Complement of Normal Subgroup is Isomorphic to Quotient Group

Definition
Let $G$ be a group with identity $e$.

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

Let $K$ be a complement of $N$.

Then the quotient group $G / N$ is isomorphic to $K$.

Proof
Let $\phi: K \to G / N$ be the mapping defined as:


 * $\forall k \in K: \map \phi k = k N$

We have:

Thus $\phi$ is shown to be a homomorphism.

By definition of $K$ and $N$ we have that:
 * $\forall g \in G: \exists k \in K, n \in N: g = k n$

That is, every element of $g$ is the product of an element of $K$ with an element of $N$.

Let $x N, y N \in G / N$.

Then:

demonstrating that $\phi$ is an injection.

Then we have:

demonstrating that $\phi$ is a surjection.

Thus $\phi$ is an injective and surjective homomorphism.

Hence, by definition, $\phi$ is an isomorphism.

Hence the result.