Fundamental theorem on homomorphisms

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 φ be the natural surjective homomorphism G->G/K. Then there exists a unique homomorphism h:G/K->H such that f = h φ. 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:

Missing image
FundHomDiag.png
image:FundHomDiag.png

Similar theorems are valid for monoids, vector spaces, modules, and rings.

See also: Fundamental theorem on homomorphisms, Abstract algebra, Commutative diagram, Group (mathematics), Group homomorphism, Homomorphism, Injective, Kernel of a homomorphism, Module (mathematics), Monoid