Publishing house / Journals and Serials / Fundamenta Mathematicae / All issues

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.