  Given a conjugacy class $\mathcal{C}$ in a group $G$ we define a new graph,
$\Gamma(\mathcal{C})$, whose vertices are elements of $\mathcal{C}$; two
vertices $g,h\in \mathcal{C}$ are connected in $\Gamma(\mathcal{C})$ 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 $\Gamma(\mathcal{C})$. This lemma allows us to give
a complete classification of all binary actions when $G=A_n$, an alternating
group on $n$ letters with $n\geq 5$.
