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On properties of (weakly) small groups

Published online by Cambridge University Press:  12 March 2014

Cédric Milliet
Cédric Milliet*
Affiliation:
Université de Lyon, Université Lyon 1, Institut Camille Jordan, UMR 5208 CNRS 43 Boulevard du 11 Novembre 1918, 69622 Villeurbanne Cedex, France
*
Université Galatasaray, Faculté de Sciences et de Lettres, Département de Mathématiques, Çiraǧan Caddesi 36, 34357 Ortaköy, Istamboul, Turquie, E-mail: milliet@math.univ-lyon1.fr
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Abstract

A group is small if it has only countably many complete n-types over the empty set for each natural number n. More generally, a group G is weakly small if it has only countably many complete 1-types over every finite subset of G. We show here that in a weakly small group, subgroups which are definable with parameters lying in a finitely generated algebraic closure satisfy the descending chain conditions for their traces in any finitely generated algebraic closure. An infinite weakly small group has an infinite abelian subgroup, which may not be definable. A small nilpotent group is the central product of a definable divisible group with a definable one of bounded exponent. In a group with simple theory, any set of pairwise commuting elements is contained in a definable finite-by-abelian subgroup. First corollary: a weakly small group with simple theory has an infinite definable finite-by-abelian subgroup. Secondly, in a group with simple theory, a solvable group A of derived length n is contained in an A-definable almost solvable group of class at most 2n – 1.

Keywords

Small groupweakly small groupCantor-Bendixson ranklocal chain conditioninfinite abelian subgroupgroup in a simple theoryinfinite finite-by-abelian subgroupnilpotent group
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 77 , Issue 1 , March 2012 , pp. 94 - 110
Copyright
Copyright © Association for Symbolic Logic 2012

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References

REFERENCES

[1] de Aldama, Ricardo, Chaîtes et Dépendance, Ph.D. thesis, Lyon, 2009.Google Scholar
[2] de Aldama, Ricardo, A result on definable groups without the independence property. The Bulletin of Symbolic Logic, to appear.Google Scholar
[3] Baer, Reinhold, The subgroup of the elements of finite order of an abelian group, The Annals of Mathematics, Second Series, vol. 37 (1936), no. 4, pp. 766–781.CrossRefGoogle Scholar
[4] Baur, Walter, Cherlin, Gregory, and Macintyre, Angus, Totally categorical groups and rings, Journal of Algebra, vol. 57 (1979), no. 2, pp. 407–440.CrossRefGoogle Scholar
[5] Berline, Chantal and Lascar, Daniel, Superstable groups, Annals of Pure and Applied Logic, vol. 30 (1986), pp. 1–43.CrossRefGoogle Scholar
[6] Borovik, Alexandre and Nesin, Ali, Groups of finite Morley rank, Oxford University Press, 1994.CrossRefGoogle Scholar
[7] Cherlin, Gregory, Groups of small Morley rank, Annals of Mathematical Logic, vol. 17 (1979), pp. 1–28.CrossRefGoogle Scholar
[8] Evans, David M. and Wagner, Frank O., Supersimple ω-categorical groups and theories, this Journal, vol. 65 (2000), no. 2, pp. 767–776.Google Scholar
[9] Felgner, Ulrich, On ℵ-categorical extra-special p-groups, Logique et Analyse, vol. 71-72 (1975), pp. 407–428.Google Scholar
[10] Fuchs, Laszló, Infinite abelian groups, Academic Press, 1970.Google Scholar
[11] Hall, Philip and Kulatilaka, C.R., A property of locally finite groups, Journal of the London Mathematical Society, vol. 39 (1964), pp. 235–239.CrossRefGoogle Scholar
[12] Kaplan, Itay, Scanlon, Thomas, and Wagner, Frank O., Artin-Schreier extensions in dependent and simple fields, to be published.Google Scholar
[13] Morley, Michael, Categoricity in power, Transactions of the American Mathematical Society, vol. 114 (1965), no. 2, pp. 514–538.CrossRefGoogle Scholar
[14] Nesin, Ali, Poly-separated and co-stable nilpotent groups, this Journal, vol. 56 (1991), no. 2, pp. 694–699.Google Scholar
[15] Neumann, Bernhard H., Groups covered by permutable subsets, Journal of the London Mathematical Society, vol. 29 (1954), pp. 236–248.CrossRefGoogle Scholar
[16] Plotkin, Jacob M., ZF and locally finite groups, Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 27 (1981), pp. 375–379.CrossRefGoogle Scholar
[17] Poizat, Bruno, Groupes stables, Nur Al-Mantiq Wal-Ma'rifah, 1987.Google Scholar
[18] Poizat, Bruno, Quelques tentatives de definir une notion générate de groupes et de corps de dimension un et de déterminer leurs propriétés algébriques, Confluentes Mathematici, à paraître, 2009.Google Scholar
[19] Poizat, Bruno, Groups of small Cantor rank, this Journal, vol. 75 (2010), no. 1, pp. 346–354.Google Scholar
[20] Puninskaya, Vera, Vaught's conjecture, Journal of Mathematical Sciences, vol. 109 (2002), no. 3, pp. 1649–1668.CrossRefGoogle Scholar
[21] Reineke, Joachim, Minimale Gruppen, Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 21 (1975), pp. 357–359.CrossRefGoogle Scholar
[22] Schlichting, Günter, Operationen mit periodischen Stabilisatoren, Archiv der Matematik, vol. 34 (1980), pp. 97–99.Google Scholar
[23] Shelah, Saharon, Dependent first order theories, continued, Israel Journal of Mathematics, vol. 173 (2009), pp. 1–60.CrossRefGoogle Scholar
[24] Wagner, Frank O., Small stable groups and generics, this Journal, vol. 56 (1991), pp. 1026–1037.Google Scholar
[25] Wagner, Frank O., Quasi-endomorphisms in small stable groups, this Journal, vol. 58 (1993), pp. 1044–1051.Google Scholar
[26] Wagner, Frank O., Simple theories, Kluwer Academic Publishers, Dordrecht, 2000.CrossRefGoogle Scholar