(a) Let be a finite group acting on a finite set . For any subset of , we define the fixed point set as . Write for . Let be the set of -orbits in . In what follows you may assume the orbit-stabiliser theorem.
where the sum is taken over a set of representatives for the orbits containing more than one element.
By considering the set , or otherwise, show also that
(b) Let be the set of vertices of a regular pentagon and let the dihedral group act on . Consider the set of functions (the integers mod . Assume that and its rotation subgroup act on by the rule
where and . It is given that . We define a necklace to be a -orbit in and a bracelet to be a -orbit in .
Find the number of necklaces and bracelets for any .