On $𝜔$-categorical groups and rings with NIP
Proceedings of the American Mathematical Society
We prove that ω-categorical rings with NIP are nilpotent-by-finite and that ω-categorical groups with NIP and fsg are nilpotent-by-finite, too. We give an easy proof that each infinite, ω-categorical 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 ∅, we show that the group is abelian. Moreover, we prove that in any group with at least one
... gly regular type all non-central elements are conjugated, and we conclude that assuming in addition ω-categoricity, such a group must be abelian. Introduction Recall that a first-order structure M in a countable language is said to be ωcategorical if, up to isomorphism, T h(M ) has at most one model of cardinality ℵ 0 . By Ryll-Nardzewski's theorem, this is equivalent to the condition that for every natural number n there are only finitely many n-types over ∅. If M is countable and ω-categorical or if M is a monster model (i.e. a model which is κ-saturated and strongly κ-homogeneous for a big cardinal κ), two finite tuples have the same type over ∅ iff they lie in the same orbit under the action of the automorphism group of M . Hence, for a countable M or for M being a monster model, ω-categoricity means that for each natural number n the automorphism group of M has only finitely many orbits on n-tuples (which implies that M is locally finite). There is a long history of results describing the structure of ω-categorical groups and rings. However, many questions in this area are still wide open. It follows easily that each countable, ω-categorical group has a finite series of characteristic (i.e. invariant under the automorphism group) subgroups in which all successive quotients are characteristically simple groups (i.e. they do not have non-trivial, proper characteristic subgroups). On the other hand, Wilson (see [23, 1]) proved Fact 0.1. For each countably infinite, ω-categorical, characteristically simple group H, one of the following holds. (i) H is an elementary abelian p-group for some prime p.