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 = ,
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