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Open Mathematics

formerly Central European Journal of Mathematics

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


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Volume 11, Issue 1

Issues

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
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/ 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
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Published Online: 2012-10-24 | DOI: https://doi.org/10.2478/s11533-012-0131-6

Abstract

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

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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.

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

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