Almost elusive permutation groups

根据19世纪Jordan的一个经典定理，我们知道每个至少2次的有限传递置换群都包含一个无不动点的元素，我们称之为错位。这一观察引出了一些自然问题，这些问题近年来得到了深入的研究，并有许多应用。艾米丽的项目将在这一领域探索一些新的方向。例如，Fein、Kantor和Schacher利用有限单群的分类证明了每个可迁群都有素数幂阶乱序，但也有不存在素数阶乱序的例子，它们被称为“难以捉摸的群”。Giudici在2002年对难以捉摸的本原群进行了分类，如果定义稍有减弱，比如允许素数阶乱列的唯一共轭类，会发生什么，这将是很有趣的。艾米丽将研究这类“几乎难以捉摸的群”，利用O‘Nan-Scott定理将问题归结为几乎简单的群，这将是她研究的主要焦点。我预计这将自然而然地导致其他几个相关问题。例如，将对错位的研究与有关最小和概率生成的问题(例如，两个错位可以生成哪些原始组？)结合起来将是很有趣的。关于原始群体中错乱的比例，还有一些有趣的未决问题，这可能构成艾米丽项目的一部分。

By a classical theorem of Jordan from the 19th century, we know that every finite transitive permutation group of degree at least two contains a fixed-point-free element, which we call a derangement. This observation leads to a number of natural questions, which have been intensively studied in recent years, with numerous applications. Emily's project will explore some new directions in this area. For example, Fein, Kantor and Schacher used the Classification of Finite Simple Groups to prove that every transitive group has a derangement of prime power order, but there are examples with no derangements of prime order; they are called "elusive groups". The elusive primitive groups were classified by Giudici in 2002 and it would be interesting to see what happens if the definition is weakened slightly, say by allowing a unique conjugacy class of derangements of prime order. Emily will investigate this class of "almost elusive groups", using the O'Nan-Scott Theorem to reduce the problem to almost simple groups, which will be the main focus of her research. I expect this will lead naturally to several other related problems. For example, it would be interesting to combine the study of derangements with problems concerning minimal and probabilistic generation (e.g, which primitive groups can be generated by two derangements?). There are also a number of interesting open problems on the proportion of derangements in primitive groups, which may form part of Emily's project.