# Sh:1109

- Paolini, G., & Shelah, S. (2019).
*Reconstructing structures with the strong small index property up to bi-definability*. Fund. Math.,**247**(1), 25–35. arXiv: 1703.10498 DOI: 10.4064/fm640-9-2018 MR: 3984277 -
Abstract:

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 \forall \exists-interpretation technique of [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. - Version 2017-04-03_2 (8p) published version (12p)

Bib entry

@article{Sh:1109, author = {Paolini, Gianluca and Shelah, Saharon}, title = {{Reconstructing structures with the strong small index property up to bi-definability}}, journal = {Fund. Math.}, fjournal = {Fundamenta Mathematicae}, volume = {247}, number = {1}, year = {2019}, pages = {25--35}, issn = {0016-2736}, mrnumber = {3984277}, mrclass = {20B27 (03C15 03C35)}, doi = {10.4064/fm640-9-2018}, note = {\href{https://arxiv.org/abs/1703.10498}{arXiv: 1703.10498}}, arxiv_number = {1703.10498} }