# Quotient Group of Reals by Integers is Circle Group

## Theorem

Let $\struct {\Z, +}$ be the additive group of integers.

Let $\struct {\R, +}$ be the additive group of real numbers.

Let $K$ be the circle group.

Then the quotient group of $\struct {\R, +}$ by $\struct {\Z, +}$ is isomorphic to $K$.

## Proof

Define $\phi: \R / \Z \to K$ by:

- $\map \phi {x + \Z} = \map \exp {2 \pi i x}$

Then $\phi$ is well-defined.

For, if $x + \Z = y + \Z$, then $y = x + n$ for some $n \in \Z$, and:

- $\map \exp {2 \pi i \paren {x + n} } = \map \exp {2 \pi i x}$

by Complex Exponential Function has Imaginary Period.

Moreover, by Exponential of Sum:

- $\map \exp {x + y + \Z} = \map \exp {2 \pi i \paren {x + y} } = \map \exp {2 \pi i x} \, \map \exp {2 \pi i y}$

meaning $\phi$ is a group homomorphism.

By Euler's Formula:

- $\map \exp {2 \pi i x} = \map \cos {2 \pi i x} + i \, \map \sin {2 \pi i x}$

so that, by Sine and Cosine are Periodic on Reals:

- $\map \phi x = 1$ if and only if $x + \Z = 0 + \Z$

Hence, by Kernel is Trivial iff Monomorphism, $\phi$ is a monomorphism.

From polar form for complex numbers, it follows that all $z \in \C$ with $\cmod z = 1$ are of the form:

- $z = \map \exp {2 \pi i x}$

for some $x \in \R$.

Hence $\phi$ is also an epimorphism.

Thus $\phi: \R / \Z \to K$ is a group isomorphism.

$\blacksquare$

## Sources

- 1971: Allan Clark:
*Elements of Abstract Algebra*... (previous) ... (next): Chapter $2$: Conjugacy, Normal Subgroups, and Quotient Groups: $\S 47 \beta$ - 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): $\S 52$. The first isomorphism theorem