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Fixers and stabilizers for Ree groups

Yilin Xie

TL;DR

This work determines large fixers for almost simple primitive groups with socle $^2G_2(q)$, connecting the fixer concept to Erdős–Ko–Rado type questions and derangements. It analyzes the Ree group structure via a Borel subgroup, unipotent subgroups, and a 7-dimensional representation, and then applies a reduction to two concrete scenarios involving maximal subfield subgroups and a $q=27$ exception. The main result provides explicit normalizer forms for large fixers in the two cases and shows nonexistence of such fixers in many other configurations, enriching the understanding of fixers in Lie type rank-1 groups. The findings have implications for combinatorial settings (EKR-type properties) and derangement-related questions in Ree group actions.

Abstract

Let $G$ be a finite permutation group on $Ω,$ a subgroup $K\leqslant G$ is called a fixer if each element in $K$ fixes some element in $Ω.$ In this paper, we characterize fixers $K$ with $|K|\geqslant |G_ω|$ for each primitive action of almost simple group $G$ with socle ${}^2G_2(q).$

Fixers and stabilizers for Ree groups

TL;DR

This work determines large fixers for almost simple primitive groups with socle , connecting the fixer concept to Erdős–Ko–Rado type questions and derangements. It analyzes the Ree group structure via a Borel subgroup, unipotent subgroups, and a 7-dimensional representation, and then applies a reduction to two concrete scenarios involving maximal subfield subgroups and a exception. The main result provides explicit normalizer forms for large fixers in the two cases and shows nonexistence of such fixers in many other configurations, enriching the understanding of fixers in Lie type rank-1 groups. The findings have implications for combinatorial settings (EKR-type properties) and derangement-related questions in Ree group actions.

Abstract

Let be a finite permutation group on a subgroup is called a fixer if each element in fixes some element in In this paper, we characterize fixers with for each primitive action of almost simple group with socle

Paper Structure

This paper contains 6 sections, 30 theorems, 96 equations.

Table of Contents

  1. Introduction
  2. Basic properties for fixers
  3. Basic properties for finite Ree groups
  4. Some property of finite field $\mathbb{F}_q$
  5. A reduction
  6. Proof of Theorem \ref{['thm:main']}

Key Result

Theorem 2

Let $G_0={}^2G_2(q)$, where $q=3^{2n+1}$, and let $G=G_0.\langle \psi\rangle$ with $\psi\in\langle \phi\rangle$. Let $G$ be a primitive permutation group on $\Omega$ which has a fixer $K$ with $|K|\geqslant|G_\omega|$. Then either

Theorems & Definitions (50)

  • Theorem 2
  • Corollary 1.1
  • Lemma 2.1
  • Example 3
  • Example 4
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Lemma 3.1
  • Lemma 3.2
  • ...and 40 more