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Subject:

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