[[Group order]]
# Torsion elements and groups
A **torsion** $a \in G$ is a group element of finite [[Group order|order]],
i.e. there exists $n \in \mathbb{N}$ such that $a^n = e$. #m/def/group 
A group containing only such elements is called a **torsion group**,
whereas a group whose only finite-order element is the identity is called **torsion-free**. 
The least common multiple of the orders of all elements, if it exists, is called the **exponent** of the group.

## Subgroups that are torsion groups
The following subgroups of abelian groups are torsion groups:

- The canonical [[Torsion subgroup of an abelian group]]
- [[Fixed order subgroup of an abelian group]]

## Special cases

- [[Torsion group with a central cyclic commutator subgroup]]

## See also

- In the case of abelian groups, these notions coincide with the more general [[Torsion]].



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