Abstract
This is a survey, intended both for group theorists and model theorists, concerning the structure of pseudofinite groups, that is, infinite models of the first-order theory of finite groups. The focus is on concepts from stability theory and generalisations in the context of pseudofinite groups, and on the information this might provide for finite group theory.
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References
Ax, J.: The elementary theory of finite fields. Ann. Math. 88, 239–271 (1968)
Baldwin, J., Saxl, J.: Logical stability in group theory. J. Aust. Math. Soc. 21, 267–276 (1976)
Barbina, S.: Reconstruction of classical geometries from their automorphism groups. J. Lond. Math. Soc. (2) 75, 298–316 (2007)
Baudisch, A.: Mekler’s construction preserves CM-triviality. Ann. Pure Appl. Logic 115, 115–173 (2002)
Bélair, L.: Types dans les corps valués munis d’applications coefficients. Ill. J. Math. 43(2), 410–425 (1999)
Bello Aguirre, R.: Generalised stability of ultraproducts of finite residue rings. arXiv:1503.04454
Bello Aguirre, R.: Model Theory of Finite and Pseudofinite Rings. Ph.D. thesis, University of Leeds, (2016)
Borovik, A., Cherlin, G.: Permutation groups of finite Morley rank. In: Chatzidakis, Z., Macpherson, H.D., Pillay, A., Wilkie, A.J. (eds.) Model Theory with Applications to Algebra and Analysis. London Math. Soc. Lecture Notes 350, pp. 59–124. Cambridge University Press, Cambridge (2008)
Borovik, A., Thomas, S.: On generic normal subgroups. In: Kaye, R., Macpherson, H.D. (eds.) Automorphisms of First-Order Structures, pp. 319–324. Clarendon Press, Oxford (1994)
Breuillard, E., Green, B., Tao, T.C.: Suzuki groups as expanders. Groups Geom. Dyn. 5, 281–299 (2011)
Bunina, E.I., Mikhalev, A.V.: Elementary properties of linear groups and related problems. J. Math. Sci. 123, 3921–3985 (2004)
Carter, R.W.: Simple Groups of Lie Type. Wiley, London (1972)
Casanovas, E.: Simple Theories and Hyperimaginaries. Lecture Notes in Logic, vol. 39. Cambridge University Press, Cambridge (2011)
Chatzidakis, Z., van den Dries, L., Macintyre, A.J.: Definable sets over finite fields. J. Reine Angew. Math. 427, 107–135 (1992)
Chatzidakis, Z.: Model theory of finite fields and pseudofinite fields. Ann. Pure Appl. Logic 88, 95–108 (1997)
Chatzidakis, Z., Hrushovski, E.: The model theory of difference fields. Trans. Am. Math. Soc. 351, 2997–3071 (1999)
Chatzidakis, Z., Hrushovski, E., Peterzil, Y.: Model theory of difference fields II. Periodic ideals and the trichotomy in all characteristics. Proc. Lond. Math. Soc. (3) 85, 257–311 (2002)
Cherlin, G., Hrushovski, E.: Finite Structures with Few Types. Annals of Math. Studies 152. Princeton University Press, Princeton (2003)
Chernikov, A.: Theories without the tree property of the second kind. Ann. Pure Appl. Logic 165, 695–723 (2014)
Chernikov, A., Kaplan, I., Simon, P.: Groups and fields with NTP2. Proc. Am. Math. Soc. 143, 395–406 (2015)
Dello Stritto, P.: Asymptotic classes of finite Moufang polygons. J. Algebra 332, 114–135 (2011)
Duret, J.L.: Les corps faiblement algébriquement clos non séparablement clos ont la propriété d’indépendence. In: Model Theory of Algebra and Arithmetic, Springer Lecture Notes vol. 834, pp. 135–157 (1980)
Elwes, R.: Asymptotic classes of finite structures. J. Symb. Logic 72, 418–438 (2007)
Elwes, R., Jaligot, E., Macpherson, H.D., Ryten, M.J.: Groups in supersimple and pseudofinite theories. Proc. Lond. Math. Soc. (3) 103, 1049–1082 (2011)
Elwes, R., Ryten, M.: Measurable groups of low dimension. Math. Log. Q. 54, 374–386 (2008)
Ershov, J.: Undecidability of the theories of symmetric and simple finite groups. Dokl. Akad. Nauk. SSSR 158, 777–779 (1964)
Felgner, U.: Pseudo-endliche Gruppen. In: Proceedings of the 8th Easter Conference on Model Theory (Wendisch-Rietz, 1990), pp. 82–96, Seminarberichte 110, Humboldt University, Berlin (1990)
Garcia, D., Macpherson, H.D., Steinhorn, C.: Pseudofinite structures and simplicity. J. Math. Logic 15(1), 1550002 (2015)
Gowers, T.: Quasirandom groups. Comb. Prob. Comput. 17, 363–378 (2008)
Granger, N.: Stability, simplicity and the model theory of bilinear forms. Ph.D. Thesis, University of Manchester (1999). www.maths.manchester.ac.uk/~mprest/
Higman, G.: A finitely generated infinite simple group. J. Lond. Math. Soc. 26, 61–64 (1951)
Higman, D.G.: Intersection matrices for finite permutation groups. J. Algebra 6, 22–42 (1967)
Hodges, W.: Model Theory. Cambridge University Press, Cambridge (1993)
Hrushovski, E.: Almost orthogonal regular types. Ann. Pure Appl. Logic 45, 139–155 (1989)
Hrushovski, E.: Pseudofinite fields and related structures. In: Bélair, L., Chatzidakis, Z., d’Aquino, P., Marker, D., Otero, M., Point, F., Wilkie, A.J. (eds.) Model Theory and Applications. Quaderni di Matematica, vol. 11, pp. 151–212. Aracne, Rome (2002)
Hrushovski, E.: The elementary theory of the Frobenius automorphisms. arXiv:math/0406514
Hrushovski, A., Pillay, A.: Definable subgroups of algebraic groups over finite fields. J. Reine Angew. Math. 462, 69–91 (1995)
Huber-Dyson, V.: A reduction of the open sentence problem for finite groups. Bull. Lond. Math. Soc. 13, 331–338 (1981)
Kassabov, M., Lubotzky, A., Nikolov, N.: Finite simple groups as expanders. Proc. Natl. Acad. Sci. USA 103(16), 6116–6119 (2006)
Kestner, C.: Measurability in modules. Arch. Math. Logic 53, 593–620 (2014)
Khukhro, E.I.: On solubility of groups with bounded centralizer chains. Glasgow Math. J. 51, 49–54 (2009)
Kiefe, C.: Sets definable over finite fields: their zeta functions. Trans. Am. Math. Soc. 223, 45–59 (1976)
Kim, B.: Simplicity Theory, Oxford Logic Guides. Clarendon Press, Oxford (2013)
Larsen, M.: Word maps have large image. Isr. J. Math. 139, 149–156 (2004)
Larsen, M., Shalev, A., Tiep, P.: The Waring problem for finite simple groups. Ann. Math. 174, 1885–1950 (2011)
Larsen, M., Shalev, A., Tiep, P.: The Waring problem for finite quasisimple groups. Int. Math. Res. Not. 10, 2323–2348 (2013)
Liebeck, M.W., Shalev, A.: Diameters of finite simple groups: sharp bounds and applications. Ann. Math. 154, 383–406 (2001)
Liebeck, M.W., Macpherson, H.D., Tent, K.: Primitive permutation groups of bounded orbital diameter. Proc. Lond. Math. Soc. (3) 100, 216–248 (2010)
Liebeck, M.W., O’Brien, E.A., Shalev, A., Tiep, P.H.: The ore conjecture. J. Eur. Math. Soc. 12, 939–1008 (2010)
Lubotzky, A.: Finite simple groups of Lie type as expanders. J. Eur. Math. Soc. 13, 1331–1341 (2011)
Macpherson, H.D., Mosley, A., Tent, K.: Permutation groups in o-minimal structures. J. Lond. Math. Soc. (2) 62, 650–670 (2000)
Macpherson, H.D., Pillay, A.: Primitive permutation groups of finite Morley rank. Proc. Lond. Math. Soc. (3) 70, 481–504 (1995)
Macpherson, H.D., Steinhorn, C.: One-dimensional asymptotic classes of finite structures. Trans. Am. Math. Soc. 360, 411–448 (2008)
Macpherson, H.D., Tent, K.: Stable pseudofinite groups. J. Algebra 312, 550–561 (2007)
Macpherson, H.D., Tent, K.: Pseudofinite groups with NIP theory and definability in finite simple groups. In: Strüngmann, L., Droste, M., Fuchs, L., Tent, K. (eds.) Groups and Model Theory. Contemp. Math. Soc., vol. 576, pp. 255–267. American Mathematical Society, Providence (2012)
Macpherson, H.D., Tent, K.: Profinite groups with NIP theory and \(p\)-adic analytic groups. arXiv:1603.02179
Mekler, A.H.: Stability of nilpotent groups of class 2 and prime exponent. J. Symb. Logic 46, 781–788 (1981)
Milliet, C.: On the definability of radicals in supersimple groups. J. Symb. Logic 78(2), 649–656 (2013)
Milliet, C.: Definable envelopes in groups having a simple theory. https://hal.archives-ouvertes.fr/hal-00657716v2/document
Nikolov, N., Pyber, L.: Product decompositions of quasirandom groups and a Jordan type theorem. J. Eur. Math. Soc. 13, 1063–1077 (2011)
Nies, A., Tent, K.: Describing finite groups by short first-order sentences. arXiv:1409.8390
Ould Houcine, A., Point, F.: Alternatives for pseudofinite groups. J. Group Theory 16, 461–495 (2013)
Pillay, A.: Geometric Stability Theory. Oxford Logic Guides. Clarendon Press, Oxford (1996)
Pillay, A., Scanlon, T., Wagner, F.: Supersimple fields and division rings. Math. Res. Lett. 5, 473–483 (1998)
Pillay, A.: Remarks on compactifications of pseudofinite groups. arXiv:1509.02895
Point, F.: Ultraproducts and chevalley groups. Arch. Math. Logic 38, 355–372 (1999)
Ryten, M.J.: Model theory of finite difference fields and simple groups. Ph.D. Thesis, University of Leeds (2007). http://www1.maths.leeds.ac.uk/Pure/staff/macpherson/ryten1.pdf
Shalev, A.: Characterisation of \(p\)-adic analytic groups in terms of wreath products. J. Algebra 145, 204–208 (1992)
Simon, P.: A Guide to NIP Theories. Lecture Notes in Logic, vol. 44. Cambridge University Press, Cambridge (2015)
Tent, K., Ziegler, M.: A Course in Model Theory. Lecture Notes in Logic, vol. 40. Cambridge University Press, Cambridge (2012)
Thompson, J.G.: Nonsolvable finite groups all of whose local subgroups are solvable (Part I). Bull. Am. Math. Soc. (NS) 74, 383–437 (1968)
Tits, J., Weiss, R.: Moufang Polygons. Springer Monographs in Mathematics. Springer, Berlin (2002)
Ugurlu, P.: Pseudofinite groups as fixed points of simple groups of finite Morley rank. J. Pure Appl. Algebra 217, 892–900 (2013)
van den Dries, L.: Lectures on the model theory of valued fields. In: Model Theory in Algebra, Analysis, and Arithmetic, Lecture Notes in Math. No. 2111, pp. 55–157. Springer, Heidelberg (2014)
Wagner, F.: Stable Groups. London Math. Soc. Lecture Notes no. 240. Cambridge University Press, Cambridge (1997)
Wagner, F.: Simple Theories. Kluwer, Dordrecht (2000)
Wagner, F.: Pseudofinite \(\tilde{\mathfrak{M}}_c\)-groups. arXiv:1511.09272
Wilson, J.S.: On pseudofinite simple groups. J. Lond. Math. Soc. (2) 51, 471–490 (1995)
Wilson, J.S.: Finite axiomatisation of finite soluble groups. J. Lond. Math. Soc. 74, 566–582 (2006)
Wilson, J.S.: First-order characterization of the radical of a finite group. J. Symb. Logic 74, 1429–1435 (2009)
Zilber, B.: Perfect infinities and finite approximation. In: Infinity and Truth, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., vol. 25, pp. 199–223. World Sci. Publ., Hackensack (2014)
Acknowledgements
I warmly thank the organisers of the ‘IPM conference on set theory and model theory’, Tehran, October 12–16 2015, a meeting which led to preparation of this paper. I also thank the referee for a number of helpful presentational suggestions.
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Research partially supported by EPSRC Grant EP/K020692/1.
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Macpherson, D. Model theory of finite and pseudofinite groups. Arch. Math. Logic 57, 159–184 (2018). https://doi.org/10.1007/s00153-017-0584-1
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DOI: https://doi.org/10.1007/s00153-017-0584-1