Hostname: page-component-8448b6f56d-jr42d Total loading time: 0 Render date: 2024-04-17T02:22:51.429Z Has data issue: false hasContentIssue false
  • English
  • Français

Supersimple ω-categorical groups and theories

Published online by Cambridge University Press:  12 March 2014

David M. Evans  and
Frank O. Wagner
David M. Evans
Affiliation:
School of Mathematics, UEA, Norwich NR4 7TJ, UK, E-mail: d.evans@uea.ac.uk
Frank O. Wagner*
Affiliation:
Mathematical Institute, University of Oxford, 24–29 ST Giles' Oxford Ox1 3LB., UK
*
Current address: Institut Girard Desargues, Université Claude Bernard, Mathématiques, bâtiment, 101, 43, boulevard du 11 novembre 1918, 69622 Villeurbanne-cedex, France, E-mail: wagner@maths.ox.ac.uk, E-mail: wagner@desargues.univ-lyonl.fr
Get access Rights & Permissions [Opens in a new window]

Abstract

An ω-categorical supersimple group is finite-by-abelian-by-finite, and has finite SU-rank. Every definable subgroup is commensurable with an acl(ø)-definable subgroup. Every finitely based regular type in a CM-trivial ω-categorical simple theory is non-orthogonal to a type of SU-rank 1. In particular, a supersimple ω-categorical CM-trivial theory has finite SU-rank.

Keywords

ω-categoricalsupersimplefinite rankgroupabelianCM-trivial
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 65 , Issue 2 , June 2000 , pp. 767 - 776
Copyright
Copyright © Association for Symbolic Logic 2000

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Baur, Walter, Cherlin, Gregory, and Macintyre, Angus, Totally categorical groups and rings, Journal of Algebra, vol. 57 (1979), pp. 407–440.CrossRefGoogle Scholar
[2]Bergman, George M. and Lenstra, Hendrik W. Jr., Subgroups close to normal subgroups, Journal of Algebra, vol. 127 (1989), pp. 80–97.CrossRefGoogle Scholar
[3]Cherlin, Gregory, Harrington, Leo, and Lachlan, Alistair, ℵ0-categoricalℵ0-stable structures, Annals of Pure and Applied Logic, vol. 28 (1985), pp. 103–135.CrossRefGoogle Scholar
[4]Hart, Bradd, Kim, Byunghan, and Pillay, Anand, Coordinatization and canonical bases in simple theories, to appear in this Journal.Google Scholar
[5]Hrushovski, Ehud, Smoothly approximated structures, unpublished notes, 1991.Google Scholar
[6]Hrushovski, Ehud, A new strongly minimal set, Annals of Pure and Applied Logic, vol. 62 (1993), pp. 147–166.CrossRefGoogle Scholar
[7]Hrushovski, Ehud, Simplicity and the Lascar group, notes, 1997.Google Scholar
[8]Kim, Byunghan, A note on Lascar strong types in simple theories, this Journal, vol. 63 (1998), pp. 926–936.Google Scholar
[9]Kim, Byunghan and Pillay, Anand, From stability to simplicity, The Bulletin of Symbolic Logic, vol. 4 (1998), pp. 17–36.CrossRefGoogle Scholar
[10]Macpherson, H. D., Absolutely ubiquitous structures and ℵ0-categorical groups, Quarterly Journal of Mathematics, Oxford, vol. 39 (1988), pp. 483–500.CrossRefGoogle Scholar
[11]Pillay, Anand, The geometry of forking and groups of finite Morley rank, this Journal, vol. 60 (1995), pp. 1251–1259.Google Scholar
[12]Poizat, Bruno, Groupes stables, Nur al mantiq wal marifah, 1988.Google Scholar
[13]Pourmahdian, Massoud, notes, 1998.Google Scholar
[14]Schlichting, G., Operationen mit periodischen Stabilisatoren, Archiv der Mathematik (Basel), vol. 34 (1980), pp. 97–99.CrossRefGoogle Scholar
[15]Wagner, Frank O., Groups in simple theories, submitted to this Journal, 1997.Google Scholar