Automorphism Groups of Models of Peano Arithmetic

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. For any structure , let Aut( ) be its automorphism group. There are groups which are not isomorphic to any model = ( N , +, ·, 0, 1, ≤) of PA. For example, it is clear that Aut( N ), being a subgroup of Aut(( , if ( A , is a linearly ordered set and G is a subgroup of Aut (( A , then there are models of PA such that Aut ( ) ≅ G . If is a structure, then its automorphism group can be considered as a topological group by letting the stabilizers of finite subsets of A be the basic open subgroups. If ′ is an expansion of , then Aut( ′) is a closed subgroup of Aut( ). Conversely, for any closed subgroup G ≤ Aut( ) there is an expansion ′ of such that Aut( ′) = G . Thus, if is a model of PA, then Aut( ) is not only a subgroup of Aut(( N , closed subgroup of Aut(( N , ′)). There is a characterization, due to Cohn [2] and to Conrad [3], of those groups G which are isomorphic to closed subgroups of automorphism groups of linearly ordered sets.