Let G be an abelian group. Prove that $\displaystyle \{g\in G:|g|< \infty\}$ 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 $\displaystyle xy^{-1}\in H$.

Let $\displaystyle y=x$, $\displaystyle |x|<\infty$, and $\displaystyle x\in H$.

By property 2, $\displaystyle xy^{-1}\in H$. Therefore, $\displaystyle xx^{-1}=e\in H$.

Since e and x are in H, we have $\displaystyle ex^{-1}=x^{-1}\in H$

Correct?

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