On 𝜔-categorical groups and rings with NIP

Abstract

We prove that ω \omega -categorical rings with NIP are nilpotent-by-finite and that ω \omega -categorical groups with NIP and fsg are nilpotent-by-finite, too. We give an easy proof that each infinite, ω \omega -categorical p p -group with NIP has an infinite, definable abelian subgroup. Assuming additionally that the group in question is characteristically simple and has a non-algebraic type which is generically stable over ∅ \emptyset , we show that the group is abelian. Moreover, we prove that in any group with at least one strongly regular type all non-central elements are conjugated, and we conclude that assuming in addition ω \omega -categoricity, such a group must be abelian.