# Split orthogonal group of degree two

## Definition

Suppose $K$ is a field. The split orthogonal group of degree two over $K$ is defined as the subgroup of the general linear group of degree two over $K$ given as follows:

$\{ A \in GL(2,K) \mid A\begin{pmatrix} 0 & 1 \\ 1 & 0 \\\end{pmatrix}A^T = \begin{pmatrix} 0 & 1 \\ 1 & 0 \\\end{pmatrix}$

For characteristic not equal to two, an alternative definition, which gives a conjugate subgroup and hence an isomorphic group, is:

$\{ A \in GL(2,K) \mid A\begin{pmatrix} 1 & 0 \\ 0 & -1 \\\end{pmatrix}A^T = \begin{pmatrix} 1 & 0 \\ 0 & -1 \\\end{pmatrix} \}$

For a finite field $K$, this group is denoted $O(+1,2,K)$ and is termed the orthogonal group of "+" type. It is also denoted $O(+1,2,q)$ where $q$ is the size of the field.

## Arithmetic functions

### Over a finite field

We consider the group $O(+1,2,q) = O(+1,2,K)$ where $K$ is a field (unique up to isomorphism) of size $q$.

Function Value Similar groups Explanation
order $2(q - 1)$