Theorem

Let $\mathbb{F}$ be any field. Any finite subgroup $G$ of $\mathbb{F}^{*}$ is cyclic.

Corollary 1

The only finite subgroups of $\mathbb{C}^{*}$ are the groups $\mu_{n}$ consisting of the $n$-th roots of unity with $n\ge 1$.

Corollary 2

If $\mathbb{F}$ is a finite field, then $\mathbb{F}^{*}$ is cyclic.