Group order

Torsion elements and groups

A torsion 𝑎 ∈𝐺 is a group element of finite order, i.e. there exists 𝑛 ∈ℕ such that 𝑎𝑛 =𝑒. 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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