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#############################################################################
##
#W grpcompl.gi GAP Library Alexander Hulpke
##
#H @(#)$Id: grpcompl.gi,v 4.13.4.3 2006/08/16 09:13:10 gap Exp $
##
#Y Copyright (C) 1997
#Y (C) 1998 School Math and Comp. Sci., University of St. Andrews, Scotland
#Y Copyright (C) 2002 The GAP Group
##
## This file contains the operations for the computation of complements in
## 'white box groups'
##
Revision.grpcompl_gi :=
"@(#)$Id: grpcompl.gi,v 4.13.4.3 2006/08/16 09:13:10 gap Exp $";
ComplementclassesSolvableWBG:=function(G,N)
local s, h, q, fpi, factorpres, com, comgens, cen, ocrels, fpcgs, ncom,
ncomgens, ncen, nlcom, nlcomgens, nlcen, ocr, generators,
modulePcgs, l, v, dimran, opfun, k, afu, i, j, jj;
# compute a series through N
s:=ChiefSeriesUnderAction(G,N);
Info(InfoComplement,1,"Series of factors:",
List([1..Length(s)-1],i->Size(s[i])/Size(s[i+1])));
# transfer probably to better group (later, AgCase)
# construct a presentation
h:=NaturalHomomorphismByNormalSubgroup(G,N);
# AH still: Try to find a more simple presentation if available.
if Source(h)=G then
q:=ImagesSource(h);
else
q:=Image(h,G);
fi;
fpi:=IsomorphismFpGroup(q);
Info(InfoComplement,2,"using a presentation with ",
Length(MappingGeneratorsImages(fpi)[2])," generators");
factorpres:=[FreeGeneratorsOfFpGroup(Range(fpi)),
RelatorsOfFpGroup(Range(fpi)),
List(MappingGeneratorsImages(fpi)[2],
i->PreImagesRepresentative(fpi,i))];
Assert(1,ForAll(factorpres[3],i->Image(h,PreImagesRepresentative(h,i))=i));
# initialize
com:=[G];
comgens:=[List(factorpres[3],i->PreImagesRepresentative(h,i))];
cen:=[s[1]];
ocrels:=false;
# step down
for i in [2..Length(s)] do
Info(InfoComplement,1,"Step ",i-1);
# we know the complements after s[i-1], we want them after s[i].
#fpcgs:=Pcgs(s[i-1]); # the factor pcgs
#fpcgs:=fpcgs mod InducedPcgsByGenerators(fpcgs,GeneratorsOfGroup(s[i]));
fpcgs:=ModuloPcgs(s[i-1],s[i]);
ncom:=[];
ncomgens:=[];
ncen:=[];
# loop over all complements so far
for j in [1..Length(com)] do
nlcom:=[];
nlcomgens:=[];
nlcen:=[];
# compute complements
ocr:=rec(group:=ClosureGroup(com[j],s[i-1]),
generators:=comgens[j],
modulePcgs:=fpcgs,
factorpres:=factorpres
);
if ocrels<>false then
ocr.relators:=ocrels;
Assert(2,ForAll(ocr.relators,
k->Product(List([1..Length(k.generators)],
l->ocr.generators[k.generators[l]]^k.powers[l]))
in s[i-1]));
fi;
OCOneCocycles(ocr,true);
ocrels:=ocr.relators;
if IsBound(ocr.complement) then
# special treatment for trivial case:
if Dimension(ocr.oneCocycles)=Dimension(ocr.oneCoboundaries) then
l:=[ExternalSet(cen[j],[Zero(ocr.oneCocycles)])];
SetStabilizerOfExternalSet(l[1],cen[j]);
else
l:=BaseSteinitzVectors(BasisVectors(Basis(ocr.oneCocycles)),
BasisVectors(Basis(ocr.oneCoboundaries)));
v:=Enumerator(VectorSpace(LeftActingDomain(ocr.oneCocycles),
l.factorspace,Zero(ocr.oneCocycles)));
dimran:=[1..Length(v[1])];
# fuse
Info(InfoComplement,2,"fuse ",Length(v)," classes; working in dim ",
Dimension(ocr.oneCocycles),"/",Dimension(ocr.oneCoboundaries));
opfun:=function(z,g)
Assert(3,z in AsList(v));
z:=ocr.cocycleToList(z);
for k in [1..Length(z)] do
z[k]:=Inverse(ocr.complementGens[k])*(ocr.complementGens[k]*z[k])^g;
od;
Assert(2,ForAll(z,k->k in s[i-1]));
z:=ocr.listToCocycle(z);
Assert(2,z in ocr.oneCocycles);
# sift z
for k in dimran do
if IsBound(l.heads[k]) and l.heads[k]<0 then
z:=z-z[k]*l.subspace[-l.heads[k]];
fi;
od;
Assert(1,z in AsList(v));
return z;
end;
l:=ExternalOrbitsStabilizers(cen[j],v,opfun);
fi;
Info(InfoComplement,2,"splits in ",Length(l)," complements");
else
l:=[];
Info(InfoComplement,2,"no complements");
fi;
for k in l do
q:=StabilizerOfExternalSet(k);
k:=ocr.cocycleToComplement(Representative(k));
Assert(3,Length(GeneratorsOfGroup(k))
=Length(MappingGeneratorsImages(fpi)[2]));
# correct stabilizer to obtain centralizer
v:=Normalizer(q,ClosureGroup(s[i],k));
afu:=function(x,g) return CanonicalRightCosetElement(s[i],x^g);end;
for jj in GeneratorsOfGroup(k) do
if ForAny(GeneratorsOfGroup(v),x->not Comm(x,jj) in s[i]) then
# we are likely very close as we centralized in the higher level
# and stabilize the cohomology. Thus a plain stabilizer
# calculation ought to work.
v:=Stabilizer(v,CanonicalRightCosetElement(s[i],jj),afu);
fi;
od;
Add(ncen,v);
Add(nlcom,k);
Add(nlcomgens,GeneratorsOfGroup(k));
od;
ncom:=Concatenation(ncom,nlcom);
ncomgens:=Concatenation(ncomgens,nlcomgens);
ncen:=Concatenation(ncen,nlcen);
od;
com:=ncom;
comgens:=ncomgens;
cen:=ncen;
Info(InfoComplement,1,Length(com)," complements in total");
od;
return com;
end;
InstallMethod(ComplementclassesSolvableNC,"using cohomology",IsIdenticalObj,
[IsGroup,IsGroup],1,
ComplementclassesSolvableWBG);
#############################################################################
##
#E grpcompl.gi
##
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