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

Let $H$ be a permutation group on a set $\\Lambda$, which is permutationally\nisomorphic to a finite alternating or symmetric group $A_n$ or $S_n$ acting on\nthe $k$-element subsets of points from $\\{1,\\ldots,n\\}$, for some arbitrary but\nfixed $k$. Suppose moreover that no isomorphism with this action is known. We\nshow that key elements of $H$ needed to construct such an isomorphism\n$\\varphi$, such as those whose image under $\\varphi$ is an $n$-cycle or\n$(n-1)$-cycle, can be recognised with high probability by the lengths of just\nfour of their cycles in $\\Lambda$.\n