Special 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 $k$ a field. Then the special orthogonal group of order $n$ over the field $k$, denoted $SO(n,k)$, is defined as the group of all matrices $A$ such that $det(A)=1$ and $AA^{t}=I$.

A group is termed a special orthogonal group if it occurs as $SO(n,k)$ for some natural number $n$ and field $k$.

Relation with other linear algebraic groups

Supergroups

• Orthogonal group
• Special affine orthogonal group
• Affine orthogonal group
• Special orthogonal similitude group
• Orthogonal similitude group

Particular cases

Finite fields

Size of field	Order of matrices	Common name for the special orthogonal group	The special orthogonal group as embedded in the special linear group
$q$	1	Trivial group	Trivial subgroup of trivial group
$2^{n}$	2	Elementary abelian group of order $2^{n}$
2	2	Cyclic group:Z2	Two-element subgroups of symmetric group:S3
3	2	Cyclic group:Z4	Cyclic four-subgroups of special linear group:SL(2,3)
4	2	Klein four-group	Klein four-subgroups of alternating group:A5
5	2	Cyclic group:Z4	Cyclic four-subgroups of special linear group:SL(2,5)
7	2	Cyclic group:Z8
8	2	Elementary abelian group of order eight
9	2	Cyclic group:Z8
17	2	Cyclic group:Z16
2	3	Cyclic group:Z6
3	3	Symmetric group:S4
4	3	Alternating group:A5
5	3	Symmetric group:S5