Orthogonal Group - Over Finite Fields

Over Finite Fields

Orthogonal groups can also be defined over finite fields F_q, 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 L_i are hyperbolic lines and W contains no singular vectors. If W = 0, then G is of plus type. If W = then G has odd dimension. If W has dimension 2, G is of minus type.

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 F_q

If −1 is a non-square in F_q