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Mister Spy & Souheyl Bypass Shell

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#W  grpcompl.gi                  GAP Library                 Alexander Hulpke
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#Y  Copyright (C)  1997
#Y  (C) 1998 School Math and Comp. Sci., University of St Andrews, Scotland
#Y  Copyright (C) 2002 The GAP Group
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##  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);

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, modulePcgs, 
      l, complement, k, v, afu, i, j, jj;

  G:=arg[1];
  N:=arg[2];
  # compute a series through N
  s:=ChiefSeriesUnderAction(G,N);
  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(s[Length(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(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:=[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);

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#E  grpcompl.gi
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