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## Commutators

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Speaking of group elements, here's a bit of theory for you. To undo UR, you have to undo the R first, so the inverse of UR is R-1U-1, not U-1R-1. Implicit in that statement is the fact that R-1U-1 and U-1R-1 are different things. In such cases, when for some x and y we have xy != yx, we say that x and y don't commute. We can even measure how much of a failure to commute we have by constructing the commutator [x,y] = xyx-1y-1. (Or is it [x,y] = x-1y-1xy? There are different conventions, but no matter, I'll use the former.)

I learned about commutators when I was studying group theory. At the time I thought they were just an uninteresting technical device, but it turns out they also have a practical application. Consider this sequence.

x = RL-1 F LR-1 D-1 RL-1 F2 LR-1

If you watch the upper front edge piece, you'll see that it moves around and returns home flipped. That's the only effect the sequence has on the top layer—on the top two layers, in fact. If we then do the inverse x-1, of course it will un-flip the upper front edge and put everything else back where it came from. However, the inverse doesn't magically know which piece it's supposed to un-flip, and it doesn't change anything else on the upper face, so if we insert a y = U to put the upper right edge in the hot seat and do a y-1 = U-1 at the end to put it back, we get a surgical strike sequence that does nothing except flip two edges! That sequence is the commutator [x,y].

Sequences like x are called monoflips; there are also monotwists and monoswaps. [MT] attributes the monoflip-commutator idea to David Seal, who developed the original monoflip that we'll see in a later subessay.

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