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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: [email protected]
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: [email protected], E-mail: [email protected]
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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

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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[5]Hrushovski, Ehud, Smoothly approximated structures, unpublished notes, 1991.Google Scholar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