Torsion subgroup

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[1]).

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 every homomorphism to the obvious induced homomorphism (which is easily seen to be well-defined).

If A is finitely generated and abelian, then it can be written as the direct sum of its torsion subgroup T and a torsion-free subgroup (but this is not true for all infinitely generated abelian groups).

This is a key step in the classification of finitely generated abelian groups.

The product over the set of all prime numbers of the restriction of these functors to the category of torsion groups, is a faithful functor from the category of torsion groups to the product over all prime numbers of the categories of p-torsion groups.