The binary actions of alternating groups

TL;DR

This paper develops a graph-theoretic tool, $\Gamma(\mathcal{C})$, built from a conjugacy class $\mathcal{C}$ to relate the binary action property of a finite group to connectivity properties of $\Gamma(\mathcal{C})$. By establishing a key lemma that connects maximal-fixity $p$-elements and stabilizers to these connectivity properties, the authors achieve a complete classification of binary actions for the alternating group $A_n$ with $n \ge 6$, showing that orbits must be trivial or regular in any binary action. The analysis hinges on detailed descriptions of $\Gamma(\mathcal{C})$ for involutions and the behavior of $p$-elements, together with reductions that exclude possibilities where the point stabilizer has even order or is a $3$-group. Overall, the work demonstrates how conjugacy-structure and stabilizer constraints rigidly limit binary actions in almost simple groups, offering a framework likely extendable to other families of finite simple groups.

Abstract

Given a conjugacy class $\mathcal{C}$ in a group $G$ we define a new graph, $Γ(\mathcal{C})$, whose vertices are elements of $\mathcal{C}$; two vertices $g,h\in \mathcal{C}$ are connected in $Γ(\mathcal{C})$ if $[g,h]=1$ and either $gh^{-1}$ or $hg^{-1}$ is in $\mathcal{C}$. We prove a lemma that relates the binary actions of the group $G$ to connectivity properties of $Γ(\mathcal{C})$. This lemma allows us to give a complete classification of all binary actions when $G=A_n$, an alternating group on $n$ letters with $n\geq 5$.

Theorems & Definitions (46)

• Lemma 1.1
• Theorem 1.2
• Example 2.1
• Example 2.2
• Lemma 2.3
• proof
• Lemma 2.4: basic criteria
• proof
• Lemma 2.5
• proof