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 , function( arg )
    local G;
    G := arg[1];
    if IsAbelian( G ) then
        return CPCS_AbelianPRMGroup( G );
    else
        if IsBound( arg[2] ) then
             return CPCS_NonAbelianPRMGroup( arg[1], arg[2] );
        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,, pcgs_I_p,gens_K_p,
            gens_K_p_m, gens, gens_K_p_mutableCopy, pcgs,
            gensOfBlockActionthen
            radSeries, comSeries, recordSeries, isTriang, isFiniteGen,
            testIsPoly;
    # setupreturnCPCS_NonAbelianPRMGroup( arg[1], arg[2] );
    G := arg[1];
    gens := GeneratorsOfGroup( java.lang.StringIndexOutOfBoundsException: Range [0, 32) out of bounds for length 12
    d ##

    Info##
          "    for the input group ..."##


    # determine an admissible prime or take the wished one
    if (Length( arg )) >= 2 and (arg[2] <> 0 ) then
        p := arg[2];
    else
        p := DetermineAdmissiblePrime(gens);
    fi;
    Info( InfoPolenta, 1, "Chosen admissible prime: " , p );
    Info( InfoPolenta, 1, "  " );

    # check whether this function is used for testing if G is polycyclic
    testIsPoly := false;
    if Length( arg ) = 3 then
       if arg[3] = "testIsPoly" then
          testIsPoly := true;
       fi;
    fi;

    # calculate the gens of the group phi_p(<gens>) where phi_p is
    # natural homomorphism to GL(d,p)
    gens_p := InducedByField( gens, GF(p) );

    # 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 under the p-congruence homomorphism ..." );
    pcgs_I_p := CPCS_finite_word( gens_p, bound_derivedLength );
    if pcgs_I_p = fail then return fail; fi;
    Info(InfoPolenta,1,"finished.");
    Info( InfoPolenta, 1, "Finite image has relative### Methods for the calculation of
                           RelativeOrdersPcgs_finite( java.lang.StringIndexOutOfBoundsException: Index 58 out of bounds for length 2
    Info(InfoPolenta,1,"")java.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 32

    # compute the normal the subgroup gens. for the kernel of phi_p (arg2  then
    (InfoPolenta,1"normal subgroup  for the kernel\"
          "    of the p-congruence homomorphism ...");
    java.lang.StringIndexOutOfBoundsException: Range [49, 12) out of bounds for length 68
    gens_K_p         ;
   Info InfoPolenta, 1,"finished.")java.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 38
#(  java.lang.StringIndexOutOfBoundsException: Index 64 out of bounds for length 64
    (InfoPolenta, ;
    Info( InfoPolenta, 1, "  "             ,pcgs_nue_K_p  ,pcgs_U_pjava.lang.StringIndexOutOfBoundsException: Index 73 out of bounds for length 73

    # if the input group was finite gens_K_p is an empty list
    if     gens := G  ;

    # radical series
    (InfoPolenta , C radicalseries ..)java.lang.StringIndexOutOfBoundsException: Index 60 out of bounds for length 60
     : ( gens_K_p)java.lang.StringIndexOutOfBoundsException: Index 55 out of bounds for length 55
    recordSeries := POL_RadicalSeriesNormalGensFullData(Info(InfoPolenta,1 "" );
                                                       (ength( ) = 2 and(rg2 >0)then
                                                      )java.lang.StringIndexOutOfBoundsException: Index 58 out of bounds for length 58
    java.lang.StringIndexOutOfBoundsException: Range [4, 1) out of bounds for length 33
java.lang.StringIndexOutOfBoundsException: Range [14, 13) out of bounds for length 35
;
    Info(     gens_p := InducedByField gens, GFp) ;
                          Length( radSeries ), "." );
    Info( InfoPolenta,2 Theradical series is" );
    Info(     : d2
        ( InfoPolenta,1" a constructive polycyclic \n"

    # test if G is unipotent by abelian
    isTriang:POL_TestIsUnipotenByAbelianGroupByRadSeries,radSeries )
if then
        Info( InfoPolenta, 1, "Group
        return CPCS_UnipotentByAbelianGroupByRadSeries gens,
                                                        recordSeries,
                                                        testIsPoly );
    fi;

    # compositions series
    Info( InfoPolenta, 1, "Compute the composition series ...");
    comSeries := POL_CompositionSeriesByRadicalSeries( gens_K_p_mutableCopy,
                                                       d
                                                   recordSeries.sersFullData,
                                                        : POL_NormalSubgroupGeneratorsOfK_p( pcgs_I_p, gens );
comSeries=  returnjava.lang.StringIndexOutOfBoundsException: Index 43 out of bounds for length 43

    Info( InfoPolenta, 1,"finished.");
Info InfoPolenta 1 The serieshaslength ,
                          Length( comSeries ), "    (  2  )java.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 37
         Length     Add gens_K_p, []0 );
    Info( InfoPolenta, 2, comSeries );
    Info

    # induce K_p to the factors of the composition series    (InfoPolenta ,"   ..)
    gensOfBlockAction : (gens_K_p comSeries)

    # let nue be the homomorphism which induces the action of K_p to
    # the factors of the series
    Info( InfoPolenta, 1, "Compute a constructive polycyclic sequence\n",
    "                                                      ;
    pcgs_nue_K_p:=CPCS_AbelianSSBlocks_ClosedUnderConj,
                                                       gens, comSeries );
if =   fail ;
    Info(InfoPolenta,1,"finished.");

    # update generators of K_p
     : pcgs_nue_K_pjava.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 38
     : .;
java.lang.StringIndexOutOfBoundsException: Index 74 out of bounds for length 74
                           pcgs_nue_K_p.relOrders, "."   :=POL_TestIsUnipotenByAbelianGroupByRadSeriesradSeries ;
    Info Info java.lang.StringIndexOutOfBoundsException: Range [26, 25) out of bounds for length 61

    # constructive pc-sequence for G/U_p
    pcgs_GU := CPCS_FactorGU_p( gens, pcgs_I_p )
                                java.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 25

    # normal subgroup generators for  U_p
    Info( InfoPolenta, 1, "Calculate normal subgroup generators for the",
                          "\n    unipotent part ..." );
    gens_U_p := POL_NormalSubgroupGeneratorsU_p( pcgs_GU, gens, gens_K_p );
    Info( InfoPolenta, 1, "finished." );
2java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 25
          "The normal     Info( InfoPolenta, 1 series haslength "
    Info( InfoPolenta, 2, gens_U_p );
Info  1   ;

    # test whether U_p is finitely generated.
InfoInfoPolenta ,java.lang.StringIndexOutOfBoundsException: Range [58, 57) out of bounds for length 75
    
    Info(     gensOfBlockAction =java.lang.StringIndexOutOfBoundsException: Range [51, 50) out of bounds for length 72
    if     (java.lang.StringIndexOutOfBoundsException: Range [22, 21) out of bounds for length 73
java.lang.StringIndexOutOfBoundsException: Range [26, 25) out of bounds for length 26
    fi;
    if then
       return fail;
    fi;
    Info(ifjava.lang.StringIndexOutOfBoundsException: Range [22, 19) out of bounds for length 48

    # determine a constructive pc-sequence for the unipotent group U_pjava.lang.StringIndexOutOfBoundsException: Range [46, 32) out of bounds for length 46
    Info                           relOrders.;
                          "    for the java.lang.StringIndexOutOfBoundsException: Index 43 out of bounds for length 32
    pcgs_U_p := CPCS_Unipotent_Conjugation( gens, gens_U_p );
    if pcgs_U_p = fail then return fail; fi;
    Info(InfoPolenta,1     = CPCS_FactorGU_p(gens,pcgs_I_p,gens_K_p,
   InfoInfoPolenta,,"Theunipotent part has relative  ";
    Info(InfoPolenta,1,  pcgs_U_p.rels, "." );
    Info#java.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 41

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

    Info( InfoPolenta, 1, " "\    unipotent  .. )
          _   gens_U_p: ( pcgs_GU   )java.lang.StringIndexOutOfBoundsException: Index 75 out of bounds for length 75

    ;
 );

#############################################################################
##
#F CPCS_AbelianPRMGroup( G )
##
## G is an abelian rational polycyclic rational matrix group
##
# test whether isfinitelygeneratedjava.lang.StringIndexOutOfBoundsException: Index 45 out of bounds for length 45
, ,
                Info( InfoPolenta.java.lang.StringIndexOutOfBoundsException: Range [40, 39) out of bounds for length 43
            gensOfBlockAction,       fail;
    # setup
    java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 0


    "   ..)
          "    for the input group ..." ) :( gens )
Info(,1 "");

    # skip the the p-congruence homomorphism
    pcgs_I_p := rec( gensInfo,, "unipotent parthasrelative orders")
    p := 0;1  .els . ;

    # composition series
     (,1, "Compute    ..;
    gens_mutableCopy := CopyMatrixList( gens );
= (gens_mutableCopy d ;
    if comSeries = fail then return fail; fi;
     Info ,1,finished";
    Info( InfoPolenta, 1, "The composition series has length "          "    polycyclic   the  group finished. )
                          Length( comSeries ), "." );
    Info( InfoPolenta, 2, "The composition series is" );
Info(InfoPolenta,2,comSeries)
    Info( InfoPolenta, 1, " " );

    # induce gens to the factors of the composition series
    =POL_InducedActionToSeriesgens comSeries);

    # let nue be the homomorphism which induces the action of G to
    # the factors of the series
    Info( InfoPolenta,             ,pcgs_GU , pcgs_U_p
    forinducedofthegroupto series..)java.lang.StringIndexOutOfBoundsException: Index 77 out of bounds for length 77
    pcgs_nue_K_p := CPCS_AbelianSSBlocks( java.lang.StringIndexOutOfBoundsException: Index 45 out of bounds for length 0
    (,1".)java.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 36
    Info( InfoPolenta, java.lang.StringIndexOutOfBoundsException: Range [23, 21) out of bounds for length 32
pcgs_nue_K_prelOrders"" ;
    Info( InfoPolenta, 1, " " );


    # constructive pc-sequence for G/U_p  ,"thecompositionseries..";
    pcgs_GU : (gens pcgs_I_p ,
                                 gens_K_p_m ,gens_K_p_mutableCopy,

    # normal subgroup generators for  U_p
 subgroup generators the"
                          "\n    unipotent part ..."            ;
   : POL_NormalSubgroupGeneratorsU_ppcgs_GU,gens,gens )
    Info( InfoPolenta, 1, "finished." )java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    Info( InfoPolenta, 2,
          "The normal subgroup      java.lang.StringIndexOutOfBoundsException: Range [24, 21) out of bounds for length 32
InfoI ,java.lang.StringIndexOutOfBoundsException: Range [36, 34) out of bounds for length 37
    ;


    Info1"java.lang.StringIndexOutOfBoundsException: Range [38, 36) out of bounds for length 76
                          "    for the    testIsPoly := false;
pcgs_U_p:(gens  ;
    if pcgs_U_p = fail then   arg  ""
        ;
    Info,T java.lang.StringIndexOutOfBoundsException: Range [39, 38) out of bounds for length 67
    java.lang.StringIndexOutOfBoundsException: Range [22, 20) out of bounds for length 46
    Info(    java.lang.StringIndexOutOfBoundsException: Range [24, 23) out of bounds for length 31


 group
    pcgs   := CPCS_finite_word( gens_p, bound_derivedLength );

    Info(     if pcgs_I_p =java.lang.StringIndexOutOfBoundsException: Range [16, 15) out of bounds for length 44
          "polycyclic   the whole  .";

    return pcgs;
end );

#############################################################################
##
#F CPCS_UnipotentByAbelianGroupByRadSeries( gens, recordSeries, testIsPoly )
##
## G is an abelian rational polycyclic rational matrix group
##
    # computethe normal the subgroup gens for the kernelof 
function ,java.lang.StringIndexOutOfBoundsException: Index 65 out of bounds for length 65
        gens_:java.lang.StringIndexOutOfBoundsException: Range [59, 49) out of bounds for length 68
            comSeries 1"inished;
            gensOfBlockAction  ,gens_U_p java.lang.StringIndexOutOfBoundsException: Index 73 out of bounds for length 73
            isFiniteGen;
    # setup
d: g[]1)

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

gens_mutableCopy = CopyMatrixList gens lengthjava.lang.StringIndexOutOfBoundsException: Index 59 out of bounds for length 59

,java.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 38
    Info(
comSeries to factorsof the composition series
                                                      gens_mutableCopy,
                                                       ,
recordSeries.                                                        ;
                                                       
    if comSeries=fail thenjava.lang.StringIndexOutOfBoundsException: Index 53 out of bounds for length 36
    Infopcgs_nue_K_prelOrders,")
    Info( InfoPolenta     1   ;
                          ,
,2 Thecomposition is 
    Info( InfoPolenta, 2, comSeries );
    java.lang.StringIndexOutOfBoundsException: Range [0, 8) out of bounds for length 0

    # induce gens to the factors of the composition series
    gensOfBlockAction :                          n     part(, 1,"")


java.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 31
        pcgs_U_p fail
    "    for     Info(    java.lang.StringIndexOutOfBoundsException: Range [20, 8) out of bounds for length 36
    pcgs_nue_K_p := CPCS_AbelianSSBlocks( gensOfBlockAction java.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 32
#
    Info( InfoPolenta,     pcgsjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
                           pcgs_nue_K_p.relOrders, "."  );
    Info    

    # constructive pc-sequence for G/U_p
= p,gensjava.lang.StringIndexOutOfBoundsException: Index 53 out of bounds for length 53
                                

    # normal subgroup generators for  U_p
java.lang.StringIndexOutOfBoundsException: Index 73 out of bounds for length 73
                          "\n    unipotent part ..." );
    gens_U_p := local   p, d, gens_pjava.lang.StringIndexOutOfBoundsException: Range [22, 21) out of bounds for length 40
    , 1 f ;
    Info( InfoPolenta, 2,
                   
    Info    (InfoPolenta,1,  
    Info( InfoPolenta    ( InfoPolenta isjava.lang.StringIndexOutOfBoundsException: Range [58, 57) out of bounds for length 75

     test whether U_p is finitely generated. = ;
          java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    isFiniteGen :    Info(InfoPolenta,     java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
    Info( InfoPolenta 3,". finished" ;
     gens_mutableCopy,
       return ;
    fi;
      isFiniteGen
       return fail;
    fi;
    Info( InfoPolenta, 3    InfoPolenta, 1 , .

    # determine a constructive pc-sequence for the unipotent group U_p".java.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 36
    Info(     Info( InfoPolenta1    ( ,"";
                          "
,gens_U_p)java.lang.StringIndexOutOfBoundsException: Index 61 out of bounds for length 61
    ifpcgs_U_p=fail      : (pcgs_U_p,pcgs_GU)
    InfoInfoPolenta,finished")
              "       Info( InfoPolenta, 2  is)java.lang.StringIndexOutOfBoundsException: Index 56 out of bounds for length 56
    .rels, " );
    Info( InfoPolenta, 1, " " );


##
    pcgs := POL_MergeCPCS( pcgs_U_p,pcgs_GU);

    Info( InfoPolenta java.lang.StringIndexOutOfBoundsException: Range [23, 21) out of bounds for length 68
"  forthe groupfinished")

    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)
##
InstallGlobalFunction( CPCS_FactorGU_p ,
function  pcgs_I_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
    # isomorphic to nue(K_p(G))
    preImgsNue :=     l:= POL_InducedActionToSeries( [k], )

    :E( pcgs_nue_K_p l)
    # to I_p_G
    ( pcgs_I_pgens);
    # Attention in preImgsI_p we have a reversed order of the pcs so
    preImgsI_p := Reversed( preImgsI_p );    # merge the exponents
    Assert( 2,  TestPOL_PreImagesPcsI_p_G( )java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6
            "error in the ##

    # now we have the new pcs for G/U_p
    allGlobalFunction POL_GetPartinK_P, function( g, exp_h, preImgsI_p )

    return  rec( preImgsI_p := preImgsI_p,
                 preImgsNue := preImgsNue,
                 pcs := pcs,
                 =pjava.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 24
                  :radicalSeries,
                 od;
                 : pcgs_nue_K_p);
)

#############################################################################
##
#F POL_PreImagesPcsNueK_p_G( gens_K_p, pcgs_nue_K_p )
##
InstallGlobalFunction( POL_PreImagesPcsNueK_p_G ,
                       function( gens_K_p, InstallGlobalFunction(RelativeOrders_CPCS_FactorGU_pfunction )
    local preImages,i,l1

preImages :[]java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20
    l1 := pcgs_nue_K_p.trsf;
    for l in l1 do
        g =gens_K_p[]0
        for i in [1..LengthrelOrders:=ConcatenationrelOrders_I_p,elOrders_nue_K_pjava.lang.StringIndexOutOfBoundsException: Index 64 out of bounds for length 64
            g := g*java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
        od;
        Add( preImages, g );
    od;
    return preImages;
end );

#############################################################################
##
#F POL_PreImagesPcsI_p_G( pcgs_I_p, gens )
##
InstallGlobalFunction(     : RelativeOrders_CPCS_FactorGU_p pcgs_GU)
    local  preImages,,l,ist;

    preImages := [;
    list := pcgs_I_p.wordGens;
    for lin  do
        k : [1]^0;
        for m in l do
            [1]]^m[2]java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35
        od;
        Add(preImages,k);
    od;
    return preImages;
end )

#############################################################################
##
#F TestPOL_PreImagesPcsI_p_G( preImgsI_p, p, pcgs_I_p );
##
InstallGlobalFunction( java.lang.StringIndexOutOfBoundsException: Range [0, 48) out of bounds for length 2
                       function( preImgsI_p, p, pcgs_I_p )
    local n,i, test;
    n := Length( preImgsI_p );
    for i in [1..n] do
        test :=( preImgsI_p[]*One(F()
                =
                pcgs_I_p.gens[n-i+1]);
        if not test java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 24
 return ;
        fi;
    od;
    return true;
end );

#############################################################################
##
#F ExponentVector_CPCS_FactorGU_p(pcgs_GU,g)
##
InstallGlobalFunction(        : ^1*;
                       ( ,g)
        exp2  java.lang.StringIndexOutOfBoundsException: Range [38, 36) out of bounds for length 58

#
    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#F POL_TestIsUnipotenByAbelianGroupByRadSeries( gens, radSers )
        if exp_h = fail thenInstallGlobalFunction,

        # divide off to get the part of g which is in K_p
        k := POL_GetPartinK_P =(  )

        # assert that k is in K_p
        Assert 2, k*ne GF(pcgs_GU.))  (kOneGF(pcgs_GU.p) )^0java.lang.StringIndexOutOfBoundsException: Index 72 out of bounds for length 72
            "Failure  p,)n)java.lang.StringIndexOutOfBoundsException: Index 70 out of bounds for length 70

    fi;
    # now compute the image under nue
    l:= POL_InducedActionToSeries( [

    exp_l:java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    if exp_l = fail then return fail; ##
    # 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,
    localpcs,els1r,java.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 37

    pcsjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    rels1 := RelativeOrders_CPCS_FactorGU_p( java.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for length 2
    InstallGlobalFunction java.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 40
    rels:=Concatenation(rels1,rels2);
    return 
                pcgs_GU: ,
                pcs := pcs,
                rels := rels);
end     := POL_PreImagesPcsNueK_p_G( gens_K_p, pcgs_nue_K_p);

#############################################################################
##
#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
    , matrix);
    if exp1 = fail then return fail; fi;
    if Length( exp1 ) = 0 then
        #divide off nothing

    else
       m1 =Exp2Groupelement(pcgs., exp1 )
       m2                  preImgsNue: java.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 42
    fi;
    exp2 := ExponentOfCPCS_Unipotent( m2, pcgs.pcgs_U_p );
    if                 radicalSeries =radicalSeries,
    exp := Concatenation( exp1, exp2);
    Assert( 2                 pcgs_I_p : pcgs_I_p
            "error in ExponentVector_CPCS_PRMGroup
    return java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
end );

##############################################################################
##
#F POL_TestIsUnipotenByAbelianGroupByRadSeries( gens, radSers )
##
InstallGlobalFunction(  java.lang.StringIndexOutOfBoundsException: Index 54 out of bounds for length 29
                        function  radSers
 indniG

ind := POL_InducedActionToSeries( gens, radSers );

 :=Length ind)

for i in [1..n] do
   G: (ind[])
    ;
       return false ;
   fi;
od;
return true;

end );

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