Suppose that $G$ is a group with the property that for each prime $p$ there is a surjective homomorphism $f_p: G \to \mathbb{Z}_p$. Prove that $G$ cannot be finite. Give an example of such a group
I believe that I have an example of one of these groups in $G = \mathbb{Z}^+$ where $f_p(x) = x \pmod p$
However I'm having a tough time proving that $G$ can't be finite. Any help here would be appreciated!
The existence of a surjective homomorphism from $G$ to $\mathbb{Z}_p$ implies $|G| \geq |\mathbb{Z}_p|$ for each $p$. But, $|Z_p|$ go to infinity as $p$ goes to infinity. So there you go.
(From the rest of your question I assume you define $\mathbb{Z}_p$ to be the field ${\mathbb{Z}}/p{\mathbb{Z}}$.)