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.

Theorem

Groups

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

Let $\ker \left({\phi}\right)$ be the kernel of $\phi$.

Then:

$\operatorname {Im} \left({\phi}\right) \cong G_1 / \ker \left({\phi}\right)$

where $\cong$ denotes group isomorphism.

Rings

Let $\phi: R \to S$ be a ring homomorphism.

Let $\ker \left({\phi}\right)$ be the kernel of $\phi$.

Then:

$\operatorname {Im} \left({\phi}\right) \cong R / \ker \left({\phi}\right)$

where $\cong$ denotes ring isomorphism.

Also known as

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.