Group homomorphism from finite abelian group to $\mathbb{C}\setminus\{0\}$

Question

Let $G$ be a finite abelian group, $g\in G$ be an element of order $n$ and $\xi = \exp (\frac{2\pi i}{n})$. I want to show that for each $0\leq i\leq n-1$, there are exactly $\frac{|G|}{n}$ group homomorphisms $G\to \mathbb{C}\setminus\{0\}$ which satisfy $g\mapsto \xi^i$.

I applied the structure theorem of finite abelian group, but it doesn't seem to help me.

Answer 1

For a finite abelian group we have $$|G|=|Hom(G,\Bbb{C}^*)|$$

Let $$f:Hom(G,\Bbb{C}^*)\to \mu_n, \qquad f(\chi)=\chi(g)$$ It is surjective so $$Hom(G,\Bbb{C}^*)/\ker(f)\cong \mu_n$$ Whence any coset has size $$|Hom(G,\Bbb{C}^*)|/|\mu_n| = |G|/n$$