Circle Group is Uncountably Infinite

Theorem

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

Proof

From Quotient Group of Reals by Integers is Circle Group, $\struct {K, \times}$ is isomorphic to the quotient group of $\struct {\R, +}$ by $\struct {\Z, +}$.

But $\dfrac {\struct {\R, +} } {\struct {\Z, +} }$ is the half-open interval $\hointr 0 1$.

A real interval is uncountable by Real Numbers are Uncountable.

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Hence the result.

$\blacksquare$

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Sources

• 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: Subgroups and Cosets: $\S 38$