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##########################################################################
#0
#F RightTransversal
##
## Define a new method for computing the set of right cosets
## of a congruence subgroup in its main group SL2Z[1/m]
##
## Input: A pair of groups (G,H) where G is SL2Z(1/m)
## and H is its congruence subgroup
## Output: A set of right cosets of H in G
##
##
InstallOtherMethod(RightTransversal,
"Right Transversal for SL2Z(1/m) and its congruence subgroup",
[IsHAPRationalSpecialLinearGroup,IsMatrixGroup],
function(G,H)
local
T,trans,NameG,i,j,m,n,p,S,d,enum;
NameG:=Name(G);
i:=Position(NameG,'/');
j:=Position(NameG,']');
m:=Int(NameG{[i+1..j-1]});
n:=H!.levels[1];
p:=H!.levels[2];
if not n=m then
return "two groups are not in the same level";
fi;
T:=[[1,0],[1,1]];
S:=[[0,-1],[1,0]];
enum:=EnumeratorByFunctions(CollectionsFamily( FamilyObj( G ) ),rec(
ElementNumber:=function(enum,n)
if n<p+1 then
return CanonicalRightCountableCosetElement(H,T^n);
else
return CanonicalRightCountableCosetElement(H,S);
fi;
end,
NumberElement:=function(enum,elm)
local i;
if elm*S^-1 in H then
return p+1;fi;
for i in [1..p] do
if elm*T^-i in H then
return i;
fi;
od;
return fail;
end,
Length:=function(enum)
return p+1;
end,
PrintObj:=function(enum)
Print("RightTransversal of ",Name(H)," in ",Name(G));
end,
group:=G,
subgroup:=H,
));
SetIsSSortedList( enum, true );
return enum;
end);
##
################### end of RightTransversal ##############################
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