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The binary actions of simple groups of Lie type of characteristic 2

Nick Gill, Pierre Guillot, Martin W. Liebeck

TL;DR

This work analyzes binary actions of finite simple groups of Lie type in characteristic $2$ via involution graphs $\\Gamma(\\mathcal{C})$ and their component groups $\\Delta(t)$. It introduces the transport group technique and accompanying randomized methods to compute $\\Delta(t)$ and the terminal variant $\\Delta_\\infty(t)$ across classical and exceptional groups, delivering a near-complete classification for actions with even-order point stabilizers (with some exclusions in the symplectic and unitary families). The main results classify when $\\Delta(t)$ is the whole group or a center of a long or short root subgroup (or Sylow subgroup in certain small cases), and they establish a partial converse showing that several TI-subgroup coset actions are binary. The approach combines deep structural analysis of involution centralizers with computer-assisted computations to manage large groups and exceptional types, yielding a precise map between root-subgroup structure and binary-action behavior in characteristic $2$.

Abstract

Let $\mathcal{C}$ be a conjugacy class of involutions in a group $G$. We study the graph $Γ(\mathcal{C})$ whose vertices are elements of $\mathcal{C}$ with $g,h\in\mathcal{C}$ connected by an edge if and only if $gh\in\mathcal{C}$. For $t\in \mathcal{C}$, we define the component group of $t$ to be the subgroup of $G$ generated by all vertices in $Γ(\mathcal{C})$ that lie in the connected component of the graph that contains $t$. We classify the component groups of all involutions in simple groups of Lie type over a field of characteristic $2$. We use this classification to partially classify the transitive binary actions of the simple groups of Lie type over a field of characteristic $2$ for which a point stabilizer has even order. The classification is complete unless the simple group in question is a symplectic or unitary group.

The binary actions of simple groups of Lie type of characteristic 2

TL;DR

This work analyzes binary actions of finite simple groups of Lie type in characteristic via involution graphs and their component groups . It introduces the transport group technique and accompanying randomized methods to compute and the terminal variant across classical and exceptional groups, delivering a near-complete classification for actions with even-order point stabilizers (with some exclusions in the symplectic and unitary families). The main results classify when is the whole group or a center of a long or short root subgroup (or Sylow subgroup in certain small cases), and they establish a partial converse showing that several TI-subgroup coset actions are binary. The approach combines deep structural analysis of involution centralizers with computer-assisted computations to manage large groups and exceptional types, yielding a precise map between root-subgroup structure and binary-action behavior in characteristic .

Abstract

Let be a conjugacy class of involutions in a group . We study the graph whose vertices are elements of with connected by an edge if and only if . For , we define the component group of to be the subgroup of generated by all vertices in that lie in the connected component of the graph that contains . We classify the component groups of all involutions in simple groups of Lie type over a field of characteristic . We use this classification to partially classify the transitive binary actions of the simple groups of Lie type over a field of characteristic for which a point stabilizer has even order. The classification is complete unless the simple group in question is a symplectic or unitary group.
Paper Structure (16 sections, 20 theorems, 18 equations, 1 table)

This paper contains 16 sections, 20 theorems, 18 equations, 1 table.

Table of Contents

  1. Introduction
  2. Binary actions and component groups
  3. Component groups
  4. Terminal component groups
  5. Computing with component groups
  6. Introducing the transport group
  7. Randomized calculations
  8. Results
  9. Proof of Theorem \ref{['t: component']}
  10. Long root elements
  11. Linear groups
  12. Unitary groups
  13. Symplectic and Orthogonal groups
  14. Exceptional groups of Lie type
  15. Proof of Theorem \ref{['t: char 2']}
  16. ...and 1 more sections

Key Result

Theorem 1

Let $G = G(q)$ be a simple group of Lie type over $\mathbb{F}_q$, where $q = 2^a$. Suppose that $G$ has a binary action on a set $\Omega$ and that there exists $\omega\in\Omega$ such that $G_\omega$, the stabilizer in $G$ of $\omega$, is a proper subgroup of $G$ of even order. Then one of the follow

Theorems & Definitions (38)

  • Theorem 1
  • Theorem 2
  • Definition 2.1
  • Lemma 2.2
  • Example 2.3
  • Definition 2.4
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • proof
  • ...and 28 more