Quelle basic.gi Sprache: unbekannt

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

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##
#F DetermineAdmissiblePrime(gensOfG)
##
## determines a prime number which does not divide  the denominators
## of the entries of the matrices in gensOfG and which does not divide the
## the entries of the inverses of the matrices in gensOfG
##
## input is a list of generators of a rational polycyclic matrix group
##
InstallGlobalFunction( DetermineAdmissiblePrime , function(gensOfG)
    local d,list1,list2,g,i,j,antiPrime,temp,temp2,p,found;
    d := Length( gensOfG[1] );
    list1:=[];
    list2:=[];

    # construct a list of all elements in gensOfG and their inverses
    for g in gensOfG do
        Add(list1,g);
        Add(list1,g^-1);
    od;

    # write denominators of all matrix entries in list
    for g in list1 do
        for i in [1..d] do
            for j in [1..d] do
                Add(list2,DenominatorRat(g[i][j]));
            od;
        od;
    od;
    antiPrime:=ConsideredPrimes(list2);

    # choose a small prime which is not in antiPrime
    found:=false;
    p:=3;
    while not found do
        if not p in antiPrime then
            return p;
        fi;
        p:=NextPrimeInt(p);
    od;
end );

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##
#F POL_NormalSubgroupGeneratorsOfK_p(pcgs,gensOfRealG)
##
## pcgs is a constructive pc-Sequence for I_p(G)
## (image of G under the p-congruence hom.).
## This function calculates  normal subgroup generators for K_p(G)
## (the kernel of the p-congruence hom.)
##
InstallGlobalFunction( POL_NormalSubgroupGeneratorsOfK_p ,
                       function( pcgs, gensOfRealG )
    local g, relations, rightSide, leftSide, preimages, revPreimages,
          preimage, genList, ftl, n, ro, i, j, exp, conj, f_i, f_j,
          r_i, pcSeq;

    n := Length(pcgs.gens);
    preimages := [];
    relations := [];

    # catch the trivial case
    if Length(pcgs.gens)=0 then
        return gensOfRealG;
    fi;

    # calcuclate all preimages of pcgs.gens
    for i in [1..n] do
        preimage := SubsWord( pcgs.wordGens[i], gensOfRealG );
        Add( preimages, preimage);
    od;

    # Attention: In pcgs.gens we have the pc-sequence in inverse order,
    # because we built up  the structure bottom up
    pcSeq := StructuralCopy(Reversed(pcgs.gens));
    revPreimages := StructuralCopy(Reversed(preimages));

    # calculate the relative orders
    ro := RelativeOrdersPcgs_finite( pcgs );

    # express the power relations in terms of gensOfRealG
    for i in [1..n] do
        f_i := pcSeq[i];
        r_i := ro[i];
        exp := ExponentvectorPcgs_finite( pcgs, f_i^r_i );
        leftSide := revPreimages[i]^r_i;
        rightSide := Exp2Groupelement(revPreimages,exp);
        Add(relations,leftSide*(rightSide^-1));
    od;

    # conjugation relations
    for i in [1..n] do
        for j in [1..(i-1)] do
            f_i := pcSeq[i];
            f_j := pcSeq[j];
            conj := (f_j^-1)*f_i*f_j;
            exp := ExponentvectorPcgs_finite( pcgs, conj);
            leftSide := (revPreimages[j]^-1)*revPreimages[i]*revPreimages[j];
            rightSide := Exp2Groupelement(revPreimages,exp);
            Add( relations, leftSide*(rightSide^-1));
        od;
    od;

    # Add  some other relations, because we changed the generating
    # set of the image under the p-congruence hom.
    for i in [1..Length(pcgs.gensOfG)] do
        exp := ExponentvectorPcgs_finite( pcgs, pcgs.gensOfG[i]);
        rightSide := Exp2Groupelement( revPreimages, exp);
        leftSide := gensOfRealG[i];
        Add( relations, leftSide*(rightSide^-1));
    od;
    return relations;
end );

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##
#F Exp2Groupelement(list,exp)
##
InstallGlobalFunction( Exp2Groupelement, function(list,exp)
    local g,i;
    g:=list[1]^0;
    for i in [1..Length(list)] do
        g:=g*list[i]^exp[i];
    od;
    return g;
end );

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##
#F CopyMatrixList(list)
##
InstallGlobalFunction( CopyMatrixList, function(list)
    local i,j,k,list2;
    list2:=[];
    for i in [1..Length(list)] do
        Add(list2,[]);
        for j in [1..Length(list[i])] do
            Add(list2[i],[]);
            for k in [1..Length(list[i][j])] do
                Add(list2[i][j],[]);
                list2[i][j][k]:= list[i][j][k];
            od;
        od;
    od;
    return list2;
end );

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##
#F POL_CopyVectorList(list)
##
InstallGlobalFunction( POL_CopyVectorList, function(list)
    local i,j,k,list2;
    list2:=[];
    for i in [1..Length(list)] do
        Add(list2,[]);
        for j in [1..Length(list[i])] do
            Add(list2[i],[]);
                list2[i][j]:= list[i][j];
        od;
    od;
    return list2;
end );

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##
#F POL_NormalSubgroupGeneratorsU_p( pcgs_GU, gens, gens_K_p )
##
## pcgs_GU  is a constructive pc-Sequence for G/U,
## this function calculates normal subgroup generators for U_p(G)
##
InstallGlobalFunction( POL_NormalSubgroupGeneratorsU_p ,
                       function( pcgs_GU, gens, gens_K_p )
    local relations,rightSide,leftSide,preimages,revPreimages,
          preimage,genList,ftl,n,ro,i,j,exp,conj,f_i,f_j,r_i,pcs, g, k;

    # setup
    pcs := pcgs_GU.pcs;
    n:=Length(pcs);
    preimages:=[];
    relations:=[];
    k := Length( pcgs_GU.pcgs_I_p.gens );

    # catch the trivial case (G/U trivial)
    if Length(pcgs_GU.pcs)=0 then
       return gens;
    fi;

    # catch the trivial case (U_p = 1)
    if Length(pcgs_GU.radicalSeries)=2 then
       return [];
    fi;

    # calculate the relative orders
    #ro:= RelativeOrdersPcgs( pcgs );
    ro := RelativeOrders_CPCS_FactorGU_p( pcgs_GU );

    # the elements stored in gens_K_p where found by evaluating
    # the pcp-relations of G/I_p. So we don't have to calculate them
    # again.
    for g in gens_K_p do
        exp := ExponentVector_CPCS_FactorGU_p( pcgs_GU, g );
        leftSide := g;
        rightSide := Exp2Groupelement( pcs, exp );
        Add( relations, leftSide*(rightSide^-1) );
    od;

    # Express the power relations in terms of gens
    for i in [ (k+1)..n ] do
        f_i:=pcs[i];
        r_i:=ro[i];
        # we have to exclude the case r_i=0 because this means that
        # the order is equal to infinity
        if not r_i=0 then
            exp:=ExponentVector_CPCS_FactorGU_p(pcgs_GU,f_i^r_i);
            leftSide:=f_i^r_i;
            rightSide:=Exp2Groupelement(pcs,exp);
            Add( relations, leftSide*(rightSide^-1) );
        fi;
    od;

    # conjugation relations
    for i in [ (k+1)..n ] do
        for j in [1..(i-1)] do
            f_i := pcs[i];
            f_j := pcs[j];
            conj := (f_j^-1)*f_i*f_j;
            exp := ExponentVector_CPCS_FactorGU_p( pcgs_GU, conj );
            leftSide := (pcs[j]^-1)*pcs[i]*pcs[j];
            rightSide := Exp2Groupelement(pcs,exp);
            Add( relations, leftSide*(rightSide^-1) );
        od;
    od;
    relations := Filtered( relations,x -> not x=x^0 );
    if Length( relations ) = 0 then relations[1] := gens[1]^0; fi;
    return relations;
end );

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##
#E

Quelle basic.gi Sprache: unbekannt

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