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:

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

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

• Special affine orthogonal group
• Orthogonal group
• Special orthogonal group

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