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#############################################################################
##
#W isom.gi POLENTA package Bjoern Assmann
##
## Methods for the calculation of
## isomorphisms from matrix groups to pcp-presentations
##
#Y 2003
##
#############################################################################
##
#F POL_IsomorphismToMatrixGroup_infinite
##
POL_IsomorphismToMatrixGroup_infinite := function( arg )
local CPCS, pcp, H, nat,G;
G := arg[1];
# calculate a constructive pc-sequence
if Length( arg ) = 2 then
CPCS := CPCS_PRMGroup( G, arg[2] );
else
CPCS := CPCS_PRMGroup( G );
fi;
if CPCS = fail then return fail; fi;
Info( InfoPolenta, 1, " " );
Info( InfoPolenta, 1,"Compute the relations of the polycyclic\n",
" presentation of the group ..." );
pcp := POL_SetPcPresentation_infinite( CPCS );
Info( InfoPolenta, 1,"finished." );
Info( InfoPolenta, 1, " " );
Info( InfoPolenta, 1,"Construct the polycyclic presented group ..." );
H := PcpGroupByCollector( pcp );
Info( InfoPolenta, 1,"finished.");
Info( InfoPolenta, 1, " " );
Info( InfoPolenta, 1,"Construct the isomorphism on the polycyclic\n",
" presented group ..." );
nat := GroupHomomorphismByImagesNC( G, H, CPCS.pcs, AsList(Pcp(H)) );
Info( InfoPolenta, 1,"finished.");
# add infos
SetIsBijective( nat, true );
SetIsIsomorphismByPolycyclicMatrixGroup( nat, true );
nat!.CPCS := CPCS;
return nat;
end;
#############################################################################
##
#F POL_IsomorphismToMatrixGroup_finite
##
POL_IsomorphismToMatrixGroup_finite := function( G )
local CPCS, pcp, H, nat, gens, d, pcs, bound_derivedLength;
Info( InfoPolenta, 1,"Determine a constructive polycyclic sequence\n",
" for the input group ...");
# 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;
CPCS := CPCS_finite_word( gens, bound_derivedLength );
if CPCS = fail then return fail; fi;
Info( InfoPolenta, 1, "finished." );
Info( InfoPolenta, 1, " " );
Info( InfoPolenta, 1,"Compute the relations of the polycyclic\n",
" presentation of the group ..." );
pcp := POL_SetPcPresentation_finite( CPCS );
Info( InfoPolenta, 1,"finished." );
Info( InfoPolenta, 1, " " );
Info( InfoPolenta, 1,"Construct the polycyclic presented group ..." );
H := PcpGroupByCollector( pcp );
Info( InfoPolenta, 1,"finished.");
Info( InfoPolenta, 1, " " );
# new generating set for G
pcs := Reversed(CPCS.gens);
Info( InfoPolenta, 1,"Construct the isomorphism on the polycyclic\n",
" presented group ..." );
nat := GroupHomomorphismByImagesNC( G, H, pcs, AsList(Pcp(H)) );
Info( InfoPolenta, 1,"finished.");
Info( InfoPolenta, 1, " " );
# add infos
SetIsBijective( nat, true );
SetIsIsomorphismByFinitePolycyclicMatrixGroup( nat, true );
nat!.CPCS := CPCS;
return nat;
end;
#############################################################################
##
#M Create isom to pcp group
##
InstallMethod( IsomorphismPcpGroup,
"for matrix groups over a finite field (Polenta)", true,
[ IsFFEMatrixGroup ], 0,
POL_IsomorphismToMatrixGroup_finite );
InstallMethod( IsomorphismPcpGroup,
"for rational matrix groups (Polenta)", true,
[ IsRationalMatrixGroup ], 0,
POL_IsomorphismToMatrixGroup_infinite );
## Enforce rationality check for cyclotomic matrix groups
RedispatchOnCondition( IsomorphismPcpGroup, true,
[ IsCyclotomicMatrixGroup ], [ IsRationalMatrixGroup ],
RankFilter(IsCyclotomicMatrixGroup) );
InstallOtherMethod( IsomorphismPcpGroup,
"for matrix groups (Polenta)", true,
[IsCyclotomicMatrixGroup, IsInt], 0,
function( G, p )
if IsRationalMatrixGroup( G ) then
if not IsPrime(p) then
Print( "Second argument must be a prime number.\n" );
return fail;
fi;
return POL_IsomorphismToMatrixGroup_infinite( G, p );
fi;
TryNextMethod();
end);
#############################################################################
##
#M Images under IsomorphismByPolycyclicMatrixGroup
##
InstallMethod( ImagesRepresentative,
"for isom by matrix groups (Polenta)",
FamSourceEqFamElm,
[IsGroupGeneralMappingByImages and IsIsomorphismByPolycyclicMatrixGroup, IsMultiplicativeElementWithInverse],
0,
function( nat, h )
local H, e, CPCS;
CPCS := nat!.CPCS;
H := Range( nat );
e := ExponentVector_CPCS_PRMGroup( h, CPCS );
if e=fail then return fail; fi;
if Length(e)=0 then return OneOfPcp( Pcp( H ) );fi;
return MappedVector( e, Pcp(H) );
end);
#############################################################################
##
#M Images under IsomorphismByFinitePolycyclicMatrixGroup
##
InstallMethod( ImagesRepresentative,
"for isom by finite matrix groups (Polenta)",
FamSourceEqFamElm,
[IsGroupGeneralMappingByImages and IsIsomorphismByFinitePolycyclicMatrixGroup, IsMultiplicativeElementWithInverse],
0,
function( nat, h )
local H, e, CPCS;
CPCS := nat!.CPCS;
H := Range( nat );
e := ExponentvectorPcgs_finite( CPCS, h );
if e=fail then return fail; fi;
if Length(e)=0 then return OneOfPcp( Pcp( H ) );fi;
return MappedVector( e, Pcp(H) );
end);
#############################################################################
##
#E
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2026-03-28
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