A question on values of 1-dim representation dicide the element of an abelian group

Question

Let $G$ be an abelian group, and let $g,h\in G$. I want to show that if $\phi(g)=\phi(h)$ for any group homomorphism $\phi:G\to\mathbb{C}^{\times}$, then $g=h$.

If $G$ is a finite group, the problem is obvious. But how about the general case?