# Nilpotent group that is powered for a set of primes

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This page describes a group property obtained as a conjunction (AND) of two (or more) more fundamental group properties: powered group for a set of primes and nilpotent group

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## Definition

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

- is a -powered group and is also a nilpotent group.
- is a nilpotent group that is both -divisible and -torsion-free.

### Equivalence of definitions

Part of the proof relies on equivalence of definitions of nilpotent group that is torsion-free for a set of primes.