Fundamental theorem on homomorphisms

The homomorphism is a mathematical theorem in the field of algebra, which applies in a corresponding form for maps between groups, vector spaces and rings. It represents each establishes a close relationship between group homomorphisms and normal subgroups, Vektorraumhomomorphismen and vector subspaces and ring homomorphisms and ideals. The homomorphism theorem is:

Group

Statement

Is a group homomorphism, then the kernel is a normal subgroup of and the factor group is isomorphic to the image. A corresponding isomorphism is given by.

Evidence

The proof boils down to showing that a group isomorphism is. If two representations of the same coset, then there are elements with. Da and the elements of the core are, is considered. It follows that

And thus is well-defined.

The map is a group homomorphism, since for all cosets and of the equation

Applies. is also injective, since

And the neutral element of the factor group.

It is also surjective, since, for every: .

It follows that a group isomorphism, and thus.

Examples

• It stands for the general linear group represented by regular matrices over a field. The determinant

• Analog shows you:

• It stands for the symmetric group. The Signum mapping defines a group homomorphism with ( alternating group ), which is surjective. After the homomorphism therefore applies to:

Vector space

Is a homomorphism of vector, ie a linear map from to, then the core is a subspace of and the factor space is isomorphic to the image.

Ring

Is a ring homomorphism, then the kernel is an ideal of and the factor ring is isomorphic to the image.

The proof is analogous to the proof for groups, it only needs to be shown:

Generalizations

• The sentence is generally in any abelian category.
• The sentence also applies for example in the category of topological groups; However, the image is then to be understood in categorical sense, it is not generally to the set-theoretic image with the induced topology. A bijective continuous homomorphism is only a categorical isomorphism, even if its inverse is continuous, that is, if he is also a homeomorphism.