  We introduce a new approach to the study of finite binary permutation groups
and, as an application of our method, we prove Cherlin's binary groups
conjecture for groups with socle a finite alternating group, and for the
$\mathcal{C}_1$-primitive actions of the finite classical groups.
  Our new approach involves the notion, defined with respect to a group action,
of a `\emph{beautiful subset}'. We demonstrate how the presence of such subsets
can be used to show that a given action is not binary. In particular, the study
of such sets will lead to a resolution of many of the remaining open cases of
Cherlin's binary groups conjecture.
