Circle Group is Infinite Abelian Group

Theorem
The circle group $\struct {K, \times}$ is an uncountably infinite abelian group.

Proof
From Circle Group is Group, we have that $\struct {K, \times}$ is a group.

As $K \subseteq \C$, it follows by definition that $\struct {K, \times}$ is a subgroup of $\struct {\C, \times}$.

From Complex Multiplication is Commutative, $\times$ is commutative on $\C$.

From Restriction of Commutative Operation is Commutative, it follows that $\times$ is commutative on $K$.

Hence, by definition, $\struct {K, \times}$ is an abelian group.

Finally we have that the Circle Group is Uncountably Infinite.