A classification of finite partial linear spaces with a primitive rank 3 automorphism group of almost simple type

0 Abstract

In this paper, we give a complete classification of all finite proper partial linear spaces admitting a primitive rank $3$ automorphism group of almost simple type.

1 Introduction

The finite non-trivial graphs having the property of 2-ultrahomogeneity are exactly the finite rank $3$ graphs, whose classification immediately follows from the classification of finite primitive rank $3$ groups. If the automorphism group is imprimitive, then the graph is isomorphic to $K_n[N_m]$. If the automorphism group $G$ is primitive, then $G$ acting on $\Omega \times \Omega$ has three orbits $\{(\omega ,\omega ):\omega \in \Omega \}$, ${\Delta }_1$ and ${\Delta }_2$ where one of ${\Delta }_i$ is the set of edges. So $G$ is a primitive rank $3$ group.

2 Preliminary definition and results

Construction of partial linear space

For any permutation group $G$ acting on $\mathcal{P}$, the induced partial linear space has only two possibilities (block or local block):

Theorem

Remark. It is different with “block design”: ^6ky4f9.

It gives us a way to construct a partial linear space.

Theorem

Remark. If there exists a partial linear space having $G$ as a rank 3 automorphism group, then its collinearity graph will be one of the two strongly regular graphs associated with $G$.

Classification of $2$-ultrahomogeneous linear space

Kantor 1985

A reduction

Lemma

Definitions of well-known families of partial linear spaces

• A partial geometry with parameters $(s,t,\alpha )$ is a partial linear space satisfying the following conditions:
  • each line is incident with $s+1$ points ($s\ge 1$),
  • each point is incident with $t+1$ lines $(t\ge 1)$,
  • any point $p$ outside a line $L$ is collinear with a constant number $\alpha$ of points of $L$.
• A generalized quadrangle $GQ(s,t)$ of order $(s,t)$ is a partial geometry with parameters $(s,t,1)$; it is also a particular type of polar space.
• A polar space is a partial linear space satisfying the following conditions:
  • each line is incident with a constant number of points,
  • each point is incident with a constant number of lines,
  • any point $p$ outside a line $L$ is collinear with $1$ or all points of $L$.
• A copolar space is a partial linear space satisfying the following conditions:
  • each line is incident with a constant number of points,
  • each point is incident with a constant number of lines,
  • any point $p$ outside a line $L$ is collinear with none or all but one points of $L$.
• A $t$-$(v,k,\lambda )$ design is a point-block geometry $(\mathcal{P},\mathcal{B})$ where $\mathcal{P}$ is a set of $v$ points and $\mathcal{B}$ is a collection of $k$-subsets of $\mathcal{P}$, with the property that every $t$-subset of $\mathcal{P}$ is contained in exactly $\lambda$ blocks of $\mathcal{B}$. For example, a linear space over ${\mathbb{F}}_p$ is a $2-(v,p,1)$ design.
• A Steiner system $S(t,k,v)$ is a $t-(v,k,1)$ design.