In how many ways can you color, up to symmetry, the vertices of a regular decagon using $q$ colors?
(We are talking here about the dihedral group of order $20$).
So I was thinking about maybe group actions, thinking about it as orbits... But I really couldn't get to anything.
I understand that I should use Burnside lemma to calculate the fixed points size which is the answer.
What I can't see is what are the fixed points for that group action?
Obviously it would be the symmetries where the coloring remains the same, and yet I'm not sure how to do that...