# Orthogonal group of degree two and type -1

## 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