# Homomorphism theorem

The **homomorphism theorem** is a mathematical theorem from the field of algebra , which applies in a corresponding form to mappings between groups , vector spaces and rings . It creates a close connection between group homomorphisms and normal divisors , vector space homomorphisms and sub-vector spaces as well as ring homomorphisms and ideals . The homomorphism theorem reads:

- Is a homomorphism and the core by , then the quotient is isomorphic to the image .

## group

### statement

If a group is homomorphic , then the kernel is a normal divisor of and the factor group is isomorphic to the image . A corresponding isomorphism is given by .

### proof

It suffices to show that the map is a group isomorphism .

is well defined and injective , there

is a group homomorphism , since for all secondary classes and the following applies:

surjective because for each applies: .

It follows from this that is a group isomorphism , and thus .

### Examples

- Let it stand for the general linear group represented by regular matrices over a body . The determinant

- is a group homomorphism whose core consists of the special linear group of matrices with determinants . According to the homomorphism theorem applies
- .

- From this it follows in particular that, in contrast to the linear group, the factor group is Abelian.

- Analogously one shows:

- where stands for the orthogonal group and for the special orthogonal group.

- It stands for the symmetrical group . The Signum mapping defines a group homomorphism with ( alternating group ), which is for surjective. According to the homomorphism theorem, we have :

## ring

If 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:

## Vector space

### statement

Is a vector space homomorphism, i. H. a linear mapping from to , then the kernel is a subspace of and the factor space is isomorphic to the image .

### example

The differential operator

is a homomorphism from the vector space of the continuously differentiable functions into the vector space of the continuous functions . Its core is the set of constant functions that is noted here as . According to the homomorphism theorem applies

The isomorphism is the induced homomorphism

- .

Its inverse homomorphism is indefinite integration

where is any antiderivative of .

## Generalizations

- Homomorphism theorem for algebraic structures:

- If and are two algebraic structures of the same kind and if a homomorphism of this kind with a kernel , then applies .

- The theorem applies generally to every Abelian category .
- The theorem also applies to the category of topological groups , for example ; however, the picture is then also to be understood in the categorical sense, so it is generally not the set-theoretical picture with the induced topology . Also, a bijective continuous homomorphism is only a categorical isomorphism if its inverse is continuous, i.e. H. if it is also a homeomorphism .

## literature

- Christian Karpfinger, Kurt Meyberg:
*Algebra. Groups - rings - bodies.*Spektrum Akademischer Verlag, Heidelberg 2009, ISBN 978-3-8274-2018-3 , p. 54, p. 167-168