On 𝜔-categorical groups and rings with NIP

Krupiński, Krzysztof

doi:10.1090/S0002-9939-2011-11102-X

On $\omega$-categorical groups and rings with NIP
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Proc. Amer. Math. Soc. 140 (2012), 2501-2512 Request permission

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.

References

A. B. Apps, On the structure of $\aleph _{0}$-categorical groups, J. Algebra 81 (1983), no. 2, 320–339. MR 700288, DOI 10.1016/0021-8693(83)90192-8
Richard Archer and Dugald Macpherson, Soluble omega-categorical groups, Math. Proc. Cambridge Philos. Soc. 121 (1997), no. 2, 219–227. MR 1426520, DOI 10.1017/S0305004196001387
John T. Baldwin and Bruce Rose, $\aleph _{0}$-categoricity and stability of rings, J. Algebra 45 (1977), no. 1, 1–16. MR 439621, DOI 10.1016/0021-8693(77)90359-3
Walter Baur, Gregory Cherlin, and Angus Macintyre, Totally categorical groups and rings, J. Algebra 57 (1979), no. 2, 407–440. MR 533805, DOI 10.1016/0021-8693(79)90230-8
Gregory Cherlin, On $\aleph _{0}$-categorical nilrings. II, J. Symbolic Logic 45 (1980), no. 2, 291–301. MR 569399, DOI 10.2307/2273189
Clifton Ealy, Krzysztof Krupiński, and Anand Pillay, Superrosy dependent groups having finitely satisfiable generics, Ann. Pure Appl. Logic 151 (2008), no. 1, 1–21. MR 2381504, DOI 10.1016/j.apal.2007.09.004
David M. Evans and Frank O. Wagner, Supersimple $\omega$-categorical groups and theories, J. Symbolic Logic 65 (2000), no. 2, 767–776. MR 1771084, DOI 10.2307/2586568
I. N. Herstein, Noncommutative rings, The Carus Mathematical Monographs, No. 15, Mathematical Association of America; distributed by John Wiley & Sons, Inc., New York, 1968. MR 0227205
Ehud Hrushovski, Ya’acov Peterzil, and Anand Pillay, Groups, measures, and the NIP, J. Amer. Math. Soc. 21 (2008), no. 2, 563–596. MR 2373360, DOI 10.1090/S0894-0347-07-00558-9
Ehud Hrushovski and Anand Pillay, On NIP and invariant measures, J. Eur. Math. Soc. (JEMS) 13 (2011), no. 4, 1005–1061. MR 2800483, DOI 10.4171/JEMS/274
Otto H. Kegel and Bertram A. F. Wehrfritz, Locally finite groups, North-Holland Mathematical Library, Vol. 3, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973. MR 0470081
Krzysztof Krupiński, Fields interpretable in rosy theories, Israel J. Math. 175 (2010), 421–444. MR 2607552, DOI 10.1007/s11856-010-0017-x
K. Krupiński, On relationships between algebraic properties of groups and rings in some model-theoretic contexts, Journal of Symbolic Logic, accepted.
Krzysztof Krupiński and Frank Wagner, Small profinite groups and rings, J. Algebra 306 (2006), no. 2, 494–506. MR 2271348, DOI 10.1016/j.jalgebra.2006.01.027
Jacques Lewin, Subrings of finite index in finitely generated rings, J. Algebra 5 (1967), 84–88. MR 200297, DOI 10.1016/0021-8693(67)90027-0
H. D. Macpherson, Absolutely ubiquitous structures and $\aleph _0$-categorical groups, Quart. J. Math. Oxford Ser. (2) 39 (1988), no. 156, 483–500. MR 975912, DOI 10.1093/qmath/39.4.483
A. Pillay, P. Tanović, Generic stability, regularity, and quasi-minimality, to appear in Categories and Models, a volume in honour of M. Makkai’s 70th birthday.
J. M. Plotkin, $\textrm {ZF}$ and locally finite groups, Z. Math. Logik Grundlagen Math. 27 (1981), no. 4, 375–379. MR 626682, DOI 10.1002/malq.19810272305
Bruno Poizat, Stable groups, Mathematical Surveys and Monographs, vol. 87, American Mathematical Society, Providence, RI, 2001. Translated from the 1987 French original by Moses Gabriel Klein. MR 1827833, DOI 10.1090/surv/087
Saharon Shelah, Dependent first order theories, continued, Israel J. Math. 173 (2009), 1–60. MR 2570659, DOI 10.1007/s11856-009-0082-1
Alexander Usvyatsov, On generically stable types in dependent theories, J. Symbolic Logic 74 (2009), no. 1, 216–250. MR 2499428, DOI 10.2178/jsl/1231082310
Frank O. Wagner, Stable groups, London Mathematical Society Lecture Note Series, vol. 240, Cambridge University Press, Cambridge, 1997. MR 1473226, DOI 10.1017/CBO9780511566080
J. S. Wilson, The algebraic structure of $\aleph _{0}$-categorical groups, Groups—St. Andrews 1981 (St. Andrews, 1981) London Math. Soc. Lecture Note Ser., vol. 71, Cambridge Univ. Press, Cambridge-New York, 1982, pp. 345–358. MR 679175