We have $f:(\mathbb{Z}/n\mathbb{Z},+)\rightarrow(U_n,.)$ is a homomorphism we need to prove it bijective. $f$ is defined to be $f(\bar{k})=z^{k}$ and $U_n$={ $z\in\mathbb{C}\setminus \{0\}$ such that $z^{n}=1$}.
proof.
$f$ is injective:
let $x,y \in\mathbb{Z}/n\mathbb{Z}$ then $x=\bar{k}$ and $y=\bar{k^\prime}$
$f(x)=f(y)$ then $f(\bar{k})=f(\bar{k^\prime})$ hence $z^{k}=z^{k^\prime}$ so $k=k^\prime$.
$f$ is surjective:
let $y\in U_n$ then $\exists$ $k\in\mathbb{Z}/n\mathbb{Z} $ such that $y=z^{k}=f(\bar{k})$
I'm not really sure if I proved it surjective. Also, could anyone tell me how to write this better.
Hint: Since both groups have the same number of elements, it suffices to prove that the map is injective.