Identifying long cycles in finite alternating and symmetric groups acting on subsets

Let H be a permutation group on a set Λ, which is permutationally isomorphic to a finite alternating or symmetric group An or Sn acting on the k-element subsets of points from {1, . . ., n}, for some arbitrary but fixed k.Suppose moreover that no isomorphism with this action is known.We show that key elements of H needed to construct such an isomorphism ϕ, such as those whose image under ϕ is an n-cycle or (n -1)-cycle, can be recognised with high probability by the lengths of just four of their cycles in Λ.