Remark

When we look at the regular pentagon in the plane, we can consider symmetries in two ways.

1. as automorphisms of the Euclidean plane (rotations, reflections, etc.) that leave invariant the pentagon;
2. as a group of permutations of the graph with vertex set {1,2,3,4,5} and edges {1,2},  {2,3},   {3,4},   {4,5},  {1,5}.

Naturally the former symmetry group (subject to more restrictions) is contained in the latter. Remarkably enough, the two groups coincide. They are both the dihedral group D₅ of order 10.

The elements of order 5 correspond to rotations around the origin with angle a multiple of 72 degrees, and the elements of order 2 to reflections in an axis through the center and one of the vertices of the pentagon.

More precisely, D₅ consists of

1. the rotations e, (1,2,3,4,5), (1,3,5,2,4), (1,4,2,5,3), (1,5,4,3,2),   and
2. the reflections (2,5)(3,4), (1,3)(4,5), (2,4)(1,5), (1,2)(3,5), (1,4)(2,3).

Each pair of neighbouring vertices of the pentagon forms a basis for D₅.