General orthogonal groups

The general orthogonal group GO(n,𝔽_q) is the group of all matrices over 𝔽_q preserving a non-degenerate quadratic form.

Two types in even dimension

Unlike Hermitian forms, which are all equivalent in a given dimension, quadratic forms over a finite field fall into two classes when the dimension is even. They are told apart by the Witt index — the dimension of a largest subspace on which the form vanishes identically:

Plus type, written GO⁺(2m,𝔽_q) — Witt index m, the largest possible. The form is an orthogonal sum of m hyperbolic planes.
Minus type, written GO⁻(2m,𝔽_q) — Witt index m − 1: hyperbolic planes together with a 2-dimensional anisotropic piece, on which the form vanishes only at 0.

In odd dimension the form is unique up to scaling, and scaling does not change the isometry group, so no type is written: GO(3,𝔽₅).

Order

|GO(2m+1,𝔽_q)| = 2 · q^m² · ∏_i=1^m (q²ⁱ − 1) for odd q, which is 2 · |Sp(2m,𝔽_q)|.

|GO^ε(2m, q)| = 2 · q^m(m−1) · (q^m − ε) · ∏_i=1^m−1 (q²ⁱ − 1), with ε = +1 for plus type and ε = −1 for minus type — so minus type is the larger group: |GO⁺(4,𝔽₂)| = 72 against |GO⁻(4,𝔽₂)| = 120.

The sign runs the other way for isotropic vectors: the quadric has (q^m − ε)(q^m−1 + ε)/(q − 1) points, so plus type has more of them and a smaller isometry group.

Subgroups

SO — determinant 1, of index 2. Over a field of even order every isometry has determinant 1, so SO = GO there.
Omega — a further index-2 subgroup of SO, the kernel of the spinor norm (of the Dickson invariant in even characteristic). Its projective quotient is where the simple groups live.

Dimension 2 is dihedral

|GO^ε(2, q)| = 2(q − ε), and the groups really are dihedral: plus type is D_q−1 (the isometries of Q(x,y) = xy, scaling one axis against the other, plus the swap) and minus type is D_q+1 (the norm-one elements of 𝔽_q² together with the Frobenius).

Odd dimension in even characteristic