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2013 | 11 | 1 | 17-26
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Collineation group as a subgroup of the symmetric group

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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.
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Tom
11
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1
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17-26
Opis fizyczny
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wydano
2013-01-01
online
2012-10-24
Bibliografia
  • [1] Ball R.W., Maximal subgroups of symmetric groups, Trans. Amer. Math. Soc., 1966, 121(2), 393–407 http://dx.doi.org/10.1090/S0002-9947-1966-0202813-2
  • [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/CBO9780511735264
  • [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-z
  • [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/CBO9780511549809
  • [5] Dixon J.D., Mortimer B., Permutation Groups, Grad. Texts in Math., 163, Springer, New York, 1996 http://dx.doi.org/10.1007/978-1-4612-0731-3
  • [6] Hodges W., Model Theory, Encyclopedia Math. Appl., 42, Cambridge University Press, Cambridge, 2008
  • [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/bdp033
  • [8] Kantor W.M., Jordan groups, J. Algebra, 1969, 12(4), 471–493 http://dx.doi.org/10.1016/0021-8693(69)90024-6
  • [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/s2-8.3.426
  • [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.2009.03.020
  • [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/s2-42.1.64
  • [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/S0002-9904-1915-02720-5
  • [13] Richman F., Maximal subgroups of infinite symmetric groups, Canad. Math. Bull., 1967, 10(3), 375–381 http://dx.doi.org/10.4153/CMB-1967-035-0
  • [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
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