In fact, this identifies the group of versors with the group of rotations of , modulo the fact that the versors and determine the same rotation. As noted above, the rotations act transitively on , and the set of versors which fix a given right versor have the form , where and are real numbers with . This is a circle subgroup. For concreteness, one can take , and then the Hopf fibration can be defined as the map sending a versor . All the quaternions , where is one of the circle of versors that fix , get mapped to the same thing (which happens to be one of the two rotations rotating to the same place as does).