Let $G$ be a finitely generated abelian group. Prove that there is no non-trivial homomophism from $\mathbb Q$ to $G$.
I believe this has to do with divisibility. $\mathbb Q$ is divisible. The image of it under a group homomorphism must be divisible. If $G$ were finite, this would imply that the image is the trivial subgroup since any divisible finite group is trivial. But $G$ is not necessarily finite. What should the argument be like instead?