[GAP Forum] generators of subgroups

[Alexander Hulpke]

Dear Bill,

> I have a matrix group G generated by a family of matrix, says
> (example from GAP documentation) and C its composition series:
>
> m1 := [ [ Z(3)^0, Z(3)^0,   Z(3) ], [   Z(3), 0*Z(3),   Z(3) ],  
> [ 0*Z(3),   Z(3), 0*Z(3) ] ];;
> m2 := [ [   Z(3),   Z(3), Z(3)^0 ], [   Z(3), 0*Z(3),   Z(3) ],  
> [ Z(3)^0, 0*Z(3),   Z(3) ] ];;
> G := Group( m1, m2 );
> C := CompositionSeries(G);
>
> The elements of C are subgroups of G.
> I would like to get a set of generators for the elements of C  
> expressed in term of the generators of G.

If your group is rather small (say up to 10^6), you can use  
`Factorization', which gives you a shortest word. (The code will be  
improved in future releases to permit groups of several magnitudes  
larger.)

gap> List(C,i->List(GeneratorsOfGroup(i),x->Factorization(G,x)));

The products are given in a free group given as the Source of
gap> EpimorphismFromFreeGroup(G);
[ x1, x2 ] ->
[ [ [ Z(3)^0, Z(3)^0, Z(3) ], [ Z(3), 0*Z(3), Z(3) ], [ 0*Z(3), Z(3),  
0*Z(3)
          ] ],
   [ [ Z(3), Z(3), Z(3)^0 ], [ Z(3), 0*Z(3), Z(3) ], [ Z(3)^0, 0*Z(3),  
Z(3) ]
      ] ]

If your group is larger, see the manual section
Expressing Group Elements as Words in Generators
on how to replace `Factorization'.

Best,

     Alexander