Quotient Theorem for Group Homomorphisms/Examples/Integers to Modulo Integers under Multiplication

Example of Use of Quotient Theorem for Group Homomorphisms
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.

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

We have that:

By Group Homomorphism Preserves Identity it is confirmed that $\eqclass 0 m$ is the identity of $\struct {\Z_m, \times}$.

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

That is:

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

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

where:
 * $\alpha: d \, \Z_c \to \Z_m$, which is a monomorphism
 * $\beta: \Z / c \, \Z \to d \, \Z_c$, which is an isomorphism
 * $\gamma: \Z \to \Z / c \, \Z$, which is an epimorphism.

Finally we have from Quotient Group of Integers by Multiples:
 * $\Z / c \, \Z = \Z_c $

and the result follows.