This paper unifies and extends several recent nonparametric identification results on auctions, adverse selection or IV regression models by showing that all these models satisfy what we call an induction property. This property states that if the structural function is identified at a given point, then it is also identified at another point. Using results from group theory, we characterize the set where the structural function is identified under this property. The nature of this set depends on the dimensionality of the problem and on a property of the underlying group action, which is called freeness. Full identification can be achieved in the one dimension case under freeness.