# Torsion-free group

From Groupprops

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism

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## Definition

A group is said to be **torsion-free** or **aperiodic** if it has no non-identity periodic element, or equivalently, if there is no non-identity element of finite order.

(The term *aperiodic* is sometimes also used with slightly different meanings, so *torsion-free* is the more unambiguous term).

## Metaproperties

Metaproperty name | Satisfied? | Proof | Statement with symbols |
---|---|---|---|

quasivarietal group property | Yes | torsion-freeness is quasivarietal | The condition of being torsion-free can be described by a collection of quasi-identities. |

subgroup-closed group property | Yes | torsion-freeness is subgroup-closed | Suppose is a torsion-free group and is a subgroup of . Then, is also a torsion-free group. |

quotient-closed group property | No | torsion-freeness is not quotient-closed | It is possible to have a torsion-free group and a normal subgroup of such that the quotient group is not a torsion-free group. |

direct product-closed group property | Yes | torsion-freeness is direct product-closed | If , are all torsion-free groups, so is the external direct product . |

## Relation with other properties

### Stronger properties

### Prime-parametrized version

- Torsion-free group for a set of primes: Given a set of primes , a -torsion-free group is a group that has no element of order for any .