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

Abstract

Let $H$ be a permutation group on a set $\Lambda$, which is permutationallyisomorphic to a finite alternating or symmetric group $A_n$ or $S_n$ actingon the $k$-element subsets of points from $\{1,\ldots,n\}$, for somearbitrary but fixed $k$. Suppose moreover that no isomorphism with thisaction is known. We show that key elements of $H$ needed to construct suchan isomorphism $\varphi$, such as those whose image under $\varphi$ is an $n$%-cycle or $(n-1)$-cycle, can be recognised with high probability by thelengths of just four of their cycles in $\Lambda$.