Special orthogonal group for the standard dot product

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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 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


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