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
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
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.
|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 .|