# Torsion-free nilpotent group

From Groupprops

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

## Contents

## Definition

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

- is a group in which every power map is injective.
- is a torsion-free group.
- For each prime number , there exists an element (possibly dependent on ) such that the equation has a unique solution for .
- The center is a torsion-free abelian group.
- Each of the successive quotients in the upper central series of is a torsion-free abelian group.
- All quotients of the form for are [[group in which every power map is injective|groups in which every power map is injective], i.e., is injective in each such quotient group for all prime numbers .

### Equivalence of definitions

`Further information: equivalence of definitions of nilpotent group that is torsion-free for a set of primes`

## 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 , we can talk of the notion of -torsion-free nilpotent group, which is a nilpotent group that has no -torsion.