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

Abstract

Let $\mathbf{K}$ be the class of countable structures $M$ with the strong small index property and locally finite algebraicity, and ${\mathbf{K}}_{\ast }$ the class of $M\in \mathbf{K}$ such that $ac{l}_M(\{a\})=\{a\}$ 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 $M,N\in {\mathbf{K}}_{\ast }$, $Aut(M)$ 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.