# Affine orthogonal group

This article defines a natural number-parametrized system of algebraic matrix groups. In other words, for every field and every natural number, we get a matrix group defined by a system of algebraic equations. The definition may also generalize to arbitrary commutative unital rings, though the default usage of the term is over fields.

View other linear algebraic groups|View other affine algebraic groups

## Contents

## Definition

Let be a field and be a natural number. The **affine orthogonal group** is defined as the semidirect product of the vector space with the orthogonal group .

This is naturally a subgroup of the general affine group , which in turn is a subgroup of the general linear group .

## As a map

### As a functor from fields to groups

For fixed , we get a functor from the category of fields to the category of groups, sending a field to the affine orthogonal group .

### As an IAPS

`Further information: Affine orthogonal IAPS`

The affine orthogonal groups form an IAPS of groups. In other words, for any natural numbers , there is an injective group homomorphism:

.

This homomorphism essentially does the left group element on the first coordinates and the right group element on the next coordinates.

### As a functor from fields to IAPSes

If we fix neither nor , we get a functor that inputs a field and outputs an IAPS of groups.

## Relation with other linear algebraic groups

### Supergroups

### Subgroups

### Group and subgroup operations

- intersection with the general linear group yields the orthogonal group.
- intersection with the special affine group yields the special affine orthogonal group.
- intersection with the special linear group yields the special orthogonal group.

## Particular cases

### Finite fields

Size of field | Order of matrices | Common name for the orthogonal group |
---|---|---|

Odd prime | 1 | Dihedral group |

Odd | 1 | Semidirect product of elementary abelian group of order by inverse map. |

1 | Elementary abelian group of order . | |

2 | 2 | Dihedral group:D8 |