Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-28T04:56:11.626Z Has data issue: false hasContentIssue false
  • English
  • Français

CM-triviality and stable groups

Published online by Cambridge University Press:  12 March 2014

Frank O. Wagner
Frank O. Wagner*
Affiliation:
Mathematical Institute, 24-29 St Giles' Oxford OX1 3LB, England. E-mail:[email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We define a generalized version of CM-triviality, and show that in the presence of enough regular types, or solubility, a stable CM-trivial group is nilpotent-by-finite. A torsion-free small CM-trivial stable group is abelian and connected. The first result makes use of a generalized version of the analysis of bad groups.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 63 , Issue 4 , December 1998 , pp. 1473 - 1495
Copyright
Copyright © Association for Symbolic Logic 1998

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] Bachmann, Friedrich, Aufbau der Geometrie aus dem Spiegelungsbegriff, Springer Verlag, Heidelberg, 1959.Google Scholar
[2] Baudisch, Andreas, A new uncountably categorical group, Transactions of the American Mathematical Society, vol. 348 (1996), pp. 3889–3940.Google Scholar
[3] Baudisch, Andreas and Wilson, John, Stable actions of torsion groups and stable soluble groups, Journal of Algebra, vol. 153 (1992), pp. 453–457.Google Scholar
[4] Berline, Chantal and Lascar, Daniel, Superstable groups, Annals of Pure and Applied Logic, vol. 30 (1986), pp. 1–43.CrossRefGoogle Scholar
[5] Corredor, Luis, Bad groups of finite Morley rank, this Journal, vol. 54 (1989), pp. 768–773.Google Scholar
[6] Derakhshan, Jamshid and Wagner, Frank O., Nilpotency in groups with chain conditions, Oxford Quarterly Journal of Mathematics, vol. 48 (1997), pp. 453–468.CrossRefGoogle Scholar
[7] Grünenwald, Claus and Haug, Frieder, On stable torsion-free nilpotent groups, Archive for Mathematical Logic, vol. 32 (1993), pp. 451–462.Google Scholar
[8] Hrushovski, Ehud, A new strongly minimal set, Annals of Pure and Applied Logic, vol. 62 (1993), pp. 147–166.CrossRefGoogle Scholar
[9] Macibtyre, Angus, On ω 1-categorical theories of fields, Fundamenta Mathematicae, vol. 71 (1971), pp. 1–25.Google Scholar
[10] Nesin, Ali, Non-soluble groups of Morley rank 3, Journal of Algebra, vol. 124 (1989), pp. 199–218.Google Scholar
[11] Pillay, Anand, The geometry of forking and groups of finite Morkey rank, this Journal, vol. 60 (1995), pp. 1251–1259.Google Scholar
[12] Poizat, Bruno, Groupes stables, Nur al-Mantiq wal-Ma'rifah, Villeurbanne, 1987.Google Scholar
[13] Suzuki, Michio, Group theory II, Springer-Verlag, New York, Berlin, Heidelberg, Tokyo, 1986.Google Scholar
[14] Wagner, Frank O., Small stable groups and generics, this Journal, vol. 56 (1991), pp. 1026–1037.Google Scholar
[15] Wagner, Frank O., More on ℜ, Notre Dame Journal of Formal Logic, vol. 33 (1992), pp. 159–174.Google Scholar
[16] Wagner, Frank O., Stable groups, mostly of finite exponent, Notre Dame Journal of Formal Logic, vol. 34 (1993), pp. 183–192.Google Scholar
[17] Wagner, Frank O., Hyperstable theories, Proceedings of the logic colloquium '93, Keele (Hodges, Wilfrid, Hyland, Martin, Steinhorn, Charles, and Truss, John, editors), Oxford University Press, Oxford, 1996, pp. 483–514.Google Scholar