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#############################################################################
##
## This file is part of GAP, a system for computational discrete algebra.
## This file's authors include Alexander Hulpke.
##
## Copyright of GAP belongs to its developers, whose names are too numerous
## to list here. Please refer to the COPYRIGHT file for details.
##
## SPDX-License-Identifier: GPL-2.0-or-later
##
## This file contains the operations for the computation of complements in
## 'white box groups'
##
BindGlobal("COCohomologyAction",function(oc,actgrp,auts,orbs)
local b, mats, orb, com, stabilizer, i,coc,u;
if not IsBound(oc.complement) then
return [];
fi;
oc.zero:=Zero(LeftActingDomain(oc.oneCocycles));
b:=BaseSteinitzVectors(BasisVectors(Basis(oc.oneCocycles)),
BasisVectors(Basis(oc.oneCoboundaries)));
if Length(b.factorspace)=0 then
u:=rec(com:=[rec(cocycle:=Zero(oc.oneCocycles),stabilizer:=actgrp)],
bas:=b);
if orbs then
u.com[1].orbit:=[Zero(oc.oneCocycles)];
fi;
return u;
fi;
Info(InfoComplement,2,"fuse ",
Characteristic(oc.zero)^Length(b.factorspace)," classes");
if Length(auts)=0 then
auts:=[One(actgrp)];
fi;
mats:=COAffineCohomologyAction(oc,oc.complementGens,auts,b);
orb:=COAffineBlocks(actgrp,auts,mats,orbs);
com:=[];
for i in orb do
coc:=i.vector*b.factorspace;
#u:=oc.cocycleToComplement(coc);
u:=rec(cocycle:=coc,
#complement:=u,
stabilizer:=i.stabilizer);
if orbs then u.orbit:=i.orbit;fi;
Add(com,u);
od;
Info(InfoComplement,1,"obtain ",Length(com)," orbits");
return rec(com:=com,bas:=b,mats:=mats);
end);
BindGlobal( "ComplementClassesRepresentativesSolvableWBG", function(arg)
local G,N,K,s, h, q, fpi, factorpres, com, comgens, cen, ocrels, fpcgs, ncom,
ncomgens, ncen, nlcom, nlcomgens, nlcen, ocr, generators,
l, complement, k, v, afu, i, j, jj;
G:=arg[1];
N:=arg[2];
# compute a series through N
if IsElementaryAbelian(N) then
s:=[N,TrivialSubgroup(N)];
else
s:=ChiefSeriesUnderAction(G,N);
fi;
if Length(arg)=2 then
K:=fail;
else
K:=arg[3];
# build a series only down to K
h:=List(s,x->ClosureGroup(K,x));
s:=[h[1]];
for i in h{[2..Length(h)]} do
if Size(i)<Size(Last(s)) then
Add(s,i);
fi;
od;
fi;
Info(InfoComplement,1,"Series of factors:",
List([1..Length(s)-1],i->Size(s[i])/Size(s[i+1])));
# #T 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(GeneratorsOfGroup(Range(fpi)),
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:=[rec(stabilizer:=cen[j],
cocycle:=Zero(ocr.oneCocycles),
complement:=ocr.complement)];
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;
#
# k:=ExternalOrbitsStabilizers(cen[j],v,opfun);
l:=COCohomologyAction(ocr,cen[j],GeneratorsOfGroup(cen[j]),false).com;
# if Length(l)<>Length(k) then Error("differ!");fi;
fi;
Info(InfoComplement,2,"splits in ",Length(l)," complements");
else
l:=[];
Info(InfoComplement,2,"no complements");
fi;
for k in l do
q:=k.stabilizer;
k:=ocr.cocycleToComplement(k.cocycle);
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;
if K<>fail then
com:=List(com,x->ClosureGroup(K,x));
fi;
return com;
end );
InstallMethod(ComplementClassesRepresentativesSolvableNC,"using cohomology",
IsIdenticalObj,
[IsGroup,IsGroup],1,
ComplementClassesRepresentativesSolvableWBG);
[ Dauer der Verarbeitung: 0.34 Sekunden
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