Split orthogonal group
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.
|Size of field||Common name for the split orthogonal group||Order of the group|
|7||1||2||direct product of S3 and Z2||12|