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

Year 2015, Volume: 2 Issue: 2, 117 - 149, 30.04.2015

Steve Linton Alice C. Niemeyer Cheryl E. Praeger

https://doi.org/10.13069/jacodesmath.28239

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 $\Lambda$.

Keywords

Symmetric and alternating groups in subset actions, Large base permutation groups, Finding long cycles

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