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

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