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.
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Definition

Let $k$ be a field and $n$ be a natural number. The affine orthogonal group $AO(n,k)$ is defined as the semidirect product of the vector space $k^{n}$ with the orthogonal group $O(n,k)$ .

This is naturally a subgroup of the general affine group $GA(n,k)$ , which in turn is a subgroup of the general linear group $GL(n+1,k)$ .

As a map

As a functor from fields to groups

For fixed $n$ , we get a functor from the category of fields to the category of groups, sending a field $k$ to the affine orthogonal group $AO(n,k)$ .

As an IAPS

Further information: Affine orthogonal IAPS

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

$\Phi _{m,n}:AO(m,k)\times AO(n,k)\to AO(m+n,k)$ .

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

As a functor from fields to IAPSes

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

Relation with other linear algebraic groups

Supergroups

Affine orthogonal similitude group

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 $p$	1	Dihedral group $D_{2p}$
Odd $q$	1	Semidirect product of elementary abelian group of order $q$ by inverse map.
$2^{n}$	1	Elementary abelian group of order $2^{n}$ .
2	2	Dihedral group:D8