On \(\omega\)-categorical groups and rings with NIP (Q2845065)

In this article the author studies \(\omega\)-categorical groups and rings and generalizes some of the results from the stable case to the NIP context: \(\omega\)-categorical rings with NIP are nilpotent by finite; \(\omega\)-categorical groups with NIP and fsg are nilpotent by finite. He also proves that every characteristically simple, \(\omega\)-categorical \(p\)-group with NIP having a non-algebraic generically stable type is abelian. In the last section, groups with at least one strongly regular type (in the sense of Pillay and Tanović) are studied. It is proved that all non-central elements of such a group are conjugated and that \(\omega\)-categorical groups of that kind are abelian.