Let $\mathit{G} $ be an abelian group. Show that the elements of finite order in $\mathit{G}$ form a subgroup of $\mathit{G}$, called the torsion subgroup of $\mathit{G}$.
let $g \in G$
I know that the order of the element is the smallest integer $n$ such that $g^{n}=e$, the identity element of the group.
My approach is that let $G'=\left \{ g_{1}, g_{2}, ..., g_{n} \right \} $, $g_{x} \in G, x\in \mathbb {N}$ be an arbitrary set of elements of finite order.
Then, show that the set $G'$ is closed under the binary operation, has an unique identity, has inverse for each element, and is associative.
However, I do realize that $G'$ does not have to be finite; moreover, I don't have any clue how to show closure with the information given by the problem (I don't know where to start).
Could you please point me in the right direction?
Thanks.