I need to prove the following, Suppose $G$ is an Abelian group. Show that $H = \{g ∈ G : |g|\ \hbox{is finite}\}$ is a subgroup. I know that to prove a subset is a subgroup one can prove $ab^{-1} \in H$ for all $a,b \in H$ but I'm not sure how to apply that for this problem.
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Note that the fact $G$ is an Abelian group. It is not difficult to check that $ab^{-1} \in H$ for any $a, b \in H$. – bing Apr 06 '17 at 01:20
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Hint:
(i)$|g|=|g^{-1}|$ for every $g\in G$
(ii) If $G$ is abelian, then $|ab|$ divides $|a||b|$ if $a$ and $b$ are of finite order.
Indeed, $H$ is known as torsion subgroup of $G$.
Wang Kah Lun
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