Circle Group is Infinite Abelian Group
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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.
$\blacksquare$