Quelle  1

#(C) Graham Ellis, 2005-2006

cnt:=0;
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InstallGlobalFunction(PrimePartDerivedFunctorViaSubgroupChain,
function(GG,R,F,n)
local
 G,C,P,P1, prime, AscChn, HP, HPrels, AddRels, Q,
        DCRS, L, S, f,fx, imfx, bool, dcrs,
 HK, HPK, HKhomHPK, HPKhomHP, HKhomHP, HKx,HPKx, 
 HKxhomHPKx, HPKxhomHP, HKxhomHP, HKhomHKx,  HKhomHP2,
 x, y, i, Cent, hh, HPpres, ord, Pone, RPone;


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P:=R!.group;
prime:=PrimePGroup(P);
C:=F(R);

if IsGroup(GG) then G:=GG; 
P1:=Normalizer(G,P);

AscChn:=AscendingChain(G,P1 : refineIndex:=10);  #Added refineIndex, December 2024
fi;
if IsList(GG) then G:=GG[Length(GG)]; 
AscChn:=GG;
P1:=Normalizer(G,P);
fi;

#HP:=GroupHomomorphismByFunction(P,P,x->x);
#HP:=EquivariantChainMap(R,R,HP);
#HP:=F(HP);
#HP:=Homology(HP,n);   #This takes too much time!!
x:=IntegralHomology("HomologyAsFpGroup",n);       #Modified December 2024
HP:=x(F(R),n);
HP:=HP.fpgroup;
#HP:=Source(HP);
if Length(AbelianInvariants(HP))=0 then return []; fi;
HPrels:=[Identity(HP)];
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AddRels:=function(Q,L)  #Here P < Q < G where P=Syl_p(G)
local i, hh, Lhh, gg, g, gg1, h, sylQQ, QQ, xx, RC, gens, bool;

QQ:=Intersection(Q,Q^L); 
sylQQ:=SylowSubgroup(QQ,prime);
if not Order(sylQQ)>1 then return; fi;
gens:=SmallGeneratingSet(sylQQ);

RC:=RightTransversal(Q,Normalizer(Q,sylQQ));
for g in RC do                #December 2024 changed 
bool:=true;
   for xx in gens do
      if not xx^g in P then bool:=false; break; fi;
   od;
if bool then gg:=g; break; fi; 
#if IsSubgroup(P,sylQQ^g) then gg:=g; break; fi;
od;



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if Order(P)/Order(sylQQ)>1 then  #NEED TO OPTIMIZETHIS CHOICE!!

S:=ResolutionGenericGroup(sylQQ,n+1);

else

#S:=ResolutionFiniteSubgroup(R,sylQQ^gg);   #WITH THIS!!
S:=R;

S!.group:=sylQQ;
gg1:=gg^-1;
S!.elts:=List(S!.elts,x->x^(gg1));
fi;
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hh:=Homology(F(S),n);
if IsInt(hh) then hh:=List([1..hh],i->0); fi;
if not Length(hh)>0 then return; fi;
f:=GroupHomomorphismByFunction(sylQQ,P,x->x^gg);
xx:=F(EquivariantChainMap(S,R,f));;
HKhomHPK:=Homology(xx,n);
HK:=Source(HKhomHPK);
HPK:=Range(HKhomHPK);
HPKhomHP:=GroupHomomorphismByImagesNC(HPK,HP,GeneratorsOfGroup(HPK),
                                                  GeneratorsOfGroup(HP));
HKhomHP:=GroupHomomorphismByFunction(HK,HP,x->
Image(HPKhomHP, Image(HKhomHPK,x) ) );


fx:=GroupHomomorphismByFunction(sylQQ,Q,g->g^(L^-1));
imfx:=Image(fx);
#hh:=false;
#RC:=ConjugateSubgroups(Q,imfx);
#i:=PositionProperty(RC,x->IsSubgroup(P,x));
#hh:=RepresentativeAction(Q,imfx,RC[i]);

RC:=RightCosets(Q,Normalizer(Q,imfx));
RC:=List(RC,Representative);
for h in RC do                   #December 2024 changed from Q to RC
if IsSubgroup(P,imfx^h) then hh:=h; break; fi;
od;
Lhh:=L^-1*hh;
#fx:=GroupHomomorphismByFunction(sylQQ,P,g->(g^(L^-1))^hh);
fx:=GroupHomomorphismByFunction(sylQQ,P,g->g^Lhh);

xx:=F(EquivariantChainMap(S,R,fx));
HKxhomHPKx:=Homology(xx,n);
HKx:=Source(HKxhomHPKx);
HPKx:=Parent(Image(HKxhomHPKx));
HPKxhomHP:=GroupHomomorphismByImagesNC(HPKx,HP,GeneratorsOfGroup(HPKx),
                                                  GeneratorsOfGroup(HP));
HKxhomHP:=GroupHomomorphismByFunction(HKx,HP,x->
Image(HPKxhomHP, Image(HKxhomHPKx,x) ) );
HKhomHKx:=GroupHomomorphismByImagesNC(HK,HKx,GeneratorsOfGroup(HK),GeneratorsOfGroup(HKx));
HKhomHP2:=GroupHomomorphismByFunction(HK,HP,a->
Image(HKxhomHP, Image(HKhomHKx,a)));

for x in GeneratorsOfGroup(HK) do
Append(HPrels, [Image(HKhomHP,x)*Image(HKhomHP2,x)^-1]);
od;

end;
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ord:=function(x,y); return Order(x)<Order(y); end; 
if Order(P1)>Order(P) then 
DCRS:=SmallGeneratingSet(P1);
   for L in DCRS do
   AddRels(P,L);
   od;
fi;
for i in [2..Length(AscChn)] do

DCRS:=List(DoubleCosetRepsAndSizes(AscChn[i],AscChn[i-1],AscChn[i-1]),
x->x[1]);
Cent:=Centralizer(AscChn[i],AscChn[i-1]);


Sort(DCRS,ord);
DCRS:=Filtered(DCRS,a->not a in Cent);  #This does not achieve much
#DCRS:=Classify(DCRS,x->Cent*x);        #And this achieves nothing!
#DCRS:=List(DCRS,x->x[1]);              #

   for L in DCRS do
   cnt:=cnt+1;
   AddRels(AscChn[i-1],L);
   od;
od;
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return AbelianInvariants(HP/NormalClosure(HP,Group(HPrels)));
end);
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