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#############################################################################
#0
#F HomotopyLowerCentralSeriesOfCrossedModule
## Input: A crossed module X
## Output: The homotopy lower central series of X
##
InstallGlobalFunction(HomotopyLowerCentralSeriesOfCrossedModule, function(X)
local
del,act,M,P,GensM,ImgGensM,A,G,nat,Gs,Ps,
nOne,XOne,i,phi,MorphismOne,
Gens,As,nTwo,a,g,natMs,Ms,XTwo,GenMs,
PreImGenMs,MorphismTwo,
ActOne,MapOne,MapTwo;
del:=X!.map;
act:=X!.action;
M:=Source(del);
P:=Range(del);
GensM:=GeneratorsOfGroup(M);
ImgGensM:=List(GensM,m->Image(del,m));
nat:=NaturalHomomorphismByNormalSubgroup(P,Image(del));
A:=Kernel(del);
G:=Range(nat);
Gs:=[G];
Ps:=[P];
nOne:=1;
while not IsTrivial(Gs[nOne]) do
nOne:=nOne+1;
Gs[nOne]:=CommutatorSubgroup(Gs[nOne-1],G);
Ps[nOne]:=PreImage(nat,Gs[nOne]);
od;
MorphismOne:=[];
if nOne>1 then
Ps:=Reversed(Ps);
XOne:=[];
for i in [1..nOne-1] do
phi:=GroupHomomorphismByImages(M,Ps[i],GensM,ImgGensM);
XOne[i]:=Objectify(HapCrossedModule,
rec(map:=phi,
action:=act
));
od;
XOne[nOne]:=X;
#########################################################
#1
MapOne:= function(i)
return function(n)
local Gens;
if n = 1 then
return IdentityMapping(M);
fi;
if n =2 then
Gens:=GeneratorsOfGroup(Ps[i]);
return GroupHomomorphismByImages(Ps[i],
Ps[i+1],Gens,Gens);
fi;
end;
end;
##
#########################################################
for i in [1..nOne-1] do
MorphismOne[i]:=Objectify(HapCrossedModuleMorphism,
rec(source:=XOne[i],
target:=XOne[i+1],
mapping:=MapOne(i)
));
od;
fi; ## end of nOne>1
G:=List(G,g->PreImagesRepresentative(nat,g));
As:=[A];
nTwo:=1;
while not IsTrivial(As[nTwo]) do
Gens:=[];
for a in As[nTwo] do
for g in G do
Add(Gens,a*act(g,a^(-1)));
od;
od;
nTwo:=nTwo+1;
As[nTwo]:=Group(Gens);
od;
MorphismTwo:=[];
if nTwo>1 then
As:=Reversed(As);
natMs:=[IdentityMapping(M)];
Ms:=[M];
XTwo:=[X];
GenMs:=[GensM];
PreImGenMs:=[GensM];
#########################################################
#1
ActOne:=function(i)
return function(p,mA)
return Image(natMs[i],act(p,
PreImagesRepresentative(natMs[i],mA)));
end;
end;
##
#########################################################
for i in [2..nTwo] do
natMs[i]:=NaturalHomomorphismByNormalSubgroup(M,As[i]);
Ms[i]:=Range(natMs[i]);
GenMs[i]:=GeneratorsOfGroup(Ms[i]);
PreImGenMs[i]:=List(GenMs[i],
m->PreImagesRepresentative(natMs[i],m));
phi:=GroupHomomorphismByImages(Ms[i],P,GenMs[i],
List(PreImGenMs[i],m->Image(del,m)));
XTwo[i]:=Objectify(HapCrossedModule,
rec(map:=phi,
action:=ActOne(i)
));
od;
#########################################################
#1
MapTwo:=function(i)
return function(n)
if n = 1 then
return GroupHomomorphismByImages(Ms[i],Ms[i+1],
GenMs[i],List(PreImGenMs[i],
m->Image(natMs[i+1],m)));
fi;
if n =2 then
return IdentityMapping(P);
fi;
end;
end;
##
#########################################################
for i in [1..nTwo-1] do
MorphismTwo[i]:=Objectify(HapCrossedModuleMorphism,
rec(source:=XTwo[i],
target:=XTwo[i+1],
mapping:=MapTwo(i)
));
od;
fi; ##end of nTwo>1
return Concatenation(MorphismOne,MorphismTwo);
end);
##
################### end of HomotopyLowerCentralSeriesOfCrossedModule ########
#############################################################################
#0
#F HomotopyLowerCentralSeries
## Input: A crossed module or a cat-1-group
## Output: The homotopy lower central series of the input
##
InstallGlobalFunction(HomotopyLowerCentralSeries, function(X)
local XC,LXC;
if IsHapCrossedModule(X) then
return HomotopyLowerCentralSeriesOfCrossedModule(X);
fi;
if IsHapCatOneGroup(X) then
XC:=CrossedModuleByCatOneGroup(X);
LXC:=HomotopyLowerCentralSeriesOfCrossedModule(XC);
return CatOneGroupByCrossedModule(LXC);
fi;
end);
##
################### end of HomotopyLowerCentralSeries #######################
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