Nilpotent group that is powered for a set of primes

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
View other group property conjunctions OR view all group properties

Definition

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

1. $G$ is a $\pi$-powered group and is also a nilpotent group.
2. $G$ is a nilpotent group that is both $\pi$-divisible and $\pi$-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.