please typ.e this answer on page BY HANDWRITING TO LOOK CLEAR

question

solution

(a) Let G be a group of order p, which is a prime number, acting on a set X. Then for every X we have either Stabc(x)= {e} or Stabc(x) = G.

To prove this, let x be an arbitrary element of X. Then by the definition of the action of G on X, we have that Gx = {gx | g G} is a subset of X. Moreover, since G is a group, we have that Gx is also a group.

Now, since G is a group of order p, we have that Gx is also a group of order p. Thus, by Lagrange's theorem, we have that |Stabc(x)| is a divisor of p.

However, since Stabc(x) is a subset of G, we have that |Stabc(x)| is also less than or equal to |G|. Thus, we have that |Stabc(x)| is either 1 or p.

Therefore, we have that either Stabc(x)= {e} or Stabc(x) = G.

(b)

Let p > 2 be prime

and let G = {id, o,op-1} 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 Stabc(a) = G?)

Since G is the group of rotations of a regular polygon with p sides, we have that each element of G fixes the vertices of the polygon. Thus, every coloring of the vertices of the polygon that is fixed by all of the elements of G is a valid coloring.

Now, since there are k colors, we have that there are k^p different colorings of the vertices of the polygon. However, since each coloring is fixed by all of the elements of G, we have that each coloring is also fixed by any rotation of the polygon.

Thus, we have that there are k^p / p different colorings of the vertices of the polygon that are fixed by all of the elements of G.

(c)

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?

Explanation:

Since we are identifying colorings that can be obtained from each other by rotations, we have that there are k^p / p different colorings of the vertices of the polygon.

To see this, note that each coloring of the vertices of the polygon is fixed by all of the elements of G. Thus, each coloring is also fixed by any rotation of the polygon.

Therefore, we have that there are k^p / p different colorings of the vertices of the polygon that are fixed by all of the elements of G.

Thus, in order to color the vertices of the polygon in k colors, we can choose any of the k^p / p different colorings of the vertices of the polygon