Torsion-free group

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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

Stronger 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 $p\in \pi$ .