Third Isomorphism Theorem/Groups/Corollary

Theorem
Let $G$ be a group.

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

Let $q: G \to \dfrac G N$ be the natural 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 can be applied directly.