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

TL;DR

This paper presents a probabilistic method to identify key elements in permutation groups acting on subsets, enabling recognition of elements corresponding to long cycles in symmetric or alternating groups without prior isomorphism knowledge.

Contribution

It introduces a novel probabilistic approach to recognize elements related to long cycles in permutation groups acting on subsets, aiding isomorphism construction.

Findings

Key elements can be recognized with high probability using cycle lengths of just four elements.

The method applies to groups permutationally isomorphic to symmetric or alternating groups.

It facilitates isomorphism detection without prior explicit knowledge of the action.

Abstract

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 $\Lambda$.