On the Reconstruction of ℵ 0 ‐Categorical Structures from their Automorphism Groups

A structure M has no algebraicity if, for every finite A ⊆ | M | and a ε | M | − A is not algebraic over A . Let K be the class of ℵ 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 ), N 〉 and 〈Aut( M ), M 〉 are isomorphic. We prove a theorem stating that if M ∈ K has a ‘weak ∀∃ 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, 〈Q, <〉, the family with 2 ℵ0 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 Г is the automorphism group of the universal countable n ‐coloured graph, then Out(Г) ≅ S n . Here S n denotes the symmetric group on n elements.