Quotient Theorem for Group Homomorphisms/Examples/Integer Power on Circle Group

Example of Use of Quotient Theorem for Group Homomorphisms
Let $K$ denote the circle group.

Let $\phi: K \to K$ be the homomorphism defined as:
 * $\forall z \in K: \map \phi z = z^n$

for some $n \in \Z_{>0}$.

Then $\phi$ can be decomposed into the form:
 * $\phi = \alpha \beta \gamma$

in the following way:


 * $\alpha: K \to K$ is defined as:
 * $\forall z \in K: \map \alpha z = z$
 * that is, $\alpha$ is the identity mapping


 * $\beta: S \to K$ is defined as:
 * $\forall z \in S: \map \phi z = z^n$
 * where $S$ denotes the set defined as:
 * $S := \set {z \in \C: z = e^{2 \pi i x}, 0 \le x < \dfrac 1 n}$


 * $\gamma: K \to S$ is defined as:
 * $\forall z \in K: \map \gamma z = z \bmod \dfrac 1 n$
 * where $\bmod$ denotes the modulo operation.

Proof
It is first demonstrated that $\phi$ is a homomorphism:

We have that $1$ is the identity element of $K$, and to confirm:


 * $\map \phi 1 = 1^n = 1$

Now we can establish what the kernel of $\phi$ is:

Thus $\map \ker \phi$ is the set of complex $n$th roots of unity:


 * $\map \ker \phi = U_n = \set {e^{2 i k \pi / n}: k \in \N_n}$

where $\N_n = \set {0, 1, 2, \ldots, n - 1}$.

Next we establish what the image of $\phi$ is.

Let $w \in \Img \phi$ such that:
 * $w = e^{x i}$

for some $x \in \R$.

Thus every element of $K$ has a preimage under $\phi$.

Hence:
 * $\Img \phi = K$

Thus, from the Quotient Theorem for Group Homomorphisms, $\phi$ can be decomposed into:
 * $\phi = \alpha \beta \gamma$

where:
 * $\alpha: K \to K$, which is the identity mapping
 * $\beta: K / \map \ker \phi \to K$, which is an isomorphism
 * $\gamma: K \to K / \map \ker \phi$, which is an epimorphism.

The result follows.