On derangements in simple permutation groups

Timothy C. Burness, Marco Fusari
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Abstract

Let $G \leqslant {\rm Sym}(\Omega)$ be a finite transitive permutation group and recall that an element in $G$ is a derangement if it has no fixed points on $\Omega$. Let $\Delta(G)$ be the set of derangements in $G$ and define $\delta(G) = |\Delta(G)|/|G|$ and $\Delta(G)^2 = \{ xy \,:\, x,y \in \Delta(G)\}$. In recent years, there has been a focus on studying derangements in simple groups, leading to several remarkable results. For example, by combining a theorem of Fulman and Guralnick with recent work by Larsen, Shalev and Tiep, it follows that $\delta(G) \geqslant 0.016$ and $G = \Delta(G)^2$ for all sufficiently large simple transitive groups $G$. In this paper, we extend these results in several directions. For example, we prove that $\delta(G) \geqslant 89/325$ and $G = \Delta(G)^2$ for all finite simple primitive groups with soluble point stabilisers, without any order assumptions, and we show that the given lower bound on $\delta(G)$ is best possible. We also prove that every finite simple transitive group can be generated by two conjugate derangements, and we present several new results on derangements in arbitrary primitive permutation groups.
论简单置换群中的变化
让 $G \leqslant {\rm Sym}(\Omega)$ 是一个有限传递置换群,并回忆一下,如果 $G$ 中的一个元素在 $\Omega$ 上没有定点,那么它就是一个反演。让$\Delta(G)$ 是$G$ 中错乱的集合,并定义$\delta(G) = |\Delta(G)|/|G|$ 和 $\Delta(G)^2 = \{ xy \,:\, x,y \in\Delta(G)\}$ 。近年来,人们开始关注简单群中的衍生研究,并取得了一些令人瞩目的成果。例如,将 Fulman 和 Guralnick 的定理与 Larsen、Shalevand Tiep 的最新研究结合起来,可以得出对于所有足够大的简单传递群 $G$,$\delta(G)\geqslant 0.016$ 和 $G = \Delta(G)^2$。在本文中,我们从几个方向扩展了这些结果。例如,我们证明了对于所有具有可溶点稳定器的有限简单基元群,不需要任何阶假设,$\delta(G)\geqslant 89/325$ 和 $G = \Delta(G)^2$。我们还证明了每一个有限简单反式群都可以由两个共轭导差生成,并提出了关于任意基元跃迁群中导差的几个新结果。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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