Möbius Transformations in the Second Symmetric Product of ℂ

Let F2(C) denote the second symmetric product of the complex plane C endowed with the Hausdorff topology, i.e., F2(C)={A⊂C:|A|≤2,A≠∅}. In this paper, we extended the concept of Möbius transformations to F2(C). More precisely, given a Möbius transformation T of C, we define the map T˜({z,w})={T(z),T(w)} within F2(C). We describe some general properties of these maps, including the structure of their generators, characteristics related to transitivity, and the geometry of the conjugacy classes.