Bounding lengths of subgroup series for finite permutation and matrix groups

该项目属于EPSRC研究领域代数。它的动机是试图估计在有限置换和矩阵群中进行实际计算的各种算法的理论复杂性。这种复杂性在很大程度上取决于所涉及的群体的一些属性，例如生成它所需的元素数量，或者群体中各种类型的子群体链的最大长度，例如组合或主要系列。在文献中有许多这类的结果，尽管它们中的许多都是以数量级而不是精确的方式来陈述的。对于某些算法的应用，有必要得到精确的结果。该项目的目的是证明定理，提供各种类型级数的精确界限。其中一些结果将估计现有结果中涉及的常数，提供数量级，其他结果将是以前未研究过的不同类型的序列的结果。学生可能会继续研究算法的具体应用，并可能设计（也可能实现，这取决于他是否擅长编写计算机代码）新的算法或改进现有的算法。应用于计算群论。许多相关的算法，如确定所涉及的群的结构，被用于其他数学分支的计算，如数论、伽罗瓦理论、代数几何和数学密码学。该项目符合EPSRC的战略，即支持代数领域的研究和培训组合的发展，以维持英国目前的地位，并建立在计算有限群论的关键优势上。

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.