General Orthogonal Group

The general orthogonal group associated with a nonsingular quadratic form on is the group

$GO_n(q,F)={A in GL_n(q):F(Av)=F(v) for every v in F_q^n}.$

Here is an matrix over , and ranges over the vectors in . Thus it is a subgroup of the general linear group, while the projective general orthogonal group is obtained only after factoring out the scalar matrices that it contains.

The notation is not uniform. In the convention used by the Atlas of Finite Groups, consists of all isometries of . Many authors denote the same group by and use for linear transformations that preserve only up to a nonzero scalar. When is odd, the determinant takes the values and its kernel is the special orthogonal group, so

$1->SO_n(q,F)->GO_n(q,F)->^(det){+/-1}->1.$

In characteristic 2, the determinant is identically 1 on the group, so the determinant-one subgroup equals the full group.

In odd dimension the choice of nonsingular quadratic form gives one isomorphism type of general orthogonal group over a finite field. In even dimension there are two types, denoted and , according to the type of the quadratic form.

References