Is there a onto group homomorphism from $\Bbb Q$ to $\Bbb Z$?

Question

I am learning group-homomorphisms. I have two questions:

1. Is there a onto group homomorphism from $\Bbb Z$ to $\Bbb Q$?
2. Is there a onto group homomorphism from $\Bbb Q$ to $\Bbb Z$?

I have the answer of the first one.

1. $\Bbb Z$ is cyclic and homomorphic image of a cyclic group is cyclic but $\Bbb Q$ is not.
2. I am stuck here. Please help me.

Answer 1

Your answer to the first question is correct.

For the second question, suppose that there were an onto homomorphism $f : \mathbb{Q} \to \mathbb{Z}$. Then there exists some $q \in \mathbb{Q}$ such that $f(q) = 1$. But then, $x = f(q/2)$ is an integer satisfying $$2x = x+x = f(q/2) + f(q/2) = f(q/2 + q/2) = f(q) = 1,$$ which is impossible. Therefore there is no such $f$.