The general question of recovering a model (or its theory, or some appropriate AEC connected to it) from its automorphism group was originally studied by Hodges, Lascar, Shelah among others. The “Small Index Property” (SIP) emerged in their work as a bridge between topological and purely algebraic properties of those groups and ultimately as a tool to understand. I will speak about the SIP from two perspectives: first, I will present aspects of Lascar and Shelah’s proof of SIP for uncountable saturated structures (succinctly, if M is such a structure and G is a subgroup of Aut(M) with index less than or equal to λ=|M|, then G is open in the “λ-topology”) and then I will present an extension of the SIP to some abstract elementary classes. This second part of the lecture is joint work with Zaniar Ghadernezhad.