
I claim that any group homomorphism will be of the form $f(x)=qx$ for some $q\in\mathbb{Q}$, now as $f$ is non zero so $q\neq 0$ now clearly kernel of $f$ will be $\{0\}$ only hence $f$ is injective, now let $f(1)=p\neq 0$ then for any $y\in\mathbb{Q}$ and $f(y/p)=y$ so $f$ is surjective too, hence bijective. am I right?