If C6 acts on a regular hexagon by rotation and each of the vertices is colored red, blue or green, use the Burnsideââ?¬â?¢s formula to determine how many possible colorings there are up to cyclic symmetry.
The number of colorings can be written as:
1/|G| Sum over all g in G of Fix(g)
Here G is the symmetry group, |G| is the number of elements in G (which is 6 in this case), and Fix(g) is the number of colorings that g will keep fixed. Let's consider all the rotations that are in G:
1) Identity. If we do nothing, then all possible colorings will be left fixed. There are 3^6 colorings in total, because there are 3 choices for each of the 6 vertices. We thus have:
Fix(Identity) = 3^6
2) Clockwise rotation over 60 degrees. Under this group element, a coloring changes such that the new color of a vertex becomes the old color of the vertex located 60 degrees counterclockwise. ...
We show in detail how Burnside's theorem can be used to obtain the number of colored hexagons up to the cyclic symmetry.