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

TL;DR

This paper explores the binary actions of finite simple groups with a single conjugacy class of involutions, establishing a link between associated graphs and strongly embedded subgroups to advance understanding of their structure.

Contribution

It introduces a novel connection between conjugacy class graphs and strongly embedded subgroups, providing new insights into the binary actions of simple groups with one involution class.

Findings

Established a link between the graph $()$ and strongly embedded subgroups.

Proved results on binary actions of simple groups with a single involution class.

Enhanced understanding of the structure of simple groups with specific conjugacy class properties.

Abstract

We continue our investigation of binary actions of simple groups. In this paper, we demonstrate a connection between the graph $\Gamma(\mathcal{C})$ 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.