# Torsion-free group

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## Contents

This article is about a standard (though not very rudimentary) definition in group theory. The article text may, however, contain more than just the basic definition
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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 $G$ is a torsion-free group and $H$ is a subgroup of $G$. Then, $H$ 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 $G$ and a normal subgroup $H$ of $G$ such that the quotient group $G/H$ is not a torsion-free group.
direct product-closed group property Yes torsion-freeness is direct product-closed If $G_i, i \in I$, are all torsion-free groups, so is the external direct product $\prod_{i \in I} G_i$.

## Relation with other properties

### Prime-parametrized version

• Torsion-free group for a set of primes: Given a set of primes $\pi$, a $\pi$-torsion-free group is a group that has no element of order $p$ for any $p \in \pi$.