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 $\text{[math]}$ .

$\text{[math]}$ .