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On the Reconstruction of ℵ 0 ‐Categorical Structures from their Automorphism Groups
Author(s) -
Rubin Matatyahu
Publication year - 1994
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/s3-69.2.225
Subject(s) - mathematics , categorical variable , combinatorics , automorphism , permutation group , countable set , automorphism group , group (periodic table) , discrete mathematics , graph automorphism , permutation (music) , graph , line graph , voltage graph , statistics , physics , chemistry , organic chemistry , acoustics
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.