(4) (a) (9 points) Suppose that a group G of order p, which is a prime num- ber, acts on a set X. Prove that for every x E X we have either StabG(x) = {e} or Staba(x) = G. (b) (9 points) Let p > 2 be prime and let G = {id, o, . . .op-} be the group of rotations of a regular polygon with p sides (here . is a rotation to the right by 360 degrees). How many colorings with k colors of the vertices of that polygon are fixed by all of the elements of G? (Namely how many colorings have StabG(x) = G?) (c) (9 points) In how many different ways can one color the vertices of a regular polygon with p > 2 (p is prime) sides, using k colors, where we identify colorings that can be obtained from each other by rotations?"