Searching a Group Homomorphism

Question

Is there a non-zero group homomorphism from the abelian group $\prod_{n=2}^\infty \mathbb Z_n$ to $\mathbb Q$?

I thought of the map taking $(1,1,...)$ to $1$, but could not make through. Thanks for any suggestion!

Answer 1

No there isn't. The image of any finite subgroup must be $\{0\}$, which is the only finite subgroup of rationals, because the image of a finite group must be finite. So there's only the zero homomorphism