31. Moreover, since composition of rotations corresponds to matrix multiplication, the rotation group is isomorphic to the special orthogonal group . 32. Every such group is a subgroup of the orthogonal group O ( 2 ), including O ( 2 ) itself. 33. One is rotation group, or more generally of double cover of the generalized special orthogonal group on spaces with metric signature. 34. The set of orthogonal matrices forms a group O ( " n " ), known as the orthogonal group . 35. The term "'rotation group "'can be used to describe either the special or general orthogonal group . 36. The set of all orthogonal matrices of size with determinant + 1 or-1 forms the ( general ) orthogonal group . 37. The properties of the spin representations depend, in a subtle way, on the dimension and signature of the orthogonal group . 38. An archetypical irreducible reductive dual pair of type I consists of an orthogonal group and a symplectic group and is constructed analogously. 39. The "'indefinite special orthogonal group "', is the subgroup of consisting of all elements with determinant 1. 40. Any subgroup containing at least one non-zero translation must be infinite, but subgroups of the orthogonal group can be finite.
