Is there $f$ in $\operatorname{Hom}(\mathbb{Q},\mathbb{Q})$ with kernel $\mathbb{Z}$? Is there a group homomorphism from $\mathbb{Q}$ (the group of rationals) to $\mathbb{Q}$ whose kernel is $\mathbb{Z}$?

The hint in in the same vein as that given by Tobias in the comments above, but a bit more explicit. Consider where $1/2$ would be sent to, given that $1/2 + 1/2 = 1$.
