# Reconstructing Structures with the Strong Small Index Property up to Bi-Definability

## TL;DR

This paper demonstrates that for certain countable structures with specific properties, automorphism groups uniquely determine the structures up to bi-definability, and shows how any finite group can be realized as an outer automorphism group of such structures.

## Contribution

It introduces the expanded automorphism group for structures with the strong small index property and no algebraicity, establishing a bi-definability reconstruction result and realizing finite groups as outer automorphism groups.

## Key findings

- Automorphism groups are isomorphic iff the structures are bi-definably equivalent.
- Every finite group can be realized as an outer automorphism group of a suitable structure.
- Reconstruction up to bi-definability holds for alategorical structures with these properties.

## 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 [7]. 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.