Rubik’s Cube Not a Group?
I take a Rubik's cube in the solved state, and I secretly assign a unique integer label to each of the cubies. I then, via an arbitrarily long series of random moves, scramble and resolve the cube without looking at the integer labels. Will the integer labels be mapped back to their original positions? Does this answer change for a 4x4x4 cube?
If not, how large is the group of states for this integer labeled cube? Surely it must be quite a bit larger than the Rubik's cube permutation group?