# First Isomorphism Theorem

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## 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 $\map \ker \phi$ be the kernel of $\phi$.

Then:

- $\Img \phi \cong G_1 / \map \ker \phi$

where $\cong$ denotes group isomorphism.

### Rings

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

Let $\map \ker \phi$ be the kernel of $\phi$.

Then:

- $\Img \phi \cong R / \map \ker \phi$

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**.