As an outsider of algebraic topology, I would like to consult your guidance to understand finite group actions over spheres. I do not imagine how a group acts on a sphere, in particular why a group acts freely on some spheres. I've read that $D_{2n}$ acts non-freely on any sphere. If someone could explain the action of $D_{2n}$ on $S^1$ and give some book examples to study group actions on spheres for those who are new to these things?
2026-03-29 23:44:11.1774827851
Dihedral group actions on Spheres
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The Dihedral group acts on $S^1$ by rotations (by $\frac{2m\pi }{n})$ and reflections about the symmetry axes of the regular $n$-gon. The action is not free in general because rotating a point on $S^1$ by some angle can have the same effect as a reflecting it about one of the axes which tells us that the action is not free. This can also be seen from the fact that the reflections have fixed points.