Quotient Theorem for Group Homomorphisms/Examples

Examples of Use of Quotient Theorem for Group Homomorphisms

Real to Complex Numbers under $e^{2 \pi i x}$

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

Let $\struct {\C_{\ne 0}, \times}$ denote the multiplicative group of complex numbers.

Let $\phi: \struct {\R, +} \to \struct {\C_{\ne 0}, \times}$ be the homomorphism defined as:

$\forall x \in \R: \map \phi x = e^{2 \pi i x}$

Then $\phi$ can be decomposed into the form:

$\phi = \alpha \beta \gamma$

in the following way:

$\alpha: \struct {K, \times} \to \struct {\C_{\ne 0}, \times}$ is defined as:

$\forall z \in K: \map \alpha z = z$

where $\struct {K, \times}$ denotes the circle group:

$K = \set {z \in \C: \cmod z = 1}$

$\times$ is the operation of complex multiplication

$\beta: \hointr 0 1 \to K$ is defined as:

$\forall x \in \hointr 0 1: \map \beta x = e^{2 \pi i x}$

where $\hointr 0 1$ denotes the right half-open real interval $\set {x \in \R: 0 \le x < 1}$

$\gamma: \R \to \hointr 0 1$ is defined as:

$\forall x \in \R: \map \gamma x = \fractpart x$

where $\fractpart x$ is the fractional part of $x$:

$\fractpart x := x - \floor x$

Inner Automorphism by Inverse Element

Let $G$ be a group.

Let $\Aut G$ denote the automorphism group of $G$.

Let $\phi: G \to \Aut G$ be the homomorphism defined as:

$\forall g \in G: \map \phi g = \kappa_{g^{-1} }$

where $\kappa_{g^{-1} }$ denotes the inner automorphism of $G$ by $g^{-1}$.

Then $\phi$ can be decomposed into the form:

$\phi = \alpha \beta \gamma$

in the following way:

$\alpha: \Inn G \to \Aut G$ is defined as:

$\forall \kappa \in \Inn G: \map \alpha \kappa = \kappa$

where $\Inn G$ denotes the inner automorphism group of $G$

$\beta: G / \map Z G \to \Inn G$ is defined as:

$\forall g \in G / \map Z G: \map \phi g = \kappa_{g^{-1} }$

where $G / \map Z G$ denotes the quotient group of $G$ by the center of $G$

$\gamma: G \to G / \map Z G$ is defined as:

$\forall g \in G: \map \gamma g = \map {q_{\map Z G} } g = g \, \map Z G$

where $q_{\map Z G}$ is the quotient epimorphism from $G$ to $G / \map Z G$.

Integers to Modulo Integers under Multiplication

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

Let $\struct {\Z_m, +}$ denote the additive group of integers modulo $m$.

Let $\phi: \struct {\Z, +} \to \struct {\Z_m, +}$ be the homomorphism defined as:

$\forall k \in \Z: \map \phi k = \eqclass {n k} m$

for some $n \in \Z$.

Let $d := \gcd \set {m, n}$, where $\gcd \set {m, n}$ denotes the GCD of $m$ and $n$.

Let $c := \dfrac m d = \dfrac m {\gcd \set {m, n} }$.

Then $\phi$ can be decomposed into the form:

$\phi = \alpha \beta \gamma$

in the following way:

$\alpha: \struct {d \, \Z_c, +} \to \struct {\Z_m, +}$ is defined as:

$\forall x \in d \, \Z_c: \map \alpha x = x$

where $d \, \Z_c := \set {0, d, 2 d, \ldots, \paren {c - 1} d}$

$\beta: \Z_c \to d \, \Z_c$ is defined as:

$\forall \eqclass x c \in \Z_c: \map \beta {\eqclass x c} = \eqclass {n x} m$

$\gamma: \Z \to \Z_c$ is defined as:

$\forall x \in \Z: \map \gamma x = \eqclass {x \bmod c} c$

where $\bmod$ denotes the modulo operation.

Integer Power on Circle Group

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.