On the reconstruction of x₀-categorical structures from their automorphism groups

A structure M has no algebraicity if, for every finite A |M| and a |M| A is not algebraic over A. Let K be the class of ₀-categorical structures without algebraicity, and M K. The structure M is group-categorical in K if for every N K the following holds: if Aut(N) Aut(M), then the permutation groups Aut(N), and Aut(M), are isomorphic. prove a theorem stating that if has a interpretation(see Definition 2.1), then it is group-categorical in K. By applying the above theorem we prove that the random graph, the universal poset, the universal tournament, <, the family with 2 many directed graphs constructed by Henson in [10], all trees belonging to K, and various other structures, are group-categorical in K. It is easy to determine the outer automorphism group of the automorphism group of a group-categorical structure. For example, if is the automorphism group of the universal countable n-coloured graph, then Out S_n. Here S_n denotes the symmetric group on n elements.