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:
 * $\alpha: \Img \phi \to G'$ is a monomorphism
 * $\beta: G / \map \ker \phi \to \Img \phi$ is an isomorphism
 * $\gamma: G \to G / \map \ker \phi$ is an epimorphism.

Monomorphism
The mapping $\alpha$ is the inclusion mapping defined as $\iota: \Img \phi \to G'$:
 * $\forall x \in \Img \phi: \map \iota x = x$

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

Isomorphism
From the First Isomorphism Theorem for Groups:
 * $\Img \phi \cong G / \map \ker \phi$

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 $\gamma: G \to G / \map \ker \phi$ is given by:
 * $\forall a \in G: \map \gamma a = a \, \map \ker \phi$

where $a \, \map \ker \phi$ is the left coset of $\map \ker \phi$ by $a$.

This is justified by Kernel is Normal Subgroup of Domain.

By definition, each (left) coset of $\map \ker \phi$ is the (left) 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:


 * $\begin{xy}\xymatrix@L+2mu@+1em{

G \ar@{-->}[r]^*{\phi} \ar[d]^*{\gamma} & G' \\ G / \map \ker \phi \ar[r]_*{\beta}^*{\cong} & \Img \phi \ar[u]^*{\alpha} }\end{xy}$

Also see

 * Quotient Theorem for Sets