{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,4,18]],"date-time":"2024-04-18T21:10:51Z","timestamp":1713474651925},"reference-count":20,"publisher":"Cambridge University Press (CUP)","issue":"4","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":4119,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[2002,12]]},"abstract":"<jats:p>Which groups are isomorphic to automorphism groups of models of Peano Arithmetic? It will be shown here that any group that has half a chance of being isomorphic to the automorphism group of some model of Peano Arithmetic actually is.<\/jats:p><jats:p>For any structure<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200009051_inline1\" \/>, let Aut(<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200009051_inline1\" \/>) be its automorphism group. There are groups which are not isomorphic to any model<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200009051_inline2\" \/>= (<jats:italic>N<\/jats:italic>, +, \u00b7, 0, 1, \u2264) of PA. For example, it is clear that Aut(<jats:italic>N<\/jats:italic>), being a subgroup of Aut((<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200009051_inline2\" \/>, &lt;)), must be torsion-free. However, as will be proved in this paper,<jats:italic>if<\/jats:italic>(<jats:italic>A<\/jats:italic>, &lt;)<jats:italic>is a linearly ordered set and G is a subgroup of Aut<\/jats:italic>((<jats:italic>A<\/jats:italic>, &lt;)),<jats:italic>then there are models<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200009051_inline2\" \/>of<\/jats:italic>PA<jats:italic>such that Aut<\/jats:italic>(<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200009051_inline2\" \/>) \u2245<jats:italic>G<\/jats:italic>.<\/jats:p><jats:p>If<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200009051_inline1\" \/>is a structure, then its automorphism group can be considered as a topological group by letting the stabilizers of finite subsets of<jats:italic>A<\/jats:italic>be the basic open subgroups. If<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200009051_inline1\" \/>\u2032 is an expansion of<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200009051_inline1\" \/>, then Aut(<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200009051_inline1\" \/>\u2032) is a closed subgroup of Aut(<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200009051_inline1\" \/>). Conversely, for any closed subgroup<jats:italic>G<\/jats:italic>\u2264 Aut(<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200009051_inline1\" \/>) there is an expansion<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200009051_inline1\" \/>\u2032 of<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200009051_inline1\" \/>such that Aut(<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200009051_inline1\" \/>\u2032) =<jats:italic>G<\/jats:italic>. Thus, if<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200009051_inline2\" \/>is a model of PA, then Aut(<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200009051_inline2\" \/>) is not only a subgroup of Aut((<jats:italic>N<\/jats:italic>, &lt;)), but it is even a<jats:italic>closed<\/jats:italic>subgroup of Aut((<jats:italic>N<\/jats:italic>, \u2032)).<\/jats:p><jats:p>There is a characterization, due to Cohn [2] and to Conrad [3], of those groups<jats:italic>G<\/jats:italic>which are isomorphic to closed subgroups of automorphism groups of linearly ordered 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