Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo

IMPACT FACTOR 2017: 0.831
5-year IMPACT FACTOR: 0.836

CiteScore 2017: 0.68

SCImago Journal Rank (SJR) 2017: 0.450
Source Normalized Impact per Paper (SNIP) 2017: 0.829

Mathematical Citation Quotient (MCQ) 2017: 0.32

ICV 2017: 161.82

Open Access
See all formats and pricing
More options …
Volume 11, Issue 1


Volume 13 (2015)

Collineation group as a subgroup of the symmetric group

Fedor Bogomolov
  • Courant Institute of Mathematical Sciences, 251 Mercer St., New York, NY, 10012, USA
  • Laboratory of Algebraic Geometry, National Research University Higher School of Economics, 7 Vavilova Str., Moscow, 117312, Russia
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Marat Rovinsky
  • Laboratory of Algebraic Geometry, National Research University Higher School of Economics, 7 Vavilova Str., Moscow, 117312, Russia
  • Institute for Information Transmission Problems of Russian Academy of Sciences, Bolshoy Karetny Per. 19, Moscow, 127994, Russia
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2012-10-24 | DOI: https://doi.org/10.2478/s11533-012-0131-6


Let ψ be the projectivization (i.e., the set of one-dimensional vector subspaces) of a vector space of dimension ≥ 3 over a field. Let H be a closed (in the pointwise convergence topology) subgroup of the permutation group $\mathfrak{S}_\psi $ of the set ψ. Suppose that H contains the projective group and an arbitrary self-bijection of ψ transforming a triple of collinear points to a non-collinear triple. It is well known from [Kantor W.M., McDonough T.P., On the maximality of PSL(d+1,q), d ≥ 2, J. London Math. Soc., 1974, 8(3), 426] that if ψ is finite then H contains the alternating subgroup $\mathfrak{A}_\psi $ of $\mathfrak{S}_\psi $. We show in Theorem 3.1 that H = $\mathfrak{S}_\psi $, if ψ is infinite.

MSC: 20B22; 20B35

Keywords: Projective group; Collineations; Symmetric groups

  • [1] Ball R.W., Maximal subgroups of symmetric groups, Trans. Amer. Math. Soc., 1966, 121(2), 393–407 http://dx.doi.org/10.1090/S0002-9947-1966-0202813-2CrossrefGoogle Scholar

  • [2] Becker H., Kechris A.S., The Descriptive Set Theory of Polish Group Actions, London Math. Soc. Lecture Note Ser., 232, Cambridge University Press, Cambridge, 1996 http://dx.doi.org/10.1017/CBO9780511735264CrossrefGoogle Scholar

  • [3] Bergman G., Shelah S., Closed subgroups of the infinite symmetric group, Algebra Universalis, 2006, 55(2–3), 137–173 http://dx.doi.org/10.1007/s00012-006-1959-zCrossrefGoogle Scholar

  • [4] Cameron P.J., Oligomorphic Permutation Groups, London Math. Soc. Lecture Note Ser., 152, Cambridge University Press, Cambridge, 1990 http://dx.doi.org/10.1017/CBO9780511549809CrossrefGoogle Scholar

  • [5] Dixon J.D., Mortimer B., Permutation Groups, Grad. Texts in Math., 163, Springer, New York, 1996 http://dx.doi.org/10.1007/978-1-4612-0731-3CrossrefGoogle Scholar

  • [6] Hodges W., Model Theory, Encyclopedia Math. Appl., 42, Cambridge University Press, Cambridge, 2008 Google Scholar

  • [7] Huisman J., Mangolte F., The group of automorphisms of a real rational surface is n-transitive, Bull. Lond. Math. Soc., 2009, 41(3), 563–568 http://dx.doi.org/10.1112/blms/bdp033CrossrefGoogle Scholar

  • [8] Kantor W.M., Jordan groups, J. Algebra, 1969, 12(4), 471–493 http://dx.doi.org/10.1016/0021-8693(69)90024-6CrossrefGoogle Scholar

  • [9] Kantor W.M., McDonough T.P., On the maximality of PSL(d+1, q), d ≥ 2, J. London Math. Soc., 1974, 8(3), 426 http://dx.doi.org/10.1112/jlms/s2-8.3.426CrossrefGoogle Scholar

  • [10] Kollár J., Mangolte F., Cremona transformations and diffeomorphisms of surfaces, Adv. Math., 2009, 222(1), 44–61 http://dx.doi.org/10.1016/j.aim.2009.03.020CrossrefWeb of ScienceGoogle Scholar

  • [11] Macpherson H.D., Neumann P.M., Subgroups of infinite symmetric groups, J. London Math. Soc., 1990, 42(1), 64–84 http://dx.doi.org/10.1112/jlms/s2-42.1.64CrossrefGoogle Scholar

  • [12] Miller G.A., Limits of the degree of transitivity of substitution groups, Bull. Amer. Math. Soc., 1915, 22(2), 68–71 http://dx.doi.org/10.1090/S0002-9904-1915-02720-5CrossrefGoogle Scholar

  • [13] Richman F., Maximal subgroups of infinite symmetric groups, Canad. Math. Bull., 1967, 10(3), 375–381 http://dx.doi.org/10.4153/CMB-1967-035-0CrossrefGoogle Scholar

  • [14] Wielandt H., Abschätzungen für den Grad einer Permutationsgruppe von vorgeschriebenem Transitivitätsgrad, Schriften des mathematischen Seminars und des Instituts für angewandte Mathematik der Universität Berlin, 1934, 2, 151–174 Google Scholar

About the article

Published Online: 2012-10-24

Published in Print: 2013-01-01

Citation Information: Open Mathematics, Volume 11, Issue 1, Pages 17–26, ISSN (Online) 2391-5455, DOI: https://doi.org/10.2478/s11533-012-0131-6.

Export Citation

© 2012 Versita Warsaw. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

Comments (0)

Please log in or register to comment.
Log in