Let G be an abelian group. Prove that is a subgroup of G. Give an explicit example where this set is not a subgroup when G is non-abelian.

First if G is infinite, then the torsion group would be empty. So suppose G is finite. I am supposed to use the subgroup criterion to show this but not entirely sure with what to do.

Subgroup criterion is a subset H of a group is a subgroup iff H is nonempty and for all x,y in H .

Let , , and .

By property 2, . Therefore, .

Since e and x are in H, we have

Correct?

Not sure about the example when G is non-abelian.