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DEFINABLY SIMPLE STABLE GROUPS WITH FINITARY GROUPS OF AUTOMORPHISMS

Published online by Cambridge University Press:  10 April 2019

ULLA KARHUMÄKI
ULLA KARHUMÄKI*
Affiliation:
SCHOOL OF MATHEMATICS UNIVERSITY OF MANCHESTER OXFORD ROAD, MANCHESTER M13 9PL, UKE-mail: ulla.karhumaki@manchester.ac.uk
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Abstract

We prove that infinite definably simple locally finite groups of finite centraliser dimension are simple groups of Lie type over locally finite fields. Then, we identify conditions on automorphisms of a stable group that make it resemble the Frobenius maps, and allow us to classify definably simple stable groups in the specific case when they admit such automorphisms.

Keywords

03C6020G99locally finite groupsstable groupsFrobenius maps

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Type
Articles
Information
The Journal of Symbolic Logic , Volume 84 , Issue 2 , June 2019 , pp. 704 - 712
Copyright
Copyright © The Association for Symbolic Logic 2019 

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References

REFERENCES

Altınel, T., Borovik, A. V., and Cherlin, G. L., Simple Groups of Finite Morley Rank, American Mathematical Society Providence, Providence, RI, 2008.10.1090/surv/145CrossRefGoogle Scholar
Baldwin, J. T. and Saxl, J., Logical stability in group theory. Journal of the Australian Mathematical Society, vol. 21 (1976), no. 3, pp. 267–276.10.1017/S1446788700018553CrossRefGoogle Scholar
Bell, J. L. and Slomson, A. B., Models and Ultraproducts: An Introduction, North Holland, Amsterdam, 1971.Google Scholar
Bestvina, M. and Feighn, M., Talk on negligible sets. Slides, Luminy, Marseille, June 2010. Available at http://andromeda.rutgers.edu/∼feighn/luminy.pdf.Google Scholar
Borovik, A. V., Simple locally finite groups of finite Morley rank and odd type, Finite and Locally Finite Groups (Istanbul, 1994) (Hartley, B., Seitz, G. M., Borovik, A. V., and Bryant, R. M., editors), NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, vol. 471, Kluwer Academic, Dordrecht, 1995, pp. 247–284.10.1007/978-94-011-0329-9_10CrossRefGoogle Scholar
Borovik, A. V. and Karhumäki, U., Locally finite groups of finite centraliser dimension. Journal of Group Theory, to appear.Google Scholar
Borovik, A. V. and Nesin, A., Groups of Finite Morley Rank, Oxford Logic Guides, vol. 26, Oxford University Press, New York, 1994.10.1093/oso/9780198534457.001.0001CrossRefGoogle Scholar
Buturlakin, A. A., Locally finite groups of finite c-dimension, preprint, 2018, arXiv:1805.00910.Google Scholar
Carter, R. W., Simple Groups of Lie Type, Wiley, New York, 1971.Google Scholar
Cherlin, G. L., Groups of small Morley rank. Annals of Mathematical Logic, vol. 17 (1979), no. 1, pp. 1–28.10.1016/0003-4843(79)90019-6CrossRefGoogle Scholar
Duret, J.-L., Les corps faiblement algebriquement clos non separablement clos ont la propriete d’independance, Model Theory of Algebra and Arithmetic: Proceedings of the Conference on Applications of Logic to Algebra and Arithmetic Held at Karpacz, Poland (Pacholski, L., Wierzejewski, J., and Wilkie, A. J., editors), Springer, Berlin, 1979, pp. 136–162.Google Scholar
Ellers, E. W., Gordeev, N., and Herzog, M., Covering numbers for Chevalley groups. Israel Journal of Mathematics, vol. 111 (1999), no. 1, pp. 339–372.10.1007/BF02810691CrossRefGoogle Scholar
Gorenstein, D., Finite Groups, second ed., AMS Chelsea Publishing, Providence, RI, 1980.Google Scholar
Gorenstein, D., Lyons, R., and Solomon, R., The Classification of the Finite Simple Groups, Mathematical Surveys and Monographs, vol. 40, American Mathematical Society, Providence, RI, 1998.Google Scholar
Hartley, B., Seitz, G. M., Borovik, A. V., and Bryant, R. M. (eds.), Finite and Locally Finite Groups, NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, vol. 471, Kluwer Academic, Dordrecht, 1995.10.1007/978-94-011-0329-9CrossRefGoogle Scholar
Hodges, W., Model Theory, Cambridge University Press, Cambridge, 1993.CrossRefGoogle Scholar
Kegel, O. H. and Wehrfritz, B. A. F., Locally Finite Groups, North-Holland, Amsterdam, 1973.Google Scholar
Kharlampovich, O. and Myasnikov, A., Definable sets in a hyperbolic group. International Journal of Algebra and Computation, vol. 23 (2013), pp. 91–110.CrossRefGoogle Scholar
Marker, D., Model Theory: An Introduction, Springer-Verlag, New York, 2002.Google Scholar
Perin, C., Pillay, A., Sklinos, R., and Tent, K., On groups and fields interpretable in torsion-free hyperbolic groups. Notre Dame Journal of Formal Logic, vol. 54 (2013), pp. 521–539.Google Scholar
Pillay, A., An Introduction to Stability Theory, Courier Corporation, New York, 2013.Google Scholar
Point, F., Ultraproducts and Chevalley groups. Archive for Mathematical Logic, vol. 38 (1999), no. 6, pp. 355–372.Google Scholar
Poizat, B., Stable Groups, Surveys and Monographs, American Mathematical Society, Providence, RI, 2001.CrossRefGoogle Scholar
Steinberg, R., Lectures on Chevalley Groups, University Lecture Series, vol. 66, Yale University Press, New Haven, CT, 1967.Google Scholar
Ryten, M. J., Results around asymptotic and measurable groups, Ph.D. thesis, University of Leeds, 2007.Google Scholar
Sela, Z., Diophantine geometry over groups viii: Stability. Annals of Mathematics. Second Series, vol. 177 (2013), pp. 787–868.Google Scholar
Tent, K. and Ziegler, M., A Course in Model Theory, Lecture Notes in Logic, Cambridge University Press, Cambridge, 2012.10.1017/CBO9781139015417CrossRefGoogle Scholar
Thomas, S. R., The classification of the simple periodic linear groups. Archiv der Mathematik, vol. 41 (1983), pp. 103–116.CrossRefGoogle Scholar
Thomas, S. R., Classification theory of simple locally finite groups, Ph.D. thesis, University of London, 1983.Google Scholar
Wagner, F. O., Stable Groups, Cambridge University Press, Cambridge, 1997.10.1017/CBO9780511566080CrossRefGoogle Scholar
Wilson, J. S., On simple pseudofinite groups. Journal of the London Mathematical Society, vol. 51 (1995), no. 2, pp. 471–490.10.1112/jlms/51.3.471CrossRefGoogle Scholar
Zilber, B. I., Groups and rings whose theory is categorical. Fundamenta Mathematicae, vol. 95 (1977), no. 3, pp. 173–188.Google Scholar
Zsigmondy, K., Zur Theorie der Potenzreste. Monatshefte für Mathematik und Physik, vol. 3 (1892), no. 1, pp. 265–284.10.1007/BF01692444CrossRefGoogle Scholar