Fundamental theorem on homomorphisms

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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 : GH be a group homomorphism; let K be the kernel of f; let φ be the natural surjective homomorphism GG/K. Then there exists a unique homomorphism h:G/KH 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 monoids, vector spaces, modules, and rings.

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