gap-libs 4r6p5-3 / usr / share / gap / lib / grppclat.gi

#############################################################################
##
#W  grppclat.gi                GAP library                   Alexander Hulpke
##
##
#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 declarations for the subgroup lattice functions for
##  pc groups.
##

#############################################################################
##
#F  InvariantElementaryAbelianSeries( <G>, <morph>, [ <N> ] )
##           find <morph> invariant EAS of G (through N)
##
InstallGlobalFunction(InvariantElementaryAbelianSeries,function(arg)
local G,morph,N,s,p,e,i,j,k,ise,fine,cor;
  G:=arg[1];
  morph:=arg[2];
  fine:=false;
  if Length(arg)>2 then
    N:=arg[3];
    e:=[G,N];
    if Length(arg)>3 then
      fine:=arg[4];
    fi;
    if fine then
      e:=ElementaryAbelianSeries(e);
    else
      e:=ElementaryAbelianSeriesLargeSteps(e);
    fi;
  else
    N:=TrivialSubgroup(G);
    e:=DerivedSeriesOfGroup(G);
    e:=ElementaryAbelianSeriesLargeSteps(e);
  fi;
  s:=[G];
  i:=2;
  while i<=Length(e) do
    # intersect all images of normal subgroup to obtain invariant one
    # as G is invariant, we dont have to deal with special cases
    ise:=[e[i]];
    cor:=e[i];
    for j in ise do
      for k in morph do
	p:=Image(k,j);
	if not IsSubset(p,cor) then
	  Add(ise,p);
	  cor:=Intersection(cor,p);
        fi;
      od;
    od;
    Assert(1,HasElementaryAbelianFactorGroup(s[Length(s)],cor));
    ise:=cor;
    Add(s,ise);
    p:=Position(e,ise);
    if p<>fail then
      i:=p+1;
    elif fine then
      e:=ElementaryAbelianSeries([G,ise,TrivialSubgroup(G)]);
      i:=Position(e,ise)+1;
    else
      e:=ElementaryAbelianSeriesLargeSteps([G,ise,TrivialSubgroup(G)]);
      i:=Position(e,ise)+1;
    fi;
    Assert(1,ise in e);
  od;
  return s;
end);

#############################################################################
##
#F  InducedAutomorphism(<epi>,<aut>)
##
InstallGlobalFunction(InducedAutomorphism,function(epi,aut)
local f;
  f:=Range(epi);
  if HasIsConjugatorAutomorphism( aut ) and IsConjugatorAutomorphism( aut ) 
     and ConjugatorOfConjugatorIsomorphism( aut ) in Source( epi ) then
    aut:= ConjugatorAutomorphismNC( f,
              Image( epi, ConjugatorOfConjugatorIsomorphism( aut ) ) );
  else
    aut:= GroupHomomorphismByImagesNC(f,f,GeneratorsOfGroup(f),
				   List(GeneratorsOfGroup(f),
	       i->Image(epi,Image(aut,PreImagesRepresentative(epi,i)))));
    SetIsInjective(aut,true);
    SetIsSurjective(aut,true);
  fi;
  return aut;
end);

#############################################################################
##
#F  InvariantSubgroupsElementaryAbelianGroup(<G>,<homs>[,<dims])  submodules
#F    find all subgroups of el. ab. <G>, which are invariant under all <homs>
#F    which have dimension in dims
##
InstallGlobalFunction(InvariantSubgroupsElementaryAbelianGroup,function(arg)
local g,op,a,pcgs,ma,mat,d,f,i,j,new,newmat,id,p,dodim,compldim,compl,dims,nm;
  g:=arg[1];
  op:=arg[2];
  if not IsElementaryAbelian(g) then
    Error("<g> must be a vector space");
  fi;
  if IsTrivial(g) then
    return [g];
  fi;
  pcgs:=Pcgs(g);
  d:=Length(pcgs);
  p:=RelativeOrderOfPcElement(pcgs,pcgs[1]);
  f:=GF(p);
  if Length(arg)=2 then
    dims:=[0..d];
  else
    dims:=arg[3];
  fi;

  if Length(dims)=0 then
    return [];
  fi;

  if Length(op)=0 then

    # trivial operation: enumerate subspaces
    # check which dimensions we'll need
    ma:=QuoInt(d,2);
    dodim:=[];
    compldim:=[];
    for i in dims do
      if i<=ma then
        AddSet(dodim,i);
      else
        AddSet(dodim,d-i);
	AddSet(compldim,d-i);
      fi;
    od;
    if d<3 then compldim:=[]; fi;
    dodim:=Maximum(dodim);

    # enumerate spaces
    id:= Immutable( IdentityMat(d, One(f)) );
    ma:=[[],[ShallowCopy(id[1])]];
    ConvertToMatrixRep(ma[2],f);
    # the complements to ma
    if d>1 then
      compl:=[ShallowCopy(id)];
    else
      compl:=[];
    fi;
    if d>2 then
      nm:=TriangulizedNullspaceMat(TransposedMat(id{[1]}));
      ConvertToMatrixRep(nm,f);
      Add(compl,nm);
    fi;
    for i in [2..d] do
      new:=[];
      for mat in ma do
	# subspaces of equal dimension
	for j in [0..p^Length(mat)-1] do
	  if j=0 then
	    # special case for subspace of higher dimension
	    if Length(mat)<dodim then
	      newmat:=Concatenation(mat,[id[i]]);
	      ConvertToMatrixRep(newmat,f);
	    else
	      newmat:=false;
	    fi;
	  else
	    # possible extension number d
	    a:=CoefficientsQadic(j,p)*One(f);
	    newmat:=List(mat,ShallowCopy);
	    for j in [1..Length(a)] do
		newmat[j][i]:=a[j];
	    od;
	    ConvertToMatrixRep(newmat,f);
	  fi;
	  if newmat<>false then
	    # we will need the space for the next level
	    Add(new,newmat);

	    # note complements if necc.
	    if Length(newmat) in compldim then
	      nm:=NullspaceMat(TransposedMat(newmat));
	      ConvertToMatrixRep(nm,f);
	      Add(compl,nm);
	      #Add(compl,List(NullspaceMat(TransposedMat(newmat*One(f))),
	      #               i->List(i,IntFFE)));
	    fi;
	  fi;
        od;
      od;
      ma:=Concatenation(ma,new);
    od;
    
    ma:=Concatenation(ma,compl);

    # take only those of right dim
    ma:=Filtered(ma,i->Length(i) in dims);

    # convert to grps (noting also the triv. one)
    new:=ma;
    for i in [1..Length(new)] do
      #a:=SubgroupNC(Parent(g),List(i,j->Product([1..d],k->pcgs[k]^j[k])));
      ma:=new[i];
      a:=SubgroupNC(Parent(g),List(ma,
	                  j->PcElementByExponentsNC(pcgs,List(j,IntFFE))));
#      a:=MySubgroupNC(Parent(g),List(i,j->PcElementByExponentsNC(pcgs,j)),
#                      IsFinite and IsSubsetLocallyFiniteGroup and
#		      IsSupersolvableGroup and IsNilpotentGroup and
#		      IsCommutative and IsElementaryAbelian);

      SetSize(a,p^Length(ma));
      new[i]:=a;
    od;
    ma:=new;

  else

    # compute representation
    ma:=[];
    for i in op do
      mat:=[];
      for j in pcgs do
	Add(mat,ExponentsOfPcElement(pcgs,Image(i,j))*One(f));
      od;
      mat:=ImmutableMatrix(f,mat);
      Add(ma,mat);
    od;

    ma:=GModuleByMats(ma,f);
    mat:=MTX.BasesSubmodules(ma);

    ma:=[];
    for i in mat do
      Add(ma,SubgroupNC(Parent(g),
		      List(i,j->PcElementByExponentsNC(pcgs,j))));
		      #List(i,j->Product([1..d],k->pcgs[k]^IntFFE(j[k])))));
    od;
  fi;
  return ma;
end);

#############################################################################
##
#F  ActionSubspacesElementaryAbelianGroup(<P>,<G>[,<dims>])
##
##  compute the permutation action of <P> on the subspaces of the
##  elementary abelian subgroup <G> of <P>. Returns
##  a list [<subspaces>,<action>], where <subspaces> is a list of all the
##  subspaces and <action> a homomorphism from <P> in a permutation group,
##  which is equal to the action homomrophism for the action of <P> on
##  <subspaces>. If <dims> is given, only subspaces of dimension <dims> are
##  considered.
##  Instead of <G> also a (modulo) pcgs may be given.
##
InstallGlobalFunction(ActionSubspacesElementaryAbelianGroup,function(arg)
local P,g,op,act,a,pcgs,ma,mat,d,f,i,j,new,newmat,id,p,dodim,compldim,compl,
      dims,Pgens,Pcgens,Pu,Pc,perms,one,par,ker,kersz;

  P:=arg[1];
  if IsModuloPcgs(arg[2]) then
    pcgs:=arg[2];
    g:=Group(NumeratorOfModuloPcgs(pcgs));
    if not IsSubset(Parent(P),g) then # for matrix groups we need a parent here.
      par:=ClosureGroup(Parent(P),g);
    else
      par:=P;
    fi;
    Pu:=AsSubgroup(par,g);
    ker:=SubgroupNC(par,DenominatorOfModuloPcgs(pcgs));
    kersz:=Size(ker);
  else
    kersz:=1;
    g:=arg[2];
    par:=Parent(g);
    Pu:=Centralizer(P,g);
    if not IsElementaryAbelian(g) then
      Error("<g> must be a vector space");
    fi;
    if IsTrivial(g) then
      return [g];
    fi;

    pcgs:=Pcgs(g);
  fi;

  d:=Length(pcgs);
  p:=RelativeOrderOfPcElement(pcgs,pcgs[1]);
  f:=GF(p);
  one:=One(f);
  if Length(arg)=2 then
    dims:=[0..d];
  else
    dims:=arg[3];
  fi;

  if Length(dims)=0 then
    return [];
  fi;

  # find representatives generating the acting factor
  Pgens:=[];
  Pc:=Pu;
  Pcgens:=GeneratorsOfGroup(Pu);
  while Size(Pu)<Size(P) do
    repeat
      i:=Random(P);
    until not i in Pu;
    Add(Pgens,i);
    Pu:=ClosureGroup(Pu,i);
  od;
  if Length(Pgens)>2 and Length(Pgens)>Length(SmallGeneratingSet(P)) then
    Pgens:=SmallGeneratingSet(P);
  fi;

  # compute representation
  op:=[];
  for i in Pgens do
    mat:=[];
    for j in pcgs do
      Add(mat,ExponentsConjugateLayer(pcgs,j,i)*One(f));
    od;
    mat:=ImmutableMatrix(f,mat);
    Add(op,mat);
  od;

  # and action on canonical bases
  #act:=function(bas,m)
  #  bas:=bas*m;
  #  bas:=List(bas,ShallowCopy);
  #  TriangulizeMat(bas);
  #  bas:=List(bas,IntVecFFE);
  #  return bas;
  #end;
  if p=2 then
    act:=OnSubspacesByCanonicalBasisGF2;
  else
    act:=OnSubspacesByCanonicalBasis;
  fi;

  # enumerate subspaces
  # check which dimensions we'll need
  ma:=QuoInt(d,2);
  dodim:=[];
  compldim:=[];
  for i in dims do
    if i<=ma then
      AddSet(dodim,i);
    else
      AddSet(dodim,d-i);
      AddSet(compldim,d-i);
    fi;
  od;
  if d<3 then compldim:=[]; fi;
  dodim:=Maximum(dodim);

  # enumerate spaces
  id:= Immutable( IdentityMat(d, one) );
  ma:=[[],[id[1]]];
  # the complements to ma
  if d>1 then
    compl:=[ShallowCopy(id)];
  else
    compl:=[];
  fi;
  if d>2 then
    Add(compl,List(TriangulizedNullspaceMat(TransposedMat(id{[1]})),
                   ShallowCopy));
  fi;
  for i in [2..d] do
    new:=[];
    for mat in ma do
      # subspaces of equal dimension
      for j in [0..p^Length(mat)-1] do
	if j=0 then
	  # special case for subspace of higher dimension
	  if Length(mat)<dodim then
	    newmat:=Concatenation(mat,[id[i]]);
	  else
	    newmat:=false;
	  fi;
	else
	  # possible extension number d
	  a:=CoefficientsQadic(j,p)*one;
	  newmat:=List(mat,ShallowCopy);
	  for j in [1..Length(a)] do
	      newmat[j][i]:=a[j];
	  od;
	fi;
	if newmat<>false then
	  # we will need the space for the next level
	  Add(new,Immutable(newmat));

	  # note complements if necc.
	  if Length(newmat) in compldim then
	    a:=List(TriangulizedNullspaceMat(MutableTransposedMat(newmat)),
	            ShallowCopy);
	    Add(compl,Immutable(a));
	  fi;
	fi;
      od;
    od;

    ma:=Concatenation(ma,new);
  od;
  
  ma:=Concatenation(ma,compl);

  # take only those of right dim
  ma:=Filtered(ma,i->Length(i) in dims);

  perms:=List(Pgens,i->());
  new:=[];
  for i in dims do
    mat:=Immutable(Set(Filtered(ma,j->Length(j)=i)));
    # compute action on mat
    if i>0 and i<d then
      for j in [1..Length(Pgens)] do
	#a:=Permutation(op[j],mat,act);
	a:=List([1..Length(mat)],k->PositionSorted(mat,act(mat[k],op[j])));
	a:=PermList(a);
	perms[j]:=perms[j]*a^MappingPermListList([1..Length(mat)],
				[Length(new)+1..Length(new)+Length(mat)]);
      od;
    fi;
    Append(new,mat);
  od;
  ma:=new;

  # convert to grps
  new:=[];
  for i in ma do
    #a:=SubgroupNC(Parent(g),List(i,j->Product([1..d],k->pcgs[k]^j[k])));
    if kersz=1 then
      a:=SubgroupNC(par,List(i,j->PcElementByExponentsNC(pcgs,j)));
    else
      a:=ClosureGroup(ker,List(i,j->PcElementByExponentsNC(pcgs,j)));
    fi;
    SetSize(a,kersz*p^Length(i));
    Add(new,a);
  od;

  ma:= GroupByGenerators( perms, () );
  #Assert(1,Group(perms)=Action(P,new));

  op:=GroupHomomorphismByImagesNC(P,ma,Concatenation(Pcgens,Pgens),
    Concatenation(List(Pcgens,i->()),perms));
#  Assert(1,Size(P)=Size(KernelOfMultiplicativeGeneralMapping(op))
#                   *Size(Image(op)));
  return [new,op];

end);

# test whether the c-conjugate of g is h-invariant, internal
HasInvariantConjugateSubgroup:=function(g,c,h)
  # This should be done better!
  g:=ConjugateSubgroup(g,c);
  return ForAll(h,i->Image(i,g)=g);
end;

#############################################################################
##
#F  SubgroupsSolvableGroup(<G>[,<opt>]) . classreps of subgrps of <G>,
##   				             <homs>-inv. with options.
##    Options are:  
##                  actions:  list of automorphisms: search for invariants
##		    funcnorm: N_G(actions) (speeds up calculation)
##                  normal:   just search for normal subgroups
##                  consider: function(A,N,B,M) indicator function, whether 
##			      complements of this type would be needed
##                  retnorm:  return normalizers
##
InstallGlobalFunction(SubgroupsSolvableGroup,function(arg)
local g,	# group
      isom,	# isomorphism onto AgSeries group
      func,	# automorphisms to be invariant under
      funcs,    # <func>
      funcnorm, # N_G(funcs)
      efunc,	# induced automs on factor
      efnorm,	# funcnorm^epi
      e,	# EAS
      len,	# Length(e)
      start,	# last index with EA factor
      i,j,k,l,
      m,kp,	# loop
      kgens,	# generators of k
      kconh,	# complemnt conjugacy storage
      opt,	# options record
      normal,	# flag for 'normal' option
      consider,	# optional 'consider' function
      retnorm,	# option: return all normalizers
      f,	# g/e[i]
      home,	# HomePcgs(f)
      epi,	# g -> f
      lastepi,  # epi of last step
      n,	# e[i-1]^epi
      fa,	# f/n = g/e[i-1]
      hom,	# f -> fa
      B,	# subgroups of n	
      ophom,	# perm action of f on B (or false if not computed)
      a,	# preimg. of group over n
      no,	# N_f(a)
#      aop,	# a^ophom
#      nohom,	# ophom\rest no
      oppcgs,	# acting pcgs
      oppcgsimg,# images under ophom
      ex,	# external set/orbits
      bs,	# b\in B normal under a, reps
      bsp,	# bs index
      bsnorms,	# respective normalizers
      b,	# in bs
      bpos,	# position in bs
      hom2,	# N_f(b) -> N_f(b)/b
      nag,	# AgGroup(n^hom2)
      fghom,	# assoc. epi
      t,s,	# dnk-transversals
      z,	# Cocycles
      coboundbas,# Basis(OneCobounds)
      field,	# GF(Exponent(n))
      com,	# complements
      comnorms,	# normalizers supergroups
      isTrueComnorm, # is comnorms the true normalizer or a supergroup
      comproj,	# projection onto complement
      kgn,
      kgim,	# stored decompositions, translated to matrix language
      kgnr,	# assoc index
      ncom,	# dito, tested
      idmat,	# 1-matrix
      mat,	# matrix action
      mats,	# list of mats
      conj,	# matrix action	
      chom,	# homom onto <conj>
      shom,	# by s induced autom
      shoms,	# list of these
      smats,	# dito, matrices 
      conjnr,	# assoc. index
      glsyl,
      glsyr,	# left and right side of eqn system
      found,	# indicator for success
      grps,	# list of subgroups
      ngrps,	# dito, new level
      gj,	# grps[j]
      grpsnorms,# normalizers of grps
      ngrpsnorms,# dito, new level
      bgids,    # generators of b many 1's (used for copro)
      opr,	# operation on complements
      xo;	# xternal orbits

  g:=arg[1];
  if Size(g)=1 then
    return [g];
  fi;
  if Length(arg)>1 and IsRecord(arg[Length(arg)]) then
    opt:=arg[Length(arg)];
  else
    opt:=rec();
  fi;

  # parse options
  normal:=IsBound(opt.normal) and opt.normal=true;
  if IsBound(opt.consider) then 
    consider:=opt.consider;
  else
    consider:=false;
  fi;

  retnorm:=IsBound(opt.retnorm) and opt.retnorm;

  isom:=fail;

  # get automorphisms and compute their normalizer, if applicable
  if IsBound(opt.actions) then
    func:=opt.actions;
    hom2:= Filtered( func,     HasIsConjugatorAutomorphism
			   and IsConjugatorAutomorphism );
    hom2:= List( hom2, ConjugatorOfConjugatorIsomorphism );

    if IsBound(opt.funcnorm) then
      # get the func. normalizer
      funcnorm:=opt.funcnorm;
      b:=g;
    else
      funcs:= GroupByGenerators( Filtered( func,
                  i -> not ( HasIsConjugatorAutomorphism( i ) and
                             IsConjugatorAutomorphism( i ) ) ),
		   IdentityMapping(g));
      IsGroupOfAutomorphismsFiniteGroup(funcs); # set filter
      if IsTrivial( funcs ) then
	b:=ClosureGroup(Parent(g),List(func,x->ConjugatorOfConjugatorIsomorphism(x)));
	func:=hom2;
      else
        if IsSolvableGroup(funcs) then
	  a:=IsomorphismPcGroup(funcs);
	else
	  a:=IsomorphismPermGroup(funcs);
	fi;
	hom:=InverseGeneralMapping(a);
	IsTotal(hom); IsSingleValued(hom); # to be sure (should be set anyway)
	b:=SemidirectProduct(Image(a),hom,g);
	hom:=Embedding(b,1);
	funcs:=List(GeneratorsOfGroup(funcs),i->Image(hom,Image(a,i)));
	isom:=Embedding(b,2);
	hom2:=List(hom2,i->Image(isom,i));
	func:=Concatenation(funcs,hom2);
	g:=Image(isom,g);
      fi;

      # get the normalizer of <func>
      funcnorm:=Normalizer(g,SubgroupNC(b,func));
      func:=List(func,i->ConjugatorAutomorphism(b,i));
    fi;

    Assert(1,IsSubgroup(g,funcnorm));

    # compute <func> characteristic series
    e:=InvariantElementaryAbelianSeries(g,func);
  else
    func:=[];
    funcnorm:=g;
    e:=ElementaryAbelianSeriesLargeSteps(g);
  fi;

  if IsBound(opt.series) then
    e:=opt.series;
  else
    f:=DerivedSeriesOfGroup(g);
    if Length(e)>Length(f) and
      ForAll([1..Length(f)-1],i->IsElementaryAbelian(f[i]/f[i+1])) then
      Info(InfoPcSubgroup,1,"  Preferring Derived Series");
      e:=f;
    fi;
  fi;

#  # check, if the series is compatible with the AgSeries and if g is a
#  # parent group. If not, enforce this
#  if not(IsParent(g) and ForAll(e,i->IsElementAgSeries(i))) then
#    Info(InfoPcSubgroup,1,"  computing better series");
#    isom:=IsomorphismAgGroup(e);
#    g:=Image(isom,g);
#    e:=List(e,i->Image(isom,i));
#    funcnorm:=Image(isom,funcnorm);
#
#    #func:=List(func,i->isom^-1*i*isom); 
#    hom:=[];
#    for i in func do
#      hom2:=GroupHomomorphismByImagesNC(g,g,g.generators,List(g.generators,
#                 j->Image(isom,Image(i,PreImagesRepresentative(isom,j)))));
#      hom2.isMapping:=true;
#      Add(hom,hom2);
#    od;
#    func:=hom;
#  else
#    isom:=false;
#  fi;

  len:=Length(e);

  if IsBound(opt.groups) then
    start:=0;
    while start+1<=Length(e) and ForAll(opt.groups,i->IsSubgroup(e[start+1],i)) do
      start:=start+1;
    od;
    Info(InfoPcSubgroup,1,"starting index ",start);
    epi:=NaturalHomomorphismByNormalSubgroup(g,e[start]);
    lastepi:=epi;
    f:=Image(epi,g);
    grps:=List(opt.groups,i->Image(epi,i));
    if not IsBound(opt.grpsnorms) then
      opt:=ShallowCopy(opt);
      opt.grpsnorms:=List(opt.groups,i->Normalizer(e[1],i));
    fi;
    grpsnorms:=List(opt.grpsnorms,i->Image(epi,i));
  else
    # search the largest elementary abelian quotient
    start:=2;
    while start<len and IsElementaryAbelian(g/e[start+1]) do
      start:=start+1;
    od;

    # compute all subgroups there
    if start<len then
      # form only factor groups if necessary
      epi:=NaturalHomomorphismByNormalSubgroup(g,e[start]);
      LockNaturalHomomorphismsPool(g,e[start]);
      f:=Image(epi,g);
    else
      f:=g;
      epi:=IdentityMapping(f);
    fi;
    lastepi:=epi;
    efunc:=List(func,i->InducedAutomorphism(epi,i));
    grps:=InvariantSubgroupsElementaryAbelianGroup(f,efunc);
    Assert(1,ForAll(grps,i->ForAll(efunc,j->Image(j,i)=i)));
    grpsnorms:=List(grps,i->f);
    Info(InfoPcSubgroup,5,List(grps,Size),List(grpsnorms,Size));

  fi;

  for i in [start+1..len] do
    Info(InfoPcSubgroup,1," step ",i,": ",Index(e[i-1],e[i]),", ",
                    Length(grps)," groups"); 
    # compute modulo e[i]
    if i<len then
      # form only factor groups if necessary
      epi:=NaturalHomomorphismByNormalSubgroup(g,e[i]);
      f:=Image(epi,g);
    else
      f:=g;
      epi:=IdentityMapping(g);
    fi;
    home:=HomePcgs(f); # we want to compute wrt. this pcgs
    n:=Image(epi,e[i-1]);

    # the induced factor automs
    efunc:=List(func,i->InducedAutomorphism(epi,i));
    # filter the non-trivial ones
    efunc:=Filtered(efunc,i->ForAny(GeneratorsOfGroup(f),j->Image(i,j)<>j));

    if Length(efunc)>0 then
      efnorm:=Image(epi,funcnorm);
    fi;

    if Length(efunc)=0 then
      ophom:=ActionSubspacesElementaryAbelianGroup(f,n);
      B:=ophom[1];
      Info(InfoPcSubgroup,2,"  ",Length(B)," normal subgroups"); 
      ophom:=ophom[2];

      ngrps:=[];
      ngrpsnorms:=[];
      oppcgs:=Pcgs(Source(ophom));
      oppcgsimg:=List(oppcgs,i->Image(ophom,i));
      ex:=[1..Length(B)];
      IsSSortedList(ex);
      ex:=ExternalSet(Source(ophom),ex,oppcgs,oppcgsimg,OnPoints);
      ex:=ExternalOrbitsStabilizers(ex);

      for j in ex do
        Add(ngrps,B[Representative(j)]);
	Add(ngrpsnorms,StabilizerOfExternalSet(j));
#	Assert(1,Normalizer(f,B[j[1]])=ngrpsnorms[Length(ngrps)]);
      od;

    else
      B:=InvariantSubgroupsElementaryAbelianGroup(n,efunc);
      ophom:=false;
      Info(InfoPcSubgroup,2,"  ",Length(B)," normal subgroups"); 

      # note the groups in B
      ngrps:=SubgroupsOrbitsAndNormalizers(f,B,false);
      ngrpsnorms:=List(ngrps,i->i.normalizer);
      ngrps:=List(ngrps,i->i.representative);
    fi;

    # Get epi to the old factor group
    # as hom:=NaturalHomomorphism(f,fa); does not work, we have to play tricks
    hom:=lastepi;
    lastepi:=epi;
    fa:=Image(hom,g);

    hom:= GroupHomomorphismByImagesNC(f,fa,GeneratorsOfGroup(f),
           List(GeneratorsOfGroup(f),i->
	     Image(hom,PreImagesRepresentative(epi,i))));
    Assert(2,KernelOfMultiplicativeGeneralMapping(hom)=n);

    # lift the known groups
    for j in [1..Length(grps)] do

      gj:=grps[j];
      if Size(gj)>1 then
	a:=PreImage(hom,gj);
	Assert(1,Size(a)=Size(gj)*Size(n));
	Add(ngrps,a);
	no:=PreImage(hom,grpsnorms[j]);

	Add(ngrpsnorms,no);

	if Length(efunc)>0 then
	  # get the double cosets
	  t:=List(DoubleCosets(f,no,efnorm),Representative);
	  Info(InfoPcSubgroup,2,"  |t|=",Length(t));
	  t:=Filtered(t,i->HasInvariantConjugateSubgroup(a,i,efunc));
	  Info(InfoPcSubgroup,2,"invar:",Length(t));
        fi;

	# we have to extend with those b in B, that are normal in a
	if ophom<>false then
	  #aop:=Image(ophom,a);
	  #SetIsSolvableGroup(aop,true);

	  if Length(GeneratorsOfGroup(a))>2 then
	    bs:=SmallGeneratingSet(a);
	  else
	    bs:=GeneratorsOfGroup(a);
	  fi;
	  bs:=List(bs,i->Image(ophom,i));

	  bsp:=Filtered([1..Length(B)],i->ForAll(bs,j->i^j=i)
	                                 and Size(B[i])<Size(n));
	  bs:=B{bsp};
	else
	  bsp:=false;
	  bs:=Filtered(B,i->IsNormal(a,i) and Size(i)<Size(n));
	fi;

        if Length(efunc)>0 and Length(t)>1 then
	  # compute also the invariant ones under the conjugates:
	  # equivalently: Take all equivalent ones and take those, whose
	  # conjugates lie in a and are normal under a
	  for k in Filtered(t,i->not i in no) do
	    bs:=Union(bs,Filtered(List(B,i->ConjugateSubgroup(i,k^(-1))),
		  i->IsSubset(a,i) and IsNormal(a,i) and Size(i)<Size(n) ));
	  od;
	fi;

	# take only those bs which are valid
	if consider<>false then
	  Info(InfoPcSubgroup,2,"  ",Length(bs)," subgroups lead to ");
	  if bsp<>false then
	    bsp:=Filtered(bsp,j->consider(no,a,n,B[j],e[i])<>false);
	    IsSSortedList(bsp);
	    bs:=bsp; # to get the 'Info' right
	  else
	    bs:=Filtered(bs,j->consider(no,a,n,j,e[i])<>false);
	  fi;
	  Info(InfoPcSubgroup,2,Length(bs)," valid ones");
	fi;

	if ophom<>false then
	  #nohom:=List(GeneratorsOfGroup(no),i->Image(ophom,i));
	  #aop:=SubgroupNC(Image(ophom),nohom);
	  #nohom:=GroupHomomorphismByImagesNC(no,aop,
	  #                                   GeneratorsOfGroup(no),nohom);

	  if Length(bsp)>0 then
	    oppcgs:=Pcgs(no);
	    oppcgsimg:=List(oppcgs,i->Image(ophom,i));
	    ex:=ExternalSet(no,bsp,oppcgs,oppcgsimg,OnPoints);
	    ex:=ExternalOrbitsStabilizers(ex);

	    bs:=[];
	    bsnorms:=[];
	    for bpos in ex do
	      Add(bs,B[Representative(bpos)]);
	      Add(bsnorms,StabilizerOfExternalSet(bpos));
#	    Assert(1,Normalizer(no,B[bpos[1]])=bsnorms[Length(bsnorms)]);
	    od;
          fi;

	else
	  # fuse under the action of no and compute the local normalizers
	  bs:=SubgroupsOrbitsAndNormalizers(no,bs,true);
	  bsnorms:=List(bs,i->i.normalizer);
	  bs:=List(bs,i->i.representative);
        fi;

Assert(1,ForAll(bs,i->ForAll(efunc,j->Image(j,i)=i)));

	# now run through the b in bs
	for bpos in [1..Length(bs)] do
	  b:=bs[bpos];
	  Assert(2,IsNormal(a,b));
	  # test, whether we'll have to consider this case

# this test has basically be done before the orbit calculation already
#	  if consider<>false and consider(a,n,b,e[i])=false then
#	    Info(InfoPcSubgroup,2,"  Ignoring case");
#	    s:=[];

	  # test, whether b is invariant
	  if Length(efunc)>0 then
	    # extend to dcs of bnormalizer
	    s:=RightTransversal(no,bsnorms[bpos]);
	    nag:=Length(s);
	    s:=Concatenation(List(s,i->List(t,j->i*j)));
	    z:=Length(s);
	    #NOCH: Fusion
	    # test, which ones are usable at all
	    s:=Filtered(s,i->HasInvariantConjugateSubgroup(b,i,efunc));
	    Info(InfoPcSubgroup,2,"  |s|=",nag,"-(m)>",z,"-(i)>",Length(s));
	  else
	    s:=[()];
	  fi;

          if Length(s)>0 then
	    nag:=InducedPcgs(home,n);
	    nag:=nag mod InducedPcgs(nag,b);
#	    if Index(Parent(a),a.normalizer)>1 then
#	      Info(InfoPcSubgroup,2,"  normalizer index ",
#	                      Index(Parent(a),a.normalizer));
#	    fi;

	    z:=rec(group:=a,
	        generators:=InducedPcgs(home,a) mod NumeratorOfModuloPcgs(nag),
	        modulePcgs:=nag);
	    OCOneCocycles(z,true);
	    if IsBound(z.complement) and 
	      # normal complements exist, iff the coboundaries are trivial
	      (normal=false or Dimension(z.oneCoboundaries)=0)
	      then
	      # now fetch the complements

	      z.factorGens:=z.generators;
	      coboundbas:=Basis(z.oneCoboundaries);
	      com:=BaseSteinitzVectors(BasisVectors(Basis(z.oneCocycles)),
	                               BasisVectors(coboundbas));
	      field:=LeftActingDomain(z.oneCocycles);
	      if Size(field)^Length(com.factorspace)>100000 then
		Info(InfoWarning,1, "Many (",
		  Size(field)^Length(com.factorspace),") complements!");
	      fi;
	      com:=Enumerator(VectorSpace(field,com.factorspace,
	                                       Zero(z.oneCocycles)));
	      Info(InfoPcSubgroup,3,"  ",Length(com),
	           " local complement classes");

	      # compute fusion
	      kconh:=List([1..Length(com)],i->[i]);
	      if i<len or retnorm then
		# we need to compute normalizers
		comnorms:=[];
	      else
		comnorms:=fail;
	      fi;

	      if Length(com)>1 and Size(a)<Size(bsnorms[bpos]) then

	        opr:=function(cyc,elm)
		      local l,i;
			l:=z.cocycleToList(cyc);
			for i in [1..Length(l)] do
			  l[i]:=(z.complementGens[i]*l[i])^elm;
			od;
			l:=CorrespondingGeneratorsByModuloPcgs(z.origgens,l);
			for i in [1..Length(l)] do
			  l[i]:=LeftQuotient(z.complementGens[i],l[i]);
			od;
			l:=z.listToCocycle(l);
			return SiftedVector(coboundbas,l);
		      end;

		xo:=ExternalOrbitsStabilizers(
		     ExternalSet(bsnorms[bpos],com,opr));

                for k in xo do
		  l:=List(k,i->Position(com,i));
		  if comnorms<>fail then
		    comnorms[l[1]]:=StabilizerOfExternalSet(k);
		    isTrueComnorm:=false;
		  fi;
		  l:=Set(l);
		  for kp in l do
		    kconh[kp]:=l;
		  od;
		od;

	      elif comnorms<>fail then
		if Size(a)=Size(bsnorms[bpos]) then
		  comnorms:=List(com,i->z.cocycleToComplement(i));
		  isTrueComnorm:=true;
		  comnorms:=List(comnorms,
			      i->ClosureSubgroup(CentralizerModulo(n,b,i),i));
	        else
		  isTrueComnorm:=false;
		  comnorms:=List(com,i->bsnorms[bpos]);
		fi;
	      fi;


              if Length(efunc)>0 then
		ncom:=[];

	        #search for invariant ones

		# force exponents corresponding to vector space

                # get matrices for the inner automorphisms
#		conj:=[];
#		for k in GeneratorsOfGroup(a) do
#		  mat:=[];
#		  for l in nag do
#		    Add(mat,One(field)*ExponentsOfPcElement(nag,l^k));
#		  od;
#		  Add(conj,mat);
#		od;
                conj:=LinearOperationLayer(a,GeneratorsOfGroup(a),nag);

                idmat:=conj[1]^0;
		mat:= GroupByGenerators( conj, idmat );
		chom:= GroupHomomorphismByImagesNC(a,mat,
		        GeneratorsOfGroup(a),conj);

		smats:=[];
		shoms:=[];

                fghom:=Concatenation(z.factorGens,GeneratorsOfGroup(n));
		bgids:=List(GeneratorsOfGroup(n),i->One(b));

		# now run through the complements
		for kp in [1..Length(com)] do

		  if kconh[kp]=fail then
		    Info(InfoPcSubgroup,3,"already conjugate");
		  else

		    k:=z.cocycleToComplement(com[kp]);
		    # the projection on the complement
		    comproj:= GroupHomomorphismByImagesNC(a,a,fghom,
			       Concatenation(GeneratorsOfGroup(k),bgids));
		    k:=ClosureSubgroup(b,k);
		    
		    # now run through the conjugating elements
		    conjnr:=1;
		    found:=false;
		    while conjnr<=Length(s) and found=false do
		      if not IsBound(smats[conjnr]) then
			# compute the matrix action for the induced, jugated
			# morphisms
			m:=s[conjnr];
			smats[conjnr]:=[];
			shoms[conjnr]:=[];
			for l in efunc do
			  # the induced, jugated morphism
			  shom:= GroupHomomorphismByImagesNC(a,a,
				  GeneratorsOfGroup(a),
				  List(GeneratorsOfGroup(a),
				   i->Image(l,i^m)^Inverse(m)));

			  mat:=List(nag,
				i->One(field)*ExponentsOfPcElement(nag,
				 Image(shom,i)));
			  Add(smats[conjnr],mat);
			  Add(shoms[conjnr],shom);
			od;
		      fi;

		      mats:=smats[conjnr];
		      # now test whether the complement k can be conjugated to
		      # be invariant under the morphisms to mats
		      glsyl:=List(nag,i->[]);
		      glsyr:=[];
		      for l in [1..Length(efunc)] do
			kgens:=GeneratorsOfGroup(k);
			for kgnr in [1..Length(kgens)] do

			  kgn:=Image(shoms[conjnr][l],kgens[kgnr]);
			  kgim:=Image(comproj,kgn);
			  Assert(2,kgim^-1*kgn in n);
			  # nt part
			  kgn:=kgim^-1*kgn;

			  # translate into matrix terms
			  kgim:=Image(chom,kgim);
			  kgn:=One(field)*ExponentsOfPcElement(nag,kgn);

			  # the matrix action
			  mat:=idmat+(mats[l]-idmat)*kgim-mats[l];
			  
			  # store action and vector
			  for m in [1..Length(glsyl)] do
			    glsyl[m]:=Concatenation(glsyl[m],mat[m]);
			  od;
			  glsyr:=Concatenation(glsyr,kgn);

			od;
		      od;

		      # a possible conjugating element is a solution of the
		      # large LGS
		      l:= SolutionMat(glsyl,glsyr);
		      if l <> fail then
			m:=Product([1..Length(l)],
				   i->nag[i]^IntFFE(l[i]));
			# note that we found one!
			found:=[s[conjnr],m];
		      fi;

		      conjnr:=conjnr+1;
		    od;

		    # there is an invariant complement?
		    if found<>false then
		      found:=found[2]*found[1];
		      l:=ConjugateSubgroup(ClosureSubgroup(b,k),found);
		      Assert(1,ForAll(efunc,i->Image(i,l)=l));
		      l:=rec(representative:=l);
		      if comnorms<>fail then
			if IsBound(comnorms[kp]) then
			  l.normalizer:=ConjugateSubgroup(comnorms[kp],found);
			else
			  l.normalizer:=ConjugateSubgroup(
			                  Normalizer(bsnorms[bpos],
				  ClosureSubgroup(b,k)), found);
			fi;
		      fi;
		      Add(ncom,l);

		      # tag all conjugates
		      for l in kconh[kp] do
		        kconh[l]:=fail;
		      od;

		    fi;

                  fi; # if not already a conjugate

		od;

		# if invariance test needed
	      else
		# get representatives of the fused complement classes
		l:=Filtered([1..Length(com)],i->kconh[i][1]=i);

		ncom:=[];
		for kp in l do
		  m:=rec(representative:=
			  ClosureSubgroup(b,z.cocycleToComplement(com[kp])));
		  if comnorms<>fail then
		    m.normalizer:=comnorms[kp];
		  fi;
		  Add(ncom,m);
		od;
	      fi; 
	      com:=ncom;

	      # take the preimages
	      for k in com do

		Assert(1,ForAll(efunc,i->Image(i,k.representative)
		                         =k.representative));
		Add(ngrps,k.representative);
		if IsBound(k.normalizer) then
		  if isTrueComnorm then
		    Add(ngrpsnorms,k.normalizer);
		  else
		    Add(ngrpsnorms,Normalizer(k.normalizer,k.representative));
		  fi;
		fi;
	      od;
	    fi;
	  fi;
	od;

      fi;
    od;

    grps:=ngrps;
    grpsnorms:=ngrpsnorms;
    Info(InfoPcSubgroup,5,List(grps,Size),List(grpsnorms,Size));
  od;

  if isom<>fail then
    grps:=List(grps,j->PreImage(isom,j));
    if retnorm then
      grpsnorms:=List(grpsnorms,j->PreImage(isom,j));
    fi;
  fi;
  
  if retnorm then
    return [grps,grpsnorms];
  else
    return grps;
  fi;
end);


#############################################################################
##
#M  LatticeSubgroups(<G>)  . . . . . . . . . .  lattice of subgroups
##
InstallMethod(LatticeSubgroups,"elementary abelian extension",true,
  [IsGroup],
  # want to be better than cyclic extension.
  1,
function(G)
local s,i,c,classes, lattice,map,GI;

  if not IsSolvableGroup(G) then #or not CanEasilyComputePcgs(G) then
    TryNextMethod();
  fi;
  if not IsPcGroup(G) or IsPermGroup(G) then
    map:=IsomorphismPcGroup(G);
    GI:=Image(map,G);
  else
    map:=fail;
    GI:=G;
  fi;
  s:=SubgroupsSolvableGroup(GI,rec(retnorm:=true));
  classes:=[];
  for i in [1..Length(s[1])] do
    if map=fail then
      c:=ConjugacyClassSubgroups(G,s[1][i]);
      SetStabilizerOfExternalSet(c,s[2][i]);
    else
      c:=ConjugacyClassSubgroups(G,PreImage(map,s[1][i]));
      SetStabilizerOfExternalSet(c,PreImage(map,s[2][i]));
    fi;
    Add(classes,c);
  od;
  Sort(classes,function(a,b) 
                 return Size(Representative(a))<Size(Representative(b));
	       end);

  # create the lattice
  lattice:=Objectify(NewType(FamilyObj(classes),IsLatticeSubgroupsRep),
		     rec());
  lattice!.conjugacyClassesSubgroups:=classes;
  lattice!.group     :=G;

  # return the lattice
  return lattice;

end);

# #############################################################################
# ##
# #M  NormalSubgroups(<G>)  . . . . . . . . . .  list of normal subgroups
# ##
# InstallMethod(NormalSubgroups,"elementary abelian extension",true,
#   [CanEasilyComputePcgs],0,
# function(G)
# local n;
#   n:=SubgroupsSolvableGroup(G,rec(
#        actions:=List(GeneratorsOfGroup(G),i->InnerAutomorphism(G,i)),
#        normal:=true));
# 
#   # sort the normal subgroups according to their size
#   Sort(n,function(a,b) return Size(a) < Size(b); end);
# 
#   return n;
# end);

#############################################################################
##
#F  SizeConsiderFunction(<size>)  returns auxiliary function for
##  'SubgroupsSolvableGroup' that allows one to discard all subgroups whose
##  size is not divisible by <size>
##
InstallGlobalFunction(SizeConsiderFunction,function(size)
  return function(c,a,n,b,m)
	   return IsInt(Size(a)/Size(n)*Size(b)*Size(m)/size);
         end;
end);

#############################################################################
##
#F  ExactSizeConsiderFunction(<size>)  returns auxiliary function for
##  'SubgroupsSolvableGroup' that allows one to discard all subgroups whose
##  size is not <size>
##
InstallGlobalFunction(ExactSizeConsiderFunction,function(size)
  return function(c,a,n,b,m)
	   return IsInt(Size(a)/Size(n)*Size(b)*Size(m)/size)
	      and not (Size(a)/Size(n)*Size(b))>size;
         end;
end);

#############################################################################
##
#E  grppclat.gi . . . . . . . . . . . . . . . . . . . . . . . . . . ends here
##