Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On transitive sets of derangements in primitive groups

Peter Müller

TL;DR

The paper investigates Thompson's Problem 8.75, asking whether a derangement can map two distinct points in a finite primitive group. It identifies a negative answer within the almost simple group $^{3}D_{4}(2)$ by constructing a primitive action of degree $4{,}064{,}256$ and proving that any element sending $\alpha$ to $\beta$ fixes a point. The authors realize $^{3}D_{4}(2)$ explicitly as a subgroup of $GL_8(\mathbb{F}_8)$, and verify key group-theoretic relations via matrix identities (a SageMath script is provided). The work both answers Thompson's question in this case and discusses potential extensions to automorphism groups and other twisted groups, highlighting connections to orbital graphs and primitivity in almost simple groups.

Abstract

We construct a primitive permutation action of the Steinberg triality group $^3D_4(2)$ of degree $4064256$ and show that there are distinct points $α,β$ such that there is no derangement $g\in{^3D_4}(2)$ with $α^g=β$. This answers a question by John G. Thompson (Problem 8.75 in the Kourovka Notebook) in the negative.

On transitive sets of derangements in primitive groups

TL;DR

The paper investigates Thompson's Problem 8.75, asking whether a derangement can map two distinct points in a finite primitive group. It identifies a negative answer within the almost simple group by constructing a primitive action of degree and proving that any element sending to fixes a point. The authors realize explicitly as a subgroup of , and verify key group-theoretic relations via matrix identities (a SageMath script is provided). The work both answers Thompson's question in this case and discusses potential extensions to automorphism groups and other twisted groups, highlighting connections to orbital graphs and primitivity in almost simple groups.

Abstract

We construct a primitive permutation action of the Steinberg triality group of degree and show that there are distinct points such that there is no derangement with . This answers a question by John G. Thompson (Problem 8.75 in the Kourovka Notebook) in the negative.
Paper Structure (4 sections, 2 theorems, 2 equations)

This paper contains 4 sections, 2 theorems, 2 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['t:3d4']}
  3. Proof of Lemma \ref{['l:mat8']}.
  4. Remarks

Key Result

Theorem 1.2

The Steinberg triality group $G = {}^{3}D_{4}(2)$ of order $211{,}341{,}312 = 2^{12}\cdot 3^{4}\cdot 7^{2}\cdot 13$ admits a primitive permutation action on a set $\Omega$ of size $4{,}064{,}256$ with the following property: There exist distinct elements $\alpha, \beta\in\Omega$ such that every $g\i

Theorems & Definitions (2)

  • Theorem 1.2
  • Lemma 2.1