Collineation group as a subgroup of the symmetric group

Abstract

Let $Ψ$ 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 $\mathfrak{S}_Ψ$ 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 \cite{KantorMcDonough} that if $Ψ$ is finite then $H$ contains the alternating subgroup $\mathfrak{A}_Ψ$ of $\mathfrak{S}_Ψ$. We show in Theorem \ref{density} below that $H=\mathfrak{S}_Ψ$, if $Ψ$ is infinite.

9 pages