FreeGrps.gd

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#W  FreeGroups.gd            FGA package                    Christian Sievers
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##  Main declaration file for the FGA package
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#Y  2003 - 2012
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#A  FreeGeneratorsOfGroup( <G> )
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##  returns a list of free generators of the group <G>.
##  This is a minimal generating set, but is also guarantied to
##  be N-reduced.
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DeclareAttribute( "FreeGeneratorsOfGroup", IsFreeGroup );


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#A  RankOfFreeGroup( <G> )
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##  returns the rank of a free group.
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DeclareAttribute( "RankOfFreeGroup", IsFreeGroup );


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#A  FreeGroupAutomaton( <G> )
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##  returns the automaton representing <G>.
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DeclareAttribute( "FreeGroupAutomaton",
                  IsFreeGroup,
                  "mutable" );


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#A  FreeGroupExtendedAutomaton( <G> )
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##  return the extended automaton representing <G>.
##  The extra information is enough for a constructive membership test
##  with respect to the given generators of <G>.
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DeclareAttribute( "FreeGroupExtendedAutomaton", IsFreeGroup );


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#O  AsWordLetterRepInFreeGenerators( <g>, <G> )
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##  returns the unique list <l> representing a word in letter representation
##  such that
##  <g> = Product( <l>, 
##                 x -> FreeGeneratorsOfGroup(<G>)[AbsInt(x)]^(SignInt(x)),
##                 One(<G>) )
##  or fail, if <g> is not in <G>.
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DeclareOperation( "AsWordLetterRepInFreeGenerators",
    [ IsElementOfFreeGroup, IsFreeGroup ] );


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#O  AsWordLetterRepInGenerators( <g>, <G> )
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##  returns a list <l> representing a word in letter representation such that
##  <g> = Product( <l>, 
##                 x -> GeneratorsOfGroup(<G>)[AbsInt(x)]^(SignInt(x)),
##                 One(<G>) )
##  or fail, if <g> is not in <G>.
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DeclareOperation( "AsWordLetterRepInGenerators",
    [ IsElementOfFreeGroup, IsFreeGroup ] );



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#O  CyclicallyReducedWord( <g> )
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##  returns the the cyclically reduced form of <g>
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DeclareOperation( "CyclicallyReducedWord",
    [ IsElementOfFreeGroup ] );


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#F  CanComputeWithInverseAutomaton( <G> )
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##  indicates whether we can use inverse automata to compute with <G>.
##  We assume this is possible if <G> is a finitely generated free group,
##  hoping that we actually can get a generating set when needed.
##  This is not always true, but generally then there is also no other way.
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DeclareSynonym( "CanComputeWithInverseAutomaton",
                 IsFreeGroup and IsFinitelyGeneratedGroup );

InstallTrueMethod( CanComputeWithInverseAutomaton, HasFreeGroupAutomaton );


InstallTrueMethod( CanEasilyTestMembership, HasFreeGroupAutomaton );

InstallTrueMethod( CanComputeSizeAnySubgroup, IsFreeGroup );


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#E