Nilpotent group that is divisible for a set of primes

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


 * 1) $$G$$ is nilpotent and $$\pi$$-divisible.
 * 2) $$G$$ is nilpotent and the abelianization of $$G$$ is $$\pi$$-divisible.
 * 3) $$G$$ is nilpotent and for every positive integer $$i$$, the quotient group $$\gamma_i(G)/\gamma_{i+1}(G)$$ of successive members of the lower central series is $$\pi$$-divisible.
 * 4) $$G$$ is nilpotent and for any two positive integers $$i < j$$, if $$\gamma_i(G),\gamma_j(G)$$ denote respectively the $$i^{th}$$ and $$j^{th}$$ members of the lower central series of $$G$$, then the quotient group $$\gamma_i(G)/\gamma_j(G)$$ is $$\pi$$-divisible.