Quotient Epimorphism is 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 $N$.

Let $q: G \to G / N$ be the natural epimorphism from $G$ to $G / N$:
 * $q: G \to G / N: q \left({x}\right) = x N$

Then $q$ is a group epimorphism whose kernel is $N$.

Proof
The proof follows from Quotient Mapping on Structure is Canonical Epimorphism.

When $N \triangleleft G$, we have:

Therefore $q$ is a homomorphism.

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

Therefore $q$ is an epimorphism.

Let $x \in G$.

... thus proving that $\ker \left({q}\right) = N$ 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.