AutGrp.gd

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#W  AutGrp.gd                FGA package                    Christian Sievers
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##  Methods for automorphism groups of free groups
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#Y  2003 - 2012
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#F  IsAutomorphismGroupOfFreeGroup( <obj> )
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##  returns true if <obj> is the automorphism group of a free group.
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DeclareFilter( "IsAutomorphismGroupOfFreeGroup" );

InstallTrueMethod( IsAutomorphismGroup,
                   IsAutomorphismGroupOfFreeGroup );


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#F  FreeGroupEndomorphismByImages( <group>, <images> )
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##  returns the endomorphism of <group> that maps the generators of <group>
##  to <images>.
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DeclareGlobalFunction( "FreeGroupEndomorphismByImages" );


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#F  FreeGroupAutomorphismsGeneratorO( <group> )
#F  FreeGroupAutomorphismsGeneratorP( <group> )
#F  FreeGroupAutomorphismsGeneratorU( <group> )
#F  FreeGroupAutomorphismsGeneratorS( <group> )
#F  FreeGroupAutomorphismsGeneratorT( <group> )
#F  FreeGroupAutomorphismsGeneratorQ( <group> )
#F  FreeGroupAutomorphismsGeneratorR( <group> )
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##  These functions return the automorphism of <group> which maps the
##  generators [<x1>, <x2>, ..., <xn>] to
##  O : [<x1>^-1 , <x2>,       ..., <xn>   ]        (n>=1)
##  P : [<x2>    , <x1>, <x3>, ..., <xn>   ]        (n>=2)
##  U : [<x1><x2>, <x2>, <x3>, ..., <xn>   ]        (n>=2)
##  S : [<x2>^-1, <x3>^-1, ..., <xn>^-1, <x1>^-1 ]  (n>=1)
##  T : [<x2>    , <x1>^-1, <x3>, ..., <xn>]        (n>=2)
##  Q : [<x2>, <x3>, ..., <xn>, <x1> ]              (n>=2)
##  R : [<x2>^-1, <x1>, <x3>, <x4>, ..., 
##       <x{n-2}>, <xn><x{n-1}>^-1, <x{n-1}>^-1]    (n>=4)
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DeclareGlobalFunction( "FreeGroupAutomorphismsGeneratorO" );
DeclareGlobalFunction( "FreeGroupAutomorphismsGeneratorP" );
DeclareGlobalFunction( "FreeGroupAutomorphismsGeneratorU" );
DeclareGlobalFunction( "FreeGroupAutomorphismsGeneratorS" );
DeclareGlobalFunction( "FreeGroupAutomorphismsGeneratorT" );
DeclareGlobalFunction( "FreeGroupAutomorphismsGeneratorQ" );
DeclareGlobalFunction( "FreeGroupAutomorphismsGeneratorR" );

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#F FGA_CheckRank( <group>, <minrank> )
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## Checks whether <group> has rank at least <minrank>, and signals an
## error otherwise (helper function for FreeGroupAutomorphismsGenerator*)
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DeclareGlobalFunction( "FGA_CheckRank" );

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#E