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
Famous quotes containing the words finite and/or fields:
“For it is only the finite that has wrought and suffered; the infinite lies stretched in smiling repose.”
—Ralph Waldo Emerson (1803–1882)
“It matters little comparatively whether the fields fill the farmer’s barn. The true husbandman will cease from anxiety, as the squirrels manifest no concern whether the woods will bear chestnuts this year or not, and finish his labor with every day, relinquishing all claim to the produce of his fields, and sacrificing in his mind not only his first but his last fruits also.”
—Henry David Thoreau (1817–1862)