Nilpotent group that is powered for a set of primes

Definition
Suppose $$G$$ is a group and $$\pi$$ is a set of primes. We say that $$G$$ is a $$\pi$$-powered nilpotent group if it satisfies the following equivalent conditions:


 * 1) $$G$$ is a $$\pi$$-powered group and is also a nilpotent group.
 * 2) $$G$$ is a nilpotent group that is both $$\pi$$-divisible and $$\pi$$-torsion-free.

Equivalence of definitions
Part of the proof relies on equivalence of definitions of nilpotent group that is torsion-free for a set of primes.