Consider for example the infinite dihedral group, which has presentation ({*x*,*y*}, {*x*² = *y*² = 1}). This group is of countable infinite order, and in particular the element *xy* has infinite order. Since the group is generated by elements *x* and *y* which have order 2, the subset of finite elements generates the entire group.

A torsion group need not be finite; for example the direct sum of a countable number of copies of the cyclic group *C*_{2} is a torsion group, every element of which has order 2. Nor need there be an upper bound on the orders of elements in a torsion group if it isn't finitely generated, as the example of the factor group **Q**/**Z** shows.

Every free abelian group is torsion free, but the converse is not true, as is shown by the additive group of the rational numbers **Q**.

If *G* is finitely generated and abelian, then it can be written as the direct sum of its torsion subgroup and a torsion free subgroup. In any decomposition of *G* as a direct sum of *a* torsion subgroup *S* and a torsion free subgroup, *S* must equal *T* (but the torsion free subgroup is not uniquely determined). This is an important first step in the classification of finitely generated abelian groups.

Even if *G* is not finitely generated, the *size* of its torsion free part is uniquely determined, as is explained in more detail in the article on rank of an abelian group.

If *G* and *H* are abelian groups with torsion subgroups T(*G*) and T(*H*), respectively, and *f* : *G* → *H* is a group homomorphism, then *f*(T(*G*)) is a subset of T(*H*). We can thus define a functor T which assigns to each abelian group its torsion subgroup and to each homomorphism its restriction to the torsion subgroups.

An abelian group *G* is torsion free if and only if it is flat as a **Z**-module, which means that whenever *K* is a subgroup of the abelian group *H*, then the natural map between the tensor products *K* ⊗ *G* and *H* ⊗ *G* is injective.