Fundamental theorem on homomorphisms
In abstract algebra, the fundamental theorem on homomorphisms, also known as the fundamental homomorphism theorem, or the first isomorphism 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.

Group theoretic version 

Given two groups G and H and a group homomorphism ,  let K be a normal subgroup in G and φ the natural surjective homomorphism  (where G/K is the quotient group of G by K). If K is a subset of ker(f) then there exists a unique homomorphism  such that .

In other words, the natural projection φ is universal among homomorphisms on G that map K to the identity element.

The situation is described by the following commutative diagram:

h is injective if and only if . Therefore, by setting  we immediately get the first isomorphism theorem.

We can write the statement of the fundamental theorem on homomorphisms of groups as "every homomorphic image of a group is isomorphic to a quotient group".

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

See also
 Quotient category

References
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Theorems in abstract algebra