GAP Manual: 42.8. Image

Image( map, elm )

In this form Image returns the image of the element elm of the source of the mapping map under map, i.e., the element of the range to which map maps elm. Note that map must be a single valued mapping, a multi valued mapping is not allowed (see Images). This can also be expressed as elm ^ map (see Operations for Mappings). 

    gap> g := Group( (1,2,3,4), (2,4), (5,6,7) );;  g.name := "g";;
    gap> p4 := MappingByFunction( g, g, x -> x^4 );
    MappingByFunction( g, g, function ( x )
        return x ^ 4;
    end )
    gap> Image( p4, (2,4)(5,6,7) );
    (5,6,7)
    gap> p5 := MappingByFunction( g, g, x -> x^5 );
    MappingByFunction( g, g, function ( x )
        return x ^ 5;
    end )
    gap> Image( p5, (2,4)(5,6,7) );
    (2,4)(5,7,6)

Image( map, elms )

In this form Image returns the image of the set of elements elms of the source of the mapping map under map, i.e., set of images of the elements in elms. elms may be a proper set (see Sets) or a domain (see Domains). The result will be a subset of the range of map, either as a proper set or as a domain. Again map must be a single valued mapping, a multi valued mapping is not allowed (see Images). 

    gap> Image( p4, Subgroup( g, [ (2,4), (5,6,7) ] ) );
    [ (), (5,6,7), (5,7,6) ]
    gap> Image( p5, [ (5,6,7), (2,4) ] );
    [ (5,7,6), (2,4) ]

Note that in the first example, the result is returned as a proper set, even though it is mathematically a subgroup. This is because p4 is not known to be a homomorphism, even though it is one.

Image( map )

In this form Image returns the image of the mapping map, i.e., the subset of element of the range of map that are actually values of map. Note that in this case the argument may also be a multi valued mapping. 

    gap> Image( p4 );
    [ (), (5,6,7), (5,7,6) ]
    gap> Image( p5 ) = g;
    true

Image firsts checks in which form it is called.

In the first case it calls map.operations.ImageElm( map, elm ) and returns this value.

The default function called this way is MappingOps.ImageElm, which checks that map is indeed a single valued mapping, calls Images( map, elm ), and returns the single element of the set returned by Images. Look in the index under Image to see for which mappings this function is overlaid.

In the second case it calls map.operations.ImageSet( map, elms ) and returns this value.

The default function called this way is MappingOps.ImageSet, which checks that map is indeed a single valued mapping, calls Images( map, elms ), and returns this value. Look in the index under Image to see for which mappings this function is overlaid.

In the third case it tests if the field map.image is bound. If this field is bound, it simply returns this value. Otherwise it calls map.operations.ImageSource( map ), remembers the returned value in map.image, and returns it.

The default function called this way is MappingOps.ImageSource, which calls Images( map, map.source ), and returns this value. This function is seldom overlaid, since all the work is done by map.operations.ImagesSet.

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