The binary actions of alternating groups

Given a conjugacy class 𝒞 in a group G we define a new graph, Γ ( 𝒞 ) , whose vertices are elements of 𝒞 ; two vertices g , h ∈ 𝒞 are connected in Γ ( 𝒞 ) if [ g , h ] = 1 and either g h - 1 or h g - 1 is in 𝒞 . We prove a lemma that relates the binary actions of the group G to connectivity properties of Γ ( 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 ≥ 5 .