# Torsion-free nilpotent group

This page describes a group property obtained as a conjunction (AND) of two (or more) more fundamental group properties: torsion-free group and nilpotent group
View other group property conjunctions OR view all group properties
This page describes a group property obtained as a conjunction (AND) of two (or more) more fundamental group properties: group in which every power map is injective and nilpotent group
View other group property conjunctions OR view all group properties

## Definition

A group $G$ is termed a torsion-free nilpotent group if $G$ is a nilpotent group and it satisfies the following equivalent conditions:

1. $G$ is a group in which every power map is injective.
2. $G$ is a torsion-free group.
3. For each prime number $p$, there exists an element $g \in G$ (possibly dependent on $p$) such that the equation $x^p = g$ has a unique solution for $x \in G$.
4. The center $Z(G)$ is a torsion-free abelian group.
5. Each of the successive quotients $Z^{i+1}(G)/Z^i(G)$ in the upper central series of $G$ is a torsion-free abelian group.
6. All quotients of the form $Z^i(G)/Z^j(G)$ for $i > j$ are [[group in which every power map is injective|groups in which every power map is injective], i.e., $x \mapsto x^p$ is injective in each such quotient group for all prime numbers $p$.

## Relation with other properties

### Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
free nilpotent group |FULL LIST, MORE INFO
rationally powered nilpotent group |FULL LIST, MORE INFO

### Prime set-parametrized version

• Nilpotent group that is torsion-free for a set of primes: For a set of primes $\pi$, we can talk of the notion of $\pi$-torsion-free nilpotent group, which is a nilpotent group that has no $\pi$-torsion.