Fundamental theorem on homomorphisms
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In abstract algebra, the fundamental theorem on homomorphisms, also known as the fundamental homomorphism theorem, relates the structure of two objects between which a homomorphism is given, and of the kernel and image of the homomorphism.
The homomorphism theorem is used to prove the isomorphism theorems.
[edit] Group theoretic version
Given two groups G and H and a group homomorphism f : G→H, let K be a normal subgroup in G and φ the natural surjective homomorphism G→G/K. If K ⊂ ker(f) then there exists a unique homomorphism h:G/K→H such that f = h φ.
The situation is described by the following commutative diagram:
By setting K = kernel(f) we immediately get the first isomorphism theorem.
[edit] Other versions
Similar theorems are valid for monoids, vector spaces, modules, and rings.
