Quotient Group of Reals by Integers is Circle Group

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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