Point Groups in Three Dimensions - Group Structure

Group Structure

SO(3) is a subgroup of E+(3), which consists of direct isometries, i.e., isometries preserving orientation; it contains those that leave the origin fixed.

O(3) is the direct product of SO(3) and the group generated by inversion (denoted by its matrix −I):

O(3) = SO(3) × { I, −I }

Thus there is a 1-to-1 correspondence between all direct isometries and all indirect isometries, through inversion. Also there is a 1-to-1 correspondence between all groups of direct isometries H and all groups K of isometries that contain inversion:

K = H × { I, −I }

H = K ∩ SO(3)

If a group of direct isometries H has a subgroup L of index 2, then, apart from the corresponding group containing inversion there is also a corresponding group that contains indirect isometries but no inversion:

M = L ∪ ( (H \ L) × { − I } )

where isometry ( A, I ) is identified with A.

Thus M is obtained from H by inverting the isometries in H \ L. This group M is as abstract group isomorphic with H. Conversely, for all isometry groups that contain indirect isometries but no inversion we can obtain a rotation group by inverting the indirect isometries. This is clarifying when categorizing isometry groups, see below.

In 2D the cyclic group of k-fold rotations C_k is for every positive integer k a normal subgroup of O(2,R) and SO(2,R). Accordingly, in 3D, for every axis the cyclic group of k-fold rotations about that axis is a normal subgroup of the group of all rotations about that axis, and also of the group obtained by adding reflections in planes through the axis.