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## Fundamenta Mathematicae

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

### Volume 247 / 2019

Fundamenta Mathematicae 247 (2019), 25-35 MSC: 20B27, 03C35, 03C15. DOI: 10.4064/fm640-9-2018 Published online: 12 April 2019

#### 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 $\mathop {\rm acl}_M(\{ a \}) = \{ a \}$ 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 $\mathop {\rm Aut}(M)$. We use this to prove that for $M, N \in \mathbf {K}_*$, $\mathop {\rm Aut}(M)$ and $\mathop {\rm Aut}(N)$ are isomorphic as abstract groups if and only if $(\mathop {\rm Aut}(M), M)$ and $(\mathop {\rm Aut}(N), N)$ are isomorphic as permutation groups. In particular, we deduce that for $\aleph _0$-categorical structures the combination of the strong small index property and no algebraicity implies reconstruction up to bi-definability, in analogy with Rubin’s (1994) well-known $\forall \exists$-interpretation technique. Finally, we show that every finite group can be realized as the outer automorphism group of $\mathop {\rm Aut}(M)$ for some countable $\aleph _0$-categorical homogeneous structure $M$ with the strong small index property and no algebraicity.

#### Authors

• Gianluca PaoliniDepartment of Mathematics “Giuseppe Peano”
University of Torino
Via Carlo Alberto 10
10123 Torino, Italy
e-mail
• Saharon ShelahEinstein Institute of Mathematics
The Hebrew University of Jerusalem
Edmond J. Safra Campus
Givat Ram, 9190401, Jerusalem
and
Department of Mathematics
Rutgers University
The State University of New Jersey
Hill Center–Busch Campus 110 