42.5 IsBijection

IsBijection( map )

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

A mapping map is a bijection if for each element img of the range there is exactly one element elm of the source that map maps to img. We also say that map is bijective.

    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> IsBijection( p4 );
    false
    gap> IsBijection( 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> IsBijection( p5 );
    true
    gap> IsBijection( InverseMapping( p5 ) );
    true    # 'p5' is a bijection 

IsBijection first tests if the flag map.isBijection is bound. If the flag is bound, it returns its value. Otherwise it calls map.operations.IsBijection( map ), remembers the returned value in map.isBijection, and returns it.

The default function called this way is MappingOps.IsBijection, which calls IsInjective and IsSurjective, and returns the logical and of the results. This function is seldom overlaid, because all the interesting work is done by IsInjective and IsSurjective.

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