Given a theory, every definable set has a canonical parameter after introducing imaginary sorts. Many classical theories eliminate imaginaries so that it suffices to work in the real sort.
In this talk, we will present a general criterion that yields the failure of elimination of imaginaries due to equivalence classes arising as cosets of subgroups. We will illustrate the main ideas by considering the specific example of the theory of proper pairs of algebraically closed fields. Pillay and Vassiliev already proved that this theory does not eliminate imaginaries and our criterion gives an alternative proof.