Actually, this question came from the exercise (Cp.10 Q5.7) in the Artin's "Algebra". The question is that:
- Let $\phi:G\to G'$ be a homomorphism between of abelian groups. Define an induced homomorphism $\hat{\phi}:\hat{G'}\to \hat{G}$ between their character groups. And prove that if $\phi$ is injective, then $\hat{\phi}$ is surjective, and conversely.
I know the induced homomorphism is: $\hat{\phi}(\chi')=\chi'\circ \phi$, and know how to prove if $\phi$ is surjective then $\hat{\phi}$ is injective (but this is not the question asking for).
I tried some approaches of proving the last part, but still not yet finished. Any ideas are welcome!