Reconstructing structures with the strong small index property up to bi-definability

by Paolini and Shelah. [PaSh:1109]

Let mathbf {K} be the class of countable structures M with the strong small index property and locally finite algebraicity, and mathbf {K}_* the class of M in mathbf {K} such that acl_M({a}) = {a} for every a in M . For homogeneous M in mathbf {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 mathbf {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 aleph_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 for all exists-interpretation technique of [7]. %[rubin]. Finally, we show that every finite group can be realized as the outer automorphism group of Aut(M) for some countable aleph_0-categorical homogeneous structure M with the strong small index property and no algebraicity.


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