Finally we turn to the original idea connected with the cube, namely to find a sequence of turns of the faces that will transform the cube back into its original state. This amounts to a decomposition of a given element of the cube group into a product of the generators. For this purpose we introduce a free group and a homomorphism of it onto the cube group.
gap> f := FreeGroup("t","l","f","r","e","b");
<free group on the generators [ t, l, f, r, e, b ]>
gap> hom := GroupHomomorphismByImages( f, cube, GeneratorsOfGroup(f),
> GeneratorsOfGroup(cube) );
[ t, l, f, r, e, b ] ->
[ (1,3,8,6)(2,5,7,4)(9,33,25,17)(10,34,26,18)(11,35,27,19),
(1,17,41,40)(4,20,44,37)(6,22,46,35)(9,11,16,14)(10,13,15,12),
(6,25,43,16)(7,28,42,13)(8,30,41,11)(17,19,24,22)(18,21,23,20),
(3,38,43,19)(5,36,45,21)(8,33,48,24)(25,27,32,30)(26,29,31,28),
(1,14,48,27)(2,12,47,29)(3,9,46,32)(33,35,40,38)(34,37,39,36),
(14,22,30,38)(15,23,31,39)(16,24,32,40)(41,43,48,46)(42,45,47,44) ]
Using this homomorphism, we can now decompose elements into generators. The method used utilizes a stabilizer chain and does not enumerate all group elements, therefore the words obtained are not the shortest possible, though they are short enough for hand solutions.
c'est juste des mouvements à apprendre par coeur
