Split orthogonal group of degree two

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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)

Particular cases

Over a finite field

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