Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-29T14:21:28.040Z Has data issue: false hasContentIssue false
  • English
  • Français

DIMENSIONAL GROUPS AND FIELDS

Part of: Special aspects of infinite or finite groups Connections with logic Model theory

Published online by Cambridge University Press:  05 October 2020

FRANK O. WAGNER
FRANK O. WAGNER*
Affiliation:
UNIVERSITÉ LYON 1, CNRS INSTITUT CAMILLE JORDAN UMR 5208 21 AVENUE CLAUDE BERNARD 69622VILLEURBANNE CEDEX, FRANCEE-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We shall define a general notion of dimension, and study groups and rings whose interpretable sets carry such a dimension. In particular, we deduce chain conditions for groups, definability results for fields and domains, and show that a pseudofinite $\widetilde {\mathfrak M}_c$ -group of finite positive dimension contains a finite-by-abelian subgroup of positive dimension, and a pseudofinite group of dimension 2 contains a soluble subgroup of dimension 2.

Keywords

dimensionchain conditiongroupdomainskew fielddefinabilitypseudofinite group of small dimension

MSC classification

Primary: 03C45: Classification theory, stability and related concepts
Secondary: 03C20: Ultraproducts and related constructions 03C60: Model-theoretic algebra 12L12: Model theory 20F24: FC-groups and their generalizations
Type
Articles
Information
The Journal of Symbolic Logic , Volume 85 , Issue 3 , September 2020 , pp. 918 - 936
Check for updates
Copyright
© The Association for Symbolic Logic 2020

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

Borovik, A. and Nesin, A., Groups of Finite Morley rank , Oxford Logic Guides, vol. 26, The Clarendon Press---Oxford University Press, New York, NY, 1994.Google Scholar
Elwes, R. and Macpherson, D., A survey of asymptotic classes and measurable structures , Model Theory with Applications to Algebra and Analysis , vol. 2 (Chatzidakis, Z., Macpherson, D., Pillay, A, and Wilkie, A., editors), London Mathematical Society Lecture Note Series, vol. 350, Cambridge University Press, Cambridge, 2008, pp. 125–160.CrossRefGoogle Scholar
Ealy, C., Krupinski, K., and Pillay, A., Superrosy dependent groups having finitely satisfiable generics . Annals of Pure and Applied Logic , vol. 151 (2008), pp. 1–21,CrossRefGoogle Scholar
Elwes, R. and Ryten, M., Measurable groups of low dimension . Mathematical Logic Quarterly , vol. 54 (2008), no. 4, pp. 374–386.CrossRefGoogle Scholar
Guralnick, R. and Malle, G., Products of conjugacy classes and fixed point spaces, preprint, arXiv:1005.3756.Google Scholar
Hempel, N., Almost group theory, preprint, 2016. Available at hal.archives-ouvertes.fr/hal-01206954.Google Scholar
Hrushovski, E., Stable group theory and approximate subgroups . Journal of the American Mathematical Society , vol. 25 (2012), pp. 189–243.CrossRefGoogle Scholar
Hrushovski, E., On pseudo-finite dimensions . Notre Dame Journal of Formal Logic , vol. 54 (2013), no. 3–4, pp. 463–495.CrossRefGoogle Scholar
Hrushovski, E. and Wagner, F. O., Counting and dimensions , Model Theory with Applications to Algebra and Analysis , vol. 2 (Chatzidakis, Z., Macpherson, D., Pillay, A, and Wilkie, A., editors), London Mathematical Society Lecture Note Series, vol. 350, Cambridge University Press, Cambridge, 2008, pp. 161–176.CrossRefGoogle Scholar
Kaplansky, I., Rings with a polynomial identity . Bulletin of the American Mathematical Society , vol. 54 (1948), pp. 575–580.CrossRefGoogle Scholar
Krupinski, K., Fields definable in rosy theories . Israel Journal of Mathematics , vol. 175 (2010), no. 1, pp. 421–444.CrossRefGoogle Scholar
Neumann, B. H., Groups with finite classes of conjugate elements . Proceedings of the London Mathematical Society , vol. 1 (1951), pp. 178–187.CrossRefGoogle Scholar
Point, F., Ultraproducts and Chevalley groups . Archive for Mathematical Logic , vol. 38 (1999), no. 6, pp. 355–372.CrossRefGoogle Scholar
Ryten, M., Model theory of finite difference fields and simple groups , PhD thesis, University of Leeds, 2007. Available at http://www1.maths.leeds.ac.uk/pmthdm/ryten1.pdf.Google Scholar
Wagner, F. O., Simple Theories , Kluwer Academic Publishers, Dordrecht, The Netherlands, 2000.CrossRefGoogle Scholar