Finite projective orthogonal derived group

For a prime power $q$ , let $V$ be a finite-dimensional vector space over $\mathbb F_q$ , and let $Q:V\to\mathbb F_q$ be a nonsingular quadratic form. The projective orthogonal derived group is

\operatorname{P}\Omega(V,Q)=\Omega(V,Q)/(\Omega(V,Q)\cap\{\lambda I:\lambda\in\mathbb F_q^\times\}), \qquad \Omega(V,Q)=[\operatorname O(V,Q),\operatorname O(V,Q)].

Here

\operatorname O(V,Q)=\{g\in\operatorname{GL}(V):Q(gv)=Q(v)\text{ for all }v\},

and brackets denote the commutator subgroup, generated by $aba^{-1}b^{-1}$ . Nonsingular means $\{v\in\operatorname{rad}B:Q(v)=0\}=\{0\}$ , where $B(v,w)=Q(v+w)-Q(v)-Q(w)$ and $\operatorname{rad}B=\{v:B(v,w)=0\text{ for every }w\}$ . Multiplication in the projective group is multiplication of scalar cosets.

Characteristic two

A quadratic form satisfies $Q(av)=a^2Q(v)$ , with bilinear polar form $B$ . In characteristic two, $B$ does not determine $Q$ . In even dimension, nonsingularity makes $B$ nondegenerate. In odd dimension, its radical has dimension one and $Q$ is nonzero on every nonzero radical vector.

Over an even field in odd dimension $2n+1$ , the action on $V/\operatorname{rad}B$ identifies the full orthogonal group with $\operatorname{Sp}_{2n}(q)$ . In the stable simple ranges this also identifies its derived group with the symplectic group. At very small parameters the derived subgroup can be proper.

Signs and low ranks

For dimension $2n$ , the plus sign means Witt index $n$ : one may take $Q=\sum_{i=1}^n x_i y_i$ . The minus sign means Witt index $n-1$ : use $n-1$ such hyperbolic planes and the anisotropic plane $Q(z)=N_{\mathbb F_{q^2}/\mathbb F_q}(z)=z^{q+1}$ . Witt index is the largest dimension of a subspace on which $Q$ vanishes identically. In odd dimension use $Q=\sum_{i=1}^n x_i y_i+z^2$ .

Taking determinant one alone is insufficient: the derived subgroup may be smaller, and in characteristic two every orthogonal transformation has determinant one. This catalogue uses the displayed derived-group convention and explicitly excludes exceptional low parameters from low-rank identifications when necessary.

References

Peter J. Cameron, Notes on Classical Groups, §6, pp. 64–74, especially Theorems 6.3 and 6.6; §7 introduction, p. 75.
Robert A. Wilson, Classical groups (lecture notes), §3.1.7, pp. 5–6; §§3.2–3.4, pp. 7–17.