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Preprint Number 1197

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1197. Gianluca Paolini and Saharon Shelah
Reconstructing Structures with the Strong Small Index Property up to Bi-Definability

Submission date: 30 March 2017


Let K be the class of countable structures M with the strong small index property and locally finite algebraicity, and K_* the class of M in K such that acl_M(a) = { a } for every a in M. For homogeneous M in K, we introduce what we call the expanded group of automorphisms of M, and show that it is second-order definable in Aut(M). We use this to prove that for M, N in K_*, Aut(M) and Aut(N) are isomorphic as abstract groups if and only if (Aut(M), M) and (Aut(N), N) are isomorphic as permutation groups. In particular, we deduce that for ℵ_0-categorical structures the combination of strong small index property and no algebraicity implies reconstruction up to bi-definability, in analogy with Rubin's well-known ∀∃-interpretation technique of [7]. Finally, we show that every finite group can be realized as the outer automorphism group of Aut(M) for some countable ℵ_0-categorical homogeneous structure M with the strong small index property and no algebraicity.

Mathematics Subject Classification: 20B27, 03C35, 03C15

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Full text arXiv 1703.10498: pdf, ps.

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