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 1 2013

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Keywords

  • Collineations
  • Projective group
  • Symmetric groups

ASJC Scopus subject areas

  • Mathematics(all)

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