of Hirsch length at most 5. Based on this, we show that for each Hirsch length n≥4 and for each m∈N there is a T-group G of Hirsch length n with |g(G)|≥m.