how to deduce that finite group $G$ has at most $|G|$ characters

Question

Let $G$ be a finite abelian group. A character of $G$ is a group homomorphism $\chi: G\longrightarrow \mathbb{C}^{\times}$. I have proved by induction that for distinct $\chi_1,\cdots,\chi_r$, they are linearly independent over $\mathbb{C}$. From this, how to deduce that $G$ has at most $|G|$ characters? Thanks.

Answer 1

The space of complex-valued functions from $G$ to $\mathbb{C}$ has dimension $|G|$ (why?). Then any linearly independent set of elements of this space has at most $|G|$ elements. Apply this to the set of all characters on $G$.