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The binary actions of simple groups with a single conjugacy class of involutions

Nick Gill, Pierre Guillot

Abstract

We continue our investigation of binary actions of simple groups. In this paper, we demonstrate a connection between the graph $Γ(\mathcal{C})$ based on the conjugacy class $\mathcal{C}$ of the group $G$, which was introduced in our previous work, and the notion of a strongly embedded subgroup of $G$. We exploit this connection to prove a result concerning the binary actions of finite simple groups that contain a single conjugacy class of involutions.

The binary actions of simple groups with a single conjugacy class of involutions

Abstract

We continue our investigation of binary actions of simple groups. In this paper, we demonstrate a connection between the graph based on the conjugacy class of the group , which was introduced in our previous work, and the notion of a strongly embedded subgroup of . We exploit this connection to prove a result concerning the binary actions of finite simple groups that contain a single conjugacy class of involutions.
Paper Structure (10 sections, 25 theorems, 42 equations)

This paper contains 10 sections, 25 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. Background on relational complexity
  3. Complexity, height & some basic criteria
  4. The graph Gamma(C)
  5. A few lemmas
  6. Groups with a single class of involutions
  7. Strongly embedded subgroups
  8. The proof
  9. Auxiliary results
  10. The binary actions of PSL(2,q)

Key Result

Theorem 1.1

Suppose that $G$ is a simple group that contains a single conjugacy class of involutions, let $H$ be a proper subgroup of $G$ of even order and let $\Omega$ be the set of right cosets of $H$ in $G$. Then the action of $G$ on $\Omega$ is binary if and only if we are in one of the following situations

Theorems & Definitions (37)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1: basic criteria
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Example 2.4
  • Theorem 2.5
  • Example 2.6
  • ...and 27 more