This chapter shows two major results. One, due to Lascar, says that countable arithmetically saturated models of PA have sequences of generic automorphisms, and consequently have a small index property. The other, from the authors of the book, states that the standard systems of countable arithmetically saturated models of PA are coded in their automorphism groups. Other results shown include proving a connection between the existence of automorphims with dense conjugacy classes and providing answers to some combinatorial questions concerning coloring of Cartesian products of digraphs; and a theorem saying that the cofinality of the automorphism group of a countable recursively saturated model of PA is uncountable if and only if the model is arithmetically saturated.
Keywords: Lascar, saturated models of PA, automorphims, Cartesian products of digraphs