We give a criterion involving existence of many generic sequences of automorphisms for a countable structure to have the small index property. We use it to show that (i) any ω-stable ω-categorical structure, and (ii) the random graph has the small index property. We also show that the automorphism group of such a structure is not the union of a countable chain of proper subgroups.

Abstract. A class of structures C is said to have the extension property for partial automorphisms (EPPA) if, whenever C 1 and C 2 are structures in C, C 1 finite, C 1 ⊆ C 2 , and p 1 , p 2 , . . . , pn are partial automorphisms of C 1 extending to automorphisms of C 2 , then there exist a finite structure C 3 in C and automorphisms α 1 , α 2 , . . . , αn of C 3 extending the p i . We will prove that some classes of structures have the EPPA and show the equivalence of these kinds of results with problems related with the profinite topology on free groups. In particular, we will give a generalisation of the theorem, due to Ribes and Zalesskiȋ stating that a finite product of finitely generated subgroups is closed for this topology.