Dear all,
I have a matrix group G and a permutation representation phi.
I want to find a generating set for the kernel. It seems to
be a hard problem with GAP and I am unsure why. Is it really
such a hard problem ?
The problem happens already when we take GL2 and the
determinant representation, which is my primary example, though
not the only one.
Any help would be appreciated.
Mathieu
PS: Below, the example that shows the problem and an attempt solution
at it in the particular case of a permutation representation to Sym(2).
I do not claim optimality, but I think it is correct.
TestGL:=function(n)
local TheG, ListGen, ListPerm, eGen, eDet, Sym2, phi, TheKer;
TheG:=GL(n, Integers);
ListGen:=GeneratorsOfGroup(TheG);
ListPerm:=[];
for eGen in ListGen
do
eDet:=DeterminantMat(eGen);
if eDet=1 then
Add(ListPerm, ());
else
Add(ListPerm, (1,2));
fi;
od;
Sym2:=SymmetricGroup(2);
Print("Before computing phi\n");
phi:=GroupHomomorphismByImagesNC(TheG, Sym2, ListGen, ListPerm);
Print("Before computing TheKer\n");
TheKer:=Kernel(phi);
Print("After computing TheKer\n");
return rec(phi:=phi, TheKer:=TheKer);
end;
GetIndexOneTwoKernelOfMapping:=function(GRP1, GRP2, ListGens1, ListGens2)
local nbGen, ListPosPlus, ListPosMinus, ListGenRet, x1, TheId, eGenMinus,
eGenPlus, List1, List2, i1, i2;
if First(ListGens2, x->x<>() and x<>(1,2))<>fail then
Error("The ListGens2 must be only () or (1,2)");
fi;
if GRP2<>SymmetricGroup(2) then
Error("The second group must be SymmetricGroup(2)");
fi;
nbGen:=Length(ListGens2);
ListPosPlus:=Filtered([1..nbGen], x->ListGens2[x]=());
ListPosMinus:=Filtered([1..nbGen], x->ListGens2[x]=(1,2));
if Length(ListPosMinus)=0 then
return GRP1;
fi;
ListGenRet:=ListGens1{ListPosPlus};
x1:=ListGens1[ListPosMinus[1]];
TheId:=Identity(GRP1);
for eGenMinus in ListGens1{ListPosMinus}
do
for eGenPlus in Concatenation(ListPosPlus, TheId)
do
List1:=[eGenMinus, Inverse(eGenMinus)];
List2:=[x1, Inverse(x1)];
for i1 in [1,2]
do
for i2 in [1,2]
do
Add(ListGenRet, List1[i1]*eGenPlus*List2[i2]);
Add(ListGenRet, List2[i2]*eGenPlus*List1[i1]);
od;
od;
od;
od;
return Group(ListGenRet);
end;
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