On the uniqueness of roots in virtually nilpotent groups

After revisiting the concept of the torsion subgroup of a group with respec t to a set of prime numbers P (as introduced by Ribenboim), we show that, f or all p in P, p-th roots are unique in a virtually nilpotent group if and only if P-roots are unique in both its Fitting subgroup and its Fitting quo tient. This more general notion of torsion also turns out to be sufficient to understand completely the kernel of the P-localization homomorphism of a virtually nilpotent group. Using this result, we characterize the finitely generated virtually nilpotent groups such that, when dividing out the P-to rsion subgroup, P-roots exist and are unique in the resulting quotient.