Prove there does not exist any epimorphism of $(\mathbb{Q}, +)$ onto $(\mathbb{Z}, +)$.

Question

Prove there does not exist any epimorphism of $(\mathbb{Q}, +)$ onto $(\mathbb{Z}, +)$. How do I proceed on this?

Answer 1

Hints:

1) $\;\Bbb Q\;$ is a divisible group, i.e.:

$$\forall\, n\in\Bbb N\,,\,\,q\in\Bbb Q\;\;\exists\,r\in\Bbb Q\;\;s.t.\;\;q=nr$$

2) Every homomorphic image of a divisible group is divisible

3) $\;\Bbb Z\;$ is not divisible.