TITLE: Show if an abelian group $G$ has a $\mathbb Q$-vector space structure, then it is unique.
QUESTION [7 upvotes]: Show if an abelian group $G$ has a $\mathbb Q$-vector space structure, then it is unique. 
Hint: show every element must have infinite order. Also, show the unique ring homomorphism $\mathbb Z\to \mathbb Q$ is an epimorphism.
I have been able to prove both hints, but I can't put two and two together? Maybe use the fact that every abelian group is a $\mathbb Z$ module in exactly one way, and since the elements must have infinite order, the module structures of $\mathbb Z$ and $\mathbb Q$ must coincide?
In that case, it would amount to show that the diagram created by the homomorphisms $\phi:\mathbb Z\to\mathbb Q$, $\sigma_1:\mathbb Z\to \mathrm{End}(G)$, and $\sigma_2:\mathbb Q\to \mathrm{End}(G)$ commutes, given that $\phi$ and $\sigma_1$ are unique.

REPLY [3 votes]: For the other hint, note that two different $\mathbb{Q}$-module structures correspond to two different ring homomorphisms  $\alpha : \mathbb{Q} \longrightarrow \text{End}_{\text{Ab}}(G) $ and $\beta : \mathbb{Q} \longrightarrow \text{End}_{\text{Ab}}(G)$. Then, the inclusion $i : \mathbb{Z} \longrightarrow \mathbb{Q}$ tells us that $(\alpha \circ i )(1) = (\beta \circ i)(1)$ so that $\alpha = \beta$.