Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Gill-Guillot commuting graph for sporadic and related groups

David A. Craven, Coen del Valle, Chris Parker

TL;DR

The paper addresses the connectivity of the Gill-Guillot graph $\mathcal{G}(\mathcal{C})$ for conjugacy classes of elements of prime order in quasisimple groups with sporadic simple quotients or exceptional central covers, linking disconnection to strongly $p$-embedded subgroups. It develops a framework of definitions and lemmas and combines algebraic arguments with computer-assisted verification to classify all disconnected cases and enumerate them in explicit tables. Central extensions are analyzed by reducing to the simple quotient and, in the $p=2$ case, describing the graph via a lexicographic product; a complete classification of disconnected lifts for triple and exceptional covers is provided (Tables 3covers and exco). The results advance understanding of commuting structures in sporadic groups and offer practical tools for studying point stabilizers in binary actions and for performing Gill-Guillot graph computations on large finite groups.

Abstract

Let $G$ be a finite group and $\mathcal{C}$ a normal subset of $G$. The Gill-Guillot graph has vertices $\mathcal C$ and $x, y \in \mathcal C$ are adjacent if and only if $x$ and $y$ commute and $\{xy^{-1},x^{-1}y\} \cap \mathcal C$ is non-empty. We study the connectivity of this graph for quasisimple groups with $G/Z(G)$ a sporadic simple group and for certain simple groups with exceptional Schur multiplier.

The Gill-Guillot commuting graph for sporadic and related groups

TL;DR

The paper addresses the connectivity of the Gill-Guillot graph for conjugacy classes of elements of prime order in quasisimple groups with sporadic simple quotients or exceptional central covers, linking disconnection to strongly -embedded subgroups. It develops a framework of definitions and lemmas and combines algebraic arguments with computer-assisted verification to classify all disconnected cases and enumerate them in explicit tables. Central extensions are analyzed by reducing to the simple quotient and, in the case, describing the graph via a lexicographic product; a complete classification of disconnected lifts for triple and exceptional covers is provided (Tables 3covers and exco). The results advance understanding of commuting structures in sporadic groups and offer practical tools for studying point stabilizers in binary actions and for performing Gill-Guillot graph computations on large finite groups.

Abstract

Let be a finite group and a normal subset of . The Gill-Guillot graph has vertices and are adjacent if and only if and commute and is non-empty. We study the connectivity of this graph for quasisimple groups with a sporadic simple group and for certain simple groups with exceptional Schur multiplier.
Paper Structure (6 sections, 23 theorems, 6 equations, 8 tables)

This paper contains 6 sections, 23 theorems, 6 equations, 8 tables.

Table of Contents

  1. Introduction
  2. Setup and preliminary lemmas
  3. Sporadic simple groups
  4. Central extensions
  5. Covers of sporadic simple groups
  6. Exceptional covers of alternating groups and groups of Lie type

Key Result

Theorem 1.1

Let $p$ be a prime, let $G$ be a quasisimple group with $Z(G)$ a (possibly trivial) $p$-group, and let $t\in G\setminus Z(G)$ have order $p$. Set Assume that $G/Z(G)$ is either a sporadic simple group, or $G$ is an exceptional cover of either an alternating group or a simple group of Lie type. If $\mathcal{G}$ is disconnected, then one of the following holds:

Theorems & Definitions (43)

  • Theorem 1.1
  • Corollary 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 33 more