The automorphism group of countable recursively saturated models of Peano arithmetic and strong cuts

In this paper, we extend the concept of a Lascar generic automorphism in the setting of models of Peano arithmetic (PA) to the subgroup of the automorphism group of a countable recursively saturated model M of PA that fixes pointwise a strong cut I of M, denoted by (Aut(M))(I). Then, we prove that: (1) (Aut(M))(I) has the small index property. (2) The cofinality of (Aut(M))(I) is uncountable. (3) Any nontrivial normal subgroup of (Aut(M))(I) is meagre in it. In particular, the infinite cyclic group Z is not a homomorphic image of (Aut(M))(I).