This is Exercise 4.3.10 of Robinson's "A Course in the Theory of Groups (Second Edition)". According to Approach0 and this search, it is new to MSE.
The Details:
The torsion-subgroup of an abelian group $G$ is the subgroup of all elements of $G$ of finite order.
The Question:
Let $G$ be an abelian group such that ${\rm Aut}(G)$ is finite, prove that $G$ has finite torsion-subgroup.
Thoughts:
Let $G$ be as defined above and $\varphi\in{\rm Aut}(G)$.
I think that, because $\lvert \varphi (x)\rvert=\lvert x\rvert$, and there are only finitely many permutations of elements of a given order due to $\lvert{\rm Aut}(G)\rvert<\infty$, there must exist only finitely many elements of finite order.
However, I doubt this is as simple as that, and even if the idea is right, I'm not sure how to make the proof more rigorous.
Something tells me I'm missing something . . .
Please help :)