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 $G$ and $H$ be groups and $f : g \to H$ a group homomorphism. Then,

The kernel of $f$ is a normal subgroup of $G$ .
The image of $f$ is a subgroup of $H$ .

$G / \ker f ≅ im f$ .