Over Finite Fields
Orthogonal groups can also be defined over finite fields Fq, where q is a power of a prime p. When defined over such fields, they come in two types in even dimension: O+(2n, q) and O−(2n, q); and one type in odd dimension: O(2n+1, q).
If V is the vector space on which the orthogonal group G acts, it can be written as a direct orthogonal sum as follows:
where Li are hyperbolic lines and W contains no singular vectors. If W = 0, then G is of plus type. If W =
In the special case where n = 1, is a dihedral group of order .
We have the following formulas for the order of O(n, q), when the characteristic is greater than two:
If −1 is a square in Fq
If −1 is a non-square in Fq
Read more about this topic: Orthogonal Group
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