The map between a finite group acting on a set big S over the stabilizer of small s, in big S, to the orbit of small s has this property. If a function of this kind is also homomorphic, it is called an isomorphism. If a function of this kind exists between two sets, they have the same cardinality, which is often used to prove they have the same number of elements. When a function of this type is applied to its own set, the permutation results. For any map from a set X to Y with this property, there must exist an inverse that additionally has this property. For 10 points, name this kind of function where every value in the domain has a unique, corresponding value in the range and vice versa, 
The map between a finite group acting on a set big S over the stabilizer of small s, in big S, to the orbit of small s has this property. If a function of this kind is also homomorphic, it is called an isomorphism. If a function of this kind exists between two sets, they have the same cardinality, which is often used to prove they have the same number of elements. When a function of this type is applied to its own set, the permutation results. For any map from a set X to Y with this property, there must exist an inverse that additionally has this property. For 10 points, name this kind of function where every value in the domain has a unique, corresponding value in the range and vice versa, making it both injective and surjective.
The map between a finite group acting on a set big S over the stabilizer of small s, in big S, to the orbit of small s has this property.
bijective function [or bijection; or one-to-one correspondent; or word forms of the preceding; or one-to-one and onto; do not accept "one-to-one" or "onto" or "injective" or "surjective"]