First isomorphism theorem

The first isomorphism theorem is a fundamental result in abstract algebra that describes the relationship between a homomorphism, its kernel, and its image. The theorem appears uniformly across algebraic structures such as groups, rings, and modules, and serves as a prototype for many structural results in algebra. Specifically, given a homeomorphism, the quotient of its domain by its kernel is isomorphic to its image.

Group theory

Statement

Let $\text{[math]}$ and $\text{[math]}$ be groups and $\text{[math]}$ a group homomorphism. Then,

The kernel of $\text{[math]}$ , $\text{[math]}$ is a normal subgroup of $\text{[math]}$ .
The image of $\text{[math]}$ , $\text{[math]}$ is a subgroup of $\text{[math]}$ .
$\text{[math]}$ .

Proof

Proof of 1

By definition, $\text{[math]}$ where $\text{[math]}$ is the identity of $\text{[math]}$ . $\text{[math]}$ is a subgroup of $\text{[math]}$ because:

Identity: Since $\text{[math]}$ is a homomorphism, $\text{[math]}$ . Therefore $\text{[math]}$ , implying $\text{[math]}$ is non-empty and has an identity.
Closure: Let $\text{[math]}$ , then

$\text{[math]}$

Thus $\text{[math]}$ .

Inverses: Let $\text{[math]}$ , then
$\text{[math]}$

Thus $\text{[math]}$ .

Therefore, $\text{[math]}$ is a subgroup of $\text{[math]}$ .

Let $\text{[math]}$ and $\text{[math]}$ , then

$\text{[math]}$

thus $\text{[math]}$ .

Therefore, $\text{[math]}$ is a normal subgroup.

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Proof of 2

By definition, $\text{[math]}$ . $\text{[math]}$ is a subgroup of $\text{[math]}$ because:

Identity: Since $\text{[math]}$ is a homomorphism, $\text{[math]}$ . Therefore $\text{[math]}$ , implying $\text{[math]}$ is non-empty and has an identity.
Closure: Let $\text{[math]}$ , then by definition, there exists $\text{[math]}$ such that $\text{[math]}$ and $\text{[math]}$ . Thus