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The small index property for \(\omega\)-stable \(\omega\)-categorical structures and for the random graph. (English) Zbl 0788.03039

This paper gives a criterion involving existence of many generic sequences of automorphisms for a countable structure to have the small index property. It is used to show that (i) any \(\omega\)-stable \(\omega\)- categorical structure, and (ii) the random graph has the small index property. The same technique is also used to show that the automorphism group of such a structure is not the union of a countable chain of proper subgroups.

MSC:

03C15 Model theory of denumerable and separable structures
05C80 Random graphs (graph-theoretic aspects)
03C45 Classification theory, stability, and related concepts in model theory
03C35 Categoricity and completeness of theories
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