# Classification of isometries of a regular polygon

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.

A symmetry must map vertex 1 to some vertex $$i$$. Vertex 2 must be mapped to an ajacent vertex, $$i+1$$ or $$i-1$$. Once you know the images of 1 and 2, the other vertices are fixed. If 2 is mapped to $$i+1$$ you have the rotations. If 2 is mapped to $$i-1$$, you have the reflections (This might require some more arguments, at the elementary level at which this question is posed.)