# 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 $k$.

## 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