First Isomorphism Theorem

Preface: This theorem applies for Groups, Rings, Modules, Algebras, and any other algebraic structure where you see the word "homomorphism".

It is a categorical result i.e. it is independent of the structure used.

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

Then $$\mathrm {Im} \left({\phi}\right) \cong G_1 / \mathrm {ker} \left({\phi}\right)$$.

Rings
Let $$\alpha: R \rightarrow S$$ be a ring homomorphism.

Then $$\mathrm {Im} \left({\alpha}\right) \cong R / \mathrm {ker} \left({\alpha}\right)$$.

Proof for Groups
Let $$K = \mathrm {ker} \left({\phi}\right)$$.

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:

$$\forall x \in G_1: \theta \left({x K}\right) = \phi \left({x}\right)$$

is well-defined.

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

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

$$ $$ $$ $$ $$

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

Since we also have that $$\phi \left({x}\right) = \phi \left({y}\right) \implies x K = y K$$, it follows that $$\theta \left({x K}\right) = \theta \left({y K}\right) \implies x K = y K$$.

So $$\theta$$ is injective.

We also note that $$\mathrm {Im} \left({\theta}\right) = \left\{{\theta \left({x K}\right): x \in G}\right\}$$.

So:

$$ $$ $$

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

$$ $$ $$ $$ $$

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

The result follows.

Comment
There is no standard numbering for the Isomorphism Theorems. Different authors use different labellings.

This particular result, for example, is also known as the Homomorphism Theorem.