Countable structures with a fixed group of automorphisms

Abstract

We prove that, given a countable groupG, the set of countable structures (for a suitable languageL)U G whose automorphism group is isomorphic toG is a complete coanalytic set and ifG ≄H thenU G is Borel inseparable fromU H . We give also a model theoretic interpretation of this result. We prove, in contrast, that the set of countable structures forL whose automorphism group is isomorphic to ℤ p ℕ ,p a prime number, is Π 1 1 &σ 1 1 -complete.