[GAP Forum] Reidemeister-Schrier Method

[Alexander Hulpke]

Dear Forum, Dear Ahmed Arikan,

> Is it the case to find  a presentation of the derived subgroup of the following Hecke groups in GAP using Reidemeister-Schrier Method? There are information in manuals but I think I need  to see an application: Let q\geq 3 be an integer and
> 
> H_q=< x,y: x^2=y^q=1> . 

The most convenient way is probably to use `IsomorphismFpGroup'. (It calls `PresentationSubgroupRRS', but does some of the bookkeeping on its own):

gap> f:=FreeGroup("x","y");
<free group on the generators [ x, y ]>
gap> h:=f/ParseRelators(f,"x2,y10");
<fp group on the generators [ x, y ]>
gap> AbelianInvariants(h);
[ 2, 2, 5 ]
gap> d:=DerivedSubgroup(h);
Group(<fp, no generators known>)
gap> hom:=IsomorphismFpGroup(d);
[ <[ [ 1, 1 ] ]|y*x*y^-1*x^-1>, <[ [ 2, 1 ] ]|y^-1*x*y*x^-1>, <[ [ 3, 1 ] ]|y^2*x*y^-2*x^-1>, 
  <[ [ 4, 1 ] ]|y^-2*x*y^2*x^-1>, <[ [ 5, 1 ] ]|y^3*x*y^-3*x^-1>, <[ [ 6, 1 ] ]|y^-3*x*y^3*x^-1>, 
  <[ [ 7, 1 ] ]|y^4*x*y^-4*x^-1>, <[ [ 8, 1 ] ]|y^-4*x*y^4*x^-1>, <[ [ 9, 1 ] ]|y^5*x*y^-5*x^-1> ] -> 
[ F1, F2, F3, F4, F5, F6, F7, F8, F9 ]
gap> r:=Image(hom);
<fp group of size infinity on the generators [ F1, F2, F3, F4, F5, F6, F7, F8, F9 ]>
gap> RelatorsOfFpGroup(r);
[  ]

So in this case the derived subgroup is free -- in general you would get relators in the new generators Fi. e.g. if we define:

gap> h:=f/ParseRelators(f,"x2,y10,[x,y]^9");

we get relators:

[ F1^9, F2^9, (F3*F1^-1)^9, (F4*F2^-1)^9, (F5*F3^-1)^9, (F6*F4^-1)^9, (F7*F5^-1)^9, (F8*F6^-1)^9, 
  (F9*F8^-1)^9, (F9*F7^-1)^9 ]

Regards,

   Alexander Hulpke