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 π be the natural surjective homomorphism G->G/K. Then there is an injective homomorphism h:G/K->K such that f = h π. Moreover, G/K is isomorphic to Im f (the image of f).

The situation is described by the (ugly, ASCII) diagram:

   G --> H
   |   7 
pi |  / h
   V /