It's a well-known result that in a field $\displaystyle F,$ everyfinitesubgroup of $\displaystyle F^{\times}$ is cyclic. A much less known but more interesting result is this:

Let $\displaystyle D$ be a division ring, i.e. a field which is not necessarily commutative. Prove that everyfinite abeliansubgroup $\displaystyle G$ of $\displaystyle D^{\times}$ is cyclic.

Suggestion:

Spoiler: