From hoey@aic.nrl.navy.mil  Sun Dec 17 02:47:48 1995
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Date: Sun, 17 Dec 95 02:47:46 EST
From: hoey@aic.nrl.navy.mil
Message-Id: <9512170747.AA27428@sun13.aic.nrl.navy.mil>
To: Cube-Lovers@life.ai.mit.edu
Subject: Re: Presenting Rubik's Cube

For the benefit of Cube-Lovers, here is rusin@washington.math.niu.edu
(Dave Rusin)'s remark on finding a presentation of Rubik's cube.

  You have a group  Rubik  generated by the 6 90-degree rotations  g_i.
  Let  F  be the free group on 6 generators  x_i and  f: F --> Rubik the
  obvious homomorphism. There is a big kernel  N  of  f. (It is actually a
  free group: subgroups of free groups are free). You wish to find the
  smallest (free) subgroup  K  of  N  such that  N  is the normal closure
  of  K  in  F. (When you give a presentation of  Rubik  in the form
          Rubik = <g_1, ..., g_6 | word_1=1, word_2=1, ...>,
  you are implicitly describing  K  as the subgroup of  F  generated by the
  corresponding words in the   x_i.)
          To give this process at least a chance of success, you abelianize it:
  Let  N_ab  be the free abelian group  N/[N,N], so that there is a natural
  map from  N  into  N_ab. Since  N  is normal in  F  and  [N,N]  is 
  characteristic in  N, the action of  F  by conjugation on  N  lifts to an
  action of  F  on  N_ab; even better, the subgroup  N < F  acts trivially on
  N_ab, so that  F/N  (i.e., the Rubik group itself) acts on  N_ab.
  We think of  N_ab  as a  Rubik-module (or better, as a  Z[Rubik]-module).
  The subgroup  K < N  also maps to a subgroup K[N,N]/[N,N]  of  N_ab;
  significantly,  N  is the F-closure of  K  iff  N=[K,F]K so that N_ab
  is generated as a  Z[Rubik]-module by  F.
          Thus, the question of what constitutes a minimal set of
  relations is the same as asking for the number of generators needed for
  a certain  Rubik-module. (Of course, while you're at it, you might as well
  ask for a whole presentation or resolution of the Rubik-module. Inevitably,
  you will be led to questions of group cohomology.)

He also included GAP's help file on the cube, which I think has been
posted here already.

Dan
Hoey@AIC.NRL.Navy.Mil