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

by Paolini and Shelah. [PaSh:1109]

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 for all 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.

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