[GAP Forum] Rubiks cube and Fp groups

[Rudolf Zlabinger]

I refer to http://www.gap-system.org/Doc/Examples/rubik.html as given by
Martin Schönert.

I tested the presentation of a random element of the cubes permutations in a
finitely presented group.

I took the free group in the sample and formed a Fp group giving the
relators f ^ 4 for all generators of the free group, as this is also given
by the original generators. Of course, its a infinite group again, as
unnecessarily also tested by the Newman Infinity Criterion. The Preimage for
the Free group gave a solution of length 120, the same performed with the Fp
group, same permutation, resulted in a chain of 84 moves. To be secure I
tested the image also in reverse order successfully, as given by the sample.

The random permutation was in both cases
(1,22,33,17,19,40,30,35,16,3,6,25,46,24,9,41,27,11,8,14,43)(2,13,31,29,15,42
,
28)(4,5,39,18,37)(7,12,10,26,47)(20,45,36,44,23,21,34)(32,38,48)
.

The algorithm mentioned in the sample was using stabilizer chains. Is it the
same for the Fp group I used for my test? Or is there help for a "better"
algorithm caused by the relators? Or is it pure random behaviour?

By the way, another question to the same sample. There are given " wreath
products of a 3 cycle (2 cycle) with S(8)". Is it right to interprete them
as wreath products of the cycles ^ 8 and S(8)?

best regards, Rudolf Zlabinger