31. The set of all rotations forms a Lie subgroup isomorphic to the ordinary rotation group SO ( 3 ). 32. The rotation group SO ( 3 ) can be described as a subgroup of direct isometries of Euclidean "'R "'3. 33. In norm quaternions, is also simply connected, so it is the covering group of the rotation group SO ( 3 ). 34. The 5D rotation group SO ( 5 ) and all higher rotation groups contain subgroups isomorphic to O ( 4 ). 35. The set of all appropriate matrices together with the operation of matrix multiplication is the rotation group SO ( 3 ). 36. Noticing that there are 8 corners and 12 edges, and that all the rotation groups are abelian, gives the above structure. 37. For example, in the rotation group SO ( 3 ) the maximal tori are given by rotations about a fixed axis. 38. The 5D rotation group SO ( 5 ) and all higher rotation groups contain subgroups isomorphic to O ( 4 ). 39. One is rotation group , or more generally of double cover of the generalized special orthogonal group on spaces with metric signature. 40. The rotation group of an object is equal to its full symmetry group if and only if the object is chiral.
