Prove that the torsion subgroup of a finitely generated nilpotent group is finite. More generally, in any group with "almost" no torsion all periodic subgroups are finite.
Here "almost" means that there is a subgroup of finite index which has no torsion elements.
I tried to "apply" the main theorem that a f.g. NP group is polycyclic. That is probably the first step, but I don't see how to proceed from there.