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ISSN 1088-6826(e) ISSN 0002-9939(p)

On $\omega$ -categorical groups and rings with NIP

Author: Krzysztof Krupiński
Journal: Proc. Amer. Math. Soc. 140 (2012), 2501-2512
MSC (2010): Primary 03C45, 03C35, 20A15
Posted: November 18, 2011
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Abstract: We prove that $\omega$ -categorical rings with NIP are nilpotent-by-finite and that $\omega$ -categorical groups with NIP and fsg are nilpotent-by-finite, too. We give an easy proof that each infinite, $\omega$ -categorical -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 $\omega$ -categoricity, such a group must be abelian.

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