Quotient Epimorphism is Epimorphism/Group

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 $q_N: G \to G / N$ be the quotient epimorphism from $G$ to $G / N$:
 * $\forall x \in G: \map {q_N} x = x N$

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

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

When $N \lhd G$, we have:

Therefore $q$ is a homomorphism.

We have that:
 * $\forall x \in G: x N \in G / N = \map {q_N} x$

so $q$ is surjective.

Therefore $q$ is an epimorphism.

Let $x \in G$.

thus proving that $\map \ker {q_N} = N$ from definition of subset.