Torsion elements in an abelian group form a subgroup

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

Note that T(G) is nonempty since the identity has order 1. Now if x,y \in T(G), we have x^a = y^b = 1 for some positive integers a and b. Then (xy^{-1})^{ab} = (x^a)^b (y^b)^{-a} = 1, so that xy^{-1} \in T. By the Subgroup Criterion, T(G) \leq G is a subgroup.

Now consider the infinite dihedral group, D_\infty. Both sr^2 and sr have order 2, but srsr^2 = r has infinite order; in this case, T is not closed under the group operator.

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