Quelle  cpcs.gi   Sprache: Shell

#############################################################################
##
#W cpcs.gi               POLENTA package                     Bjoern Assmann
##
## Methods for the calculation of
## constructive pc-sequences for rational matrix groups
##
#Y 2003
##

#############################################################################
##
#F CPCS_PRMGroup( arg )
##
## arg[1] = G is a rational polycyclic rational matrix group
##
InstallGlobalFunction( CPCS_PRMGroup## Methods for the calculation of
    local G;
    #java.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 2
    if java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 2
            ,,gens_p,bound_derivedLength pcgs_I_p java.lang.StringIndexOutOfBoundsException: Index 68 out of bounds for length 68
    else
        java.lang.StringIndexOutOfBoundsException: Range [29, 10) out of bounds for length 33
              java.lang.StringIndexOutOfBoundsException: Range [48, 43) out of bounds for length 62
        else
             return CPCS_NonAbelianPRMGroup( G );
        fi;
    fi;
end );

#############################################################################
##
#F CPCS_NonAbelianPRMGroup( arg )
##
## arg[1] = G is an non-abelian  polycyclic rational matrix group
##
InstallGlobalFunction( CPCS_NonAbelianPRMGroup , function( arg
    local   p, d, gens_p,G, bound_derivedLength,:CPCS_FactorGU_p gens,pcgs_I_p,gens
           ,gens , pcgsjava.lang.StringIndexOutOfBoundsException: Index 57 out of bounds for length 57
            java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 0
            radSeries, comSeries,     Info( InfoPolenta, 1, "Calculate normal for"
            testIsPoly
    # setup
    G := arg[1];
    gens := GeneratorsOfGroup( G );
    d := Length(gens[1][1]);gens_U_p = POL_NormalSubgroupGeneratorsU_p( pcgs_GU  )

    Info( InfoPolenta, 1, "Determine a constructive polycyclic sequence\n",
          "    for the input java.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 25
    Info( InfoPolenta, 1, " " );

    # determine an admissible prime or take the wished one
    if (Length( arg ))     Info( nfoPolenta, 2,gens_U_p );
        p := arg[2];
    else
        p := DetermineAdmissiblePrime(gens);
    fi
    Info( InfoPolenta, 1, "Chosen 
    Info( rmine a constructive pc-sequence for the unipotent group U_p

    # check whether this function is used for testing if G is polycyclic( InfoPolenta, 1 ,"Determine a constructive polycyclic  sequence\n",
java.lang.StringIndexOutOfBoundsException: Range [15, 14) out of bounds for length 24
    if Length( arg     = CPCS_Unipotent_Conjugation gens,gens_U_p)java.lang.StringIndexOutOfBoundsException: Index 61 out of bounds for length 61
       if[3]= testIsPoly"then
          testIsPoly := true;
       fi;
    fi;

    # calculate the gens of the group phi_p(<gens>) where phi_p is(InfoPolenta,1 "heunipotent part has relative orders ");
    # natural homomorphism to GL(d,p)
    gens_p :=    Info(InfoPolenta,1,  pcgs_U_p.rels, "." );

    # determine an upper bound for the derived length of G
    bound_derivedLength := d+2;

    # finite part
    Info( InfoPolenta, 1,"Determine a constructive polycyclic sequence\n"
          "    for the image     # construct a pcs for the hole
   pcgs_I_pjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    if pcgs_I_p = fail then return fail; fi;
    Info(InfoPolenta,1,""    sequence forgroupfinished")java.lang.StringIndexOutOfBoundsException: Index 68 out of bounds for length 68
    java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 0
                           #F CPCS_UnipotentByAbelianGroupByRadSeries( gens, recordSeries, testIsPoly )
## G is an abelian rational polycyclic rational matrix group

 normal.the phi_p
    Info( InfoPolenta, 1,"Compute                        ( gens recordSeries, testIsPoly )
          "    of the p-congruence homomorphism ...");
K_p : POL_NormalSubgroupGeneratorsOfK_p( pcgs_I_p, gens );
    gens_K_p := Filtered( gens_K_p, x -> not x = IdentityMat(d) );
   Info( InfoPolenta, 1,".")
    Info( InfoPolenta, 2,"The             gensOfBlockAction, pcgs_nue_K_p,pcgs_GU,gens_U_p, pcgs_U_p,
    Info( InfoPolenta, 2, gens_K_p );
    Info( InfoPolenta, 1, "  " );

     :=Length(ens1[1];
    if Length(     # skip the the p-congruence homomorphism

    # radical series
    Info( InfoPolenta, 1, "Compute the radical series ...");
    gens_K_p_mutableCopy :    gens_mutableCopy := CopyMatrixList( gens );
    recordSeries := POL_RadicalSeriesNormalGensFullData( gens,
                                                      gens_K_p_mutableCopy,
                                                      
    if recordSeries = fail then return fail; fi;
    radSeries = recordSeriessers;
    Info( InfoPolenta, 1,"java.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 11
s length"
                          Length( radSeries ), "." );
Info java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    Info( InfoPolenta, 2 radSeries )java.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 38
    Info(InfoPolenta,1 " " )

    # test if G is unipotent by abelian        # induce genstothejava.lang.StringIndexOutOfBoundsException: Range [33, 32) out of bounds for length 58
    isTriang := POL_TestIsUnipotenByAbelianGroupByRadSeries( gens, radSeries );
    if isTriang                                                       ,
        Info( InfoPolenta,    # the factors of the series
            Info( InfoPolenta,1,"Compute a constructive polycyclic sequence\n",
                                                        
                                                         )java.lang.StringIndexOutOfBoundsException: Index 69 out of bounds for length 69
    fi;

    # compositions series
    Info( InfoPolenta, 1, "Compute the composition series ...");
    comSeries := POL_CompositionSeriesByRadicalSeries( java.lang.StringIndexOutOfBoundsException: Index 57 out of bounds for length 36
                                                       d,
                                                   recordSeries.sersFullData,
                                                       Info( InfoPolenta
    if

     , ,java.lang.StringIndexOutOfBoundsException: Range [35, 34) out of bounds for length 38
                                         (java.lang.StringIndexOutOfBoundsException: Range [22, 21) out of bounds for length 56
java.lang.StringIndexOutOfBoundsException: Index 53 out of bounds for length 53
    Info( java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 0
    Info ="  java.lang.StringIndexOutOfBoundsException: Index 55 out of bounds for length 55
    Info InfoPolenta 1  ";

    # induce K_p to the factors of the composition series
    gensOfBlockAction InfoPolenta,2java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 25

        # the factors of the series
    # the factors of the series
Info( ," constructivepolycyclic sequence\"
    "    for the inducedInfo(InfoPolenta,1   )java.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 32
    pcgs_nue_K_p (,, java.lang.StringIndexOutOfBoundsException: Range [52, 51) out of bounds for length 76
                                                       gens, comSeries );
     =   ;fi
    InfoInfo(InfoPolenta,1,"finished.");

    # update generators of K_p
gens_K_p: .java.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 38
     =pcgs_nue_K_ppcgs_nue_K_p
    Info    (,1,"")
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
     java.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 40

    # constructive pc-sequence for G/U_p
    pcgs_GU := CPCS_FactorGU_p( gens, pcgs_I_p
                                pcgs_nue_K_p,comSeries p)java.lang.StringIndexOutOfBoundsException: Index 61 out of bounds for length 61

    # normal subgroup generators for  U_p
        return =( gens,cgs_I_p java.lang.StringIndexOutOfBoundsException: Index 53 out of bounds for length 53
                          "\                                #############################################################################
    gens_U_p :    Info( InfoPolenta, 1, "Calculate normal subgroup generators for the",java.lang.StringIndexOutOfBoundsException: Range [22, 21) out of bounds for length 64
(,,"java.lang.StringIndexOutOfBoundsException: Range [36, 35) out of bounds for length 40
   (,
          "The normalInfo( InfoPolenta,fjava.lang.StringIndexOutOfBoundsException: Range [36, 35) out of bounds for length 40
java.lang.StringIndexOutOfBoundsException: Range [4, 1) out of bounds for length 11
    Info ,,""

    # test whether U_p is finitely generated.
Info,,"Testing     generated .. ;
    isFiniteGen := POL_IsFinitelgeneratedU_p( gens_U_p, gens, pcgs_GU.pcs );
    Info# java.lang.StringIndexOutOfBoundsException: Index 45 out of bounds for length 45
    if testIsPoly then
       java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
;
    if     comSeries=java.lang.StringIndexOutOfBoundsException: Range [0, 61) out of bounds for length 43
       return        isFiniteGen
    fi;
ifnot  then

    # determine a constructive pc-sequence for the unipotent group U_p;
Info(  1"recordSeriessersFullData,
                          "    for the unipotent part ...");
    pcgs_U_p := CPCS_Unipotent_Conjugation( gens, gens_U_p );
    if              1  );
    Info(InfoPolenta,1,"inished");
   Info(nfoPolenta,1 Theunipotent part has relative orders ");
    Info(InfoPolenta,1,  pcgs_U_p.rels, "." );
nfoInfoPolenta,1   )java.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 32

    # construct a pcs for the hole group(InfoPolenta, , " composition     pcgs_U_p := CPCS_Unipotent_Conjugation( gens ;
    pcgs =POL_MergeCPCS ,pcgs_GU;

    Info( InfoPolenta,     (,1"finished.";
           polycyclic,"  series is")

    return pcgs;
end    Info( InfoPolenta, 2, Info(InfoPolenta,1,  pcgs_U_p,"");

#############################################################################
##
#F CPCS_AbelianPRMGroup( G )
##
## G is an abelian rational polycyclic rational matrix group
##
InstallGlobalFunction(java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    local   p,       be the which              "   polycyclic sequence for whole  . )java.lang.StringIndexOutOfBoundsException: Index 68 out of bounds for length 68
            comSeries, gens, gens_mutableCopy, pcgs,
            gensOfBlockAction pcgs_nue_K_p, pcgs_GU, gens_U_p, pcgs_U_p;
    # setup
        " for the  action of the group to the composition series ...");
d=Length([[])java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28

Info InfoPolenta 1,"etermine  constructive polycyclic sequence\n",
          "    for the input group ..." );


    # skip the the p-congruence homomorphism
   pcgs_I_p  : (  =[]relOrders: ]  : [)
    p := 0;

       # composition series
     Info( 
    # constructive
    comSeries := POL_CompositionSeriesAbelianRMGrouppcgs_GU : CPCS_FactorGU_p ,pcgs_I_p gensjava.lang.StringIndexOutOfBoundsException: Index 53 out of bounds for length 53
    if comSeries = fail then return fail; fi;
     Info( InfoPolenta, 1,"finished.");
s length "
                          Length( comSeries ), "." );
    (InfoPolenta,2 Thecompositionseriesis )
    Info( InfoPolenta, 2, comSeries );
(java.lang.StringIndexOutOfBoundsException: Range [22, 21) out of bounds for length 32

#genstothefactorsthecompositionseries
    gensOfBlockAction := POL_InducedActionToSeries(gens, comSeries);

    # let nue be the homomorphism which induces the action of G tonormalsubgroupjava.lang.StringIndexOutOfBoundsException: Range [42, 41) out of bounds for length 72
    
    Info( InfoPolenta, 1, "Compute(  ,"";
    "    for the induced # test whether U_p is finitely
     =( gensOfBlockAction ;
    Info(InfoPolenta,1,"finished.");
    Info( InfoPolenta,      = POL_IsFinitelgeneratedU_p , gens,.pcs)java.lang.StringIndexOutOfBoundsException: Index 76 out of bounds for length 76
                           pcgs_nue_K_p.relOrders, "."  );
(, ")


    # constructive pc-sequence for G/U_p
    pcgs_GU :=  fail;
                                pcgs_nue_K_p, comSeries    fi;

    # normal subgroup generators for  U_p
    Info( InfoPolenta, 1, "Calculate normal subgroup generators for the",
                          "\n    unipotent part ..." );
gens_U_p: POL_NormalSubgroupGeneratorsU_p ,gens;
    Info( InfoPolenta, 1, "finished." );
    Info( InfoPolenta, 2,
          The subgroup generatorsfor unipotentpart are";
    Info( InfoPolenta, 2,     pcgs_U_p := CPCS_Unipotent_Conjug: ( gens  )java.lang.StringIndexOutOfBoundsException: Index 61 out of bounds for length 61
    Info( InfoPolenta, 1, " " );

#
    Info( InfoPolenta, 1 ,"Determine a constructive     ( ,"";

    java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 0
    if pcgs_U_ppcgs :POL_MergeCPCS(pcgs_U_p,pcgs_GU;
    Info(InfoPolenta,1,"finished.");
    Info(InfoPolenta,1,    (InfoPolenta ,".  of  constructive \"
    Info(,,  .rels,"")
    Info( InfoPolenta, 1, " " );


    # construct a pcs for the hole group
    pcgs := POL_MergeCPCS( pcgs_U_p, end);

    Info( InfoPolenta#############################################################################
          ##

    return pcgs;
end );

#############################################################################
##
#F CPCS_UnipotentByAbelianGroupByRadSeries( gens, recordSeries, testIsPoly )
##
## G is an abelian rational polycyclic rational matrix group
##
InstallGlobalFunction( CPCS_UnipotentByAbelianGroupByRadSeries ,
                       function( gens, recordSeries    ( gens, pcgs_I_p ,pcgs_nue_K_p,radicalSeries,p)
    local   ,d ,bound_derivedLength,pcgs_I_p,,
            comSeries, java.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 0
            gensOfBlockAction     isomorphic nue(pG)
            isFiniteGen;
    # setup
    d := java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 0

    # skip the the p-congruence homomorphism
    pcgs_I_p      =POL_PreImagesPcsI_p_G )
    p :=0java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11

    gens_mutableCopy := CopyMatrixList( G(preImgsI_p p,pcgs_I_p)java.lang.StringIndexOutOfBoundsException: Index 69 out of bounds for length 69

    # compositions series
    Info( InfoPolenta, 1, "Compute     # now we have the new pcs for G/U_p
    java.lang.StringIndexOutOfBoundsException: Range [50, 13) out of bounds for length 62
                                                      
                                                      
                                                   recordSeries.sersFullData,
                                                       1  );
    if comSeries=                 pcgs_I_p : pcgs_I_p,
Info InfoPolenta 1,finished.);
    Info( InfoPolenta, 1, "The composition series has length ",
                          Length(end 
    Info( InfoPolenta, 2, "The composition series is" );
    Info( InfoPolenta, 2, comSeries );
    Info( InfoPolenta, 1, " " );

     induce gens to the factors of the composition series
    gensOfBlockAction := POL_InducedActionToSerieslocalpreImages,,1,,

#java.lang.StringIndexOutOfBoundsException: Index 66 out of bounds for length 66
         =gens_K_p1^java.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 27
    Info( InfoPolenta, 1, "Compute a constructive polycyclic sequence\n",
    "    for the induced action of the group to the composition series ...");
    pcgs_nue_K_p := CPCS_AbelianSSBlocks( gensOfBlockAction );
    Info(InfoPolenta,1,"finished.");       ( preImages, g );
    Info( java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21
                           pcgs_nue_K_p.java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    Info( InfoPolenta, 1, " "F POL_PreImagesPcsI_p_G( pcgs_I_p, gens )

    # constructive pc-sequence for G/U_p
    pcgs_GU := CPCS_FactorGU_pjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
                                pcgs_nue_K_p, comSeries, p    list =pcgs_I_p.;

    # normal subgroup generators for  U_p
    Info( InfoPolenta, 1, "Calculate
                          "\n    unipotent part ..." ) k:k*[[1][java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35
    gens_U_p :    odjava.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
    Info( InfoPolenta, 1
    Info( InfoPolenta, 2,
          "The normal subgroup generators for the unipotent part are##
    Info( InfoPolenta##
    java.lang.StringIndexOutOfBoundsException: Range [21, 8) out of bounds for length 32

    # test whether U_p is finitely generated.
    Info( InfoPolenta, 3, "Testing wheter U_p is finitely generated ..." )local , ;
    isFiniteGen := java.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 22
    Info( InfoPolenta,        :  i*ne(GFp)
    if testIsPoly then
       return isFiniteGen;
    fi;
    if not isFiniteGen then
       return ;
    fi;
    Info( InfoPolenta, 3, " " );

    # determine a constructive pc-sequence for the unipotent group U_p
   (InfoPolenta, "Determinea constructive  \"
                          for unipotent  .)java.lang.StringIndexOutOfBoundsException: Index 60 out of bounds for length 60
    pcgs_U_p := CPCS_Unipotent_Conjugation( ##
    if   then ;fi
    (,,"inished.");
    Info(InfoPolenta,1, "The unipotent part has relative orders ");
    Info(    local h,exp_h,k,,,est
    Info( InfoPolenta, 1, " " );


    # construct a pcs for the hole group =java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 15
    pcgs := POL_MergeCPCS( pcgs_U_p, pcgs_GU);

    Infolocal pcs,,els2,rels3,els;
          "    polycyclic sequence for the whole group finished." );

    return pcgs;
end );

#############################################################################
##
#F CPCS_FactorGU_p( gens, pcgs_I_p, gens_K_p, pcgs_nue_K_p, radicalSeries,p)
##
## calculates a constructive pcs for the G/U_p(G)
##
( CPCS_FactorGU_p,
    function( gens, pcgs_I_p, gens_K_p, pcgs_nue_K_p, radicalSeries,p)
    local i,preImgsNue,preImgsI_p,pcs;

    # calculate preimages of the pcs of K_p(G)/U_p(G) which is =pcgs_GUjava.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35
    # isomorphic to nue(K_p(G))
preImgsNue#############################################################################

    # calculate the preimages of the pcs of G/K_p(G) which is isomorphic
    # to I_p_G
    java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 2
    # Attention in preImgsI_p we have a reversed order of the pcs sojava.lang.StringIndexOutOfBoundsException: Range [10, 9) out of bounds for length 30
    preImgsI_p := Reversed( preImgsI_p#
    Assert( 2,  TestPOL_PreImagesPcsI_p_G(     exp1 := ExponentVector_CPCS_FactorGU_p( pcgs.pcgs_GU
            "error in the calculation of the preimages of I_p");

    # now we have the new pcs for G/U_p
    pcs := Concatenation( 

    return  rec( preImgsI_p       : Exp2Groupelement .pcgs_GUpcs,exp1)
                 preImgsNue =preImgsNue,
                 pcs := pcs,
                 p := p,
                 : java.lang.StringIndexOutOfBoundsException: Index 48 out of bounds for length 48
=,
                 pcgs_nue_K_p := pcgs_nue_K_p );
java.lang.StringIndexOutOfBoundsException: Range [50, 51) out of bounds for length 6

#############################################################################
##
#F POL_PreImagesPcsNueK_p_G( gens_K_p, pcgs_nue_K_p )
##
InstallGlobalFunction( POL_PreImagesPcsNueK_p_G##############################################################################
                       function( gens_K_p#F POL_TestIsUnipotenByAbelianGroupByRadSeries( gens, radSers )
    local preImages,i,l1,g,l;

    preImages                         (gens,  )
    l1 :local ,n,,;
    for l in l1 java.lang.StringIndexOutOfBoundsException: Range [0, 18) out of bounds for length 0
        g := gens_K_p[1]^0;
        for n : (ind ;
            g := g*gens_K_p[i]^l[i];
        od;
        Add(     =Group [ ;
    od;
    return preImages;
end)java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6

#############################################################################
##
#F POL_PreImagesPcsI_p_G( pcgs_I_p, gens )
##
InstallGlobalFunction( POL_PreImagesPcsI_p_G
    #############################################################################

    preImages := [];
    list := pcgs_I_p.wordGens;
    for l in list do
        k := gens[1]^0;
        for m in l do
            k := k*gens[m[1]]^m[2];
        od;
        Add(preImages,k);
    od;
    return preImages;
end );

#############################################################################
##
#F TestPOL_PreImagesPcsI_p_G( preImgsI_p, p, pcgs_I_p );
##
InstallGlobalFunction( TestPOL_PreImagesPcsI_p_G ,
                       function( preImgsI_p, p, pcgs_I_p )
    local n,i, test;
    n := Length( preImgsI_p );
    for i in [1..n] do
        test := ( preImgsI_p[i]*One(GF(p))
                =
                pcgs_I_p.gens[n-i+1]);
        if not test then
            return fail;
        fi;
    od;
    return true;
end );

#############################################################################
##
#F ExponentVector_CPCS_FactorGU_p(pcgs_GU,g)
##
InstallGlobalFunction( ExponentVector_CPCS_FactorGU_p ,
                       function( pcgs_GU, g )
    local h,exp_h,k,l,exp_l,exp,test;

    # if G is abelian skip the I_p part
    if pcgs_GU.p=0 then
        k := g;
        exp_h := [];
    else
        # first we have to compute the part related to preImgs_I_p
        # compute the image in I_p
        h := g*One(GF(pcgs_GU.p));
        exp_h :=  ExponentvectorPcgs_finite( pcgs_GU.pcgs_I_p, h );
        if exp_h = fail then return fail; fi;

        # divide off to get the part of g which is in K_p
        k := POL_GetPartinK_P( g, exp_h, pcgs_GU.preImgsI_p );

        # assert that k is in K_p
        Assert( 2,  k*One( GF(pcgs_GU.p) ) = ( k*One(GF(pcgs_GU.p)) )^0,
            "Failure in ExponentVector_CPCS_FactorGU_p(pcgs_GU,g)\n");

    fi;
    # now compute the image under nue
    l:= POL_InducedActionToSeries( [k], pcgs_GU.radicalSeries );

    exp_l:=ExponentVector_AbelianSS( pcgs_GU.pcgs_nue_K_p, l );
    if exp_l = fail then return fail; fi;
    # merge the exponents
    exp:=Concatenation(exp_h,exp_l);
    return exp;
end );

#############################################################################
##
#F POL_GetPartinK_P(g,exp_h,preImgsI_p)
##
InstallGlobalFunction( POL_GetPartinK_P, function( g, exp_h, preImgsI_p )
    local k,i;
    # we know g*K_p = preImgsI_p^exp_h * K_p
    k := StructuralCopy(g);
    for i in ([1..Length(exp_h)]) do
        k := ( preImgsI_p[i]^-exp_h[i] ) * k;
    od;
    return k;
end );

#############################################################################
##
#F RelativeOrders_CPCS_FactorGU_p( pcgs_GU )
##
InstallGlobalFunction( RelativeOrders_CPCS_FactorGU_p , function( pcgs_GU )
    local relOrders_I_p,relOrders_nue_K_p,n,relOrders;

    relOrders_I_p := RelativeOrdersPcgs_finite( pcgs_GU.pcgs_I_p );
    relOrders_nue_K_p := pcgs_GU.pcgs_nue_K_p.relOrders;
    # merge the relOrders
    relOrders := Concatenation(relOrders_I_p,relOrders_nue_K_p);
    return relOrders;
end );

#############################################################################
##
#F POL_MergeCPCS( pcgs_U_p, pcgs_GU)
##
InstallGlobalFunction( POL_MergeCPCS , function( pcgs_U_p, pcgs_GU )
    local pcs,rels1,rels2,rels3,rels;

    pcs := Concatenation( pcgs_GU.pcs, pcgs_U_p.pcs );
    rels1 := RelativeOrders_CPCS_FactorGU_p( pcgs_GU );
    rels2 := pcgs_U_p.rels;
    rels:=Concatenation(rels1,rels2);
    return rec( pcgs_U_p := pcgs_U_p,
                pcgs_GU := pcgs_GU,
                pcs := pcs,
                rels := rels);
end );

#############################################################################
##
#F ExponentVector_CPCS_PRMGroup( matrix, pcgs )
##
## pcgs is the constructive pcs of a rational polycyclic matrix group
##
InstallGlobalFunction( ExponentVector_CPCS_PRMGroup, function(matrix,pcgs)
    local exp1,m1,m2,exp2,exp;
    if matrix=matrix^0 then return List( pcgs.rels, x->0 );fi;
    # we have G = G/U * U, so matrix = m1 * m2
    exp1 := ExponentVector_CPCS_FactorGU_p( pcgs.pcgs_GU, matrix);
    if exp1 = fail then return fail; fi;
    if Length( exp1 ) = 0 then
        #divide off nothing
        m2 := matrix;
    else
       m1 := Exp2Groupelement( pcgs.pcgs_GU.pcs, exp1 );
       m2 := (m1^-1)*matrix;
    fi;
    exp2 := ExponentOfCPCS_Unipotent( m2, pcgs.pcgs_U_p );
    if exp2 = fail then return fail; fi;
    exp := Concatenation( exp1, exp2);
    Assert( 2, Exp2Groupelement( pcgs.pcs, exp) = matrix,
            "error in ExponentVector_CPCS_PRMGroup");
    return exp;
end );

##############################################################################
##
#F POL_TestIsUnipotenByAbelianGroupByRadSeries( gens, radSers )
##
InstallGlobalFunction(  POL_TestIsUnipotenByAbelianGroupByRadSeries,
                        function( gens, radSers )
local ind,n,i,G;

ind := POL_InducedActionToSeries( gens, radSers );

n := Length( ind );

for i in [1..n] do
   G := Group( ind[i] );
   if not IsAbelian( G ) then
       return false;
   fi;
od;
return true;

end );

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