A Presentation for Rubik's cube group? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T13:28:36Z http://mathoverflow.net/feeds/question/72465 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/72465/a-presentation-for-rubiks-cube-group A Presentation for Rubik's cube group? Martin Brandenburg 2011-08-09T10:01:08Z 2011-08-10T08:56:25Z <p>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</p> <p>$G = \langle L,R,D,U,F,B ~:~ ?\rangle$.</p> <p>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).</p> <p>From "<a href="http://www.permutationpuzzles.org/rubik/webnotes/rubik.pdf" rel="nofollow">The Mathematics of Rubik's cube</a>" 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 $(a,x,b,y) \mapsto (\text{sign}(a) \text{sign}(b),\sum_i x_i,\sum_j y_j)$.</p> http://mathoverflow.net/questions/72465/a-presentation-for-rubiks-cube-group/72482#72482 Answer by Igor Rivin for A Presentation for Rubik's cube group? Igor Rivin 2011-08-09T16:02:33Z 2011-08-09T16:02:33Z <p>This discussion: <a href="http://www.math.niu.edu/~rusin/known-math/95/rubik" rel="nofollow">http://www.math.niu.edu/~rusin/known-math/95/rubik</a> 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:</p> <p><a href="http://cubezzz.dyndns.org/drupal/?q=node/view/177" rel="nofollow">http://cubezzz.dyndns.org/drupal/?q=node/view/177</a></p>