A dihedral group is the group of symmetries of a regular polygon, including both rotations and reflections. Here comes my question:
How can I show that rotations and reflections are the only symmetries of a regular polygon?
I have found no proof for such a statement when the dihedral groups are introduced in lots of textbooks. It might be a starting point to show first that an isometry (symmetry) on a regular polygon maps the vertices to vertices and then the center must be a fixed point of the isometry. But I get stuck with showing that reflections are the only possible isometries besides the rotations.