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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“The finite is annihilated in the presence of the infinite, and becomes a pure nothing. So our spirit before God, so our justice before divine justice.”
—Blaise Pascal (1623–1662)
“Nature will not let us fret and fume. She does not like our benevolence or our learning much better than she likes our frauds and wars. When we come out of the caucus, or the bank, or the abolition-convention, or the temperance-meeting, or the transcendental club, into the fields and woods, she says to us, “so hot? my little Sir.””
—Ralph Waldo Emerson (1803–1882)