Split orthogonal group

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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