Collineation group as a subgroup of the symmetric group

Fedor Bogomolov; Marat Rovinsky

Open Mathematics (2013)

  • Volume: 11, Issue: 1, page 17-26
  • ISSN: 2391-5455

Abstract

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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 𝔖 ψ 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 𝔄 ψ of 𝔖 ψ . We show in Theorem 3.1 that H = 𝔖 ψ , if ψ is infinite.

How to cite

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Fedor Bogomolov, and Marat Rovinsky. "Collineation group as a subgroup of the symmetric group." Open Mathematics 11.1 (2013): 17-26. <http://eudml.org/doc/269715>.

@article{FedorBogomolov2013,
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.},
author = {Fedor Bogomolov, Marat Rovinsky},
journal = {Open Mathematics},
keywords = {Projective group; Collineations; Symmetric groups; projective groups; collineation groups; symmetric groups; highly transitive actions; dense subgroups},
language = {eng},
number = {1},
pages = {17-26},
title = {Collineation group as a subgroup of the symmetric group},
url = {http://eudml.org/doc/269715},
volume = {11},
year = {2013},
}

TY - JOUR
AU - Fedor Bogomolov
AU - Marat Rovinsky
TI - Collineation group as a subgroup of the symmetric group
JO - Open Mathematics
PY - 2013
VL - 11
IS - 1
SP - 17
EP - 26
AB - 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.
LA - eng
KW - Projective group; Collineations; Symmetric groups; projective groups; collineation groups; symmetric groups; highly transitive actions; dense subgroups
UR - http://eudml.org/doc/269715
ER -

References

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  1. [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 Zbl0136.27805
  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 Zbl0949.54052
  3. [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 Zbl1111.20004
  4. [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 Zbl0813.20002
  5. [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. [6] Hodges W., Model Theory, Encyclopedia Math. Appl., 42, Cambridge University Press, Cambridge, 2008 
  7. [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 Zbl1170.14042
  8. [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. [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 Zbl0299.20010
  10. [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 Zbl1188.14008
  11. [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 Zbl0668.20005
  12. [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 Zbl45.1237.04
  13. [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 Zbl0149.27101
  14. [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 Zbl0011.01005

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