# Circle Group is Infinite Abelian Group

## Theorem

The circle group $\struct {K, \times}$ is an uncountably infinite abelian group under the operation of complex multiplication.

## Proof 1

First we note that $K \subseteq \C$.

So to show that $K$ is a group it is sufficient to show that $K$ is a subgroup of the multiplicative group of complex numbers $\struct {\C_{\ne 0}, \times}$.

From Complex Multiplication Identity is One, the identity element $1 + 0 i$ is in $K$.

Thus $K \ne \O$.

We now show that $z, w \in K \implies z w \in K$:

 $\displaystyle z, w$ $\in$ $\displaystyle K$ $\displaystyle \leadsto \ \$ $\displaystyle \cmod z$ $=$ $\displaystyle 1 = \cmod w$ $\displaystyle \leadsto \ \$ $\displaystyle \cmod {z w}$ $=$ $\displaystyle \cmod z \cmod w$ $\displaystyle \leadsto \ \$ $\displaystyle z w$ $\in$ $\displaystyle K$

Next we see that $z \in K \implies z^{-1} \in K$:

$\cmod z = 1 \implies \cmod {\dfrac 1 z} = 1$

Thus by the Two-Step Subgroup Test:

$K \le \C_{\ne 0}$

Thus $K$ is a group.

From Complex Multiplication is Commutative it also follows from Subgroup of Abelian Group is Abelian that $K$ is an abelian group.

Finally we have that the Circle Group is Uncountably Infinite.

$\blacksquare$

## Proof 2

We note that $K \ne \varnothing$ as the identity element $1 + 0 i \in K$.

Since all $z \in K$ have modulus $1$, they have, for some $\theta \in \hointr 0 {2 \pi}$, the polar form:

$z = \map \exp {i \theta} = \cos \theta + i \sin \theta$

Conversely, if a complex number has such a polar form, it has modulus $1$.

Observe the following property of the complex exponential function:

$\forall a, b \in \C: \map \exp {a + b} = \map \exp a \map \exp b$

We must show that if $x, y \in K$ then $x \cdot y^{-1} \in K$.

Let $x, y \in K$ be arbitrary.

Choose suitable $s, t \in \hointr 0 {2 \pi}$ such that:

$x = \map \exp {i s}$
$y = \map \exp {i t}$

We compute:

 $\displaystyle \map \exp {i t} \map \exp {-i t}$ $=$ $\displaystyle \map \exp {i \paren {t - t} }$ $\displaystyle$ $=$ $\displaystyle \map \exp 0$ $\displaystyle$ $=$ $\displaystyle 1$

So $y^{-1} = \map \exp {-i t}$.

We note that this lies in $K$.

Furthermore, we have:

 $\displaystyle x y$ $=$ $\displaystyle \map \exp {i s} \map \exp {-2 \pi i t}$ $\displaystyle$ $=$ $\displaystyle \map \exp {i \paren {s - t} }$

We conclude that $x y \in K$.

By the Two-Step Subgroup Test, $K$ is a subgroup of $\C$ under complex multiplication.

That the operation $\times$ on $K$ is commutative follows from Complex Multiplication is Commutative and Restriction of Commutative Operation is Commutative.

That is, $\times$ is commutative on $K$ because it is already commutative on $\C$.

Finally we have that the Circle Group is Uncountably Infinite.

$\blacksquare$

## Proof 3

Taking the group axioms in turn:

### $\text G 0$: Closure

 $\displaystyle z, w$ $\in$ $\displaystyle K$ $\displaystyle \leadsto \ \$ $\displaystyle \cmod z$ $=$ $\displaystyle 1 = \cmod w$ $\displaystyle \leadsto \ \$ $\displaystyle \cmod {z w}$ $=$ $\displaystyle \cmod z \cmod w$ $\displaystyle \leadsto \ \$ $\displaystyle z w$ $\in$ $\displaystyle K$

So $\struct {\mathbb S, \cdot}$ is closed.

$\Box$

### $\text G 1$: Associativity

$\Box$

### $\text G 2$: Identity

From Complex Multiplication Identity is One we have that the identity element of $K$ is $1 + 0 i$.

$\Box$

### $\text G 3$: Inverses

We have that:

$\cmod z = 1 \implies \dfrac 1 {\cmod z} = \cmod {\dfrac 1 z} = 1$

But:

$z \times \dfrac 1 z = 1 + 0 i$

So the inverse of $z$ is $\dfrac 1 z$.

$\Box$

### $\text C$: Commutative

We have that Complex Multiplication is Commutative.

We also have from Restriction of Commutative Operation is Commutative that $\times$ is likewise commutative on $K$.

$\Box$

### Uncountably Infinite

$\Box$

All the criteria are satisfied, and the result follows.

$\blacksquare$

## Proof 4

Consider the complex modulus function $\left\vert{\cdot}\right\vert: \C \to \R, z \mapsto \left\vert{z}\right\vert$.

By Complex Modulus is Norm, we have that $\left\vert{z}\right\vert \ge 0$ for all $z \in \C$, and:

$\left\vert{z}\right\vert = 0 \iff z = 0$

From Group of Units of Field and Complex Numbers form Field, we have $\C^\times = \C \setminus \left\{{0}\right\}$.

By above observation, the modulus has a restriction to $\C^\times$:

$\left\vert{\cdot}\right\vert: \C^\times \to \R_{>0}, z \mapsto \left\vert{z}\right\vert$

From $\left\vert{1}\right\vert = 1$ and Modulus of Product, it follows that it is in fact a group homomorphism:

$\phi: \left({C^\times, \cdot}\right) \to \left({\R_{>0}, \cdot}\right), z \mapsto \left\vert{z}\right\vert$

Now $K$ is by definition the kernel of $\phi$.

Hence, by Kernel of Group Homomorphism is Subgroup, $\left({K, \times}\right)$ is a subgroup of $\left({\C^\times, \cdot}\right)$.

Finally, note that $\left({K, \times}\right)$ is uncountable from Circle Group is Uncountably Infinite.

$\blacksquare$