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Concept
Torsion subgroup
Summary
In the theory of abelian groups, the torsion subgroup AT of an abelian group A is the subgroup of A consisting of all elements that have finite order (the torsion elements of A). An abelian group A is called a torsion group (or periodic group) if every element of A has finite order and is called torsion-free if every element of A except the identity is of infinite order.
The proof that AT is closed under the group operation relies on the commutativity of the operation (see examples section).
If A is abelian, then the torsion subgroup T is a fully characteristic subgroup of A and the factor group A/T is torsion-free. There is a covariant functor from the to the category of torsion groups that sends every group to its torsion subgroup and every homomorphism to its restriction to the torsion subgroup. There is another covariant functor from the category of abelian groups to the category of torsion-free groups that sends every group to its quotient by its torsion subgroup, and sends ev
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