43.4 IsIsomorphism

IsIsomorphism( map )

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

A mapping is an isomorphism if it is a bijective homomorphism (see IsBijection, 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> IsIsomorphism( p4 );
    false
    gap> p5 := MappingByFunction( g, g, x -> x^5 );
    MappingByFunction( g, g, function ( x )
        return x ^ 5;
    end )
    gap> IsIsomorphism( p5 );
    true 

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

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

Previous Up Top Next
Index