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

Group-theoretic version

Given two groups G and H and a group homomorphism, let N be a normal subgroup in G and \phi the natural surjective homomorphism (where G / N is the quotient group of G by N). If N is a subset of \ker(f) (where \ker represents a kernel) then there exists a unique homomorphism such that. In other words, the natural projection \phi 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 N = \ker(f). Therefore, by setting N = \ker(f), 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:

Proof of 1

The operation that is preserved by \phi is the group operation. If a, b \in \text{im}(\phi), then there exist elements such that \phi(a')=a and \phi(b')=b. For these a and b, we have (since \phi preserves the group operation), and thus, the closure property is satisfied in \text{im}(\phi). The identity element e \in H is also in because \phi maps the identity element of G to it. Since every element a' in G has an inverse (a')^{-1} such that (because \phi preserves the inverse property as well), we have an inverse for each element in \text{im}(\phi), therefore, is a subgroup of H.

Proof of 2

Construct a map by \psi(a\ker(\phi)) = \phi(a). This map is well-defined, as if a\ker(\phi) = b\ker(\phi), then and so which gives \phi(a) = \phi(b). This map is an isomorphism. \psi is surjective onto by definition. To show injectiveness, if, then \phi(a) = \phi(b), which implies so a\ker(\phi) = b\ker(\phi). Finally, hence \psi preserves the group operation. Hence \psi is an isomorphism between and \text{im}(\phi), which completes the proof.

Applications

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

Integers modulo n

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

N / C theorem

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

Proof

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