Divisible nilpotent group

Definition
A group $$G$$ is termed a divisible nilpotent group if it satisfies the following equivalent conditions:


 * 1) $$G$$ is a divisible group.
 * 2) The abelianization of $$G$$ is a divisible abelian group.
 * 3) For every positive integer $$i$$, the quotient group $$\gamma_i(G)/\gamma_{i+1}(G)$$ of successive members of the lower central series is a divisible abelian group.
 * 4) 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 a divisible group.

Prime set-parametrized version

 * Nilpotent group that is divisible for a set of primes: Given a set of primes $$\pi$$, we can talk of the notion of a $$\pi$$-divisible nilpotent group.