I am trying to prove that the homomorphism of $\chi:(\mathbb{Z}/q\mathbb{Z})^\times \to \mathbb{C}^*$ is an isomorphism, using the definition of Dirichlet characters for $\chi$
I know it has to do with using the fact that $(\mathbb{Z}/q\mathbb{Z})^\times$ is abelian and can therefore be split into products of cyclic groups, but i get lost there.
Do I use an induction and start with the trivial group first?