THE SMALL INDEX PROPERTY FOR OMEGA-STABLE OMEGA-CATEGORICAL STRUCTURES AND FOR THE RANDOM GRAPH

We give a criterion involving existence of many generic sequences of a utomorphisms for a countable structure to have the small index propert y. We use it to show that (i) any omega-stable omega-categorical struc ture, and (ii) the random graph have 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.