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 N be a normal subgroup in G and φ the natural surjective homomorphism  (where G/N is the quotient group of G by N). If N 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 N 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".

Proof 
The proof follows from two basic facts about homomorphisms, namely their preservation of the group operation, and their mapping of the identity element to the identity element. We need to show that if  is a homomorphism of groups, then:  is a subgroup of . is isomorphic to .

Proof of 1 
The operation that is preserved by  is the group operation. If , then there exist elements  such that  and . For these  and , we have  (since  preserves the group operation), and thus, the closure property is satisfied in . The identity element  is also in  because  maps the identity element of  to it. Since every element  in  has an inverse such that  (because  preserves the inverse property as well), we have an inverse for each element  in , therefore,  is a subgroup of .

Proof of 2 
Construct a map  by . This map is well-defined, as if , then  and so  which gives . This map is an isomorphism.  is surjective onto  by definition. To show injectiveness, if , then , which implies 
so . 
Finally, hence  preserves the group operation. Hence  is an isomorphism between  and , which completes the proof.

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