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

Let $H$ be a permutation group on a set $Lambda$, which is permutationally isomorphic to a finite alternating or symmetric group $A_n$ or $S_n$ acting on the $k$-element subsets of points from ${1,ldots,n}$, 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 $varphi$, such as those whose image under $varphi$ is an $n$-cycle or $(n-1)$-cycle, can be recognised with high probability by the lengths of just four of their cycles in $Lambda$.