[GAP Forum] (no subject)

[Stefan Kohl]

On Tue, September 8, 2015 11:48 am, fahime babaee wrote:
>
>  By Schreiers formula we know that the rank of a  finite index subgroup of
> a free group of finite rank is finite. How can I find this set of
> generators in gap? For example if we take natural homorphism from the free
> group f(x,y) to the  symmetric group S_n, we know that the kernel has rank
> n!+1, How can I find a set of  generators of the kernel?

You can do this as follows (for illustration, put n := 4):

gap> F := FreeGroup("x","y");
<free group on the generators [ x, y ]>
gap> S4 := SymmetricGroup(4);
Sym( [ 1 .. 4 ] )
gap> phi := GroupHomomorphismByImages(F,S4,GeneratorsOfGroup(F),GeneratorsOfGroup(S4));
[ x, y ] -> [ (1,2,3,4), (1,2) ]
gap> K := Kernel(phi);
Group(<free, no generators known>)
gap> gens := GeneratorsOfGroup(K);
[ y^-2, x*y^-2*x^-1, x^-4, x^-1*y^-2*x, x^2*y^-2*x^-2, (y*x)^2*y^-1*x,
  y*(x*y^-1)^2*x, y*x^-4*y^-1, (y*x^-1)^2*y^-1*x^-1, y*(x^-1*y^-1)^2*x^-1,
  x*y*x^-4*y^-1*x^-1, x*(y*x^-1)^2*y^-1*x^-2, x*y*(x^-1*y^-1)^2*x^-2,
  x^-1*(y*x)^2*y^-1*x^-2, x^-1*y*(x*y^-1)^2*x^-2, x^-1*y*x^-4*y^-1*x,
  y*x^2*y^-2*x^-2*y^-1, x^2*y*x^-4*y^-1*x^-2, x*y*x^2*y^-2*x^-2*y^-1*x^-1,
  x^-1*y*x^2*y^-2*x^-2*y^-1*x, y*x^2*y*x*y^-1*x^-2*y^-1*x,
  y*x^2*y*(x^-1*y^-1*x^-1)^2, x^2*y*x^2*y^-1*(y^-1*x^-2)^2,
  x*y*x^2*y*(x^-1*y^-1*x^-1)^2*x^-1, x^-1*y*x^2*y*x*(y^-1*x^-2)^2 ]
gap> Length(gens);
25

Hope this helps,

    Stefan