43.3 IsEpimorphism

IsEpimorphism( map )

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

A mapping is an epimorphism if it is an surjective homomorphism (see IsSurjective, IsHomomorphism).

    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> IsEpimorphism( p4 );
    false
    gap> p5 := MappingByFunction( g, g, x -> x^5 );
    MappingByFunction( g, g, function ( x )
        return x ^ 5;
    end )
    gap> IsEpimorphism( p5 );
    true 

IsEpimorphism first test if the flag map.isEpimorphism is bound. If the flag is bound, it returns this value. Otherwise it calls map.operations.IsEpimorphism( map ), remembers the returned value in map.isEpimorphism, and returns it.

The default function called this way is MappingOps.IsEpimorphism, which calls the functions IsSurjective and IsHomomorphism, and returns the logical and of the results. This function is seldom overlaid, because all the interesting work is done in IsSurjective and IsHomomorphism.

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