PΩ-(2n,q) - Knowlpedia

For integers $n\geq4$ and prime powers $q\geq 2$ , $\operatorname{P}\Omega_{2n}^{-}(q)$ is the projective orthogonal derived group of $Q(x,y,z)=\sum_{i=1}^{n-1}x_i y_i+z^{q+1}$ , with $x,y\in\mathbb F_q^{n-1}$ and $z\in\mathbb F_{q^2}$ :

\Omega=\operatorname O(Q)',\qquad \operatorname{P}\Omega_{2n}^{-}(q)=\Omega/(\Omega\cap\{\lambda I:\lambda\in\mathbb F_q^\times\}).

Here $\operatorname O(Q)$ is the group of invertible linear maps preserving $Q$ , and its prime denotes the commutator subgroup. Multiplication is multiplication of scalar cosets. The displayed quadratic form, not merely its polar form, is part of the definition.

Order and simplicity

Its order is

|\operatorname{P}\Omega_{2n}^{-}(q)|=\frac{q^{n(n-1)}(q^{n}+1)\prod_{i=1}^{n-1}(q^{2i}-1)}{\gcd(4,q^{n}+1)}.

It is a nonabelian simple group for every admitted parameter.

Rank and form convention

The ambient quadratic-space dimension is $2n$ , and the absolute rank is $n$ . This is type ${}^2D_{n}$ . The plus sign has Witt index $n$ ; the minus sign has Witt index $n-1$ , because the norm plane is anisotropic. The order denominator is $\gcd(4,q^n-1)$ in plus type and $\gcd(4,q^n+1)$ in minus type. There is no additional irreducible $D_1$ family; the ranks two and three have the separate low-rank descriptions.

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.