Third Isomorphism Theorem/Groups/Corollary

Corollary to Third Isomorphism Theorem for Groups
Let $G$ be a group.

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

Let $q: G \to \dfrac G N$ be the quotient epimorphism from $G$ to the quotient group $\dfrac G N$.

Let $K$ be the kernel of $q$.

Then:
 * $\dfrac G N \cong \dfrac {G / K} {N / K}$

Proof
From Kernel is Normal Subgroup of Domain we have that $K$ is a normal subgroup of $G$.

Thus the Third Isomorphism Theorem for Groups can be applied directly.

Also known as
Some sources refer to this as the first isomorphism theorem.