A Presentation for Rubik's cube group?

2011-08-09T10:01:08Z

Let $G$ be Rubik's cube group. It is generated by the rotations by 90 degrees $L,R,D,U,F,B$ (left, right, down, up, front, behind), but what relations beyond $L^4=R^4=...=B^4=1$ do they satisfy? Thus I would like to know a presentation of the group as

$G = \langle L,R,D,U,F,B ~:~ ?\rangle$.

After playing aroumd I'have also found the relations $LR=RL$, $(LU)^{105}=1$, $(LRFB)^{12}=1$, $(LRFBFB)^4=1$, $(LRLRFBFB)^2=1$ (of course together with the symmetric relations).

From "The Mathematics of Rubik's cube" by W. D. Joyner I know that $G$ is generated by two elements and presentations are known, but I have not found one. Besides, I'm only interested in the standard generating set above. Remark that there is a well-known abstract group-theoretic description of $G$, it is the kernel of the homomorphism $(S_{12} \ltimes (\mathbb{Z}/2)^{12}) \times (S_{8} \ltimes (\mathbb{Z}/3)^{8}) \to \mathbb{Z}/2 \times \mathbb{Z}/2 \times \mathbb{Z}/3$ which maps $(a,x,b,y) \mapsto (\text{sign}(a) \text{sign}(b),\sum_i x_i,\sum_j y_j)$.

Answer by Igor Rivin for A Presentation for Rubik's cube group?

2011-08-09T16:02:33Z

This discussion: http://www.math.niu.edu/~rusin/known-math/95/rubik seems to culminate in a presentation (due to Dan Hoey). I did not read it carefully, I must admit. The presentation is quite complicated. For the 2x2x2 group there is this:

http://cubezzz.dyndns.org/drupal/?q=node/view/177

A Presentation for Rubik's cube group? - MathOverflow

A Presentation for Rubik's cube group?

Answer by Igor Rivin for A Presentation for Rubik's cube group?