41. Operators in the orthogonal group that also preserve the orientation of vector tuples form the special orthogonal group, or the group of rotations. 42. Operators in the orthogonal group that also preserve the orientation of vector tuples form the special orthogonal group , or the group of rotations. 43. The special orthogonal groups have additional spin representations that are not tensor representations, and are " typically " not spherical harmonics. 44. When " G " is a classical group, such as a symplectic group or orthogonal group , this is particularly transparent. 45. An example is the orthogonal group , defined by the relation M T M = I where M T is the transpose of M. 46. Reflectional spherical symmetry is isomorphic with the orthogonal group O ( 3 ) and has the 3-dimensional discrete point groups as subgroups. 47. These rotations form the special orthogonal group SO ( ), which can be represented by the group of orthogonal matrices with determinant 1. 48. If " K " does not have characteristic 2 this is just the group of elements of the orthogonal group of determinant 1. 49. B r has an associated centerless compact groups the odd special orthogonal groups , SO ( 2 " r " + 1 ). 50. The matrix is a member of the three-dimensional special orthogonal group ,, that is it is an orthogonal matrix with determinant 1.
