Affine orthogonal group

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 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
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.

Supergroups

 * Supergroup::Affine orthogonal similitude group

Subgroups

 * Subgroup::Special affine orthogonal group
 * Subgroup::Orthogonal group
 * Subgroup::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.