Let be a group. An element is called *torsion* if it has finite order. The set of all torsion elements in is denoted . Prove that if is abelian, then is a subgroup of . Show by example that need not be a subgroup if is not abelian.

Note that is nonempty since the identity has order 1. Now if , we have for some positive integers and . Then , so that . By the Subgroup Criterion, is a subgroup.

Now consider the infinite dihedral group, . Both and have order 2, but has infinite order; in this case, is not closed under the group operator.