# 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.

### Equivalence of definitions

Further information: Equivalence of definitions of nilpotent group that is divisible for a set of primes