Separability of the subgroups of residually nilpotent groups in the class of finite π-groups

Given a nonempty set pi of primes, call a nilpotent group pi-bounded whenever it has a central series whose every factor F is such that: In every quotient group of F all primary components of the torsion subgroup corresponding to the numbers in pi are finite. We establish that if G is a residually pi-bounded torsion-free nilpotent group, while a subgroup H of G has finite Hirsh-Zaitsev rank then H is pi'-isolated in G if and only if H is separable in G in the class of all finite nilpotent pi-groups. By way of example, we apply the results to study the root-class residuality of the free product of two groups with amalgamation.