Orthogonal group

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You might be referring to:

  • Orthogonal group for the standard dot product: This is probably what you mean if you are working with the real numbers, the rational numbers, or some subfield of the reals, though the concept applies to all fields.
  • Orthogonal group for a symmetric bilinear form: A slightly more general notion. This is probably what you mean if you are working with arbitrary fields, particular finite fields and fields of prime characteristic.
  • Split orthogonal group: This is the orthogonal group for a hyperbolic orthogonal space. For the reals, it corresponds to a situation where the signature form has an equal number of 1s and -1s. The space must have even dimension. Note that for the reals, this differs from the orthogonal group for the standard dot product, but for finite fields whose size is congruent to 1 modulo 4, the two groups are isomorphic because the forms are equivalent.