Submission date: 4 July 2010.

Abstract:

We show that \omega-categorical rings with NIP are nilpotent-by-finite. We prove that an \omega-categorical group with NIP and fsg is nilpotent-by-finite. We also notice that an \omega-categorical group with at least one strongly regular type is abelian. Moreover, we get that each \omega-categorical, characteristically simple p-group with NIP has an infinite, definable abelian subgroup. Assuming additionally the existence of a non-algebraic, generically stable over \emptyset type, such a group is abelian.

Mathematics Subject Classification: 03C45, 03C35, 20A15

Keywords and phrases: \omega-categorical group, \omega-categorical ring, non independence property

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