CoCalc -- mapping.tst

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GAP 4.8.9 installation with standard packages -- copy to your CoCalc project to get it
Project: cocalc-sagemath-dev-slelievre
Path: gap4r8 / tst / testinstall / mapping.tst
Views: ⁴¹⁸³⁴⁶
#############################################################################
##
#W  mapping.tst                 GAP library                     Thomas Breuer
##
##
#Y  Copyright (C)  1996,  Lehrstuhl D für Mathematik,  RWTH Aachen,  Germany
##
##  To be listed in testinstall.g
##
gap> START_TEST("mapping.tst");
gap> M:= GF(3);
GF(3)
gap> tuples:= List( Tuples( AsList( M ), 2 ), DirectProductElement );;
gap> Print(tuples,"\n");
[ DirectProductElement( [ 0*Z(3), 0*Z(3) ] ), DirectProductElement( [ 0*Z(3), 
    Z(3)^0 ] ), DirectProductElement( [ 0*Z(3), Z(3) ] ), 
  DirectProductElement( [ Z(3)^0, 0*Z(3) ] ), DirectProductElement( [ Z(3)^0, 
    Z(3)^0 ] ), DirectProductElement( [ Z(3)^0, Z(3) ] ), 
  DirectProductElement( [ Z(3), 0*Z(3) ] ), 
  DirectProductElement( [ Z(3), Z(3)^0 ] ), 
  DirectProductElement( [ Z(3), Z(3) ] ) ]
gap> map:= GeneralMappingByElements( M, M, [] );
<general mapping: GF(3) -> GF(3) >
gap> IsInjective( map );
true
gap> IsSingleValued( map );
true
gap> IsSurjective( map );
false
gap> IsTotal( map );
false
gap> map:= GeneralMappingByElements( M, M, tuples{ [ 1, 2, 4, 7 ] } );
<general mapping: GF(3) -> GF(3) >
gap> IsInjective( map );
false
gap> IsSingleValued( map );
false
gap> IsSurjective( map );
false
gap> IsTotal( map );
true
gap> inv:= InverseGeneralMapping( map );
InverseGeneralMapping( <general mapping: GF(3) -> GF(3) > )
gap> Print(AsList( UnderlyingRelation( inv ) ),"\n");
[ DirectProductElement( [ 0*Z(3), 0*Z(3) ] ), DirectProductElement( [ 0*Z(3), 
    Z(3)^0 ] ), DirectProductElement( [ 0*Z(3), Z(3) ] ), 
  DirectProductElement( [ Z(3)^0, 0*Z(3) ] ) ]
gap> IsInjective( inv );
false
gap> IsSingleValued( inv );
false
gap> IsSurjective( inv );
true
gap> IsTotal( inv );
false
gap> comp:= CompositionMapping( inv, map );
CompositionMapping( 
InverseGeneralMapping( <general mapping: GF(3) -> GF(3) > ),
 <general mapping: GF(3) -> GF(3) > )
gap> Print(AsList( UnderlyingRelation( comp ) ),"\n");
[ DirectProductElement( [ 0*Z(3), 0*Z(3) ] ), DirectProductElement( [ 0*Z(3), 
    Z(3)^0 ] ), DirectProductElement( [ 0*Z(3), Z(3) ] ), 
  DirectProductElement( [ Z(3)^0, 0*Z(3) ] ), DirectProductElement( [ Z(3)^0, 
    Z(3)^0 ] ), DirectProductElement( [ Z(3)^0, Z(3) ] ), 
  DirectProductElement( [ Z(3), 0*Z(3) ] ), 
  DirectProductElement( [ Z(3), Z(3)^0 ] ), 
  DirectProductElement( [ Z(3), Z(3) ] ) ]
gap> IsInjective( comp );
false
gap> IsSingleValued( comp );
false
gap> IsSurjective( comp );
true
gap> IsTotal( comp );
true
gap> anticomp:= CompositionMapping( map, inv );
CompositionMapping( <general mapping: GF(3) -> GF(3) >,
 InverseGeneralMapping( <general mapping: GF(3) -> GF(3) > ) )
gap> Print(AsList( UnderlyingRelation( anticomp ) ),"\n");
[ DirectProductElement( [ 0*Z(3), 0*Z(3) ] ), DirectProductElement( [ 0*Z(3), 
    Z(3)^0 ] ), DirectProductElement( [ Z(3)^0, 0*Z(3) ] ), 
  DirectProductElement( [ Z(3)^0, Z(3)^0 ] ) ]
gap> IsInjective( anticomp );
false
gap> IsSingleValued( anticomp );
false
gap> IsSurjective( anticomp );
false
gap> IsTotal( anticomp );
false
gap> t1:= DirectProductElement( [ (), () ] );;  t2:= DirectProductElement( [ (1,2), (1,2) ] );;
gap> g:= Group( (1,2) );;
gap> t:= TrivialSubgroup( g );;
gap> map1:= GeneralMappingByElements( g, g, [ t1, t2 ] );;
gap> map2:= GeneralMappingByElements( t, t, [ t1 ] );;
gap> IsMapping( map1 );
true
gap> IsMapping( map2 );
true
gap> com:= CompositionMapping( map2, map1 );;
gap> Source( com );
Group([ (1,2) ])
gap> Images( com, (1,2) );
[  ]
gap> IsTotal( com );
false
gap> IsSurjective( com );
true
gap> IsSingleValued( com );
true
gap> IsInjective( com );
true
gap> map:= GeneralMappingByElements( M, M, tuples{ [ 1, 4 ] } );
<general mapping: GF(3) -> GF(3) >
gap> IsInjective( map );
false
gap> IsSingleValued( map );
true
gap> IsSurjective( map );
false
gap> IsTotal( map );
false
gap> inv:= InverseGeneralMapping( map );
InverseGeneralMapping( <general mapping: GF(3) -> GF(3) > )
gap> AsList( UnderlyingRelation( inv ) );
[ DirectProductElement( [ 0*Z(3), 0*Z(3) ] ), DirectProductElement( [ 0*Z(3),
    Z(3)^0 ] ) ]
gap> IsInjective( inv );
true
gap> IsSingleValued( inv );
false
gap> IsSurjective( inv );
false
gap> IsTotal( inv );
false
gap> comp:= CompositionMapping( inv, map );
CompositionMapping( 
InverseGeneralMapping( <general mapping: GF(3) -> GF(3) > ),
 <general mapping: GF(3) -> GF(3) > )
gap> IsInjective( comp );
false
gap> IsSingleValued( comp );
false
gap> IsSurjective( comp );
false
gap> IsTotal( comp );
false
gap> ImagesSource( map );
[ 0*Z(3) ]
gap> PreImagesRange( map );
[ 0*Z(3), Z(3)^0 ]
gap> comp:= CompositionMapping( IdentityMapping( Range( map ) ), map );
<general mapping: GF(3) -> GF(3) >
gap> comp = IdentityMapping( Source( map ) ) * map;
true
gap> map = comp;
true
gap> comp = map;
true
gap> map = inv;
false
gap> inv = map;
false
gap> map < inv;
true
gap> inv < map;
false
gap> conj:= map ^ inv;
CompositionMapping( 
InverseGeneralMapping( <general mapping: GF(3) -> GF(3) > ),
 CompositionMapping( <general mapping: GF(3) -> GF(3) >,
 <general mapping: GF(3) -> GF(3) > ) )
gap> IsSubset( UnderlyingRelation( conj ), UnderlyingRelation( map ) );
true
gap> IsSubset( UnderlyingRelation( map ), UnderlyingRelation( conj ) );
false
gap> One( map );
IdentityMapping( GF(3) )
gap> Z(3) / IdentityMapping( GF(3) );
Z(3)
gap> map:= GeneralMappingByElements( M, M, tuples{ [ 1, 4, 8 ] } );
<general mapping: GF(3) -> GF(3) >
gap> IsInjective( map );
false
gap> IsSingleValued( map );
true
gap> IsSurjective( map );
false
gap> IsTotal( map );
true
gap> ImageElm( map, Z(3) );
Z(3)^0
gap> ImagesElm( map, Z(3) );
[ Z(3)^0 ]
gap> ImagesSet( map, [ 0*Z(3), Z(3) ] );
[ 0*Z(3), Z(3)^0 ]
gap> ImagesSet( map, GF(3) );
[ 0*Z(3), Z(3)^0 ]
gap> ImagesRepresentative( map, Z(3) );
Z(3)^0
gap> (0*Z(3)) ^ map;
0*Z(3)
gap> map:= InverseGeneralMapping( map );
InverseGeneralMapping( <mapping: GF(3) -> GF(3) > )
gap> Print(AsList( UnderlyingRelation( map ) ),"\n");
[ DirectProductElement( [ 0*Z(3), 0*Z(3) ] ), DirectProductElement( [ 0*Z(3), 
    Z(3)^0 ] ), DirectProductElement( [ Z(3)^0, Z(3) ] ) ]
gap> IsInjective( map );
true
gap> IsSingleValued( map );
false
gap> IsSurjective( map );
true
gap> IsTotal( map );
false
gap> PreImageElm( map, Z(3) );
Z(3)^0
gap> PreImagesElm( map, Z(3) );
[ Z(3)^0 ]
gap> PreImagesSet( map, [ 0*Z(3), Z(3) ] );
[ 0*Z(3), Z(3)^0 ]
gap> PreImagesSet( map, GF(3) );
[ 0*Z(3), Z(3)^0 ]
gap> PreImagesRepresentative( map, Z(3) );
Z(3)^0
gap> map:= GeneralMappingByElements( M, M, tuples{ [ 2, 6, 7 ] } );
<general mapping: GF(3) -> GF(3) >
gap> IsInjective( map );
true
gap> IsSingleValued( map );
true
gap> IsSurjective( map );
true
gap> IsTotal( map );
true
gap> ImageElm( map, Z(3) );
0*Z(3)
gap> ImagesElm( map, Z(3) );
[ 0*Z(3) ]
gap> ImagesSet( map, [ 0*Z(3), Z(3) ] );
[ 0*Z(3), Z(3)^0 ]
gap> ImagesSet( map, GF(3) );
[ 0*Z(3), Z(3)^0, Z(3) ]
gap> ImagesRepresentative( map, Z(3) );
0*Z(3)
gap> map:= InverseGeneralMapping( map );
InverseGeneralMapping( <mapping: GF(3) -> GF(3) > )
gap> Print(AsList( UnderlyingRelation( map ) ),"\n");
[ DirectProductElement( [ 0*Z(3), Z(3) ] ), DirectProductElement( [ Z(3)^0, 
    0*Z(3) ] ), DirectProductElement( [ Z(3), Z(3)^0 ] ) ]
gap> IsInjective( map );
true
gap> IsSingleValued( map );
true
gap> IsSurjective( map );
true
gap> IsTotal( map );
true
gap> PreImageElm( map, Z(3) );
0*Z(3)
gap> PreImagesElm( map, Z(3) );
[ 0*Z(3) ]
gap> PreImagesSet( map, [ 0*Z(3), Z(3) ] );
[ 0*Z(3), Z(3)^0 ]
gap> PreImagesSet( map, GF(3) );
[ 0*Z(3), Z(3)^0, Z(3) ]
gap> PreImagesRepresentative( map, Z(3) );
0*Z(3)
gap> ImagesSource( map );
[ 0*Z(3), Z(3)^0, Z(3) ]
gap> PreImagesRange( map );
[ 0*Z(3), Z(3)^0, Z(3) ]
gap> g := Group((1,2),(3,4));;
gap> i := IdentityMapping( g );;
gap> i2 := AsGroupGeneralMappingByImages(i);;
gap> j:=GroupGeneralMappingByImages(g,g,AsSSortedList(g),AsSSortedList(g));;
gap> i2 = j;
true
gap> A:=[[0,1,0],[0,0,1],[1,0,0]];;
gap> B:=[[0,0,1],[0,1,0],[-1,0,0]];;
gap> C:=[[E(4),0,0],[0,E(4)^(-1),0],[0,0,1]];;
gap> g2:=GroupWithGenerators([A,B,C]);;
gap> nice := NiceMonomorphism (g2);;
gap> d  := DerivedSubgroup (g2);;
gap> res := RestrictedMapping (nice, d);;
gap> IsGroupHomomorphism(res);
true
gap> IsInjective(res);        
true

# The following test verifies we handle pcgs with large primes in them efficiently.
# See also GitHub pull requests #576 and #578
gap> V := AsVectorSpace (GF(2), GF(2^17));;
gap> h := BlownUpMat(Basis(V), [[Z(2^17)]]);;
gap> ConvertToMatrixRep(h);;
gap> G := CyclicGroup (2^17-1);;
gap> H := Group (h);;
gap> hom := GroupHomomorphismByImagesNC (G, H, [G.1], [h]);;
gap> ImagesRepresentative(hom, G.1^2) = h^2;
true
gap> STOP_TEST( "mapping.tst", 13280000);

#############################################################################
##
#E
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