%
\Chapter{Arbitrary functions on groups: EndoMappings}
%
An *endomapping* is a mapping with equal source and range, say $G$,
where $G$ is a group. An endomapping on $G$ then acts on $G$ by
*transforming* each element of $G$ into (precisely one) element of $G$.
Endomappings are special cases of Mappings.
Endomappings are created by the constructor functions
`EndoMappingByPositionList', `EndoMappingByFunction', `IdentityEndoMapping',
`ConstantEndoMapping', and are represented as mappings.
The functions described in
this section can be found in the file `grptfms.g?'.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Defining endo mappings}
\>EndoMappingByPositionList ( <G>, <list> )
The constructor function `EndoMappingByPositionList' returns the
the endomapping that maps the i-th element of the group (in the
ordering given by AsSortedList)
to the i-th element of list.
\beginexample
gap> G := GTW4_2;
4/2
gap> t1 := EndoMappingByPositionList ( G, [1, 2, 4, 4] );
<mapping: 4/2 -> 4/2 >
\endexample
\>EndoMappingByFunction( <G>, <fun> )
The constructor function `EndoMappingByFunction' returns the
function <fun> that maps elements of the group <G> into <G> as an
endomapping.
\beginexample
gap> t2 := EndoMappingByFunction ( GTW8_2, g -> g^-1 );
<mapping: 8/2 -> 8/2 >
gap> IsGroupHomomorphism ( t2 );
true
gap> t3 := EndoMappingByFunction ( GTW6_2, g -> g^-1 );
<mapping: 6/2 -> 6/2 >
gap> IsGroupHomomorphism ( t3 );
false
\endexample
`EndoMappings' and `GroupGeneralMappings' are different
kinds of objects in {\GAP}: `GroupGeneralMappings' model homomorphisms between
two different groups, whereas `EndoMappings' model nonlinear functions
on one group.
However, `GroupGeneralMappings' can be transformed into
`Endomappings' if they have equal source and range.
\>AsEndoMapping( <map> )
The constructor function `AsEndoMapping' returns the mapping
as an endomapping.
\beginexample
gap> G1 := Group ((1,2,3), (1, 2));
Group([ (1,2,3), (1,2) ])
gap> G2 := Group ((2,3,4), (2, 3));
Group([ (2,3,4), (2,3) ])
gap> f1 := IsomorphismGroups ( G1, G2 );
[ (1,2,3), (1,2) ] -> [ (2,3,4), (2,3) ]
gap> f2 := IsomorphismGroups ( G2, G1 );
[ (2,3,4), (2,3) ] -> [ (1,2,3), (1,2) ]
gap> AsEndoMapping ( CompositionMapping ( f1, f2 ) );
<mapping: Group( [ (2,3,4), (2,3) ] ) -> Group( [ (2,3,4), (2,3)
] ) >
\endexample
`EndoMappings' and `GroupGeneralMappings' are two completely different
kinds of objects in {\GAP}, but they can be transformed into one
another.
\>AsGroupGeneralMappingByImages( <endomap> )
`AsGroupGeneralMappingByImages' returns the
`GroupGeneralMappingByImages' that acts on the group the same way as
the endomapping <endomap>. It only makes sense to use this function
for endomappings that are group endomorphisms.
\beginexample
gap> m := IdentityEndoMapping ( GTW6_2 );
<mapping: 6/2 -> 6/2 >
gap> AsGroupGeneralMappingByImages ( m );
[ (1,2), (1,2,3) ] -> [ (1,2), (1,2,3) ]
\endexample
\>IsEndoMapping( <obj> )
`IsEndoMapping' returns `true' if the object <obj> is an endomapping
and `false' otherwise.
\beginexample
gap> IsEndoMapping ( InnerAutomorphisms ( GTW6_2 ) [3] );
true
\endexample
\>IdentityEndoMapping( <G> )
`IdentitEndoMapping' is the counterpart to the {\GAP} standard
library function `IdentityMapping'. It returns the identity
transformation on the group <G>.
\beginexample
gap> AsList ( UnderlyingRelation ( IdentityEndoMapping ( Group ((1,2,3,4)) ) ) );
[ DirectProductElement( [ (), () ] ), DirectProductElement( [ (1,2,3,4), (1,2,
3,4) ] ), DirectProductElement( [ (1,3)(2,4), (1,3)(2,4) ] ),
DirectProductElement( [ (1,4,3,2), (1,4,3,2) ] ) ]
\endexample
\>ConstantEndoMapping( <G>, <g> )
`ConstantEndoMapping' returns the endomapping on the group
which maps everything to the group element <g> of <G>.
\beginexample
gap> C3 := CyclicGroup (3);
<pc group of size 3 with 1 generator>
gap> m := ConstantEndoMapping (C3, AsSortedList (C3) [2]);
MappingByFunction( <pc group of size 3 with
1 generator>, <pc group of size 3 with
1 generator>, function( x ) ... end )
gap> List (AsList (C3), x -> Image (m, x));
[ f1, f1, f1 ]
\endexample
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Properties of endo mappings}
\>IsIdentityEndoMapping( <endomap> )
`IsIdentityEndoMapping' returns `true' if <endomap> is the identity
function on a group.
\beginexample
gap> IsIdentityEndoMapping (EndoMappingByFunction (
> AlternatingGroup ( [1..5] ), x -> x^31));
true
\endexample
\>IsConstantEndoMapping( <endomap> )
`IsConstantEndoMapping' returns `true' if the endomapping
<endomap> is constant and `false' otherwise.
\beginexample
gap> C3 := CyclicGroup ( 3 );
<pc group of size 3 with 1 generator>
gap> IsConstantEndoMapping ( EndoMappingByFunction ( C3, x -> x^3 ));
true
\endexample
\>IsDistributiveEndoMapping( <endomap> )
A mapping $t$ on an (additively written) group $G$ is called
*distributive* if for all elements $x$ and $y$ in $G$:java.lang.NullPointerException
$t(x+y) = t(x) + t(y)$.
The function `IsDistributiveEndoMapping' returns the according
boolean value `true' or `false'.
\beginexample
gap> G := Group ( (1,2,3), (1,2) );
Group([ (1,2,3), (1,2) ])
gap> IsDistributiveEndoMapping ( EndoMappingByFunction ( G, x -> x^3));
false
gap> IsDistributiveEndoMapping ( EndoMappingByFunction ( G, x -> x^7));
true
\endexample
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Operations for endo mappings}
While the composition operator `\*' is applicable to mappings and
transformations, the operation `+' (pointwise addition of the images) can
only be applied to transformations.
The product operator `\*' returns the transformation which is obtained
from the transformations <t1> and <t2> by composition of <t1> and <t2>
(i.e. performing <t2> *after* <t1>).
\beginexample
gap> t1 := ConstantEndoMapping ( GTW2_1, ());
MappingByFunction( 2/1, 2/1, function( x ) ... end )
gap> t2 := ConstantEndoMapping (GTW2_1, (1, 2));
MappingByFunction( 2/1, 2/1, function( x ) ... end )
gap> List ( AsList ( GTW2_1 ), x -> Image ( t1 * t2, x ));
[ (1,2), (1,2) ]
\endexample
The add operator `+' returns the endomapping which is obtained
from the endomappings <t1> and <t2> by pointwise addition
of <t1> and <t2>. (Note that in this context addition means that
for every place $x$ in the source of <t1> and <t2>,
{\GAP} performs the operation `p\ \*\ q', where
`p' is the image of at $x$ and `q' is the image of <t2> at $x$.)
The subtract operator `-' returns the endomapping which is
obtained from the endomappings <t1> and <t2> by pointwise
subtraction of <t1> and <t2>. (Note that in this context subtraction
means performing the {\GAP} operation `p\ \*\ q\^{}(-1)',
where
`p' is the image of at a place $x$ and `q' is the image of <t2> at $x$.)
\beginexample
gap> G := SymmetricGroup ( 3 );
Sym( [ 1 .. 3 ] )
gap> invertingOnG := EndoMappingByFunction ( G, x -> x^-1 );
<mapping: SymmetricGroup( [ 1 .. 3 ] ) -> SymmetricGroup(
[ 1 .. 3 ] ) >
gap> identityOnG := IdentityEndoMapping (G);
<mapping: SymmetricGroup( [ 1 .. 3 ] ) -> SymmetricGroup(
[ 1 .. 3 ] ) >
gap> AsSortedList ( G );
[ (), (2,3), (1,2), (1,2,3), (1,3,2), (1,3) ]
gap> List ( AsSortedList (G),
> x -> Image ( identityOnG * invertingOnG, x ));
[ (), (2,3), (1,2), (1,3,2), (1,2,3), (1,3) ]
gap> List ( AsSortedList (G),
> x -> Image ( identityOnG + invertingOnG, x ));
[ (), (), (), (), (), () ]
gap> IsIdentityEndoMapping ( - invertingOnG );
true
gap> - invertingOnG = identityOnG;
true
\endexample
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Nicer ways to print a mapping}
\> GraphOfMapping( <mapping> )
`GraphOfMapping' returns the set of all pairs (x,m(x)), where
x lies in the source of the mapping. In particular, it returns
List (Source (m), x -> [x, Image (m, x)]);
\beginexample
gap> G := SymmetricGroup ( 3 );
Sym( [ 1 .. 3 ] )
gap> m := ConstantEndoMapping (G, (1,2,3)) + IdentityEndoMapping( G );
MappingByFunction( Sym( [ 1 .. 3 ] ), Sym( [ 1 .. 3 ] ), function( g ) ... end )
gap> PrintArray( GraphOfMapping( m ) );
[ [ (), (1,2,3) ],
[ (2,3), (1,3) ],
[ (1,2), (2,3) ],
[ (1,2,3), (1,3,2) ],
[ (1,3,2), () ],
[ (1,3), (1,2) ] ]
\endexample
\> PrintAsTerm( <mapping> )
If <mapping> is a polynomial function on its source then `PrintAsTerm'
prints a polynomial that induces the mapping <mapping>.
\beginexample
gap> G := SymmetricGroup ( 3 );
Sym( [ 1 .. 3 ] )
gap> p := Random( PolynomialNearRing( G ) );
<mapping: SymmetricGroup( [ 1 .. 3 ] ) -> SymmetricGroup( [ 1 .. 3 ] ) >
gap> PrintAsTerm( p );
g1 - x - 2 * g1 - g2 - x - g1 - g2 + g1 - x - 2 * g1 -
g2 - x - g1 - g2 - 3 * x + g1
gap> GeneratorsOfGroup( G );
[ (1,2,3), (1,2) ]
\endexample
The expressions `g1' and `g2' stand for the first and secong generator
of the group <G> respectively. The result is not necessarily a
polynomial of minimal length.
%%% Local Variables:
%%% mode: latex
%%% TeX-master: t
%%% End:
¤ Dauer der Verarbeitung: 0.12 Sekunden
(vorverarbeitet)
¤
*© Formatika GbR, Deutschland
|
|
