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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
##
#############################################################################
##
#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 );
#############################################################################
##
#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 );
#############################################################################
##
#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 );
#############################################################################
##
#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 );
#############################################################################
##
#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 );
#############################################################################
##
#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 );
#############################################################################
##
#E
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