First Isomorphism Theorem/Groups

Theorem
Let $\phi: G_1 \to G_2$ be a group homomorphism.

Let $K$ be the kernel of $\phi$.

Then: $\operatorname {Im} \left({\phi}\right) \cong G_1 / K$ where $\cong$ denotes group isomorphism.

Proof
By Kernel is Normal Subgroup of Domain, $G_1 / K$ exists.

We need to establish that the mapping $\theta: G_1 / K \to G_2$ defined as: $\qquad \forall x \in G_1: \theta \left({x K}\right) = \phi \left({x}\right) \qquad$ is well-defined.

That is, we need to ensure that: $\forall x, y \in G: x K = y K \implies \theta \left({x K}\right) = \theta \left({y K}\right)$

Let $x, y \in G: x K = y K$. Then:

Thus we see that $\theta$ is well-defined.

We now prove $\theta$ is injective: $\theta \left({x K}\right) = \theta \left({y K}\right) \iff \phi \left({x}\right) = \phi \left({y}\right) \implies x K = y K$.

We now prove $\theta$ is surjective: We  note that $\operatorname {Im} \left({\theta}\right) = \left\{{\theta \left({x K}\right): x \in G}\right\}$.

We also prove that $\theta$ is a homomorphism:

Thus $\theta$ is a monomorphism whose image equals $\operatorname {Im} \left({\phi}\right)$.

Also known as
Some sources call this the homomorphism theorem.

Others combine this result with Group Homomorphism Preserves Subgroups, Kernel of Group Homomorphism is Subgroup and Kernel is Normal Subgroup of Domain.

Still others do not assign a special name to this theorem at all.

Also see

 * Isomorphism Theorems