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 _______________________________________________ Forum mailing list Forum@mail.gap-system.org http://mail.gap-system.org/mailman/listinfo/forum