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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
DOI: https://doi.org/10.1090/S0002-9939-2011-11102-X
Published electronically: November 18, 2011
MathSciNet review: 2898712
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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 $ 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 $ \omega $-categoricity, such a group must be abelian.


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Additional Information

Krzysztof Krupiński
Affiliation: Instytut Matematyczny, Uniwersytet Wrocławski, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland
Email: kkrup@math.uni.wroc.pl

DOI: https://doi.org/10.1090/S0002-9939-2011-11102-X
Keywords: $𝜔$-categorical group, $𝜔$-categorical ring, non-independence property
Received by editor(s): June 24, 2010
Received by editor(s) in revised form: February 25, 2011
Published electronically: November 18, 2011
Additional Notes: This research was supported by the Polish government grant N N201 545938 and by EPSRC grant EP/F009712/1.
Communicated by: Julia Knight
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.