values taken by Dirichlet character = complex roots of unity

Question

I'm still fairly new to the field of Dirichlet characters modulo $N$, denoted as maps $\chi: G_N=(\mathbb{Z}/N \mathbb{Z})^* \rightarrow \mathbb{C}^*$ and I came across this explanation:

"Since $G_N$ is finite, the values taken by any Dirichlet character are complex roots of unity and so the inverse of a Dirichlet character is its complex conjugate defined by the rule $\bar{\chi}(n)=\overline{\chi(n)}$ for all $n \in G_N$."

What troubles me is the first part of the sentence (second part is perfectly clear): How do I get from the simple fact that $G_N$ is finite that my values are already complex roots of unity? I also read this statement in several other articles but none of them mentioned any "proof" for this, thus I assumed it must be a fairly easy argument, which I unfortunately fail to see.

Answer 1

This follows through three observations:

• Every element of a finite group is torsion
• Group homomorphisms map torsion to torsion
• Roots of unity in $\mathbb{C}^\times$ are, by definition, precisely the torsion elements