Show if an abelian group $G$ has a $\mathbb Q$-vector space structure, then it is unique.

We know that the $\mathbb{Z}$-module structure is unique, because we must have $1x=x$, $2x=x+x$, $3x=x+x+x$, etc., and similarly with negatives.

Now we know that $\frac{1}{n}x$ must be an element $w$ satisfying $nw = x$. If there were 2 such elements $w$ and $w'$, then $n(w-w')=0$ and multiplying both sides by $\frac{1}{n}$ gives $w=w'$. Therefore there is only one element that can be $\frac{1}{n}x$. And of course $\frac{m}{n}x$ is $\frac{1}{n}x$ added to itself $m$ times (for $m>0$), and similarly with negative rationals.

So the $\mathbb{Q}$-module structure is unique.

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 $\alpha = \beta$.