Collineation group as a subgroup of the symmetric group

Fedor Bogomolov, Marat Rovinsky

Research output: Contribution to journalArticle

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 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, if ψ is infinite.

Original languageEnglish (US)
Pages (from-to)17-26
Number of pages10
JournalCentral European Journal of Mathematics
Volume11
Issue number1
DOIs
StatePublished - Oct 2013

Fingerprint

Collineation
Symmetric group
Dimension of a vector space
Subgroup
Pointwise Convergence
Collinear
Permutation group
Bijection
Subspace
Topology
Closed
Arbitrary
Theorem

Keywords

  • Collineations
  • Projective group
  • Symmetric groups

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Collineation group as a subgroup of the symmetric group. / Bogomolov, Fedor; Rovinsky, Marat.

In: Central European Journal of Mathematics, Vol. 11, No. 1, 10.2013, p. 17-26.

Research output: Contribution to journalArticle

@article{9ec24fbe6e4340418994541a1cc9cdd0,
title = "Collineation group as a subgroup of the symmetric group",
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 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, if ψ is infinite.",
keywords = "Collineations, Projective group, Symmetric groups",
author = "Fedor Bogomolov and Marat Rovinsky",
year = "2013",
month = "10",
doi = "10.2478/s11533-012-0131-6",
language = "English (US)",
volume = "11",
pages = "17--26",
journal = "Open Mathematics",
issn = "1895-1074",
publisher = "Walter de Gruyter GmbH & Co. KG",
number = "1",

}

TY - JOUR

T1 - Collineation group as a subgroup of the symmetric group

AU - Bogomolov, Fedor

AU - Rovinsky, Marat

PY - 2013/10

Y1 - 2013/10

N2 - 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, if ψ is infinite.

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 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, if ψ is infinite.

KW - Collineations

KW - Projective group

KW - Symmetric groups

UR - http://www.scopus.com/inward/record.url?scp=84868031052&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84868031052&partnerID=8YFLogxK

U2 - 10.2478/s11533-012-0131-6

DO - 10.2478/s11533-012-0131-6

M3 - Article

AN - SCOPUS:84868031052

VL - 11

SP - 17

EP - 26

JO - Open Mathematics

JF - Open Mathematics

SN - 1895-1074

IS - 1

ER -