On automorphisms of models of Peano arithmetic

When studying the automorphism group Aut(M) of a model M, one is interested to what extent M is recoverable from Aut(M). We show that if M is a countable arithmetically saturated of Peano Arithmetic, then Aut(M) can recognize if a maximal open subgroup is a stabilizer of a nonstandard element, which is smaller than any nonstandard definable element. ^ We use that fact to show that if M1, M2 are countable arithmetically saturated models of Peano Arithmetic such that Aut(M1) ≅ Aut( M2), then (ω, Rep(Th(M1))) ⊨RTn 2 iff (ω, Rep(Th(M2)) ⊨RTn 2 . Here: RTn2 is Ramsey\u27s Theorem stating that every 2-coloring of [ω] n has an infinite homogeneous set; and if T ⊃ PA is a complete and consistent theory, then we define Rep(T) = {ω ∩ X : X is a definable set in a prime model of T}. ^ Using this result we show the existence of countable arithmetically saturated models M0, M1, M2, M3 of Peano Arithmetic such that they have the same standard system and Aut(Mi) ≇ Aut(Mj), where i \u3c j \u3c 4. ^ We also show the similar results for saturated models of Peano Arithmetic of cardinality ℵ1 .