# Nilpotent group that is divisible for a set of primes

## Definition

Suppose is a group and is a set of primes. We say that is a -**divisible nilpotent group** if it satisfies the following equivalent conditions:

- is nilpotent and -divisible.
- is nilpotent and the abelianization of is -divisible.
- is nilpotent and for every positive integer , the quotient group of successive members of the lower central series is -divisible.
- is nilpotent and for any two positive integers , if denote respectively the and members of the lower central series of , then the quotient group is -divisible.

### Equivalence of definitions

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