TY - JOUR

T1 - Collineation group as a subgroup of the symmetric group

AU - Bogomolov, Fedor

AU - Rovinsky, Marat

N1 - Funding Information:
Fedor Bogomolov is supported by NSF grant DMS-1001662 and by AG Laboratory GU-HSE grant RF government ag. 11 11.G34.31.0023. Marat Rovinsky is supported by AG Laboratory NRU-HSE grant RF government ag. 11 11.G34.31.0023 and by RFBR grant 10-01-93113-CNRSL-a “Homological Methods in Geometry”.

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

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U2 - 10.2478/s11533-012-0131-6

DO - 10.2478/s11533-012-0131-6

M3 - Article

AN - SCOPUS:84868031052

SN - 1895-1074

VL - 11

SP - 17

EP - 26

JO - Central European Journal of Mathematics

JF - Central European Journal of Mathematics

IS - 1

ER -