Rotations in 4-dimensional Euclidean space: Difference between revisions

Revision as of 02:14, 30 September 2005

Revision as of 19:14, 5 October 2005

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The odd-dimensional rotation groups do not contain the [[Inversion |central inversion]] and are [[simple group]]s.

The even-dimensional rotation groups do contain the central inversion -I and have the group C2 = {I, -I} as their [[center of a group |centre]]. From SO(6) onwards they are almost-simple in the sense that the [[factor group]]s of their centrecentres are simple groups.

SO(4) is different: there is no [[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 O the subgroups S3L and S3R are mutually conjugate and so are not 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(2N), let alone normal subgroups.