Rubik cube theory and commutative matrices 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?
(This answer only discusses under the context of the MIT paper OP referred to.)

I have done a research on "why the heck did they put commutator and conjugation in Rubik's Cube notes" (written in a more polite language of course) and read tons of materials related to Rubik's Cube.

Basically, there is no theoretical significance now. It is useful for practical reason and stays there for historic and educational reasons.

For people who has attempted solving the Cube, I'm sure it happens at some point of time that you say to yourself "let's put this cubie away using move $A$ so the next move $B$ (which usually does something helpful) does not affect it, and we can bring it back later with a reverse move $A^{-1}$". This is conjugation $ABA^{-1}$. Commutator has similar practical meaning. 

They are useful when people want to figure out by themselves how to solve the Cube because they are simple and the effect is obvious to eyes, so they are used extensively in last century when people do not have the computational power to look at the mathematical essence of the Cube. I believe Singmaster's Notes on Rubik's Magic Cube and possibly one issue of Hofstadter's Metamagical Themas that introduces Rubik's Cube have significant influence on this. 

Right now the academia is not really concerned about how to solve the Cube anymore since you can easily find some algorithms online. Rubik's Cube is mostly used for demonstrating or introducing concepts in group theory, and given that historically people often see commutator and conjugation in books about Rubik's Cube, it is natural to pick them as two of the concepts that are demonstrated by Rubik's Cube.
(This answer only discusses under the context of the MIT paper OP referred to.)

I have done a research on "why the heck did they put commutator and conjugation in Rubik's Cube notes" (written in a more polite language of course) and read tons of materials related to Rubik's Cube.

Basically, there is no theoretical significance now. It is useful for practical reason and stays there for historic and educational reasons.

For people who has attempted solving the Cube, I'm sure it happens at some point of time that you say to yourself "let's put this cubie away using move $A$ so the next move $B$ (which usually does something helpful) does not affect it, and we can bring it back later with a reverse move $A^{-1}$". This is conjugation $ABA^{-1}$. Commutator has similar practical meaning. 

They are useful when people want to figure out by themselves how to solve the Cube because they are simple and the effect is obvious to eyes, so they are used extensively in last century when people do not have the computational power to look at the mathematical essence of the Cube. I believe Singmaster's Notes on Rubik's Magic Cube and possibly one issue of Hofstadter's Metamagical Themas that introduces Rubik's Cube have significant influence on this. 

Right now the academia is not really concerned about how to solve the Cube anymore since you can easily find some algorithms online. Rubik's Cube is mostly used for demonstrating or introducing concepts in group theory, and given that historically people often see commutator and conjugation in books about Rubik's Cube, it is natural to pick them as two of the concepts that are demonstrated by Rubik's Cube.