TITLE: Is there a onto group homomorphism from $\Bbb Q$ to $\Bbb Z$?
QUESTION [8 upvotes]: I am learning group-homomorphisms. I have two questions:

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

I have the answer of the first one. 

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

REPLY [21 votes]: 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 $f$.