Dear Forum,
I am interested in obtaining Schreier system of representatives of right
cosets of a subgroup of a finitely presented group.
http://eom.springer.de/S/s083400.htm
Let G = F/R, H< G. Could I be sure that
RightTransversal(G,H)
will give me such system?
I guess that there could be silent assumption that generators(alphabet) are
the generators of a free group F.
If I couldn't, is there other way to be sure?
It is working like that for any example which I tried.
For Example:
F:=FreeGroup("a","b","x");;
G:=F/[F.1^2,F.2^4,F.3^2,(F.1*F.2)^3,F.3*F.1*F.3*(F.1*F.2*F.1*F.2^-1*F.1),F.3*F.2*F.3*(F.2^-1*F.1*F.2*F.2*F.2^-1*F.1*F.2)];;
AssignGeneratorVariables(G);
K:=FreeGroup("x1","e1","c0","c1","c2");;
L:=K/[K.1^2, K.3^2, K.4^2, K.5^2, K.1*K.2, (K.3*K.4)^2, (K.4*K.5)^4,
K.1*K.5*K.1*K.3];;
AssignGeneratorVariables(L);
theta:=GroupHomomorphismByImages(L,G,[x1,e1,c0,c1,c2],[a,a,x,x*b^-1*a*b,x*b^-1*a*b^2]);;
J:=Kernel(theta);;
C:=AsList(RightTransversal(L,J));
which gives me Schreier system:
[<identity ...>, x1, c0, c1, c2, x1*c0, x1*c1, x1*c2, c0*c1, c0*c2, c1*x1,
c1*c2, c2*c0, c2*c1, x1*c0*c1, x1*c0*c2, x1*c1*x1, x1*c1*c2, x1*c2*c0,
x1*c2*c1, c0*c1*x1, c0*c1*c2, c0*c2*c0, c0*c2*c1, c1*x1*c0, c1*x1*c1,
c1*c2*c0, c1*c2*c1, c2*c0*c1, c2*c1*x1, c2*c1*c2, x1*c0*c1*x1, x1*c0*c1*c2,
x1*c0*c2*c0, x1*c0*c2*c1, x1*c1*x1*c0, x1*c1*x1*c1, x1*c2*c0*c1, x1*c2*c1*x1,
x1*c2*c1*c2, c0*c1*x1*c0, c0*c1*x1*c1, c0*c1*c2*c0, c0*c2*c0*c1,
c1*x1*c0*c1, c1*c2*c1*c2, c2*c0*c1*x1, x1*c0*c2*c0*c1 ]
Best regards
Bartosz Putrycz
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