权益分类	功能权益	普通用户	{{item.name}}会员
{{category.name}}	{{benefitItem.name}}

Actions of Linear Groups

线性群的作用

基本信息

批准号：
1832533
负责人：
金额：
--
依托单位：
Imperial College London
依托单位国家：
英国
项目类别：
Studentship
财政年份：
2016
资助国家：
英国
起止时间：
2016 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=studentship-1832533
关键词：
Actions Linear Groups

项目摘要

Let G be a permutation group on a finite set X. A subset B of X is said to be a base for G if its pointwise stabilizer in G is just the identity. Bases have been studied since the early years of permutation group theory, particularly in connection with orders of primitive groups and, more recently, with computational group theory. The latter connection has arisen because a group element is completely determined by its effect on a base - so the existence of a small base allows one to store group elements efficiently. There is a great deal of recent theory on bases of primitive permutation groups. Some of the focus has been on two major conjectures in the area: the Cameron-Kantor conjecture, which (roughly stated) says that a vast collection of types of primitive groups have bases of bounded size; and the Pyber conjecture, which relates the size of a minimal base of a primitive group to the order of the group in a remarkably close way. This proposal has more of the flavour of the former conjecture. Namely, for linear groups -- that is, subgroups of GL(n,q) for some dimension n and field GF(q), acting as a permutation group on the vectors of the underlying vector space V(n,q) -- we aim to determine the groups that have "small" bases; here, "small" initially will mean "size 2" (the smallest interesting size for a base), but will be extended to larger sizes as the project progresses. Clearly GL(n,q) itself does not have a small base unless n is small, but many of its subgroups do, and the aim is to classify these in as precise a way as possible. There is a substantial literature on this problem alone, but many new avenues to explore. An initial case will be to study the subgroups of GL(n,q) that are themselves simple groups of Lie type acting irreducibly on the underlying vector space V(n,q). Doing this will involve detailed use of the structure and representations of such simple groups, itself a very big subject.

设G是有限集合X上的一个置换群，如果它在G中的点向稳定子是恒等元，则称X的子集B是G的基。从置换群论的早期开始，特别是在与原始群的顺序和最近的计算群论的联系中，基就被研究了。后一种联系的出现是因为群元素完全取决于它对基的影响，所以小基的存在允许人们有效地存储群元素。近年来有大量关于原始置换群的理论。一些焦点集中在该领域的两个主要猜想上：Cameron-Kantor猜想，该猜想（粗略地说）认为大量类型的原始群具有有限大小的基；以及Pyber猜想，该猜想将原始群的最小基的大小与群的顺序非常密切地联系起来。这个建议更像前一个猜想。也就是说，对于线性群——即对于某些维数n和域GF(q)的GL（n,q）的子群，作为基础向量空间V（n,q）的向量上的置换群——我们的目标是确定具有“小”基的群；在这里，“小”最初是指“尺寸2”（基地的最小尺寸），但随着项目的进展，将扩展到更大的尺寸。显然，除非n很小，否则GL（n,q）本身的基数并不小，但它的许多子群都有，目的是尽可能精确地对这些子群进行分类。关于这个问题有大量的文献，但还有许多新的途径需要探索。一个初始的例子是研究GL（n,q）的子群，它们本身是作用于底层向量空间V（n,q）上不可约的Lie型单群。要做到这一点，需要详细使用这些简单群体的结构和表示，这本身就是一个非常大的主题。