Fundamental theorem on homomorphisms
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For some 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:

Similar theorems are valid for vector spaces, modules, and ringss.

The original article can be found at: http://en.wikipedia.org/wiki/Fundamental_theorem_on_homomorphisms.
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