Commutators

If $P$ and $Q$ are any permutations, then the commutator of $P$ and $Q$ is $PQP^{-1}Q^{-1}$. It's just a conjugate with one additional operation of $Q^{-1}$ tagged onto the end.

Here's an example of a commutator in action. Suppose that you want to find an operation that flips two edge cubies on the same face in place without affecting any of the other cubies. It's not hard to find a series of moves that leaves one face completely fixed except for flipping a single cubie on it but perhaps hopelessly jumbles the rest of the cube. Call the operation that does this $P$. Now let $Q$ be a single twist of that face that puts another cubie in the same slot where the flipped cubie was. What does $PQP^{-1}Q^{-1}$ do?

$P$ flips the cubie (but trashes the rest of the cube that's not on the face). $Q$ moves a different cubie to that slot. $P^{-1}$ then undoes all of the damage caused by $P$ on the rest of the cube, but flips the new cubie. $Q^{-1}$ just rotates the face in question back to its original condition. The operation in Section 9.4 is just such a commutator.