Bounding lengths of subgroup series for finite permutation and matrix groups
Lead Research Organisation:
University of Warwick
Department Name: Mathematics
Abstract
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.
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.
Organisations
Studentship Projects
| Project Reference | Relationship | Related To | Start | End | Student Name |
|---|---|---|---|---|---|
| EP/N509796/1 | 01/10/2016 | 30/09/2021 | |||
| 1935389 | Studentship | EP/N509796/1 | 02/10/2017 | 30/06/2021 | Benjamin Stratford |
| Description | I have continued the classification of the primitive permutation groups of finite degree to include those of degree up to 8191. I have created a function in a computational group theory program; MAGMA. This function will give a deterministic answer to the question of whether a group is semilinear or not, provided that the input group is of a certain form. |
| Exploitation Route | The classification of the primitive permutation groups of degree up to 8191 will be added to the primitive group databases on both MAGMA and GAP (computational group theory programs). This database is used to help other group theorists compute results about finite permutation groups. The function for determining semilinearity will also be added to MAGMA (and possibly GAP). This will help with the ongoing collaborative effort to find ways to identify matrix groups. |
| Sectors | Digital/Communication/Information Technologies (including Software),Other |