Partial linear spaces with a rank 3 affine primitive group of automorphisms

0 Abstract

In this paper, we classify the finite proper partial linear spaces that admit rank $3$ affine primitive automorphism groups, except for certain families of small groups, including subgroups of ${\mathrm{A\Gamma L}}_1(q)$.

We also provide a more detailed version of the classification of the rank $3$ affine primitive permutation groups, which may be of independent interest.

1 Introduction

Highly symmetric linear spaces

Highly symmetric linear spaces have been studied extensively.

• Kantor classified the $2$-transitive linear spaces ^qcyz95
  • they have automorphism groups acting transitively on ordered paired of distinct points, see ^7nqmm2.
  • include the projective space ${\mathrm{PG}}_n(q)$ and the affine space ${\mathrm{AG}}_n(q)$
  • Remark. Kantor gives a classification of $2$-transitive design with $\lambda =1$. Linear spaces are designs and so they are classified. See here.
• The flag-transitive linear space $\mathcal{S}$ has be classified in buekenhoutLinearSpacesFlagtransitive1990.
  • their automorphism group acts transitively on point-line incident pairs
  • the classification has an exception: $\mathcal{S}$ has $q$ points and $G\le {\mathrm{A\Gamma L}}_1(q)$ for some prime power $q$

Motivation

As a generalization of $2$-transitive partial linear space, consider those partial linear spaces for which some automorphism group acts transitively on ordered pairs of distinct collinear points, as well as ordered pairs of distinct non-collinear points. It is defined by 2-ultrahomogeneity. Note that such partial linear spaces are flag-transitive. Furthermore, if they have non-empty line sets and are not linear spaces, then their automorphism groups are of rank $3$ on point set.

Example 1.1

Let $(G,a,r)$ satisfy the following properties: $G$ is an affine primitive permutation group of rank 3 with socle $V:=V_d(p)$ such that $G_0⩽\Gamma L_a(r)$ and $G_0$ has two orbits ${\Delta }_1$ and ${\Delta }_2$ on the points of ${\mathrm{PG}}_{a-1}(r)$, where $r^a=p^d,a⩾2$ and $r>2$. Let ${\mathcal{L}}_i:=\left\{⟨u⟩+v:⟨u⟩\in {\Delta }_i,v\in V\right\}$ for $i\in \{1,2\}$. Then ${\mathcal{S}}_i:=\left(V,{\mathcal{L}}_i\right)$ is a proper partial linear space such that $G⩽\mathrm{Aut}\left({\mathcal{S}}_i\right)$ for $i\in \{1,2\}$.

As an example, take $V={\mathbb{F}}_3^2$ and $G_0=⟨\lambda I:\lambda \in {\mathbb{F}}_3,\phi ,\psi ⟩$, where $\phi :V\to V,(a,b)\mapsto (b,a)$ and $\psi :V\to V,(a,b)\mapsto (a,-b)$. Then ${\mathrm{PG}}_1(3)=\{(0,1),(1,1),(1,2),(1,0)\}$, and ${\mathcal{L}}_1=\{(0,1)+v,(1,0)+v:v\in V\}$, ${\mathcal{L}}_2=\{(1,1)+v,(1,2)+v:v\in V\}$. The two corresponding partial linear spaces are grid of $3\times 3$ and its complement.

Example 1.2

Let $U:=V_2(q)$ and $W:=V_m(q)$ where $q^{2m}=p^d$ and $m⩾2$. Let $V:=U\otimes W$. Let ${\Sigma }_U:=\left\{U\otimes w:w\in W^{\ast }\right\}$ and ${\Sigma }_W:=\left\{u\otimes W:u\in U^{\ast }\right\}$. For $X\in \{U,W\}$, let ${\mathcal{L}}_X:=\{Y+v:Y\in {\Sigma }_X,v\in V\}$. Then ${\mathcal{S}}_X:=\left(V,{\mathcal{L}}_X\right)$ is a proper partial linear space, and ${\mathcal{S}}_U\simeq {\mathcal{S}}_W$ when $m=2$.

Example 1.3

Let $d=2n$ where $p^n\ne 2$. The $p^n\times p^n$ grid is a proper partial linear space with point set $V:=V_n(p)\oplus V_n(p)$ whose line set is the union of $\left\{\left\{(v,w):v\in V_n(p)\right\}:w\in V_n(p)\right\}$ and $\left\{\left\{(w,v):v\in V_n(p)\right\}:w\in V_n(p)\right\}$. The $p^n\times p^n$ grid has line-size $p^n$ and point-size 2 , and its full automorphism group is $S_{p^n}\wr S_2$, which contains the rank 3 affine primitive group $V:({\mathrm{GL}}_n(p)\wr S_2)$.