Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-10T08:20:04.273Z Has data issue: false hasContentIssue false
  • English
  • Français

On the definability of radicals in supersimple groups

Published online by Cambridge University Press:  12 March 2014

Cédric Milliet
Cédric Milliet*
Affiliation:
Université Galatasaray, Faculté de Sciences et de Lettres, Département de Mathématiques, Çirağan Caddesi N. 36, 34357 Ortaköy, Istamboul, Turquie, E-mail:milliet@math.univ-lyon1.fr
Get access Rights & Permissions [Opens in a new window]

Abstract

If G is a group with a supersimple theory having a finite SU-rank, then the subgroup of G generated by all of its normal nilpotent subgroups is definable and nilpotent. This answers a question asked by Elwes, Jaligot, Macpherson and Ryten. If H is any group with a supersimple theory, then the subgroup of H generated by all of its normal soluble subgroups is definable and soluble.

Keywords

Supersimple groupFitting subgroupsoluble radical
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 78 , Issue 2 , June 2013 , pp. 649 - 656
Copyright
Copyright © Association for Symbolic Logic 2013

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]de Aldama, Ricardo, Definable nilpotent and soluble envelopes in groups without the independence property, Mathematical Logic Quarterly, to appear.Google Scholar
[2]Altinel, Tuna and Baginski, Paul, Definable envelopes of nilpotent subgroups of groups with chain conditions on centralizers, Transactions of the American Mathematical Society, to appear.Google Scholar
[3]Baudish, Andreas, On superstable groups, Journal of the London Mathematical Society, vol. 42 (1980), pp. 452–463.Google Scholar
[4]Berline, Chantale and Lascar, Daniel, Superstable groups. Annals of Pure and Applied Logic, vol. 30 (1986), pp. 1–43.CrossRefGoogle Scholar
[5]Duiguild, A.M. and McLain, D.H., FC-nilpotent and FC-soluble groups. Proceedings of the Cambridge Philosophical Society, vol. 52 (1956), pp. 391–398.Google Scholar
[6]Elwes, Richard, Jaligot, Eric, Macpherson, Dugald, and Ryten, Mark, Groups in supersimple and pseudofinite theories, Proceedings of the London Mathematical Society, vol. 103 (2011), pp. 1049–1082.CrossRefGoogle Scholar
[7]Elwes, Richard and Ryten, Mark, Measurable groups of low dimension. Mathematical Logic Quarterly, vol. 54 (2008), pp. 374–386.CrossRefGoogle Scholar
[8]Hall, Philip, Some sufficient conditions for a group to be nilpotent, Illinois Journal of Mathematics, vol. 2 (1958), pp. 787–801.CrossRefGoogle Scholar
[9]Hrushovski, Ehud, Pseudo-finite and related structures. Model theory and applications (Bélair, L., Chatzidakis, Z., D'Aquino, P., Marker, D., Otero, M., Point, F., and Wilkie, A., editors), Quaderni di Matematica, vol. 11, Caserta, 2003.Google Scholar
[10]Humphreys, James, Linear algebraic groups, Springer, 1981.Google Scholar
[11]Khukhro, Evgenii, p-automorphisms of finite p-groups, London Mathematical Society Lecture Note Series, Cambridge University Press, 1998.CrossRefGoogle Scholar
[12]Milliet, Cédric, Definable envelopes in groups with simple theory, preprint.Google Scholar
[13]Neumann, Bernhard H., Groups covered by permutable subsets, Journal of the London Mathematical Society, vol. 29 (1954), pp. 236–248.CrossRefGoogle Scholar
[14]Poizat, Bruno, Groupes stables, 1987.CrossRefGoogle Scholar
[15]Robinson, Derek, A course in the theory of groups, Springer, 1996.CrossRefGoogle Scholar
[16]Schlichting, Günter, Operationen mit periodischen Stabilisatoren, Archiv der Mathematik, vol. 34 (1980), pp. 97–99.CrossRefGoogle Scholar
[17]Shelah, Saharon, Dependent first order theories, continued, Israel Journal of Mathematics, vol. 173 (2009), pp. 1–60.CrossRefGoogle Scholar
[18]Wagner, Frank O., Fitting subgroup of a stable group, Journal of Algebra, vol. 174 (1995), pp. 599–609.CrossRefGoogle Scholar
[19]Wagner, Frank O., Simple theories, Kluwer Academic Publishers, Dordrecht, NL, 2000.CrossRefGoogle Scholar
[20]Wagner, Frank O., Groups in simple theories, Logic colloquium 2001 (Baaz, Matthias, Friedman, Sy-David, and Krajícek, Jan, editors), Lecture Notes in Logic, vol. 20, Association of Symbolic Logic and AK Peters, 2005, pp. 440–467.Google Scholar