# 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

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topFedor 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 -

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