Split orthogonal group

Definition

This group is the orthogonal group for a symmetric bilinear form where the symmetric bilinear form gives a hyperbolic space.

Let $m$ be a natural number and $k$ be any field. The split orthogonal group of degree $2m$ over $k$ can be defined as the group, under matrix multiplication:

$\{A\in GL(2m,k)\mid A{\begin{pmatrix}0&I\\I&0\\\end{pmatrix}}A^{T}={\begin{pmatrix}0&I\\I&0\\\end{pmatrix}}\}$.

Here, the $0$ and $I$ are $n\times n$ block matrices.

When the characteristic of $k$ is not equal to two, this is isomorphic to the group (in fact, they are conjugate in $GL(2m,k)$):

$\{A\in GL(2m,k)\mid A{\begin{pmatrix}I&0\\0&-I\\\end{pmatrix}}A^{T}={\begin{pmatrix}I&0\\0&-I\\\end{pmatrix}}\}$.

For a finite field, the split orthogonal group is also sometimes known as the orthogonal group of $+$ type or the $+1$ type.

Particular cases

Finite fields

Size of field	$m$	$2m$	Common name for the split orthogonal group	Order of the group
3	1	2	Klein four-group	4
5	1	2	dihedral group:D8	8
7	1	2	direct product of S3 and Z2	12
9	1	2	dihedral group:D16	16