On two questions concerning the automorphism groups of countable recursively saturated models of PA

We prove that the automorphism group of a countable recursively saturated model of PA is not divisible. Using similar techniques we show that every countable recursively saturated model has a strong initial segment whose setwise stabilizer is not maximal, and we generalize a result of Vladimir Kanovei concerning definability in structures of the form ( M; I ), where M is a model of PA and I is its initial segment.