### 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 language | English (US) |
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Pages (from-to) | 17-26 |

Number of pages | 10 |

Journal | Central European Journal of Mathematics |

Volume | 11 |

Issue number | 1 |

DOIs | |

State | Published - Oct 2013 |

### Fingerprint

### Keywords

- Collineations
- Projective group
- Symmetric groups

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Central European Journal of Mathematics*,

*11*(1), 17-26. https://doi.org/10.2478/s11533-012-0131-6

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

Research output: Contribution to journal › Article

*Central European Journal of Mathematics*, vol. 11, no. 1, pp. 17-26. https://doi.org/10.2478/s11533-012-0131-6

}

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 -