Conjugates

If $P$ and $Q$ are any permutations, then the permutation $PQP^{-1}$ is called a conjugate of $P$. In group theory, a conjugate operation is very much like a change in coordinate system.

Here's a concrete example from Rubik's Cube. Suppose that you know how to swap two corner pieces that are on the same edge (see Section 9.1; let's call this operation $Q$), but you're faced with a cube where the corners you would like to swap are not on the same edge. No problem--find a simple operation (call it $P$) that brings the two corners of interest to be on the same edge. If you perform $P$, then $Q$, and then ``undo'' $P$ (in other words, perform $P^{-1}$, the net effect will be to move the corners to the same edge, swap the corners on that edge, and then move the corners back to where they began. Doing $P$, then $Q$, then $P^{-1}$ is the same as doing $PQP^{-1}$--a conjugate of $Q$.