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Theorem relating a group with the image and kernel of a homomorphism

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.

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 1= H is isomorphic to 1= \text{im}(\phi) .

The operation that is preserved by is the group operation. If 1= a, b \in \text{im}(\phi), then there exist elements such that and 1= \phi(b')=b. For these and 1= b , we have (since preserves the group operation), and thus, the closure property is satisfied in 1= \text{im}(\phi) 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 1= \text{im}(\phi) , therefore, is a subgroup of 1= H .

The group theoretic version of fundamental homomorphism theorem can be used to show that two selected groups are isomorphic. Two examples are shown below.

For each 1= n \in \mathbb{N} , consider the groups and and a group homomorphism defined by (see modular arithmetic). Next, consider the kernel of 1= f , 1= \text{ker} (f) = n \mathbb{Z} , which is a normal subgroup in 1= \mathbb{Z} There exists a natural surjective homomorphism defined by 1= m \mapsto m+n\mathbb{Z} The theorem asserts that there exists an isomorphism between and 1= \mathbb{Z}/n\mathbb{Z} , or in other words 1= \mathbb{Z} n \cong \mathbb{Z}/n \mathbb{Z} The commutative diagram is illustrated below.

Let be a group with subgroup 1= H Let 1= C G(H) , and be the centralizer, the normalizer and the automorphism group of in 1= G , respectively. Then, the N/C theorem states that is isomorphic to a subgroup of 1= \text{Aut}(H) .

We are able to find a group homomorphism defined by 1= g \mapsto ghg^{ 1} , for all 1= h \in H Clearly, the kernel of is 1= C G(H) Hence, we have a natural surjective homomorphism defined by 1= g \mapsto gC(H) The fundamental homomorphism theorem then asserts that there exists an isomorphism between and 1= \varphi(N G(H)) , which is a subgroup of 1= \text{Aut}(H) .

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