# Circle Group is Uncountably Infinite

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## 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 (some result).

Hence the result.

$\blacksquare$

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## Sources

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