On the group of rotations of a cube I would try to prove that the rotational symmetries of the cube are isomorphic to the group of permutations on 4 elements. My attempt is the following:

*

*the number of possibile rotation is 24. Easy

*the group of rotations Acts faithfully on the set of vertices so it's embedded
in $S_8$.

*we notice that this subgroup is transitive
So, i know that i have a transitive subgroup of order 24 in $S_8$, can i conclude that this is $S_4$? Do i need more information on the subgroup?

 A: Look down on the cube and number the vertices of the upper face as $1, 2, 3$ and $4$ working counterclockwise from the top right. Number each diagonal of the cube $1, 2, 3$ and $4$ according to which vertex on the top face it passes through. With a $90^\circ$ rotation about the vertical axis you can achieve the permutation $g = (1\,2\,3\,4)$ of the diagonals, while with a $90^\circ$ rotation about one of the horizontal axes you can achieve the permutation $h=(1\,4\,2\,3)$. You then have $g^2h = (1\,2)$ and $gh^2 = (1\,3)$. By symmetry, you can achieve any transposition of two diagonals and hence any permutation of the diagonals.
(Alternatively, as you have already calculated that there are $24$ rotations, you can observe that the action of the rotations on the diagonals induces an injective homomorphism of the group of rotations into $S^4$ which must be an isomorphism since both groups have $24$ elements.)
A: Perhaps the easiest way to show that the cube rotation group is the group of permutations of 4 elements is to directly show it in fact permutes 4 elements.  This is easy: The 4 elements being permuted are the cube's major diagonals (i.e the diagonals through the cube's center).
Any permutation of these diagonals can be achieved using at most three cube rotations:  First, assuming the diagonals are numbered 1-4, rotate the cube around any axis through face centers to move diagonal x into the position of diagonal 1; then rotate the cube about diagonal x to move diagonal y into the position of diagonal 2; finally, rotate the cube about the axis through the center of the cube perpendicular to the plane containing diagonals x and y (hence also through two parallel edges), flipping x any y end-over-end but otherwise leaving x mapped to x and y to y, to swap diagonal z into the position of diagonal 3.
