PΩ+(4,q) - Knowlpedia

For $n=2$ and prime powers $q\geq 3$ , $\operatorname{P}\Omega_{4}^{+}(q)$ is the projective orthogonal derived group of $Q(x,y)=\sum_{i=1}^{2}x_i y_i$ on $\mathbb F_q^{4}$ :

\Omega=\operatorname O(Q)',\qquad \operatorname{P}\Omega_{4}^{+}(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_{4}^{+}(q)|=\frac{q^{2(2-1)}(q^{2}-1)\prod_{i=1}^{2-1}(q^{2i}-1)}{\gcd(4,q^{2}-1)}.

It is never simple: it is a direct product of two nontrivial copies of PSL(2,q).

Rank and form convention

The ambient quadratic-space dimension is $4$ , and the absolute rank is $2$ . 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.

Low-rank identification

$\operatorname{P}\Omega_4^+(q)\cong\operatorname{PSL}_2(q)\times\operatorname{PSL}_2(q)$ for $q\geq3$ . This is the reducible root-system case $D_2=A_1\sqcup A_1$ . The lower parameters are excluded here because the full orthogonal commutator subgroup can differ from the usual small Chevalley-group convention.

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.