Orthogonal group for the standard dot product

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

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 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 orthogonal group $O(n,k)$.

As an IAPS of groups

Further information: orthogonal IAPS

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.

Relation with other linear algebraic groups

Supergroups

• Orthogonal similitude group: This is the group of matrices $A$ such that $AA^t$ is a nonzero scalar matrix.
• 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.

Facts based on the nature of the field

Property of the field Meaning What we can deduce about the standard dot product
ordered field there exists a total ordering It is positive-definite, hence anisotropic.
real-closed field a lot like the real numbers It is positive-definite, hence anisotropic. All symmetric positive-definite bilinear forms are equivalent to it.
Pythagorean field a sum of squares is a square The Gram-Schmidt process works (not necessarily uniquely)
ordered Pythagorean field ordered and Pythagorean The Gram-Schmidt process works uniquely
quadratically closed field every element is a square Up to equivalence, it is the only symmetric nondegenerate biliner form
finite field of order congruent to 1 mod 4 equivalent to the hyperbolic (split orthogonal) case
finite field of order congruent to -1 mod 4 inequivalent to the hyperbolic (split orthogonal) case

Particular cases

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.

Size of field Order of matrices Common name for the orthogonal group Order What type of orthogonal group is it?
odd $q$ 1 Cyclic group:Z2 2 the only type; there is only one orthogonal group for odd degree, odd characteristic
$2^n$ 1 Trivial group 1 the only type
$2^n$ 2 elementary abelian group of order $2^n$ $2^n$
3 2 Dihedral group:D8 8 not split, i.e., has a two-dimensional anisotropic component
4 2 Klein four-group 4
5 2 Dihedral group:D8 8 the split orthogonal group, i.e., the space is hyperbolic
7 2 Dihedral group:D16 16 not split, i.e., has a two-dimensional anisotropic component
8 2 Elementary abelian group of order eight 8
9 2 Dihedral group:D16 16 the split orthogonal group, i.e., the space is hyperbolic
17 2 Cyclic group:Z16 16 the split orthogonal group, i.e., the space is hyperbolic
2 3 Cyclic group:Z6 6
4 3 Alternating group:A5 60