Orthogonal group of degree two and type -1

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Definition

Consider the finite field \mathbb{F}_q with q elements, q an odd prime power. Suppose \alpha is an element of \mathbb{F}_q^\ast that is not a square. The orthogonal group O(-1,2,q) or \Omega(-1,2,q) is defined as the orthogonal group for the symmetric bilinear form given by the matrix:

\begin{pmatrix} 1 & 0 \\ 0 & \alpha \\\end{pmatrix}

In other words, as a subgroup of the general linear group of degree two, it is defined as follows:

\Omega(-1,2,q) := \left \{ \begin{pmatrix} a & b \\ c & d \\\end{pmatrix} \mid a^2 + b^2\alpha = 1, c^2 + d^2\alpha = \alpha, ac + bd \alpha = 0 \right \}

Particular cases

Size q of the field Characteristic prime p Othogonal group \Omega(-1,2,q)
3 3 dihedral group:D8
5 5 dihedral group:D8
7 7 dihedral group:D16