I'm making a few old exams to practice for a group theory exam. In every old exam I practiced, there is a question such as the following:
Suppose we have a cube and on each face, we draw an arrow starting from the middle of the face, towards one of the four vertices of that face.
Then first, I have to find the number of ways to do this up to rotation symmetry. Thus this is a kind of 'standard' exercise where you should use Burnside's counting lemma.
But then, they ask for the following: there is a twodimensional printout for a cube given: and then they ask to draw arrows on this printout such that the stabilizer$^*$ is isomorphic to $S_3$. Now I know that the rotation of a cube is $S_4$ and that we can classify these rotations as follows:
- The identity permutation
- $6$ rotations over $90$ degrees through a line through two opposite faces.
- $3$ rotations over $180$ degrees through a line through two opposite faces.
- $8$ rotations over $120$ degrees through a main diagonal of the cube.
- $6$ rotations over $180$ degrees through the middle of two opposite edges.
but I really don't see how I should come up with this solution. I even don't see why this solution is correct. There are also similar exercises where you have to fill in the printout such that the stabilizer is isomorphic to $V_4$ and $A_4$.
I cannot find anything silimar on the internet and this is never explained in class. Any help is much appreciated!
$^*$: of this configuration of arrows, the rotation group of the cube $S_4$ acts on the set of arrow configurations