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The Burness-Giudici Conjecture on Primitive Groups with Socle PSU(3,q)

Huye Chen, Shaofei Du, Weicong Li

TL;DR

We address the Burness–Giudici Saxl-graph conjecture for primitive groups with rank-one Lie-type socles, focusing on PSU$(3,q)$. The authors prove the conjecture for the natural primitive action with socle PSU$(3,q)$ and point stabilizer $M= ext{PSO}(3,q)$, using a geometric description in unitary space based on Baer subplanes and frames to construct common neighbors in the Saxl graph. They also extend the result to the almost-simple extension $G= ext{PSU}(3,q) times ext{Z}_{m_1}$, obtaining a BG-conjecture validation under explicit lower-bounds on $ oot elax q$ (namely $ oot elax q\ge 17$ when $d=(3,q+1)=1$ and $ oot elax q\\ge 45$ when $d=3$). A central technical achievement is the estimation of regular and non-regular suborbits through a blend of algebraic, geometric, and analytic methods, including Weil bounds and intricate suborbit counting arguments. The results substantially advance the Rank-one case of the conjecture and contribute to the broader understanding of Saxl graphs for Lie-type primitive groups, with implications for base-two classification programs and permutation-group combinatorics.

Abstract

Let $G$ be a transitive permutation group on a set $Ω$, and suppose $G_α\cap G_β=1$ for some distinct $α, β\inΩ$. The Saxl graph $Σ(G)$ of $(G, Ω)$ is defined as the graph with vertex set $Ω$, where two vertices $α', β'$ are adjacent if and only if $G_{α'}\cap G_{β'}=1$. Burness and Giudici conjectured that for every primitive permutation group $G$, its Saxl graph has the property that any two vertices share a common neighbor. We focus on proving the conjecture for all primitive groups $G$ whose socle is a simple group of Lie-type of rank $1$; that is, $soc(G)\in \{PSL(2,q),PSU(3,q), Ree(q),Sz(q)\}$. The case $soc(G)=PSL(2,q)$ has been treated in two earlier papers. The purpose of the present paper is to settle the case $soc(G)=PSU(3,q)$. To finsh this work, we draw on methods from abstract- and permutation- group theory, finite unitary geometry, probabilistic approach, number theory (employing Weil's bound), and, most importantly, algebraic combinatorics, which provides us some key ideas.

The Burness-Giudici Conjecture on Primitive Groups with Socle PSU(3,q)

TL;DR

We address the Burness–Giudici Saxl-graph conjecture for primitive groups with rank-one Lie-type socles, focusing on PSU. The authors prove the conjecture for the natural primitive action with socle PSU and point stabilizer , using a geometric description in unitary space based on Baer subplanes and frames to construct common neighbors in the Saxl graph. They also extend the result to the almost-simple extension , obtaining a BG-conjecture validation under explicit lower-bounds on (namely when and when ). A central technical achievement is the estimation of regular and non-regular suborbits through a blend of algebraic, geometric, and analytic methods, including Weil bounds and intricate suborbit counting arguments. The results substantially advance the Rank-one case of the conjecture and contribute to the broader understanding of Saxl graphs for Lie-type primitive groups, with implications for base-two classification programs and permutation-group combinatorics.

Abstract

Let be a transitive permutation group on a set , and suppose for some distinct . The Saxl graph of is defined as the graph with vertex set , where two vertices are adjacent if and only if . Burness and Giudici conjectured that for every primitive permutation group , its Saxl graph has the property that any two vertices share a common neighbor. We focus on proving the conjecture for all primitive groups whose socle is a simple group of Lie-type of rank ; that is, . The case has been treated in two earlier papers. The purpose of the present paper is to settle the case . To finsh this work, we draw on methods from abstract- and permutation- group theory, finite unitary geometry, probabilistic approach, number theory (employing Weil's bound), and, most importantly, algebraic combinatorics, which provides us some key ideas.
Paper Structure (14 sections, 46 theorems, 173 equations, 1 table)

This paper contains 14 sections, 46 theorems, 173 equations, 1 table.

Table of Contents

  1. Introduction
  2. $G=\hbox{\rm PSU}(3,q)$ and $M=\hbox{\rm PSO}(3,q)$
  3. A geometrical description of the action
  4. The proof of Theorem
  5. $\hbox{\rm PSL}(3,q)\lvertneqq G$ and $\hbox{\rm SO}(3,q)\lvertneqq M$
  6. A lower bound of $|\Gamma_{r}(\alpha)|$
  7. A lower bound of $|\Gamma_{nr}(\alpha)\cap \Gamma_{nr}(\alpha')|$
  8. Measure ${\hm A}$
  9. Measure ${\hm B}$
  10. Upper bounds of ${\bf B}_1$ and ${\bf B}_2$
  11. Upper bound of $|{\bf H}_1(\mathcal{D},g)|$ with $\mathcal{D}\in F$
  12. Upper bounds of $|{\bf L}_1(\mathcal{D},g)|$ ($\mathcal{D}\in F_1$) and ${\bf f}_2$
  13. Upper bounds of $\hm B_3$ and $\hm B_4$
  14. Upper bounds of ${\hm C}$ and ${\hm D}$

Key Result

Theorem 1.2

Let $G$ be a primitive permutation group with socle $\hbox{\rm PSU}(3,q)$, where $q=p^m\ge 7$ for an odd prime $p$. Suppose a point-stabilizer $M$ of $G$ contains $\hbox{\rm PSO}(3,q)$. Then $b(G)=2$ and either (i) $G=\hbox{\rm PSU}(3,q)$ and $M=\hbox{\rm SO}(3,q)$; or (ii) $G=\hbox{\rm PSU}(3,q)\ma

Theorems & Definitions (83)

  • Conjecture 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 2.1
  • Proposition 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • ...and 73 more