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#############################################################################
##
#W solvalble.gi POLENTA package Bjoern Assmann
##
## Methods for testing if a matrix group
## is solvable or polycyclic
##
#Y 2003
##
#############################################################################
##
#F POL_IsSolvableRationalMatGroup_infinite( G )
##
POL_IsSolvableRationalMatGroup_infinite := function( G )
local p, d, gens_p, bound_derivedLength, pcgs_I_p, gens_K_p,
homSeries, gens_K_p_m, gens, gens_K_p_mutableCopy, pcgs,
gensOfBlockAction, pcgs_nue_K_p, pcgs_GU, gens_U_p, pcgs_U_p;
# handle trivial case
if IsAbelian( G ) then
return true;
fi;
# setup
gens := GeneratorsOfGroup( G );
d := Length(gens[1][1]);
# determine an admissible prime
p := DetermineAdmissiblePrime(gens);
Info( InfoPolenta, 1, "Chosen admissible prime: " , p );
Info( InfoPolenta, 1, " " );
# 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 false; fi;
Info( InfoPolenta, 1, "Finite image has relative orders ",
RelativeOrdersPcgs_finite( pcgs_I_p ), "." );
Info( InfoPolenta, 1, " " );
# compute the normal the subgroup gens. for the kernel of phi_p
Info( InfoPolenta, 1,"Compute normal subgroup generators for the kernel\n",
" of the p-congruence homomorphism ...");
gens_K_p := POL_NormalSubgroupGeneratorsOfK_p( pcgs_I_p, gens );
gens_K_p := Filtered( gens_K_p, x -> not x = IdentityMat(d) );
Info( InfoPolenta, 1,"finished.");
Info( InfoPolenta, 2,"The normal subgroup generators are" );
Info( InfoPolenta, 2, gens_K_p );
Info( InfoPolenta, 1, " " );
# homogeneous series
Info( InfoPolenta, 1, "Compute the homogeneous series ... ");
gens_K_p_mutableCopy := CopyMatrixList( gens_K_p );
homSeries := POL_HomogeneousSeriesNormalGens( gens,
gens_K_p_mutableCopy,
d );
if homSeries = fail then
return false;
else
Info( InfoPolenta, 1,"finished.");
Info( InfoPolenta, 1, "The homogeneous series has length ",
Length( homSeries ), "." );
Info( InfoPolenta, 2, "The homogeneous series is" );
Info( InfoPolenta, 2, homSeries );
Info( InfoPolenta, 1, " " );
return true;
fi;
end;
#############################################################################
##
#F POL_IsSolvableFiniteMatGroup( G )
##
POL_IsSolvableFiniteMatGroup := function( G )
local gens, d, CPCS, bound_derivedLength;
# handle trivial case
if IsAbelian( G ) then
return true;
fi;
# calculate a constructive pc-sequence
gens := GeneratorsOfGroup( G );
d := Length(gens[1][1]);
# determine an upper bound for the derived length of G
bound_derivedLength := d+2;
Info( InfoPolenta, 1,"Determine a constructive polycyclic sequence\n",
" for the finite input group ..." );
CPCS := CPCS_finite_word( gens, bound_derivedLength );
if CPCS = fail then
return false;
else
Info(InfoPolenta,1,"finished.");
return true;
fi;
end;
#############################################################################
##
#M IsSolvableGroup( G )
##
## G is a matrix group over the rationals.
##
##
InstallMethod( IsSolvableGroup, "for rational matrix groups (Polenta)", true,
[ IsRationalMatrixGroup ], 0,
POL_IsSolvableRationalMatGroup_infinite );
## Enforce rationality check for cyclotomic matrix groups
RedispatchOnCondition( IsSolvableGroup, true,
[ IsCyclotomicMatrixGroup ], [ IsRationalMatrixGroup ],
RankFilter(IsCyclotomicMatrixGroup) );
#############################################################################
##
#M IsSolvableGroup( G )
##
## G is a matrix group over a finite field.
##
InstallMethod( IsSolvableGroup, "for matrix groups over a finte field (Polenta)",
true, [ IsFFEMatrixGroup ], 0,
POL_IsSolvableFiniteMatGroup );
#############################################################################
##
#F POL_IsPolycyclicRationalMatGroup( G )
##
POL_IsPolycyclicRationalMatGroup := function( G )
local test;
if not IsFinitelyGeneratedGroup( G ) then
return false;
fi;
if IsAbelian( G ) then
return true;
fi;
test := CPCS_NonAbelianPRMGroup( G, 0, "testIsPoly" );
if test=false or test=fail then
return false;
else
return true;
fi;
end;
#############################################################################
##
#M IsPolycyclicGroup( G )
##
## G is a finitely generated subgroup of GL(n,Z), hence G is polycycylic
## if and only if G is solvable and finitely generated.
##
InstallMethod( IsPolycyclicGroup, "for integer matrix groups (Polenta)", true,
[ IsIntegerMatrixGroup ], 0,
function( G )
return IsFinitelyGeneratedGroup( G ) and IsSolvableGroup( G );
end );
#############################################################################
##
#M IsPolycyclicGroup( G )
##
## G is a matrix group over the rationals
##
InstallMethod( IsPolycyclicGroup, "for rational matrix groups (Polenta)", true,
[ IsRationalMatrixGroup ], 0,
function( G )
if IsIntegerMatrixGroup(G) then
return IsFinitelyGeneratedGroup( G ) and IsSolvableGroup( G );
fi;
return POL_IsPolycyclicRationalMatGroup( G );
end );
## Enforce rationality check for cyclotomic matrix groups
RedispatchOnCondition( IsPolycyclicGroup, true,
[ IsCyclotomicMatrixGroup ], [ IsRationalMatrixGroup ],
RankFilter(IsCyclotomicMatrixGroup) );
#############################################################################
##
#M IsPolycyclicGroup( G )
##
## G is a matrix group over a finite field
##
InstallMethod( IsPolycyclicGroup,
"for matrix groups over a finite field (Polenta)", true,
[ IsFFEMatrixGroup ], 0,
function( G )
local F;
if not IsFinitelyGeneratedGroup( G ) then
return false;
fi;
if IsAbelian( G ) then
return true;
fi;
return IsSolvableGroup( G );
end );
#############################################################################
##
#M IsPolycyclicMatGroup( G )
##
## G is a matrix group, test whether it is polycyclic.
##
## TODO: Mark this as deprecated and eventually remove it; code using it
## should be changed to use IsPolycyclicGroup.
##
InstallMethod( IsPolycyclicMatGroup, [ IsMatrixGroup ], IsPolycyclicGroup);
#############################################################################
##
#F POL_IsTriangularizableRationalMatGroup_infinite( G )
##
POL_IsTriangularizableRationalMatGroup_infinite := function( G )
local p, d, gens_p, bound_derivedLength, pcgs_I_p, gens_K_p,
gens_K_p_m, gens, gens_K_p_mutableCopy, pcgs,
gensOfBlockAction, pcgs_nue_K_p, pcgs_GU, gens_U_p, pcgs_U_p,
radSeries, comSeries, recordSeries, isTriang;
if IsAbelian( G ) then
return true;
fi;
# setup
gens := GeneratorsOfGroup( G );
d := Length(gens[1][1]);
# determine an admissible prime or take the wished one
#if Length( arg ) = 2 then
# p := arg[2];
#else
p := DetermineAdmissiblePrime(gens);
#fi;
Info( InfoPolenta, 1, "Chosen admissible prime: " , p );
Info( InfoPolenta, 1, " " );
# 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 false; fi;
Info(InfoPolenta,1,"finished.");
Info( InfoPolenta, 1, "Finite image has relative orders ",
RelativeOrdersPcgs_finite( pcgs_I_p ), "." );
Info( InfoPolenta, 1, " " );
# compute the normal the subgroup gens. for the kernel of phi_p
Info( InfoPolenta, 1,"Compute normal subgroup generators for the kernel\n",
" of the p-congruence homomorphism ...");
gens_K_p := POL_NormalSubgroupGeneratorsOfK_p( pcgs_I_p, gens );
gens_K_p := Filtered( gens_K_p, x -> not x = IdentityMat(d) );
Info( InfoPolenta, 1,"finished.");
Info( InfoPolenta, 2,"The normal subgroup generators are" );
Info( InfoPolenta, 2, gens_K_p );
Info( InfoPolenta, 1, " " );
# radical series
Info( InfoPolenta, 1, "Compute the radical series ...");
gens_K_p_mutableCopy := CopyMatrixList( gens_K_p );
recordSeries := POL_RadicalSeriesNormalGensFullData( gens,
gens_K_p_mutableCopy,
d );
if recordSeries=fail then return false; fi;
radSeries := recordSeries.sers;
Info( InfoPolenta, 1,"finished.");
Info( InfoPolenta, 1, "The radical series has length ",
Length( radSeries ), "." );
Info( InfoPolenta, 2, "The radical series is" );
Info( InfoPolenta, 2, radSeries );
Info( InfoPolenta, 1, " " );
# test if G is unipotent by abelian
isTriang := POL_TestIsUnipotenByAbelianGroupByRadSeries( gens, radSeries );
return isTriang;
end;
#############################################################################
##
#M IsTriangularizableMatGroup( G )
##
##
InstallMethod( IsTriangularizableMatGroup, "for matrix groups over Q (Polenta)", true,
[ IsRationalMatrixGroup ], 0,
POL_IsTriangularizableRationalMatGroup_infinite );
## Enforce rationality check for cyclotomic matrix groups
RedispatchOnCondition( IsTriangularizableMatGroup, true,
[ IsCyclotomicMatrixGroup ], [ IsRationalMatrixGroup ],
RankFilter(IsCyclotomicMatrixGroup) );
#############################################################################
##
#E
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