Orthogonal group for the standard dot product

Definition with symbols
Let $$n$$ be a natural number and $$k$$ be a field. The orthogonal group for the standard dot product, sometimes simply the orthogonal group, of degree $$n$$ over $$k$$ (sometimes denoted $$O(n,k)$$) is defined as the group of all matrices $$A$$ of degree $$n$$ over $$k$$ such that $$\! AA^t = I$$.

In the context of finite fields and more general treatments of fields as well as in the context of the study of simple groups of Lie type, the term orthogonal group is typically used for the notion of orthogonal group for a symmetric bilinear form. The orthogonal group for the standard dot product is then a special case where the symmetric bilinear form is the standard dot product.

Another extremely important orthogonal group that often comes up is the split orthogonal group which is the orthogonal group for a hyperbolic space.

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 orthogonal group $$O(n,k)$$.

As an IAPS of groups
The orthogonal groups form an IAPS of groups. In other words, for every $$m$$ and $$n$$, there is an injective map:

$$\Phi_{m,n}: O(m,k) \times O(n,k) \to O(m+n,k)$$

which takes a matrix $$A$$ of order $$m$$ and a matrix $$B$$ of order $$n$$ and outputs a block diagonal matrix with blocks $$A$$ and $$B$$.

As a functor from fields to IAPSes
If we fix neither $$n$$ nor $$k$$, we get a functor that takes as input a field and outputs an IAPS of groups.

Supergroups

 * Supergroup::Orthogonal similitude group: This is the group of matrices $$A$$ such that $$AA^t$$ is a nonzero scalar matrix.
 * Supergroup::Affine orthogonal group: The semidirect product of the vector space with the orthogonal group. In other words, the group generated by translations and orthogonal maps.

Subgroups

 * Subgroup::Special orthogonal group

Group and subgroup operations

 * Intersection with the special linear group yields the special orthogonal group.
 * Normalizer in the whole general linear group is the orthogonal similitude group.
 * Centralizer in the whole general linear group is (for $$n \ge 3$$) equal to the center of the general linear group.

Finite fields
The final column describes which of the orthogonal groups over a finite field is given by a standard dot product. For odd degree and odd characteristic, there is only one orthogonal group. For even degree and odd characteristic, there are two orthogonal groups. The standard dot product gives the split orthogonal group if the size of the field is 1 modulo 4, and it gives the non-split one otherwise.

Note that for odd characteristic and degree two (or higher), dihedral group:D8 always arises as a subgroup. This is because the representation of the dihedral group of order eight simply involves the elements $$0, \pm 1$$ whose multiplication facts work the same over all fields of characteristic not equal to two. See linear representation theory of dihedral group:D8 for more.