Hostname: page-component-84b7d79bbc-c654p Total loading time: 0 Render date: 2024-07-30T20:53:14.507Z Has data issue: false hasContentIssue false
  • English
  • Français

Small stable groups and generics

Published online by Cambridge University Press:  12 March 2014

Frank O. Wagner
Frank O. Wagner*
Affiliation:
Mathematical Institute, Oxford University, Oxford OX1 3LB, England Department of Mathematics and Statistics, McGill University, Montréal, Québec H3A 2K6, Canada
*
Mathematisches Institut, Universität Freiburg, Albertstr. 23b, W-7800 Freiburg, Germany
Get access Rights & Permissions [Opens in a new window]

Abstract

We define an ℜ-group to be a stable group with the property that a generic element (for any definable transitive group action) can only be algebraic over a generic. We then derive some corollaries for ℜ-groups and fields, and prove a decomposition theorem and a field theorem. As a nonsuperstable example, we prove that small stable groups are ℜ-groups.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 56 , Issue 3 , September 1991 , pp. 1026 - 1037
Copyright
Copyright © Association for Symbolic Logic 1991

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

[BL] Berline, C. and Lascar, D., Superstable groups, Annals of Pure and Applied Logic, vol. 30 (1986), pp. 1–43.CrossRefGoogle Scholar
[BP] Baldwin, J. T. and Pillay, A., Semisimple stable and superstable groups, Annals of Pure and Applied Logic, vol. 45 (1989), pp. 105–127.CrossRefGoogle Scholar
[CS] Cherlin, G. and Shelah, S., Superstable fields and groups, Annals of Mathematical Logic, vol. 18 (1980), pp. 227–270.CrossRefGoogle Scholar
[Hru1] Hrushovski, E., Contributions to stable model theory, Ph.D. thesis, University of California, Berkeley, California, 1986.Google Scholar
[Hru2] Hrushovski, E., Locally modular regular types, Classification theory (proceedings, Chicago, 1985; Baldwin, J. T., editor), Lecture Notes in Mathematics, vol. 1292, Springer-Verlag, Berlin, 1987, pp. 132–164.Google Scholar
[Hru3] Hrushovski, E., On superstable fields with automorphisms, The model theory of groups (Nesin, A. and Pillay, A., editors), Notre Dame Mathematical Lecture Notes, vol. 11, Notre Dame University Press, Notre Dame, Indiana, 1989, pp. 186–191.Google Scholar
[M] Macintyre, A., The complexity of types in field theory, Logic year 1979–80 (Lerman, M. et al., editors), Lecture Notes in Mathematics, vol. 859, Springer-Verlag, Berlin, 1981, pp. 143–156.CrossRefGoogle Scholar
[Po] Poizat, B., Groupes stables, Nur al-Mantiq wal-Ma'rifah, Villeurbanne, 1987.Google Scholar