In abstract algebra, the fundamental theorem on homomorphisms, also known as the fundamental homomorphism theorem, or the first isomorphism theorem, relates the structure of two objects between which a homomorphism is given, and of the kernel and of the homomorphism.
The homomorphism theorem is used to prove the isomorphism theorems.
Group theoretic version
Given two groups G and H and a group homomorphism f : G → H, let N be a normal subgroup in G and φ the natural surjective homomorphism G → G/N (where G/N is the quotient group of G by N). If N is a subset of ker(f) then there exists a unique homomorphism h: G/N → H such that f = h∘φ.
In other words, the natural projection φ is universal among homomorphisms on G that map N to the identity element.
The situation is described by the following commutative diagram:
h is injective if and only if