I was reading the same paper on solving the Rubik cube. The 20 pages or so are mostly proofs and introduction to group theory which preface an algorithm for solving near the end. On pages 13-14 they introduce the commutator $PMP^{-1}M^{-1}$, then conjugation of $M$ by $P$, $PMP^{-1}$, then prove that conjugacy is an equivalence relation. $(PMP^{-1})M^{-1}$ looks so close to $PMP^{-1}$, and also to the Jordan decomposition of matrices $A = PJP^{-1}$, but they do not appear to discuss whether any of that is meaningful. The significance of the commutator and conjugacy as an equivalence relation also does not seem to be developed any further when they finally present the algorithm toward the end.
Is there some connection here between the conjugacy equivalence relation, the commutator, complete commutivity on a group, and the Jordan decomposition of matrices? Is conjugacy somehow useful to identify commutative groups, for example, a subset of matrices which form an abelian group?