42.4 IsSurjective

IsSurjective( map )

IsSurjective returns true if the mapping map is surjective and false otherwise. Signals an error if map is a multi valued mapping.

A mapping map is surjective if for each element img of the range there is at least one element elm of the source that map maps to img.

    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> IsSurjective( p4 );
    false
    gap> IsSurjective( InverseMapping( p4 ) );
    Error, <map> must be a single valued mapping
    gap> p5 := MappingByFunction( g, g, x -> x^5 );
    MappingByFunction( g, g, function ( x )
        return x ^ 5;
    end )
    gap> IsSurjective( p5 );
    true
    gap> IsSurjective( InverseMapping( p5 ) );
    true    # 'p5' is a bijection 

IsSurjective first tests if the flag map.isSurjective is bound. If the flag is bound, it returns this value. Otherwise it calls map.operations.IsSurjective( map ), remembers the returned value in map.isSurjective, and returns it.

The default function called this way is MappingOps.IsSurjective, which compares the sizes of the range and image of map, and returns true if they are equal (see Image). Look in the index under IsSurjective to see for which mappings this function is overlaid.

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