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ON THE RADICALS OF A GROUP THAT DOES NOT HAVE THE INDEPENDENCE PROPERTY

Published online by Cambridge University Press:  12 August 2016

CÉDRIC MILLIET
CÉDRIC MILLIET*
Affiliation:
UNIVERSITÄT KONSTANZ FACHBEREICH MATEMATIK UND STATISTIK 78457 KONSTANZ, GERMANY PÔLE DE MATHÉMATIQUES DE L’INSA DE LYON BÂTIMENT LÉONARD DE VINCI – 21 AVENUE JEAN CAPELLE 69621 VILLEURBANNE, FRANCE E-mail: [email protected]
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Abstract

We give an example of a pure group that does not have the independence property, whose Fitting subgroup is neither nilpotent nor definable and whose soluble radical is neither soluble nor definable. This answers a question asked by E. Jaligot in May 2013.

Keywords

Model theoryindependence propertyFitting subgroup and soluble radicalultraproducts
Type
Articles
Information
The Journal of Symbolic Logic , Volume 81 , Issue 4 , December 2016 , pp. 1444 - 1450
Copyright
Copyright © The Association for Symbolic Logic 2016 

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References

REFERENCES

Altınel, T. and Baginski, P., Definable Envelopes of Nilpotent Subgroups of Groups with Chain Conditions on Centralizers . Proceedings of the American Mathematical Society, vol. 142 (2014), no. 5, pp. 1497–1506.Google Scholar
Baudisch, A., On superstable groups . Journal of the London Mathematical Society (2), vol. 42 (1990), no. 3, pp. 452–464.Google Scholar
Brenner, J., The linear homogeneous group . Annals of Mathematics (2), vol. 39 (1938), no. 2, pp. 472–493.Google Scholar
Dieudonné, J., La Géométrie Des Groupes Classiques, Ergebnisse der Mathematik und ihrer Grenzgebiete (N.F.), Heft 5, Springer-Verlag, Berlin, 1955.Google Scholar
Derakhshan, J. and Wagner, F. O., Nilpotency in groups with chain conditions . Quarterly Journal of Mathematics, vol. 48 (1997), no. 192, pp. 453–466.Google Scholar
Keisler, H. J., The ultraproduct construction , Ultrafilters Across Mathematics, Contemporary Mathematics, vol. 530, American Mathematical Society, Providence, RI, 2010, pp. 163–179.CrossRefGoogle Scholar
Klingenberg, W., Linear groups over local rings . Bulletin of the American Mathematical Society, vol. 66 (1960), pp. 294–296.Google Scholar
Matthews, L., The Independence Property in Unstable Algebraic Structures I: p-Adically Closed Fields , unpublished preprint, 1993.Google Scholar
Macpherson, D. and Tent, K., Pseudofinite groups with NIP theory and definability in finite simple groups , Groups and Model Theory, Contemporary Mathematics, vol. 576, American Mathematical Society, Providence, RI, 2012, pp. 255–267.Google Scholar
Houcine, A. O., A remark on the definability of the Fitting subgroup and the soluble radical . Mathematical Logic Quarterly, vol. 59 (2013), no. 1–2, pp. 62–65.Google Scholar
Wagner, F. O., The Fitting subgroup of a stable group . Journal of Algebra, vol. 174 (1995), no. 2, pp. 599–609.Google Scholar
Wagner, F. O., Nilpotency in groups with the minimal condition on centralizers . Journal of Algebra, vol. 217 (1999), no. 2, pp. 448–460.Google Scholar