Understanding the Dihedral group, $D_n$

Question

I know what the dihedral group is, the group of symmetries (length-preserving) functions of a regular n-gon. However, when it is referring to the symmetries of the regular n-gon what is it referring to? Is it referring to the symmetries of: The set of vertices on the n-gon, the set of points in the n-gon or the set of points on the edges of the n-gon ?

Also is it considering these points as points in the complex plane?

Answer 1

Here we are talking about isometries from the one-dimensional figure (the union of the edges of the $n$-gon) into itself. Of course, the restriction of such an isometry to the vertices of the $n$-gon will also be an isometry and every isometry from the set of the vertices of the $n$-gon into itself can be extended to one and only one isometry from the one-dimensional figure into itself.

Answer 2

All points of the polygon (interior and boundary) are defined by the vertices. So a symmetry of the polygon is a permutation of the vertices.

But not every permutation of the vertices is a symmetry of the polygon. For instance, swapping two adjacent vertices of a square isn't a symmetry of the square.