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.

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 $$q$$.