# 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.
• 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.
• Current 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}
}