TITLE: Compute $\operatorname{Hom}_{\mathbb{Z}} ( \mathbb{Q}, \mathbb{Z})$
QUESTION [11 upvotes]: I think this question could be a little idiot, however I could not solve this after some hours. I need to find all homomorphism between the additive group $\mathbb{Q}$ and the additive group $\mathbb{Z}$, i.e., $\operatorname{Hom}_{\mathbb{Z}}  ( \mathbb{Q}, \mathbb{Z})$. I tried to find a good set of generators for $\mathbb{Q}$, however none of them  satisfied a general homomorphism.

REPLY [9 votes]: $\,\Bbb Q\,$ is a divisible group , so is any of its homomorphic images. But $\,\Bbb Z\,$ is not a divisible group (why?) , so it must be that the image of any group homomorphism  $\,\Bbb Q\to\Bbb Z\,$ is the trivial subgroup (which, BTW, is the only finite group which is divisible and the only subgroup of $\,\Bbb Z\,$ that is divisible), and thus $\,\operatorname{Hom}_{\Bbb Z}(\Bbb Q,\Bbb Z)=0\,$