Cyclic group
Rotational symmetries

The set of rotational symmetries of a polygon forms a finite cyclic group. If there are n different ways of moving the polygon to itself by a rotation (including the null rotation) then this symmetry group is isomorphic to Z / n Z. In three or higher dimensions there exist other finite symmetry groups that are cyclic, but which are not all rotations around an axis, but instead rotoreflections.

The group of all rotations of a circle (the circle group, also denoted S) is not cyclic, because there is no single rotation whose integer powers generate all rotations. In fact, the infinite cyclic group C is countable, while S is not. The group of rotations by rational angles is countable, but still not cyclic.
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