First isomorphism theorem: Difference between revisions

Revision as of 20:56, 10 April 2026

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 $f$ , $im f < G$ is a subgroup of $H$ .

$G / \ker f ≅ im f$ .

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 Let <math>G</math> and <math>H</math> be [[Group|groups]] and <math>f\colon g\to H</math> a group homomorphism. Then,
-* The kernel of <math>f</math> is a normal subgroup of <math>G</math>.
+* The kernel of <math>f</math>, <math>\ker f\trianglelefteq G</math> is a normal subgroup of <math>G</math>.
-* The image of <math>f</math> is a subgroup of <math>H</math>.
+* The image of <math>f</math>, <math>\operatorname{im}f<G</math> is a subgroup of <math>H</math>.
 * <math>G/\ker f \cong \operatorname{im} f</math>.