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If $G$ and $H$ are finitely generated residually nilpotent groups, then $G$and $H$ are in the same nilpotent genus if they have the same lower centralquotients (up to isomorphism). A stronger condition is that $H$ is para-$G$ ifthere exists a monomorphism of $G$ into $H$ which induces isomorphisms betweenthe corresponding quotients of their lower central series. We first considerresidually nilpotent groups and find sufficient conditions on the monomorphismso that $H$ is para-$G.$ We then prove that for certain polycyclic groups, if$H$ is para-$G$, then $G$ and $H$ have the same Hirsch length. We also provethat the pro-nilpotent completions of these polycyclic groups are locallypolycyclic.

The nilpotent genus of finitely generated residually nilpotent groups