Bounding lengths of subgroup series for finite permutation and matrix groups

有限排列和矩阵群的子群级数的有界长度

• 批准号: 1935389
• 金额: --
• 依托单位: University of Warwick
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2017
• 资助国家: 英国
• 起止时间: 2017 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-1935389
• 关键词: Bounding; lengths; subgroup; series; finite

This project lies in the EPSRC research area Algebra.It is motivated by attempts to estimate the theoretical complexity of various algorithms for carrying out practical computations in finite permutation and matrix groups. This complexity is to a large extent determined by a number of properties of the group involved, such as the number of elements required to generate it, or the maximum lengths of various types of chains of subgroups of the group, such as a composition or chief series. There are many results of this type in the literature, although many them are stated in terms of orders of magnitude rather being precise. For some of the applications to algorithms it is necessary to have precise results.The aims of the project are to prove theorems providing precise bounds on various types of series. Some of these results will be estimating constants involved in existing results providing orders of magnitude, and others will be results on different types of series that have not been previously investigated. It is possible that the student will go on to investigate specific applications to algorithms and possibly design (and perhaps implement, depending on whether he is good at writing computer code) new algorithms or improve existing ones.The applications are to computational group theory. Many of the associated algorithms, such as determining the structure of the groups involved, are used in computations in other branches of mathematics, such as number theory, Galois theory, algebraic geometry, and mathematical cryptography.The project is in keeping with the EPSRC strategy of supporting the development of a research and training portfolio in the area of Algebra that sustains the UK's current position, and it builds on key strengths in computational finite group theory.

该项目属于 EPSRC 研究领域代数。它的动机是尝试估计各种算法的理论复杂性，以在有限排列和矩阵群中进行实际计算。这种复杂性在很大程度上是由所涉及的群的许多属性决定的，例如生成该群所需的元素数量，或者该群的子群的各种类型的链的最大长度，例如组合或主级数。文献中有许多此类结果，尽管许多结果都是以数量级而非精确的方式表述的。对于算法的某些应用，必须有精确的结果。该项目的目的是证明为各种类型的级数提供精确界限的定理。其中一些结果将估计现有结果中涉及的常数，提供数量级，而其他结果将是先前未研究过的不同类型系列的结果。学生有可能继续研究算法的具体应用，并可能设计（也可能实现，取决于他是否擅长编写计算机代码）新算法或改进现有算法。这些应用是计算群论。许多相关算法（例如确定所涉及群的结构）用于其他数学分支的计算，例如数论、伽罗瓦理论、代数几何和数学密码学。该项目符合 EPSRC 支持代数领域研究和培训组合发展的战略，以维持英国目前的地位，并且它建立在计算方面的关键优势之上 有限群论。