Infinite Isometry Groups
We restrict ourselves to isometry groups that are closed as topological subgroups of O(3). This excludes for example the group of rotations by an irrational number of turns about an axis.
The whole O(3) is the symmetry group of spherical symmetry; SO(3) is the corresponding rotation group. The other infinite isometry groups consist of all rotations about an axis through the origin, and those with additionally reflection in the planes through the axis, and/or reflection in the plane through the origin, perpendicular to the axis. Those with reflection in the planes through the axis, with or without reflection in the plane through the origin, perpendicular to the axis, are the symmetry groups for the two types of cylindrical symmetry.
See also rotational symmetry with respect to any angle.
Read more about this topic: Point Groups In Three Dimensions
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