Rotations in 4-dimensional Euclidean space: Difference between revisions

The even-dimensional rotation groups do contain the central inversion {{math|−''I''}} and have the group {{nowrap|1=C₂ = <nowiki>{</nowiki>{{math|''I''}}, {{math|−''I''}}<nowiki>}</nowiki>}} as their [[center of a group|centre]]. FromFor even n ≥ 6, SO(6n) onwards they areis almost simple in the sense that the [[factor group]]s SO(n)/C₂ of theirSO(n) centresby areits centre is a simple groupsgroup.

SO(4) is different: there is no [[Conjugation of isometries in Euclidean space|conjugation]] by any element of SO(4) that transforms left- and right-isoclinic rotations into each other. [[Reflection (mathematics)|Reflection]]s transform a left-isoclinic rotation into a right-isoclinic one by conjugation, and vice versa. This implies that under the group O(4) of ''all'' isometries with fixed point {{mvar|O}} the distinct subgroups {{math|''S''³_L}} and {{math|''S''³_R}} are mutually conjugate to each other, and so arecannot notbe normal subgroups of O(4). The 5D rotation group SO(5) and all higher rotation groups contain subgroups isomorphic to O(4). Like SO(4), all even-dimensional rotation groups contain isoclinic rotations. But unlike SO(4), in SO(6) and all higher even-dimensional rotation groups any pair of isoclinic rotations through the same angle is conjugate. The sets of all isoclinic rotations are not even subgroups of SO(2{{math|''N''}}), let alone normal subgroups.