# Split orthogonal group

From Groupprops

## Definition

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

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

.

Here, the and are block matrices.

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

.

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

## Particular cases

### Finite fields

Size of field | 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 |