Message #3399

From: apturner@mit.edu
Subject: Re: Cube puzzles and math
Date: Mon, 27 Jun 2016 23:14:09 -0700

Dear Sid,

Yes, the object you defined is simply a nonabelian (non-commutative) group.

The comment you made about mapping to Joel’s group and back is interesting, and accurate, since you never actually mentioned taking the quotient. The set R = S* (with * the Kleene star) does have an infinite number of elements that correspond to each physical state of the Rubik’s cube. With the group operation you described and no other relations in the presentation of the group, you have the free group over the set S. To actually get the Rubik’s Cube group (the group that Joel describes), you have to take the quotient by the equivalence relation that relates all sequences of moves resulting in the same physical state, R/~. An equivalent approach would be to add relations to the presentation of the group (such as L^4 = e, because rotating the left face four quarter turns is equivalent to the identity). Really, it’s R/~ that is isomorphic to Joel’s group, not the free group as you described it. In all my previous posts I had assumed you were modding out the equivalent move sequences.

Thanks for the discussion!

Cheers,
Andrew