TITLE: Rubik's Cube Not a Group?
QUESTION [34 upvotes]: I read online that

although the 3x3x3 is a great example of a mathematical group, larger cubes aren't groups at all.

How can that be true? There is obviously an identity and it is closed, so that must mean that some moves aren't invertible. But this seems unlikely to me.

REPLY [4 votes]: Arguments that the 3x3x3 is a group while the 4x4x4 isn't, based on the permutability of the center pieces in the latter case, are confused nonsense.  By that standard, the 3x3x3 isn't a group either.  Reason: although it is true that one cannot move the centers to other locations in the 3x3x3 case, one can rotate them in place.  Consider this twist sequence: $(RLFR^-L^-F^2)^2$.  If you use a cube with marked stickers, you will see that the net effect of this sequence is to rotate the $F$ center by 180 degrees.  There is another sequence which rotates one center 90 degrees clockwise and another center 90 degrees counterclockwise.  I see no reason to discount center rotations as 'movement'.
The real question is, when we say "the cube's group", what are we referring to?  From all the articles I've read, the universal presumption is that the group is the set of all 'sticker' permutations (including permutations of the stickers' corners) achievable by any sequence of face/slice twists, regardless of visual distinguishability.  In group-theoretic terms, there are three generally acknowledged constructs here:

The free group generated by { $R, L, F, B, U, D $ }.
The cube group which is the free group modulo the subgroup of all words which act as the identity on the sticker set - i.e the words which end up leaving all stickers exactly in their original position and orientation, not just in a perceptually indistinguishable state.
The perceived state of the physical cube itself, which is the result of the action of the most recently applied element of the cube group.

I agree that the third construct isn't a group, but (to my knowledge) no one serious about cube theory ever thinks of that construct when referring to the cube as representing a group.