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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?
Based on the comments, it seems like the question is whether the orientation of the center squares is fixed. The Wikipedia page for Rubik's cube says that only half the center orientations ($4^6/2$) are achievable.
Here are instructions for 2 types of center rotations: (1) rotate one center piece clockwise and an adjacent center piece counterclockwise; (2) rotate a single center by 180 degrees. (Corners and edges stay where they are.) Together these generate a subgroup of index 2 inside $(\mathbb{Z}/4\mathbb{Z})^6$ (the subgroup where the sum of all entries is even).