Quotient Theorem for Group Homomorphisms

Theorem
Let $\phi: G \to G'$ be a (group) homomorphism between two groups $G$ and $G'$.

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

where:
 * $\alpha: \operatorname{Im} \left({\phi}\right) \to G'$ is a monomorphism
 * $\beta: G / \ker \left({\phi}\right) \to \operatorname{Im} \left({\phi}\right)$ is an isomorphism
 * $\gamma: G \to G / \ker \left({\phi}\right)$ is an epimorphism.

Monomorphism
The mapping $\alpha$ is the inclusion mapping defined as $\iota: \operatorname{Im} \left({\phi}\right) \to G'$:
 * $\forall x \in \operatorname{Im} \left({\phi}\right): \iota \left({x}\right) = x$

From Inclusion Mapping is Monomorphism, it follows that $\alpha$ is a monomorphism.

Isomorphism
From the First Isomorphism Theorem for Groups:
 * $\operatorname {Im} \left({\phi}\right) \cong G / \ker \left({\phi}\right)$

for any homomorphism $\phi$.

That is, the image of $\phi$ is isomorphic to the quotient group of $G$ by the kernel of $\phi$.

Thus $\beta$ is such an isomorphism.

Epimorphism
The mapping $\gamma: G \to G / \ker \left({\phi}\right)$ is given by:
 * $\forall a \in G: \gamma \left({a}\right) = a \ker \left({\phi}\right)$

where $a \ker \left({\phi}\right)$ is the left coset of $\ker \left({\phi}\right)$ by $a$.

This is justified by Kernel is Normal Subgroup of Domain.

By definition, each coset of $\ker \left({\phi}\right)$ is the coset of each of its elements.

So $\gamma$ is a surjection.

$\gamma$ is shown to be a homomorphism thus:

thus demonstrating the morphism property.

A homomorphism which is surjective is an epimorphism.

Hence the result.

This theorem can be illustrated by means of the following commutative diagram:


 * HomomorphismDecomposition.png

Also see

 * Quotient Theorem for Sets