Subspace Topologies in Central Extensions

The possibilities for the topology induced on the centre by the profinite topology of nilpotent groups of class two and finitely generated centre-by-metabelian groups are examined. As an application, it is shown that any finite abelian group can be embedded in the profinite completion of some countable torsion-free residually finite nilpotent group of class two and that of some finitely generated torsion-free residually finite centre-by-metabelian group.