# Torsion-free abelian 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 abelian group

View other group property conjunctions OR view all group properties

## Contents

## Definition

A **torsion-free abelian group** or **aperiodic abelian group** or **locally free abelian group** is a group satisfying the following equivalent conditions:

- It is both an abelian group and a torsion-free group, i.e., no non-identity element of the group has finite order.
- Every finitely generated subgroup of it is a free abelian group.

## Relation with other properties

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

Free abelian group | |FULL LIST, MORE INFO |

### Weaker properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

Torsion-free group | |FULL LIST, MORE INFO | |||

Locally reduced free group | |FULL LIST, MORE INFO | |||

Abelian group | |FULL LIST, MORE INFO |