In abstract algebra, for a number of algebraic structures, the fundamental theorem on homomorphisms relates the structure of two objects between which a homomorphism is given, and of the kernel and image of the homomorphism.

For groups, the theorem states:

Let G and H be groups; let f : G?H be a group homomorphism; let K be the kernel of f; let f be the natural surjective homomorphism G?G/K. Then there exists a unique homomorphism h:G/K?H such that f = h f. Moreover, h is injective and provides an isomorphism between G/K and the image of f.
The situation is described by the following commutative diagram: