= Finite orientation-preserving plane isometry groups are cyclic
A finite <group> of orientation-preserving <isometries> of the <Euclidean plane> or <hyperbolic plane> is a <cyclic group>. Take one orbit and its unique minimum-radius enclosing disc. Every group element preserves the orbit and therefore the disc's centre. The orientation-preserving stabilizer of a point acts as planar rotations, so the group embeds in the circle group. In a nontrivial finite subgroup, the least positive rotation angle generates all angles by division with remainder, proving cyclicity. Orientation reversal changes the conclusion: reflection groups need not be cyclic.
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