Collineation group as a subgroup of the symmetric group

Abstract

Let $\Psi$ be the projectivization (i.e., the set of one-dimensional vector subspaces) of a vector space of dimension $\ge 3$ over a field. Let $H$ be a closed (in the pointwise convergence topology) subgroup of the permutation group ${\mathfrak{S}}_{\Psi }$ of the set $\Psi$. Suppose that $H$ contains the projective group and an arbitrary self-bijection of $\Psi$ transforming a triple of collinear points to a non-collinear triple. It is well-known from \cite{KantorMcDonough} that if $\Psi$ is finite then $H$ contains the alternating subgroup ${\mathfrak{A}}_{\Psi }$ of ${\mathfrak{S}}_{\Psi }$. We show in Theorem \ref{density} below that $H={\mathfrak{S}}_{\Psi }$, if $\Psi$ is infinite.

9 pages

References in corpus (1)

• The group of automorphisms of a real rational surface is n-transitive