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 $f$, $\ker f⊴G$ is a normal subgroup of $G$.
• The image of $f$, $\mathrm{im}f<G$ is a subgroup of $H$.

• $G/\ker f\cong \mathrm{im}f$.