Can someone help me with this question? I have no clue. Thank you so much!
Let G denote the group of rotational symmetries of a regular dodecahedron. This problem invloves considering the action of G on three different sets: F={faces of the dodecahedron}, V={vertices}, and E={edges}.
(a)Determine the orbits and stabilizers of an arbitrary element of each of F, V and E. (b)Show that all subgroups of G of order 5 are conjugate. Do the same for all subgroups of order 3. Ditto for all subgroups of order 2. (c)Using part (a) and (b) determine the number k of different conjugacy classes of G and their respective sizes n1,...,nk. (d)Using the class equation |G|=n1+n2+...+nk obtained in part (c), show that G is a simple group. Therefore show A5 is simple.
Just to give you a start let's consider the faces - do you know how many there are?
Now look at one face. The orbit of that face under the action of the group is simply a record of where it is possible for that face to go. So can you rotate that face into each of the other faces, or are there some you miss out?
The stabiliser of the face is the set of rotations which leave it where it is. You should find a rotation axis through the middle of the face.
Then check that the size of the orbit multiplied by the size of the stabiliser gives you the order of the whole group. Well you may not know the order yet, but you should get the same answer for faces, vertices and edges, which is a check on your arithmetic and visualisation.
For part $b$ you need to consider what a subgroup of order $5$ looks like geometrically. If you have done the faces part properly you might see how to rotate one face into another, do a rotation which fixes the face and then rotate back so that the face is in its original position. That - in group language - is conjugation. You should be able to do the same for edges and vertices.
For part $c$ you have to be quite careful. You should be able to count the elements of each order, for a start. But you will need to analyse carefully beyond that.
For part $d$ note that a normal subgroup is a union of complete conjugacy classes, and the order of a normal subgroup is a factor of the order of the whole group.
Comment: you really need to work this out yourself if you can. It brings a lot of simple ideas together and shows how they work, and in doing so should deepen your understanding of why some of the concepts we use in group theory reflect geometric realities.
Think more when you have sorted this out: Can you see how to use the vertices to inscribe five cubes in the dodecahedron. How does the group $A_5$ relate to those cubes.