#############################################################################
##
#W  ghomperm.gi                 GAP library       Akos Seress, Heiko Thei"sen
##
#H  @(#)$Id: ghomperm.gi,v 4.86.2.1 2002/09/18 21:24:47 gap Exp $
##
#Y  Copyright (C)  1997,  Lehrstuhl D fuer Mathematik,  RWTH Aachen, Germany
#Y  (C) 1998 School Math and Comp. Sci., University of St.  Andrews, Scotland
#Y  Copyright (C) 2002 The GAP Group
##
Revision.ghomperm_gi :=
    "@(#)$Id: ghomperm.gi,v 4.86.2.1 2002/09/18 21:24:47 gap Exp $";

#############################################################################
##
#F  AddGeneratorsGenimagesExtendSchreierTree( <S>, <newlabs>, <newlims> ) . .
##
InstallGlobalFunction( AddGeneratorsGenimagesExtendSchreierTree,
    function( S, newlabs, newlims )
    local   old,        # genlabels before extension
            len,        # initial length of the orbit of <S>
            img,        # image during orbit algorithm
	    pos,        # label positions
	    ind,        # index of new labels
            i,  j;      # loop variables
    
    # check duplicates

    # Put in the new labels and labelimages.
    old := ShallowCopy( S.genlabels );
    UniteSet( S.genlabels, Length( S.labels ) + [ 1 .. Length( newlabs ) ] );
    Append( S.labels,      newlabs );  Append( S.generators,  newlabs );
    Append( S.labelimages, newlims );  Append( S.genimages,   newlims );

    # Extend the orbit and the transversal with the new labels.
    len := Length( S.orbit );
    i := 1;
    while i <= Length( S.orbit )  do
      for j  in S.genlabels  do
	# Use new labels for old points, all labels for new points.
	if i > len  or  not j in old  then
	  img := S.orbit[ i ] / S.labels[ j ];
	  if not IsBound( S.translabels[ img ] )  then
	    S.translabels[ img ] := j;
	    Add( S.orbit, img );
	    if not IsBound(S.transversal[img]) then
	      S.transversal[ img ] := S.labels[ j ];
	      S.transimages[ img ] := S.labelimages[ j ];
	    fi;
	  fi;
	fi;
      od;
      i := i + 1;
    od;
end );

#############################################################################
##
#F  ImageSiftedBaseImage( <S>, <bimg>, <h> )   sift base image and find image
##
InstallGlobalFunction( ImageSiftedBaseImage, function( S, bimg, img, opr )
    local   base;
    
    base := BaseStabChain( S );
    while bimg <> base  do
        while bimg[ 1 ] <> base[ 1 ]  do
            img  := opr     ( img,  S.transimages[ bimg[ 1 ] ] );
            bimg := OnTuples( bimg, S.transversal[ bimg[ 1 ] ] );
        od;
        S := S.stabilizer;
        base := base{ [ 2 .. Length( base ) ] };
        bimg := bimg{ [ 2 .. Length( bimg ) ] };
    od;
    return img;
end );

#############################################################################
##
#R  IsCoKernelGensIteratorRep . . . . . . . iterator over cokernel generators
##
DeclareRepresentation( "IsCoKernelGensIteratorRep",
    IsComponentObjectRep,
    [ "level", "pointNo", "genlabelNo", "levelNo", "base", "bimg", "img" ] );

#############################################################################
##
#F  CoKernelGensIterator( <hom> ) . . . . . . . . . . . . .  make this animal
##
InstallGlobalFunction( CoKernelGensIterator, function( hom )
local   S,  iter,mgi;

  S := StabChainMutable( hom );
  iter := Objectify
	  ( NewType( IteratorsFamily,
			  IsIterator
		      and IsMutable
		      and IsCoKernelGensIteratorRep ),
	    rec( level := S,
		pointNo := 1,
	    genlabelNo := 1,
		levelNo := 1,
		  base := BaseStabChain( S ) ) );
  iter!.img  := S.idimage;
  iter!.bimg := iter!.base;
  mgi:=MappingGeneratorsImages(hom);
  iter!.trivlist:=mgi[2]{Filtered([1..Length(mgi[1])],i->IsOne(mgi[1][i]))};
  return iter;
end );

InstallMethod( IsDoneIterator,
    "for `IsCoKernelGensIteratorRep'",
    true,
    [ IsIterator and IsCoKernelGensIteratorRep ], 0,
    iter -> IsEmpty( iter!.level.genlabels ) and IsEmpty(iter!.trivlist));

InstallMethod( NextIterator,
    "for `IsCoKernelGensIteratorRep'",
    true,
    [ IsIterator and IsMutable and IsCoKernelGensIteratorRep ], 0,
    function( iter )
    local   gen,  stb,  bimg,  rep,  pnt,  img,  j,  k;
    
    # do we have to take care of a trivlist?
    if not IsEmpty(iter!.trivlist) then
      j:=Length(iter!.trivlist);
      gen:=iter!.trivlist[j];
      Unbind(iter!.trivlist[j]);
      return gen;
    fi;

    # Make the current cokernel generator.
    stb := iter!.level;
    k := stb.genlabels[ iter!.genlabelNo ];
    gen := ImageSiftedBaseImage( stb,
                   OnTuples( iter!.bimg, stb.labels[ k ] ),
                   iter!.img * stb.labelimages[ k ], OnRight );
    
    # Move on the iterator: Next generator.
    iter!.genlabelNo := iter!.genlabelNo + 1;
    if iter!.genlabelNo > Length( stb.genlabels )  then
        iter!.genlabelNo := 1;
        
        # Next basic orbit point.
        iter!.pointNo := iter!.pointNo + 1;

        if iter!.pointNo > Length( stb.orbit )  then
            iter!.pointNo := 1;
            
            # Next level of the stabilizer chain.
            iter!.levelNo := iter!.levelNo + 1;
            iter!.level := stb.stabilizer;
            stb := iter!.level;
            
            # Return prematurely if the iterator is done.
            if IsEmpty( stb.genlabels )  then
                return gen;
            fi;
            
        fi;
        pnt := stb.orbit[ iter!.pointNo ];
        rep := [  ];
        img := stb.idimage;
        while pnt <> stb.orbit[ 1 ]  do
            Add( rep, stb.transversal[ pnt ] );
            img := LeftQuotient( stb.transimages[ pnt ], img );
            pnt := pnt ^ stb.transversal[ pnt ];
        od;
        bimg := iter!.base{ [ iter!.levelNo .. Length( iter!.base ) ] };
        for k  in Reversed( [ 1 .. Length( rep ) ] )  do
            for j  in [ 1 .. Length( bimg ) ]  do
                bimg[ j ] := bimg[ j ] / rep[ k ];
            od;
        od;
        iter!.img  := img;
        iter!.bimg := bimg;
        
    fi;
    
    return gen;
end );

InstallMethod( ShallowCopy,
    "for `IsCoKernelGensIteratorRep'",
    true,
    [ IsIterator and IsCoKernelGensIteratorRep ], 0,
    function( iter )
    iter:= Objectify( Subtype( TypeObj( iter ), IsMutable ),
                rec( level      := StructuralCopy( iter!.level ),
                     pointNo    := iter!.pointNo,
                     genlabelNo := iter!.genlabelNo,
                     levelNo    := iter!.levelNo,
                     base       := ShallowCopy( iter!.base ),
                     img        := iter!.img ) );
    iter!.bimg:= iter!.base;
#T what is this good for??
    return iter;
    end );


#############################################################################
##
#F  CoKernelGensPermHom( <hom> )  . . . . . . . . generators for the cokernel
##
InstallGlobalFunction( CoKernelGensPermHom, function( hom )
    local   C,  sch;

    C := [  ];
    for sch  in CoKernelGensIterator( hom )  do
      if not (sch=One(sch) or sch in C) then
	AddSet( C, sch );
      fi;
    od;
    return C;
end );


#############################################################################
##
#F  RelatorsPermGroupHom( <hom, gens> ) . .  relators for a permutation group
##
##  `RelatorsPermGroupHom' is an internal function which is called by the
##  operation `IsomorphismFpGroupByGeneratorsNC' in case of a permutation
##  group. It implements John Cannon's multi-stage relations-finding
##  algorithm as described in
##
##  Joachim Neubueser: An elementary introduction to coset table methods
##  in computational group theory, pp. 1-45 in "Groups-St.Andrews 1981,
##  Proceedings of a conference, St.Andrews 1981", edited by Colin M.
##  Campbell and Edmund F. Robertson, London Math. Soc. Lecture Note Series
##  71, Cambridge University Press, 1982.
##
##  Warning: The arguments are not checked for being consistent.
##
InstallGlobalFunction( RelatorsPermGroupHom, function ( hom, gensG )

    local actcos, actgen, app, c, col, cosets, cont, defs1, defs2, F, fgensH,
          G, g, g1, gen0, geners, gens, gensF, gensF2, gensS, H, i, idword,
          index, inv0, iso, j, map, ndefs, next, ngens, ngens2, ni, orbit,
          order, P, perm, perms, range, regular, rel, rel2, rels, relsG,
          relsGen, relsH, relsP, S, sizeS, stabG, stabS, table, tail, tail1,
          tail2, tietze, tzword, undefined, wordsH;

    # get the involved groups
    G := PreImage( hom );
    F := Image( hom );
    gensF := GeneratorsOfGroup( F );
    ngens := Length( gensG );

    # special case: G is the identity group
    if Size( G ) = 1 then
      return gensF;
    fi;

    # apply the two-stage relations finding algorithm to recursively
    # construct a presentation for each stabilizer in a stabilizer chain of
    # G (if G is not regular), and finally for G itself
    regular := IsRegular( G );
    if regular then
      orbit := Orbits( G,MovedPoints(G) )[1];
      sizeS := 1;
    else

      # get a stabilizer chain for hom
      stabG := StabChainMutable( hom );
      orbit := stabG.orbit;

      # get the first stabilizer S
      stabS := stabG.stabilizer;
      S := Subgroup( G, stabS.labels{ stabS.genlabels } );
      sizeS := Size( S );
    fi;

    # initialize some local variables
    index := Length( orbit );
    ngens2 := ngens * 2;
    table := [];
    range := [ 1 .. index ];
    idword := One( gensF[1] );
    gensF2 := [];
    undefined := 0;
    ndefs := 0;
    defs1 := ListWithIdenticalEntries( ngens * index, 0 );
    defs2 := ListWithIdenticalEntries( ngens * index, 0 );

    # initialize a presentation for G
    P := PresentationFpGroup( F / [ ], 0 );
    tietze := P!.tietze;
    TzOptions( P ).protected := ngens;

    if sizeS > 1 then
      # construct recursively a presentation for S and lift the relators
      # of S to relators of G
      gensS := GeneratorsOfGroup( S );
      iso := IsomorphismFpGroupByGeneratorsNC( S, gensS, "x" :
        infolevel := 2 );

      H := Image( iso );
      fgensH := FreeGeneratorsOfFpGroup( H );
      relsH := RelatorsOfFpGroup( H );
      wordsH := stabS.genimages;

      for rel in relsH do
        AddRelator( P, MappedWord( rel, fgensH, wordsH ) );
      od;
    fi;

    # make the permutations act on the points 1 to index
    map := MappingPermListList( orbit, range );
    perms := List( gensG, gen -> PermList( OnTuples( orbit, gen * map ) ) );

    # get a coset table from the permutations and introduce appropriate
    # order relators for the involutory generators
    for i in [ 1 .. ngens ] do
      Add( gensF2, gensF[i] );
      Add( gensF2, gensF[i]^-1 );
      perm := perms[i];
      col := -OnTuples( range, perm );
      undefined := undefined + index;
      Add( table, col );
      order := Order( gensG[i] );
      if order <= 2 then
        rel := gensF[i]^order;
        if sizeS > 1 then
          # lift the tail of the relator from S to G
          tail := MappedWord( rel, gensF, gensG );
          if tail <> () then
            tail1 := UnderlyingElement( tail^iso );
            tail2 := UnderlyingElement( (tail^-1)^iso );
            rel2 := rel * MappedWord( tail2, fgensH, wordsH );
            rel := rel * MappedWord( tail1, fgensH, wordsH )^-1;
            if Length( rel ) > Length( rel2 ) then
              rel := rel2;
            fi;
          fi;
        fi;
        AddRelator( P, rel );
      else
        col := -OnTuples( range, perm^-1 );
        undefined := undefined + index;
      fi;
      Add( table, col );
    od;
    tietze[TZ_MODIFIED] := true;
    while tietze[TZ_MODIFIED] and tietze[TZ_TOTAL] > 0 do
      TzSearch( P );
    od;

    # reconvert the Tietze relators to abstract words
    relsP := tietze[TZ_RELATORS];
    relsG := [ ];
    for tzword in relsP do
      if tzword <> [ ] then
        Add( relsG, AbstractWordTietzeWord( tzword, gensF ) );
      fi;
    od;

    # make the rows for the relators and distribute over relsGen
    relsGen := RelsSortedByStartGen( gensF, relsG, table, true );

    # make the structure that is passed to `MakeConsequencesPres'
    app := ListWithIdenticalEntries( 8, 0 );
    app[1] := table;
    app[2] := defs1;
    app[3] := defs2;

    # define an appropriate ordering of the cosets,
    # enter the coset definitions in the table,
    # and construct the Schreier vector,
    cosets := ListWithIdenticalEntries( index, 0 );
    actcos := ListWithIdenticalEntries( index, 0 );
    actgen := ListWithIdenticalEntries( index, 0 );
    cosets[1] := 1;
    actcos[1] := 1;
    j := 1;
    i := 0;
    while i < index do
      i := i + 1;
      c := cosets[i];
      g := 0;
      while g < ngens2 do
        g := g + 1;
        next := -table[g][c];
        if next > 0 and actcos[next] = 0 then
          g1 := g + 2*(g mod 2) - 1;
          table[g][c] := next;
          undefined := undefined - 1;
          if table[g1][next] < 0 then
            table[g1][next] := c;
            undefined := undefined - 1;
          fi;
          actcos[next] := c;
          actgen[next] := g;
          ndefs := ndefs + 1;
          defs1[ndefs] := c;
          defs2[ndefs] := g;
          j := j + 1;
          cosets[j] := next;
          if j = index then
            g := ngens2;
            i := index;
          fi;
        fi;
      od;
    od;

    # run through the coset table and find the next undefined entry
    ni := 0;
    while ni < index and undefined > 0 do
      ni := ni + 1;
      i := cosets[ni];
      j := 0;
      while j < ngens2 and undefined > 0 do
        j := j + 1;
        if table[j][i] <= 0 then

          # define the entry appropriately
          g := j + 2*(j mod 2) - 1;
          c := -table[j][i];
          table[j][i] := c;
          undefined := undefined - 1;
          if table[g][c] < 0 then
            table[g][c] := i;
            undefined := undefined - 1;
          fi;
          ndefs := ndefs + 1;
          defs1[ndefs] := i;
          defs2[ndefs] := j;

          # construct the associated relator
          rel := idword;
          while c <> 1 do
            g := actgen[c];
            rel := rel * gensF2[g]^-1;
            c := actcos[c];
          od;
          rel := rel^-1 * gensF2[j]^-1;
          c := i;
          while c <> 1 do
            g := actgen[c];
            rel := rel * gensF2[g]^-1;
            c := actcos[c];
          od;
          if sizeS > 1 then
            # lift the tail of the relator from S to G
            tail := MappedWord( rel, gensF, gensG );
            if tail <> () then
              tail1 := UnderlyingElement( tail^iso );
              tail2 := UnderlyingElement( (tail^-1)^iso );
              rel2 := rel * MappedWord( tail2, fgensH, wordsH );
              rel := rel * MappedWord( tail1, fgensH, wordsH )^-1;
              if Length( rel ) > Length( rel2 ) then
                rel := rel2;
              fi;
            fi;
          fi;

          if Length( rel ) > 0 then

            # add the new relator to the Tietze presentation and reduce it
            AddRelator( P, rel );
            if tietze[TZ_MODIFIED] then
              while tietze[TZ_MODIFIED] and tietze[TZ_TOTAL] > 0 do
                TzSearch( P );
              od;

              # reconvert the Tietze relators to abstract words
              rels := relsG;
              relsG := [ ];
              relsP := tietze[TZ_RELATORS];
              for tzword in relsP do
                Add( relsG, AbstractWordTietzeWord( tzword, gensF ) );
              od;

              # reconstruct the rows for the relators if necessary
              if relsG <> rels then
                relsGen := RelsSortedByStartGen( gensF, relsG, table, true );
              fi;
            fi;
          fi;

          # continue the enumeration and find all consequences
          if undefined > 0 then
            app[4] := undefined;
            app[5] := ndefs;
            app[6] := relsGen;
            undefined := MakeConsequencesPres( app );
          fi;
        fi;
      od;
    od;

    # reduce the resulting presentation
    TzGoGo( P );

    # reconvert the reduced relators and return them
    relsP := tietze[TZ_RELATORS];
    relsG := [ ];
    for tzword in relsP do
      if tzword <> [ ] then
        Add( relsG, AbstractWordTietzeWord( tzword, gensF ) );
      fi;
    od;
    return relsG;

end );


#############################################################################
##
#M  StabChainMutable( <hom> ) . . . . . . . . . . . . . . for perm group homs
##
# new 
InstallOtherMethod( StabChainMutable, "perm mapping by images",  true,
        [ IsPermGroupGeneralMappingByImages ], 0,
    function( hom )
    local   S,
            rnd,        # list of random elements of '<hom>.source'
            rne,        # list of the images of the elements in <rnd>
            rni,        # index of the next random element to consider
            elm,        # one element in '<hom>.source'
            img,        # its image
            size,       # size of the stabilizer chain constructed so far
            stb,        # stabilizer in '<hom>.source'
            bpt,        # base point
            two,        # power of two
            trivgens,   # trivial generators and their images, must be
            trivimgs,   #   entered into every level of the chain
	    mapi,
            i, T,  # loop variables
	    orb,
	    orbf,	# indicates with which generator the image was obtained
	    dict,
	    cnt,
	    short,
	    FillTransversalShort,
	    BuildOrb,
	    maxstor,
	    l;	# position

    # build short words by an orbit algorithm on genimg
    BuildOrb:=function(genimg)
    local orb,dict,orbf,T,elm,img,i,n;
      dict:=NewDictionary(genimg[1][1],false);
      AddDictionary(dict,One(genimg[1][1]));
      orb:=[Immutable([One(genimg[1][1]),One(genimg[2][1])])];
      orbf:=[0];
      i:=1;
      n:=Length(genimg[1]);
      while Length(orb)<maxstor and i<=Length(orb) do
	for T in [1..n] do
	  if orbf[i]<>-T then
	    elm:=orb[i][1]*genimg[1][T];
	    if not KnowsDictionary(dict,elm) then
	      # new rep found
	      img:=orb[i][2]*genimg[2][T];
	      AddDictionary(dict,elm);
	      Add(orb,Immutable([elm,img]));
	      Add(orbf,T);
	    fi;
	  fi;
	  if orbf[i]<>T then
	    elm:=orb[i][1]/genimg[1][T];
	    if not KnowsDictionary(dict,elm) then
	      # new rep found
	      img:=orb[i][2]/genimg[2][T];
	      AddDictionary(dict,elm);
	      Add(orb,Immutable([elm,img]));
	      Add(orbf,-T);
	    fi;
	  fi;
	od;
	i:=i+1;
      od;
      return orb;
    end;

    mapi:=MappingGeneratorsImages(hom);

    # do products build up? (Must we prefer short words?)
    short:=IsFreeGroup(Range(hom)) or IsFpGroup(Range(hom));

    if short then
      # compute how many perms we permit to store?
      maxstor:=LargestMovedPoint(Source(hom))+1;
      if maxstor>65535 then
        maxstor:=maxstor*2; # perms need twice as much memory
      fi;
      maxstor:=40*1024^2/maxstor; # allocate at most 40MB to the perms
      # but don't be crazy 
      maxstor:=Minimum(maxstor,
                 Size(Source(hom))/10,
		 50*LogInt(Size(Source(hom)),2),
		 2000); 
#Print(maxstor,"<<\n");

      # fill transversal with elements that are short words
      # This is similar to Minkwitz' approach and produces much shorter
      # words when decoding.
      FillTransversalShort:=function(stb)
      local l,i,bpt,m;
	bpt:=stb.orbit[1];
	stb.norbit:=ShallowCopy(stb.orbit);
	# fill transversal with short words
	for l in stb.orb do
	  i:=bpt/l[1];
	  if not i in stb.norbit then
	    Add(stb.norbit,i);
	    stb.transversal[i]:=l[1];
	    stb.transimages[i]:=l[2];
	  fi;
	  i:=bpt^l[1];
	  if not i in stb.norbit then
	    Add(stb.norbit,i);
	    stb.transversal[i]:=Inverse(l[1]);
	    stb.transimages[i]:=Inverse(l[2]);
	  fi;
	od;
	stb.stabilizer.orb:=Filtered(stb.orb,i->bpt^i[1]=bpt);
	Unbind(stb.orb);
	stb:=stb.stabilizer;
	# is there too little left? If yes, extend!
	if Length(stb.orb)*20<maxstor then
	  stb.orb:=BuildOrb([List(stb.orb,i->i[1]),
	                     List(stb.orb,i->i[2])]);
	fi;
#Print(bpt,":",Length(stb.orb),"\n");
      end;
    else
      FillTransversalShort:=Ignore;
    fi;

    # initialize the random generators
    two := 16;
    rnd := ShallowCopy( mapi[1] );
    for i  in [Length(rnd)..two]  do
        Add( rnd, One( Source( hom ) ) );
    od;
    rne := ShallowCopy( mapi[2] );
    for i  in [Length(rne)..two]  do
        Add( rne, One( Range( hom ) ) );
    od;
    rni := 1;

    # initialize the top level
    bpt := SmallestMovedPoint( Source( hom ) );
    if bpt = infinity  then
        bpt := 1;
    fi;
    S := EmptyStabChain( [  ], One( Source( hom ) ),
                         [  ], One( Range( hom ) ) );
    InsertTrivialStabilizer( S, bpt );
    if short then
      S.orb:=BuildOrb(mapi);
    fi;
    # the short words usable on this level
    FillTransversalShort(S);
    
    # Extend  orbit and transversal. Store  images of the  identity for other
    # levels.
    AddGeneratorsGenimagesExtendSchreierTree( S, mapi[1], mapi[2] );
    trivgens := [  ];  trivimgs := [  ];
    for i  in [ 1 .. Length( mapi[1] ) ]  do
        if mapi[1][ i ] = One( Source( hom ) )  then
            Add( trivgens, mapi[1][ i ] );
            Add( trivimgs, mapi[2][ i ] );
        fi;
    od;

    # get the size of the stabilizer chain
    size := Length( S.orbit );

    # create new elements until we have reached the size
    while size <> Size( PreImagesRange( hom ) )  do

	# try random elements
	elm := rnd[rni];
	img := rne[rni];
	i := Random( [ 1 .. Length( mapi[1] ) ] );
	rnd[rni] := rnd[rni] * mapi[1][i];
	rne[rni] := rne[rni] * mapi[2][i];
	rni := rni mod two + 1;

        # divide the element through the stabilizer chain
        stb := S;
        bpt := BasePoint( stb );
        while     bpt <> false
              and elm <> stb.identity
              and Length( stb.genlabels ) <> 0  do
            i := bpt ^ elm;
            if IsBound( stb.translabels[ i ] )  then
                while i <> bpt  do
                    img := img * stb.transimages[ i ];
                    elm := elm * stb.transversal[ i ];
                    i := bpt ^ elm;
                od;
                stb := stb.stabilizer;
                bpt := BasePoint( stb );
            else
                bpt := false;
            fi;
        od;

        # if the element was not in the stabilizer chain
        if elm <> stb.identity  then

	  # if this stabilizer is trivial add an new level
	  if not IsBound( stb.stabilizer )  then
	    l:=SmallestMovedPoint(elm);
	    InsertTrivialStabilizer( stb, l );
	    AddGeneratorsGenimagesExtendSchreierTree( stb,
		    trivgens, trivimgs );
	    # the short words usable on this level
	    FillTransversalShort(stb);
	  fi;

#	  if short then
#	    l:=LookupDictionary(dict,elm);
#	    if l<>fail then
#	      img:=l;
#	    fi;
#	  fi;

	  # extend the Schreier trees above level `stb'
	  T := S;
	  repeat
	    T := T.stabilizer;
	    size := size / Length( T.orbit );
	    AddGeneratorsGenimagesExtendSchreierTree( T, [elm], [img] );
	    size := size * Length( T.orbit );
	  until T.orbit[ 1 ] = stb.orbit[ 1 ];

        fi;

    od;
    
    return S;
end );

#############################################################################
##
#M  CoKernelOfMultiplicativeGeneralMapping( <hom> ) . . . for perm group homs
##
InstallMethod( CoKernelOfMultiplicativeGeneralMapping,
    true, [ IsPermGroupGeneralMappingByImages ], 0,
function( hom )
local is;
  # This seems not to be useful: Otherwise we call `MakeMapping' which
  # enumerates the codomain anyhow, so a stupid `NormalClosure' cannot do
  # more harm. AH
  #if not CanComputeSizeAnySubgroup(Range(hom)) then
  #  TryNextMethod();
  #fi;

  #as ImagesSource might call ImagesSet which would require the co kernel
  #again, one has to be a careful a bit.
  if HasImagesSource(hom) then
    is:=ImagesSource(hom);
  else
    is:=SubgroupNC(Range(hom),List(GeneratorsOfGroup(Source(hom)),
                                   i->ImagesRepresentative(hom,i)));
  fi;

  return NormalClosure( is, SubgroupNC
                   ( Range( hom ), CoKernelGensPermHom( hom ) ) );
end );

#############################################################################
##
#M  IsSingleValued( <hom> ) . . . . . . . . . . . . . . . for perm group homs
##
InstallMethod( IsSingleValued, true,
        [ IsPermGroupGeneralMappingByImages ], 0,
    function( hom )
    local   sch;
    
    for sch in CoKernelGensIterator( hom )  do
        if sch <> One( sch )  then
            return false;
        fi;
    od;
    return true;
end );

#############################################################################
##
#M  ImagesRepresentative( <hom>, <elm> )  . . . . . . . . for perm group homs
##
InstallMethod( ImagesRepresentative, "perm group hom",FamSourceEqFamElm,
        [ IsPermGroupGeneralMappingByImages,
          IsMultiplicativeElementWithInverse ], 0,
function( hom, elm )
local   S,img,img2;
  if not ( HasIsTotal( hom ) and IsTotal( hom ) )
      and not elm in PreImagesRange( hom )  then
      return fail;
  else
    S := StabChainMutable( hom );
    img := ImageSiftedBaseImage( S, OnTuples( BaseStabChain( S ), elm ),
		    S.idimage, OnRight );
		    
    if IsPerm(img) then 
      TRIM_PERM(img,LargestMovedPoint(Range(hom)));
    elif IsAssocWord(img) or IsElementOfFpGroup(img) then
      # try the inverse as well -- it might be better
      img2:= ImageSiftedBaseImage( S, List(BaseStabChain(S),i->i/elm),
		    S.idimage, OnRight );
      if Length(UnderlyingElement(img2))<Length(UnderlyingElement(img)) then
        return img2;
      fi;
    fi;
    return img^-1;
  fi;
end );

#############################################################################
##
#M  CompositionMapping2( <hom1>, <hom2> ) . . . . . . . . for perm group homs
##
InstallMethod( CompositionMapping2, "group hom. with perm group hom.",
  FamSource1EqFamRange2, [ IsGroupHomomorphism,
          IsPermGroupGeneralMappingByImages and IsGroupHomomorphism ], 0,
    function( hom1, hom2 )
    local   prd,  stb,  levs,  S;

    stb := StructuralCopy( StabChainMutable( hom2 ) );
    levs := [  ];
    S := stb;
    while IsBound( S.stabilizer )  do
        S.idimage := One( Range( hom1 ) );
        if not ForAny( levs, lev -> IsIdenticalObj( lev, S.labelimages ) )  then
            Add( levs, S );
            S.labelimages := List( S.labelimages, g ->
                                   ImagesRepresentative( hom1, g ) );
        fi;
        S.generators  := S.labels     { S.genlabels };
        S.genimages   := S.labelimages{ S.genlabels };
        S.transimages := [  ];
        S.transimages{ S.orbit } := S.labelimages{ S.translabels{ S.orbit } };
        S := S.stabilizer;
    od;
    prd := GroupHomomorphismByImagesNC( Source( hom2 ), Range( hom1 ),
                   stb.generators, stb.genimages );
    SetStabChainMutable( prd, stb );
    return prd;
end );

#############################################################################
##
#M  PreImagesRepresentative( <hom>, <elm> ) . . . . . .  for perm group range
##
InstallMethod( PreImagesRepresentative, FamRangeEqFamElm,
        [ IsToPermGroupGeneralMappingByImages,
          IsMultiplicativeElementWithInverse ], 0,
    function( hom, elm )
    return ImagesRepresentative( InverseGeneralMapping( hom ), elm );
end );

#############################################################################
##
#F  StabChainPermGroupToPermGroupGeneralMappingByImages( <hom> )  . . . local
##
InstallGlobalFunction( StabChainPermGroupToPermGroupGeneralMappingByImages,
    function( hom )
    local   options,    # options record for stabilizer construction
            n,  
            k,
            i,
            longgens,
            longgroup,
            conperm,
            conperminv,
	    mapi,
            op;
    
#    if IsTrivial( PreImagesRange( hom ) )
#       then n := 0;
#       else n := LargestMovedPoint( PreImagesRange( hom ) );  fi;
#    if IsTrivial( ImagesSource( hom ) )
#       then k := 0;
#       else k := LargestMovedPoint( ImagesSource( hom ) );  fi;
    
    if IsTrivial( Source( hom ) )
       then n := 0;
       else n := LargestMovedPoint( Source( hom ) );  fi;
    if IsTrivial( Range( hom ) )
       then k := 0;
       else k := LargestMovedPoint( Range( hom ) );  fi;
    
    # collect info for options
    options := rec();
    
    # random or deterministic
    if   IsBound( StabChainOptions( Parent( Source( hom ) ) ).random )  then
        options.randomSource :=
          StabChainOptions( Parent( Source( hom ) ) ).random;
    elif IsBound( StabChainOptions( Source( hom ) ).random )  then
        options.randomSource := StabChainOptions( Source( hom ) ).random;
    elif IsBound( StabChainOptions( PreImagesRange( hom ) ).random )  then
        options.randomSource := StabChainOptions( PreImagesRange( hom ) ).random;
    else
        options.randomSource := DefaultStabChainOptions.random;
    fi;
    if   IsBound( StabChainOptions( Parent( Range( hom ) ) ).random )  then
        options.randomRange :=
          StabChainOptions( Parent( Range( hom ) ) ).random;
    elif IsBound( StabChainOptions( Range( hom ) ).random )  then
        options.randomRange := StabChainOptions( Range( hom ) ).random;
    elif HasImagesSource(hom) 
      and IsBound( StabChainOptions( ImagesSource( hom ) ).random )  then
        options.randomRange := StabChainOptions( ImagesSource( hom ) ).random;
    else
        options.randomRange := DefaultStabChainOptions.random;
    fi;
    options.random := Minimum(options.randomSource,options.randomRange);

    # if IsMapping, try to extract info from source
    if Tester( IsMapping )( hom )  and  IsMapping( hom )  then
        if   HasSize( Source( hom ) )  then
            options.size := Size( Source( hom ) );
        elif HasSize( PreImagesRange( hom ) )  then
            options.size := Size( PreImagesRange( hom ) );
        fi;
        if not IsBound( options.size )
           and HasSize( Parent( Source( hom ) ) )  then
            options.limit := Size( Parent( Source( hom ) ) );
        fi;
        if   IsBound( StabChainOptions( Source( hom ) ).knownBase )  then
            options.knownBase := StabChainOptions( Source( hom ) ).knownBase;
        elif IsBound( StabChainOptions( PreImagesRange( hom ) ).knownBase )
          then
            options.knownBase := StabChainOptions( PreImagesRange( hom ) ).
                                 knownBase;
        elif HasBaseOfGroup( Source( hom ) )  then
            options.knownBase := BaseOfGroup( Source( hom ) );
        elif HasBaseOfGroup( PreImagesRange( hom ) )  then
            options.knownBase := BaseOfGroup( PreImagesRange( hom ) );
        elif IsBound( StabChainOptions( Parent( Source( hom ) ) ).knownBase )
          then
            options.knownBase :=
              StabChainOptions( Parent( Source( hom ) ) ).knownBase;
        elif HasBaseOfGroup( Parent( Source( hom ) ) )  then
            options.knownBase := BaseOfGroup( Parent( Source( hom ) ) );
        fi;

    # if not IsMapping, settle for less
    else
        if   HasSize( Source( hom ) )  then
            options.limitSource := Size( Source( hom ) );
        elif HasSize( PreImagesRange( hom ) )  then
            options.limitSource := Size( PreImagesRange( hom ) );
        elif HasSize( Parent( Source( hom ) ) )  then
            options.limitSource := Size( Parent( Source( hom ) ) );
        fi;
        if   IsBound( StabChainOptions( Source( hom ) ).knownBase )  then
            options.knownBaseSource :=
              StabChainOptions( Source( hom ) ).knownBase;
        elif IsBound( StabChainOptions( PreImagesRange( hom ) ).knownBase )
          then
            options.knownBaseSource :=
              StabChainOptions( PreImagesRange( hom ) ).knownBase;
        elif IsBound( StabChainOptions( Parent( Source( hom ) ) ).knownBase )
          then
            options.knownBaseSource :=
                StabChainOptions( Parent( Source( hom ) ) ).knownBase;
        fi;

        # if we have info about source, try to collect info about range
        if IsBound( options.limitSource ) then 
            if   HasSize( Range( hom ) )  then
                options.limitRange := Size( Range( hom ) );
            elif HasImagesSource(hom) and HasSize( ImagesSource( hom ) )  then
                options.limitRange := Size( ImagesSource( hom ) );
            elif HasSize( Parent( Range( hom ) ) )  then
                options.limitRange := Size( Parent( Range( hom ) ) );
            fi;
            if IsBound( options.limitRange ) then 
                options.limit := options.limitSource * options.limitRange;
            fi;
        fi;
        if IsBound( options.knownBaseRange ) then 
            if   IsBound( StabChainOptions( Range( hom ) ).knownBase )  then
                options.knownBaseRange :=
                  StabChainOptions( Range( hom ) ).knownBase;
            elif IsBound( StabChainOptions( PreImagesRange( hom ) ).
                    knownBase )  then
                options.knownBaseRange :=
                  StabChainOptions( PreImagesRange( hom ) ).knownBase;
            elif IsBound( StabChainOptions( Parent( Range( hom ) ) )
                    .knownBase )
              then
                options.knownBaseRange :=
                    StabChainOptions( Parent( Range( hom ) ) ).knownBase;
            fi;
            if IsBound( options.knownBaseRange ) then 
                options.knownBase := Union( options.knownBaseSource,
                                            options.knownBaseRange + n );
            fi;
        fi;

    fi; # if IsMapping

    options.base := [1..n];

    # create concatenation of perms in hom.generators, hom.genimages
    longgens := [];
    conperm := MappingPermListList([1..k],[n+1..n+k]);
    conperminv := conperm^(-1);
    mapi:=MappingGeneratorsImages(hom);
    for i in [1..Length(mapi[1])] do
        longgens[i] := mapi[1][i] * (mapi[2][i] ^ conperm); 
    od;
    longgroup :=  GroupByGenerators( longgens, () );
    for op  in [ PreImagesRange, ImagesSource ]  do
        if      Tester(op)(hom) and HasIsSolvableGroup( op( hom ) )
           and not IsSolvableGroup( op( hom ) )  then
            SetIsSolvableGroup( longgroup, false );
            break;
        fi;
    od;

    MakeStabChainLong( hom, StabChainOp( longgroup, options ),
           [ 1 .. n ], (), conperminv, hom,
           CoKernelOfMultiplicativeGeneralMapping );
    
    if    not HasInverseGeneralMapping( hom )
       or not HasStabChainMutable( InverseGeneralMapping( hom ) )
       or not HasKernelOfMultiplicativeGeneralMapping( hom )  then
        MakeStabChainLong( InverseGeneralMapping( hom ),
                StabChainOp( longgroup, [ n + 1 .. n + k ] ),
                [ n + 1 .. n + k ], conperminv, (), hom,
                KernelOfMultiplicativeGeneralMapping );
    fi;

    return StabChainMutable( hom );
end );

#############################################################################
##
#F  MakeStabChainLong( ... )  . . . . . . . . . . . . . . . . . . . . . local
##
InstallGlobalFunction( MakeStabChainLong,
    function( hom, stb, ran, c1, c2, cohom, cokername )
    local   newlevs,  S,  i,  len,  rest,  trans;
    
    # Construct the stabilizer chain for <hom>.
    S := CopyStabChain( stb );
    SetStabChainMutable( hom, S );
    newlevs := [  ];
    repeat
        len := Length( S.labels );
        if len = 0  or  IsPerm( S.labels[ len ] )  then
            Add( S.labels, rec( labels := [  ], labelimages := [  ] ) );
            len := len + 1;
            for i  in [ 1 .. len - 1 ]  do
                rest := RestrictedPerm( S.labels[ i ], ran );
                Add( S.labels[ len ].labels, rest ^ c1 );
                Add( S.labels[ len ].labelimages,
                     LeftQuotient( rest, S.labels[ i ] ) ^ c2 );
            od;
            Add( newlevs, S.labels );
        fi;
        S.labels{ [ 1 .. len - 1 ] } := S.labels[ len ].labels;
        S.labelimages := S.labels[ len ].labelimages;
        S.generators  := S.labels{ S.genlabels };
        S.genimages   := S.labelimages{ S.genlabels };
        S.idimage     := ();
        if BasePoint( S ) in ran  then
            trans := S.translabels{ S.orbit };
            S.orbit := S.orbit - ran[ 1 ] + 1;
            S.translabels := [  ];
            S.translabels{ S.orbit } := trans;
            S.transversal := [  ];
            S.transversal{ S.orbit } := S.labels{ trans };
            S.transimages := [  ];
            S.transimages{ S.orbit } := S.labelimages{ trans };
            S := S.stabilizer;
            stb := stb.stabilizer;
        else
            RemoveStabChain( S );
	    S.genimages:=[];
            S.labelimages := [  ];
            S := false;
        fi;
    until S = false;
    for S  in newlevs  do
        Unbind( S[ Length( S ) ] );
    od;
    
    # Construct the cokernel.
    if not IsEmpty( stb.genlabels )  then
        if not Tester( cokername )( cohom )  then
            S := EmptyStabChain( [  ], () );
            ConjugateStabChain( stb, S, c2, c2 );
            Setter( cokername )
              ( cohom, GroupStabChain( Range( hom ), S, true ) );
        fi;
    else 
        Setter( cokername )( cohom, TrivialSubgroup( Range( hom ) ) );
    fi;
    
end );

#############################################################################
##
#M  StabChainMutable( <hom> ) . . . . . . . . . . for perm to perm group homs
##
InstallMethod( StabChainMutable, "perm to perm mapping by images",true,
        [ IsPermGroupGeneralMappingByImages and
          IsToPermGroupGeneralMappingByImages ], 0,
        StabChainPermGroupToPermGroupGeneralMappingByImages );

#############################################################################
##
#M  KernelOfMultiplicativeGeneralMapping(<hom>) . for perm to perm group homs
##
InstallMethod( KernelOfMultiplicativeGeneralMapping, true,
        [ IsPermGroupGeneralMappingByImages and
          IsToPermGroupGeneralMappingByImages ], 0,
    function( hom )
    StabChainPermGroupToPermGroupGeneralMappingByImages( hom );
    return KernelOfMultiplicativeGeneralMapping( hom );
end );

#############################################################################
##
#M  CoKernelOfMultiplicativeGeneralMapping(<hom>) for perm to perm group homs
##
InstallMethod( CoKernelOfMultiplicativeGeneralMapping, true,
        [ IsPermGroupGeneralMappingByImages and
          IsToPermGroupGeneralMappingByImages ], 0,
    function( hom )
    StabChainPermGroupToPermGroupGeneralMappingByImages( hom );
    return CoKernelOfMultiplicativeGeneralMapping( hom );
end );

#############################################################################
##
#M  ImagesRepresentative( <hom>, <elm> )  . . . . . . . . . . . for const hom
##
InstallMethod( ImagesRepresentative,"Constituent homomorphism",
  FamSourceEqFamElm,
        [ IsConstituentHomomorphism, IsMultiplicativeElementWithInverse ], 0,
    function( hom, elm )
    local   D;

    D := Enumerator( UnderlyingExternalSet( hom ) );
    if Length( D ) = 0  then
        return ();
    else 
        return PermList( OnTuples( [ 1 .. Length( D ) ],
                       elm ^ hom!.conperm ) );
    fi;
end );

#############################################################################
##
#M  ImagesSet( <hom>, <H> ) . . . . . . . . . . . . . . . . . . for const hom
##
InstallMethod( ImagesSet,"constituent homomorphism", CollFamSourceEqFamElms,
	# this method should *not* be applied if the group to be mapped has
	# no stabilizer chain (for example because it is very big).
        [ IsConstituentHomomorphism, IsPermGroup and HasStabChainMutable], 0,
function( hom, H )
local   D,  I,G;
  
  D := Enumerator( UnderlyingExternalSet( hom ) );
  I := EmptyStabChain( [  ], One(Range(hom)) );
  RemoveStabChain( ConjugateStabChain( StabChainOp( H, D ), I,
	  hom, hom!.conperm,
	  S -> BasePoint( S ) <> false
	    and BasePoint( S ) in D ) );
  #GroupStabChain might give too many generators
  if Length(I.generators)<10 then
    return GroupStabChain( Range( hom ), I, true );
  else
    G:=SubgroupNC(Range(hom),
      List(GeneratorsOfGroup(H),i->Permutation(i,D)));
    SetStabChainMutable(G,I);
    return G;
  fi;
end );

#############################################################################
##
#M  Range( <hom>, <H> ) . . . . . . . . . . . . . . . . . . for const hom
##
RanImgSrcSurjTraho:=function(hom)
local   D,H,I,G;
  H:=Source(hom);
  # only worth if the source has a stab chain to utilize
  if not HasStabChainMutable(H) then
    TryNextMethod();
  fi;
  D := Enumerator( UnderlyingExternalSet( hom ) );
  I := EmptyStabChain( [  ], () );
  RemoveStabChain( ConjugateStabChain( StabChainOp( H, D ), I,
	  hom, hom!.conperm,
	  S -> BasePoint( S ) <> false
	    and BasePoint( S ) in D ) );
  #GroupStabChain might give too many generators
  if Length(I.generators)<10 then
    return GroupStabChain( I );
  else
    G:=Group(List(GeneratorsOfGroup(H),i->Permutation(i,D)),());
    SetStabChainMutable(G,I);
    return G;
  fi;
end;

InstallMethod( Range,"surjective constituent homomorphism",true,
  [ IsConstituentHomomorphism and IsActionHomomorphism and IsSurjective ],0,
  RanImgSrcSurjTraho);

InstallMethod( ImagesSource,"constituent homomorphism",true,
  [ IsConstituentHomomorphism and IsActionHomomorphism ],0,
  RanImgSrcSurjTraho);

#############################################################################
##
#M  PreImagesRepresentative( <hom>, <elm> )
##
InstallMethod( PreImagesRepresentative,"constituent homomorphism",
  FamRangeEqFamElm,[IsConstituentHomomorphism,IsPerm], 0,
function( hom, elm )
local D,DP;
  if not HasStabChainMutable(Source(hom)) then
    # do not enforce a stabchain if not neccessary -- it could be big
    TryNextMethod();
  fi;
  D:=Enumerator(UnderlyingExternalSet(hom));
  DP:=Permuted(D,elm^-1);
  return RepresentativeAction(Source(hom),D,DP,OnTuples);
end);

#############################################################################
##
#M  PreImagesSet( <hom>, <I> )  . . . . . . . . . . . . . . . . for const hom
##
InstallMethod( PreImagesSet, "constituent homomorphism",CollFamRangeEqFamElms,
        [ IsConstituentHomomorphism, IsPermGroup ], 0,
    function( hom, I )
    local   H,          # preimage of <I>, result
            K,          # kernel of <hom>
            S,  T,  name;

    # compute the kernel of <hom>
    K := KernelOfMultiplicativeGeneralMapping( hom );

    # create the preimage group
    H := EmptyStabChain( [  ], One( Source( hom ) ) );
    S := ConjugateStabChain( StabChainMutable( I ), H, x ->
                 PreImagesRepresentative( hom, x ), hom!.conperm ^ -1 );
    T := H;
    while IsBound( T.stabilizer )  do
        AddGeneratorsExtendSchreierTree( T, GeneratorsOfGroup( K ) );
        T := T.stabilizer;
    od;
        
    # append the kernel to the stabilizer chain of <H>
    K := StabChainMutable( K );
    for name  in RecNames( K )  do
        S.( name ) := K.( name );
    od;
    
    return GroupStabChain( Source( hom ), H, true );
end );

#############################################################################
##
#M  KernelOfMultiplicativeGeneralMapping( <hom> ) . . . . . . . for const hom
##
InstallMethod( KernelOfMultiplicativeGeneralMapping,
    "for constituent homomorphism",
    true, [ IsConstituentHomomorphism ], 0,
function( hom )
  return Stabilizer( Source( hom ), Enumerator( UnderlyingExternalSet( hom ) ),
		  OnTuples );
end );

#############################################################################
##
#M  StabChainMutable( <hom> ) . . . . . . . . . . . . . . . .  for blocks hom
##
InstallMethod( StabChainMutable,
    "for blocks homomorphism",
    true, [ IsBlocksHomomorphism ], 0,
    function( hom )
    local   img;
    
    img := ImageKernelBlocksHomomorphism( hom, Source( hom ),false );
    if not HasImagesSource( hom )  then
        SetImagesSource( hom, img );
    fi;
    return StabChainMutable( hom );
end );

#############################################################################
##
#M  ImagesRepresentative( <hom>, <elm> )  . . . . . . . . . .  for blocks hom
##
InstallMethod( ImagesRepresentative, "blocks homomorphism", FamSourceEqFamElm,
        [ IsBlocksHomomorphism, IsMultiplicativeElementWithInverse ], 0,
    function( hom, elm )
    local   img,  D,  i;
    
    D := Enumerator( UnderlyingExternalSet( hom ) );
    
    # make the image permutation as a list
    img := [  ];
    for i  in [ 1 .. Length( D ) ]  do
        img[ i ] := hom!.reps[ D[ i ][ 1 ] ^ elm ];
    od;

    # return the image as a permutation
    return PermList( img );
end );

#############################################################################
##
#F  ImageKernelBlocksHomomorphism( <hom>, <H> ) . . . . . .  image and kernel
##
InstallGlobalFunction( ImageKernelBlocksHomomorphism, function( hom, H,par )
    local   D,          # the block system
            I,          # image of <H>, result
            S,          # block stabilizer in <H>
            T,          # corresponding stabilizer in <I>
            full,       # flag: true if <H> is (identical to) the source
            B,          # current block
            i,  j;      # loop variables
    
    D := Enumerator( UnderlyingExternalSet( hom ) );
    S := CopyStabChain( StabChainMutable( H ) );
    full := IsIdenticalObj( H, Source( hom ) );
    if full  then
        SetStabChainMutable( hom, S );
    fi;
    if par<>false then
      I := EmptyStabChain( [  ], One(par) );
    else
      I := EmptyStabChain( [  ], () );
    fi;
    T := I;

    # loop over the blocks
    for i  in [ 1 .. Length( D ) ]  do
        B := D[ i ];

        # if <S> does not already stabilize this block
        if     IsBound( B[1] )
           and ForAny( S.generators, gen -> hom!.reps[ B[ 1 ] ^ gen ] <> i )
           then
            ChangeStabChain( S, [ B[ 1 ] ] );

            # Make the next level of <T> and go down to `<T>.stabilizer'.
            T := ConjugateStabChain( S, T, hom, hom!.reps,
                         S -> BasePoint( S ) = B[ 1 ] );

            # Make <S> the stabilizer of the block <B>.
            InsertTrivialStabilizer( S.stabilizer, B[ 1 ] );
            j := 1;
            while                                j < Length( B )
                  and Length( S.stabilizer.orbit ) < Length( B )  do
                j := j + 1;
                if IsBound( S.translabels[ B[ j ] ] )  then
                    AddGeneratorsExtendSchreierTree( S.stabilizer,
                            [ InverseRepresentative( S, B[ j ] ) ] );
                fi;
            od;
            S := S.stabilizer;
                    
        fi;
    od;

    # if <H> is the full group this also gives us the kernel
    if full  and  not HasKernelOfMultiplicativeGeneralMapping( hom )  then
        SetKernelOfMultiplicativeGeneralMapping( hom,
            GroupStabChain( Source( hom ), S, true ) );
    fi;
    
    if par<>false then
      return GroupStabChain( par, I, true );
    else
      return GroupStabChain(I);
    fi;

end );

#############################################################################
##
#M  ImagesSet( <hom>, <H> ) . . . . . . . . . . . . . . . . .  for blocks hom
##
InstallMethod( ImagesSet, "for blocks homomorphism and perm. group",
    CollFamSourceEqFamElms, [ IsBlocksHomomorphism, IsPermGroup ], 0,
function(hom,U)
  return ImageKernelBlocksHomomorphism(hom,U,Range(hom));
end);

RanImgSrcSurjBloho:=function(hom)
local gens,imgs,ran,dom;
  if ValueOption("onlyimage")=fail and HasStabChainMutable(Source(hom)) 
    and NrMovedPoints(Source(hom))<20000 then
    # transfer stabchain information if not too expensive
    ran:=ImageKernelBlocksHomomorphism(hom,Source(hom),false);
  else
    gens:=GeneratorsOfGroup( Source( hom ) );
    imgs:=List(gens,gen->ImagesRepresentative( hom, gen ) );
    ran:=GroupByGenerators(imgs,());
    SetMappingGeneratorsImages(hom,[gens,imgs]);
  fi;
  return ran;
end;

InstallMethod( Range, "surjective blocks homomorphism",true,
  [ IsBlocksHomomorphism and IsSurjective ], 0,
  RanImgSrcSurjBloho);

InstallMethod( ImagesSource, "blocks homomorphism",true,
  [ IsBlocksHomomorphism ], 0,
  RanImgSrcSurjBloho);

#############################################################################
##
#M  PreImagesRepresentative( <hom>, <elm> ) . . . . . . . . .  for blocks hom
##
InstallMethod( PreImagesRepresentative, "blocks homomorphism",
        FamRangeEqFamElm,
        [ IsBlocksHomomorphism, IsMultiplicativeElementWithInverse ], 0,
    function( hom, elm )
    local   D,          # the block system
            pre,        # preimage of <elm>, result
            S,          # stabilizer in chain of <hom>
            B,          # the image block <B>
            b,          # number of image block <B>
            pos;        # position of point hit by preimage
    
    D := Enumerator( UnderlyingExternalSet( hom ) );
    S := StabChainMutable( hom );
    pre := One( Source( hom ) );

    # loop over the blocks and their iterated set stabilizers
    while Length( S.genlabels ) <> 0  do

        # Find the image block <B> of the current block.

	# test if the point is in no block (transitive action)
	# if not we can simply skip this step in the stabilizer chain.
	if IsBound(hom!.reps[S.orbit[1]]) then
	  b := hom!.reps[ S.orbit[ 1 ] ] ^ elm;
	  if b > Length( D )  then
	      return fail;
	  fi;
	  B := D[ b ];
	  
	  # Find a point in <B> that can be hit by the preimage.
	  pos := PositionProperty( B, pnt ->
			 IsBound( S.translabels[ pnt/pre ] ) );
	  if pos = fail  then
	      return fail;
	  else
	      pre := LeftQuotient( InverseRepresentative( S, B[ pos ] / pre ),
			     pre );
	  fi;

	fi;

        S := S.stabilizer;
    od;

    # return the preimage
    return pre;
end) ;

#############################################################################
##
#M  PreImagesSet( <hom>, <I> )  . . . . . . . . . . . . . . .  for blocks hom
##
InstallMethod( PreImagesSet, CollFamRangeEqFamElms,
        [ IsBlocksHomomorphism, IsPermGroup ], 0,
    function( hom, I )
    local   H;          # preimage of <I> under <hom>, result
    
    H := PreImageSetStabBlocksHomomorphism( hom, StabChainMutable( I ) );
    return GroupStabChain( Source( hom ), H, true );
end );

#############################################################################
##
#F  PreImageSetStabBlocksHomomorphism( <hom>, <I> ) . . .  recursive function
##
InstallGlobalFunction( PreImageSetStabBlocksHomomorphism, function( hom, I )
    local   H,          # preimage of <I> under <hom>, result
            pnt,        # rep. of the block that is the basepoint <I>
            gen,        # one generator of <I>
            pre;        # a representative of its preimages

    # if <I> is trivial then preimage is the kernel of <hom>
    if IsEmpty( I.genlabels )  then
        H := CopyStabChain( StabChainMutable(
                 KernelOfMultiplicativeGeneralMapping( hom ) ) );

    # else begin with the preimage $H_{block[i]}$ of the stabilizer  $I_{i}$,
    # adding preimages of the generators of  $I$  to those of  $H_{block[i]}$
    # gives us generators for $H$. Because $H_{block[i][1]} \<= H_{block[i]}$
    # the stabilizer chain below $H_{block[i][1]}$ is already complete, so we
    # only have to care about the top level with the basepoint $block[i][1]$.
    else
        pnt := Enumerator( UnderlyingExternalSet( hom ) )[ I.orbit[ 1 ] ][1];
        H := PreImageSetStabBlocksHomomorphism( hom, I.stabilizer );
        ChangeStabChain( H, [ pnt ], false );
        for gen  in I.generators  do
            pre := PreImagesRepresentative( hom, gen );
            if not IsBound( H.translabels[ pnt ^ pre ] )  then
                AddGeneratorsExtendSchreierTree( H, [ pre ] );
            fi;
        od;
    fi;

    # return the preimage
    return H;
end );

#############################################################################
##
#M  KernelOfMultiplicativeGeneralMapping( <hom> ) . . . . . .  for blocks hom
##
InstallMethod( KernelOfMultiplicativeGeneralMapping,"blocks homomorphism",
    true,
    [ IsBlocksHomomorphism ], 0,
    function( hom )
    local   img;
    
    img := ImageKernelBlocksHomomorphism( hom, Source( hom ),false);
    if not HasImagesSource( hom )  then
        SetImagesSource( hom, img );
    fi;
    return KernelOfMultiplicativeGeneralMapping( hom );
end );

DeclareRepresentation("IsBlocksOfActionHomomorphism",
  IsActionHomomorphismByBase,[]);

#############################################################################
##
#M  CompositionMapping2( <hom1>, <hom2> ) blocks of action
##
InstallMethod( CompositionMapping2,
    "for action homomorphism with blocks homomorphism",
    FamSource1EqFamRange2,
    [ IsGroupHomomorphism and IsBlocksHomomorphism, 
      IsGroupHomomorphism and IsActionHomomorphism ], 0,
function(map2,map1)
local e1,e2,d1,d2,i,ac,act,hom,xset;
  e1:=UnderlyingExternalSet(map1);
  d1:=HomeEnumerator(e1);
  if not IsPlistRep(d1) then
    TryNextMethod();
  fi;
  #sort:=CanEasilySortElements(d1[1]);
  ac:=FunctionAction(e1);
  act:=function(set,g)
         set:=List(set,i->ac(i,g));
	 Sort(set);
         return set;
       end;
  e2:=UnderlyingExternalSet(map2);
  d2:=HomeEnumerator(e2);
  d2:=List(d2,i->d1{i});
  for i in d2 do
    Sort(i);
    IsSSortedList(i);
  od;
  MakeImmutable(d2);
  IsSSortedList(d2);
  xset:=ExternalSet(Source(map1),d2,act);
  xset!.basePermImage:=BaseStabChain(StabChainMutable(ImagesSource(map2)));
  SetBaseOfGroup(xset,d2{xset!.basePermImage});

  if HasImagesSource(map1) and HasIsSurjective(map2) 
     and ImagesSource(map1)=Source(map2) then
    hom:=ActionHomomorphismConstructor(xset,true,
           IsBlocksOfActionHomomorphism);
    SetRange(hom,Range(map2));
    SetImagesSource(hom,Range(map2));
  else
    hom:=ActionHomomorphismConstructor(xset,false,
           IsBlocksOfActionHomomorphism);
  fi;
  hom!.innerAct:=ac;
  if HasMappingGeneratorsImages(map1)
    and MappingGeneratorsImages(map1)[2]=MappingGeneratorsImages(map2)[1] then
    SetMappingGeneratorsImages(hom,[MappingGeneratorsImages(map1)[1],
                                    MappingGeneratorsImages(map2)[2]]);
  fi;
  return hom;
end);

# seems to be not worth doing
# #############################################################################
# ##
# #M  ImagesRepresentative( <hom>, <elm> )
# ##
# InstallMethod( ImagesRepresentative,
#   "action blocks, using `RepresentativeAction'",
#   FamSourceEqFamElm, [ IsBlocksOfActionHomomorphism and HasImagesSource,
#           IsMultiplicativeElementWithInverse ], 0,
# function( hom, elm )
# local   xset,  D,  imgs, i, a;
# 
#   TryNextMethod();
#   xset := UnderlyingExternalSet( hom );
#   D := HomeEnumerator( xset );
#   imgs:=[];
#   for i in BaseOfGroup(xset) do
#     a:=hom!.innerAct(i[1],elm);
#     Add(imgs,PositionProperty(D,j->a in j));
#   od;
#   Error();
# 
#   return RepresentativeActionOp( ImagesSource( hom ),
# 		  xset!.basePermImage, imgs, OnTuples );
# end );



#############################################################################
##
#F  IsomorphismPermGroup( <G> )
##
InstallMethod( IsomorphismPermGroup,
    "perm groups",
    true,
    [ IsPermGroup ], 0,
    IdentityMapping );


#############################################################################
##
#M  IsConjugatorIsomorphism( <hom> )
##
InstallOtherMethod( IsConjugatorIsomorphism,
    "perm group homomorphism",
    true,
    [ IsGroupGeneralMapping ],
  # There is no filter to test whether a homomorphism goes from a perm group
  # to a perm group. So we have to test explicitly and make this method
  # higher ranking than the default one in `ghom.gi'.
  1,
    function( hom )
    local s, genss, rep,dom,insn,stb,E,bpt,fix,pnt,idom,sliced,
      o,oimgs,i,pi,sto,stbs,stbi, r, sym;

    s:= Source( hom );
    if not IsPermGroup( s ) then
      TryNextMethod();
    elif not ( IsGroupHomomorphism( hom ) and IsBijective( hom ) ) then
      return false;
    fi;
    genss:= GeneratorsOfGroup( s );

    if IsEndoGeneralMapping( hom ) then

  # test in transitive case whether we can realize in S_n
  # we do not yet compute the permutation here because we will still have to
  # test first whether it is in fact an inner automorphism:
  # ConjugatorAutomorphisms are guaranteed to conjugate with an inner
  # element if possible!
  insn:=false;
  dom:=MovedPoints(s);
  if IsTransitive(s,dom) then
    bpt := dom[ 1 ];
    stb:=Stabilizer(s,bpt);
    E:=Image(hom,stb);
    if Number(dom,i->ForAll(GeneratorsOfGroup(E),j->i^j=i))=
       Number(dom,i->ForAll(GeneratorsOfGroup(stb),j->i^j=i)) then
#T why not with NrMovedPoints?
#T why not compare orbit lengths of point stabilizer and its image?
      insn:=true;
    else
      # we cannot realize in S_n
      return false;
    fi;
  else
    # compute the orbits and their image orbits
    o:=OrbitsDomain(s,dom);
    oimgs:=[];
    stbs:=[];
    stbi:=[];
    i:=1;
    while i<=Length(o) do
      stb:=Stabilizer(s,o[i][1]);
      sto:=Collected(List(OrbitsDomain(stb,o[i]),Length)); # stb orbit lengths
      E:=Image(hom,stb);
      Add(stbs,stb);
      Add(stbi,E);
      pi:=Filtered(o,j->Length(j)=Length(o[i])); # possible images by length
      # possible images by stabilizer orbit lengths
      pi:=Filtered(pi,j->Collected(List(OrbitsDomain(E,j),Length))=sto);
      if Length(pi)=0 then
        return false; # image cannot be stabilizer
      elif Length(pi)=1 then
        Add(oimgs,pi[1]);
      else
        # orbit image not unique. We would have to backtrack. For the time
	# being, give up
#T why not inspect other orbits, and hope for a cheap `false' answer?
        i:=Length(o)+10;
      fi;
      i:=i+1;
    od;
    if Length(oimgs)=Length(o) then
      insn:=2; # conjugation in S_n established on multiple orbits
    fi;
  fi;

  # try first to find an element in the group itself
  rep:=RepresentativeAction(s, genss,
         List( genss, i -> ImagesRepresentative( hom, i ) ), OnTuples );

  if rep<>fail then
    # we found the automorphism is in fact inner
    Assert( 1, ForAll( genss, i -> ImagesRepresentative( hom, i ) = i^rep ) );
    SetIsInnerAutomorphism(hom,true);
  else
    if insn=true then
      hom:=AsGroupGeneralMappingByImages(hom);
      fix := First( dom, p -> ForAll( GeneratorsOfGroup( E ),
		    gen -> p ^ gen = p ) );
      
      # The automorphism <aut> maps <d>_bpt to <e>_fix, so permutes the points.
      # Find an element in <G> with the same action.
      idom := [  ];
      for pnt  in dom  do
	  sliced := [  ];
	  while pnt <> bpt  do
	      Add( sliced, StabChainMutable( hom ).transimages[ pnt ] );
	      pnt := pnt ^ StabChainMutable( hom ).transversal[ pnt ];
	  od;
	  Add( idom, PreImageWord( fix, sliced ) );
      od;
      
      rep:=MappingPermListList( dom, idom );
    elif insn=2 then
      dom:=[];
      idom:=[];
      for i in [1..Length(o)] do
	# compute the images for orbit o[i]
	stb:=stbs[i]; # pnt stabilizer and its image
	E:=stbi[i];
	# base point and image
	bpt:=o[i][1];
	fix:=First(oimgs[i],p->ForAll(GeneratorsOfGroup(E),
		    gen -> p ^ gen = p ) );

	# we could try to use stabilizer chains, but the homomorphism does
	# not necessarily have one which acts in every orbit. So we use the
	# time-homoured transversal
	sliced:=RightTransversal(s,stb);
	for pnt in sliced do
	  Add(dom,bpt^pnt);
	  Add(idom,fix^ImageElm(hom,pnt));
	od;
        
      od;
      rep:=MappingPermListList( dom, idom );
    else
      # we got 
      rep:=RepresentativeAction(OrbitStabilizingParentGroup(s),
            genss,
	    List( genss, i -> ImagesRepresentative( hom, i ) ), OnTuples );
      if rep<>fail then
        Assert(1,ForAll(genss,i->ImagesRepresentative(hom,i)=i^rep));
      fi;
    fi;
  fi;

    else

      r:= Range( hom );
      if not IsPermGroup( r ) then
        return false;
      fi;
      sym:= SymmetricGroup( Union( MovedPoints( s ), MovedPoints( r ) ) );

      # Simply compute a conjugator in the enveloping symmetric group.
      # (Note that all checks whether source and range
      # can fit together under conjugation
      # should better be left to `RepresentativeAction'.)
      rep:= RepresentativeAction( sym, genss, List( genss,
                      i -> ImagesRepresentative( hom, i ) ), OnTuples );
      if rep<>fail then
	Assert(1,ForAll(genss,i->ImagesRepresentative(hom,i)=i^rep));
      fi;

    fi;

    # Return the result.
    if rep <> fail then
      SetConjugatorOfConjugatorIsomorphism( hom, rep );
      return true;
    else
      return false;
    fi;
end );

#############################################################################
##
#E

