Definition:Quotient Epimorphism

Theorem
Let $G$ be a group.

Let $N$ be a normal subgroup of $G$.

Let $G / N$ be the quotient group of $G$ by $N$.

Then the mapping $\phi: G \to G / N$, defined as:


 * $\phi: G \to G / N: \phi \left({x}\right) = x N$

is a group epimorphism, and its kernel is $N$.

This epimorphism is known as: from $G$ to $G / N$.
 * the natural (epi)morphism or natural homomorphism
 * the quotient (epi)morphism
 * the canonical (epi)morphism

Proof
The proof follows from Quotient Mapping Canonical Epimorphism.


 * When $N \triangleleft G$, we have:

Therefore $\phi$ is a homomorphism.


 * $\forall x \in G: x N \in G / N = \phi \left({x}\right)$, so $\phi$ is surjective.

Therefore $\phi$ is an epimorphism.


 * Let $x \in G$.

... thus proving that $\ker \left({\phi}\right) = N$ from definition of subset.

Comment
In Kernel is Normal Subgroup of Domain it was shown that the kernel of a group homomorphism is a normal subgroup of its domain. In this result it has been shown that every normal subgroup is a kernel of at least one group homomorphism of the group of which it is the subgroup.

We see that when a subgroup is normal, its cosets make a group using the product rule defined as in this result. However, it is not possible to make the left or right cosets of a non-normal subgroup into a group using the same sort of product rule. Otherwise there would be a group homomorphism with that subgroup as the kernel, and we have seen that this can not be done unless the subgroup is normal.