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.
Similar theorems are valid for vector spaces, modules, and rings.
and a group homomorphism
the natural surjective homomorphism
is the quotient group of
represents a kernel) then there exists a unique homomorphism
In other words, the natural projection
is universal among homomorphisms on
The situation is described by the following commutative diagram:
, 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".
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.
is a homomorphism of groups, then: The operation that is preserved by
is the group operation.
preserves the group operation), and thus, the closure property is satisfied in
maps the identity element of
preserves the inverse property as well), we have an inverse for each element
ψ ( a ker ( ϕ ) ) = ψ ( b ker ( ϕ ) )
preserves the group operation.
The group theoretic version of fundamental homomorphism theorem can be used to show that two selected groups are isomorphic.
and a group homomorphism
There exists a natural surjective homomorphism
The theorem asserts that there exists an isomorphism
The commutative diagram is illustrated below.
be a group with subgroup
be the centralizer, the normalizer and the automorphism group of
We are able to find a group homomorphism
Hence, we have a natural surjective homomorphism
The fundamental homomorphism theorem then asserts that there exists an isomorphism between
