Nilpotent group that is torsion-free for a set of primes

Definition
Suppose $$\pi$$ is a set of prime numbers. A group $$G$$ is termed a $$\pi$$-torsion-free nilpotent group if it satisfies the following equivalent conditions:


 * 1) $$G$$ is a $$\pi$$-powering-injective group, i.e., $$x \mapsto x^p$$ is injective and each $$p \in \pi$$.
 * 2) $$G$$ is a $$\pi$$-torsion-free group.
 * 3) For each $$p \in \pi$$, there exists an element $$g \in G$$ (possibly dependent on $$p$$) such that the equation $$x^p = g$$ has a unique solution for $$x \in G$$.
 * 4) The center is a $$\pi$$-torsion-free group.
 * 5) Each of the successive quotients $$Z^{i+1}(G)/Z^i(G)$$ in the upper central series of $$G$$ is a $$\pi$$-torsion-free group.
 * 6) All quotients of the form $$Z^i(G)/Z^j(G)$$ for $$i > j$$ are $$\pi$$-powering-injective groups, i.e., $$x \mapsto x^p$$ is injective in each such quotient group and each $$p \in \pi$$.

Note that if we take $$\pi$$ to be the set of all primes, this just becomes the same as torsion-free nilpotent group.