Almost elusive permutation groups

根据世纪Jordan的一个经典定理，我们知道，任何次数至少为2的有限可迁置换群都包含一个不动点自由元，我们称之为一个无序元。这一观察导致了一些自然问题，这些问题近年来得到了深入研究，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使用有限单群的分类来证明每个传递群都有一个素数幂序的乱序，但也有一些例子没有素数序的乱序，它们被称为“难以捉摸的群”。难以捉摸的本原群在2002年被Giudici分类，如果定义稍微减弱会发生什么，比如允许一个唯一的共轭类的素数阶乱序，这将是很有趣的。艾米丽将调查这类“几乎难以捉摸的群体”，使用奥南-斯科特定理，以减少问题，几乎简单的群体，这将是她的研究的主要重点。我想这自然会导致其他几个相关的问题。例如，将联合收割机的研究与最小生成和概率生成的问题结合起来（例如，哪些本原群可以由两个错位生成？）是很有趣的。还有一些有趣的开放性问题是关于原始群体中无序排列的比例，这可能是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.