Date: Sun, 17 Dec 95 02:47:46 -0500 (EST)
From: Dan Hoey <hoey@aic.nrl.navy.mil >
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