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#############################################################################
##
#W subgroups.gi POLENTA package Bjoern Assmann
##
## Methods for the calculation of
## certain subgroups of matrix groups
##
#Y 2004
##
#############################################################################
##
POL_Group := function( subGens, G )
if Length( subGens ) = 0 then
return TrivialSubgroup( G );
else
return Group( subGens );
fi;
end;
#############################################################################
##
#F POL_TriangNSGFI_NonAbelianPRMGroup( arg )
##
##
##
## IN: arg[1] ..... G is an non-abelian polycyclic rational matrix group
## arg[2] ..... optional prime p
##
## OUT: Normal subgroup of finite index,
## actually the p-congruence subgroup
##
InstallGlobalFunction( POL_TriangNSGFI_NonAbelianPRMGroup , function( arg )
local p, d, gens_p,G, bound_derivedLength, pcgs_I_p, gens_K_p,
comSeries, gens_K_p_m, gens, gens_K_p_mutableCopy, pcgs,
gensOfBlockAction, pcgs_nue_K_p, pcgs_GU, gens_U_p, pcgs_U_p,
recordSeries, radSeries, isTriang, H;
# setup
G := arg[1];
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 fail; 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 fail; 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 );
if isTriang then
return G;
fi;
# compositions series
Info( InfoPolenta, 1, "Compute the composition series ...");
comSeries := POL_CompositionSeriesByRadicalSeries( gens_K_p_mutableCopy,
d,
recordSeries.sersFullData,
1 );
if comSeries=fail then return fail; fi;
Info( InfoPolenta, 1,"finished.");
Info( InfoPolenta, 1, "The composition series has length ",
Length( comSeries ), "." );
Info( InfoPolenta, 2, "The composition series is" );
Info( InfoPolenta, 2, comSeries );
Info( InfoPolenta, 1, " " );
# induce K_p to the factors of the composition series
gensOfBlockAction := POL_InducedActionToSeries(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",
" for the induced action of the kernel to the composition series ...");
pcgs_nue_K_p := CPCS_AbelianSSBlocks_ClosedUnderConj( gens_K_p,
gens, comSeries );
if pcgs_nue_K_p = fail then return fail; fi;
Info(InfoPolenta,1,"finished.");
# update generators of K_p
gens_K_p := pcgs_nue_K_p.gens_K_p;
pcgs_nue_K_p := pcgs_nue_K_p.pcgs_nue_K_p;
Info( InfoPolenta, 1, "This polycyclic sequence has relative orders ",
pcgs_nue_K_p.relOrders, "." );
Info( InfoPolenta, 1, " " );
return POL_Group( gens_K_p, G );
end );
#############################################################################
##
#F POL_TriangNSGFI_PRMGroup( arg )
##
## arg[1] = G is a rational polycyclic rational matrix group
##
InstallGlobalFunction( POL_TriangNSGFI_PRMGroup , function( arg )
local G;
G := arg[1];
if IsAbelian( G ) then
return G;
else
if IsBound( arg[2] ) then
return POL_TriangNSGFI_NonAbelianPRMGroup( arg[1], arg[2] );
else
return POL_TriangNSGFI_NonAbelianPRMGroup( G );
fi;
fi;
end );
# this code has to be reviewed. In the current form we can only assure,
# that it returns normal subgroup generators for K_p.
#############################################################################
##
#M TriangNormalSubgroupFiniteInd( G )
##
## G is a matrix group over the Rationals.
## Returned is triangularizable normal subgroup of finite index
##
##
#InstallMethod( TriangNormalSubgroupFiniteInd, "for polycyclic matrix groups",
# true, [ IsMatrixGroup ], 0,
#function( G )
# local test;
# test := POL_IsMatGroupOverFiniteField( G );
# if IsBool( test ) then
# TryNextMethod();
# elif test = 0 then
# return POL_TriangNSGFI_PRMGroup(G );
# else
# TryNextMethod();
# fi;
#end) ;
#
#InstallOtherMethod( TriangNormalSubgroupFiniteInd,
# "for polycyclic matrix groups", true,
# [ IsMatrixGroup, IsInt], 0,
#function( G, p )
# local test;
# test := POL_IsMatGroupOverFiniteField( G );
# if IsBool( test ) then
# TryNextMethod();
# elif test = 0 then
# if not IsPrime(p) then
# Print( "Second argument must be a prime number.\n" );
# return fail;
# fi;
# return POL_TriangNSGFI_PRMGroup(G );
# else
# TryNextMethod();
# fi;
#
#end );
#############################################################################
##
#M SubgroupsUnipotentByAbelianByFinite( G )
##
## G is a matrix group over the Rationals.
## Returned is triangularizable normal subgroup K of finite index
## and an unipotent normal subgroup U of K such that K/U is abelian.
##
InstallMethod( SubgroupsUnipotentByAbelianByFinite,
"for polycyclic matrix groups (Polenta)",
true, [ IsMatrixGroup ], 0,
function( G )
local cpcs, U_p, K_p;
if not IsRationalMatrixGroup( G ) then
TryNextMethod( );
fi;
cpcs := CPCS_PRMGroup( G );
if cpcs = fail then return fail; fi;
if IsAbelian( G ) then
U_p := cpcs.pcgs_U_p.pcs;
return rec( T := G , U := POL_Group( U_p, G ));
else
U_p := cpcs.pcgs_U_p.pcs;
# check if G is triangularizable
if Length( cpcs.pcgs_GU.pcgs_I_p.gens ) = 0 then
#G triangularizable
return rec( T := G, U := POL_Group( U_p, G ));
else
#G not triangularizable
K_p := cpcs.pcgs_GU.preImgsNue;
K_p := Concatenation( K_p, U_p );
return rec( T := POL_Group( K_p, G ), U := POL_Group( U_p, G ));
fi;
fi;
end );
InstallOtherMethod( SubgroupsUnipotentByAbelianByFinite ,
"for polycyclic matrix groups (Polenta)", true,
[ IsMatrixGroup, IsInt], 0,
function( G,p )
local cpcs, U_p, K_p;
if not IsRationalMatrixGroup( G ) then
TryNextMethod( );
fi;
cpcs := CPCS_PRMGroup( G,p );
if cpcs = fail then return fail; fi;
if IsAbelian( G ) then
U_p := cpcs.pcgs_U_p.pcs;
return rec( T := G , U := POL_Group( U_p, G ));
else
U_p := cpcs.pcgs_U_p.pcs;
# check if G is triangularizable
if Length( cpcs.pcgs_GU.pcgs_I_p.gens ) = 0 then
#G triangularizable
return rec( T := G, U := POL_Group( U_p, G ));
else
#G not triangularizable
K_p := cpcs.pcgs_GU.preImgsNue;
K_p := Concatenation( K_p, U_p );
return rec( T := POL_Group( K_p, G ), U := POL_Group( U_p, G ));
fi;
fi;
end );
#############################################################################
##
#E
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