The automorphism subgroup fixing a strong cut in countable recursively saturated PA models has the small index property, uncountable cofinality, and meagre nontrivial normal subgroups.
Kaye,A Galois correspondence for countable recursively saturated models of Peano arithmetic,Automorphisms of first-order structures, Oxford University Press, New York (1991), pp
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The automorphism group of countable recursively saturated models of Peano arithmetic and strong cuts
The automorphism subgroup fixing a strong cut in countable recursively saturated PA models has the small index property, uncountable cofinality, and meagre nontrivial normal subgroups.