Fundamental theorem on homomorphisms
In abstract algebra, the fundamental theorem on homomorphisms, also known as the fundamental homomorphism theorem, relates the structure of two objects between which a homomorphism is given, and of the kernel and image of the homomorphism.
The homomorphism theorem is used to prove the isomorphism theorems.Given two groups G and H and a group homomorphism f : G→H, let K be a normal subgroup in G and φ the natural surjective homomorphism G→G/K. If K is a subset of ker then there exists a unique homomorphism h:G/K→H such that f = h φ.
In other words, the natural projection φ is universal among homomorphisms on G that map K to the identity element.
The situation is described by the following commutative diagram:
By setting K = ker we immediately get the first isomorphism theorem.Similar theorems are valid for monoids, vector spaces, modules, and rings.
