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 with symbols
Let be a natural number and a field. Then the special orthogonal group of order over the field , denoted , is defined as the group of all matrices such that and .
A group is termed a special orthogonal group if it occurs as for some natural number and field .
Relation with other linear algebraic groups
- Orthogonal group
- Special affine orthogonal group
- Affine orthogonal group
- Special orthogonal similitude group
- Orthogonal similitude group
|Size of field||Order of matrices||Common name for the special orthogonal group||The special orthogonal group as embedded in the special linear group|
|1||Trivial group||Trivial subgroup of trivial group|
|2||Elementary abelian group of order|
|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)|
|8||2||Elementary abelian group of order eight|