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Groups of prime-power order and coclass theory

素幂阶群和余类理论

基本信息

批准号：
386837064
负责人：
Professorin Dr. Bettina Eick
金额：
--
依托单位：
Institut für Analysis und Algebra
依托单位国家：
德国
项目类别：
Research Grants
财政年份：
2017
资助国家：
德国
起止时间：
2016-12-31 至 2020-12-31
项目状态：
已结题

来源：
https://gepris.dfg.de/gepris/projekt/386837064?language=en
关键词：
Groups prime power order coclass

项目摘要

Groups play a central role in algebra. They formalize the concept of symmetries and thus have applications in many different areas such as crystallography, coding theory, graph theory, or Galois theory.The classification and structural investigation of finite groups is an interesting problem in algebra. The basic building blocks of all finite groups, the finite simple groups, have been classified successfully. Such a classification is a wide open problem for finite groups in general.The groups of prime-power order (p-groups) are in some ways an extreme counterpart to the finite simple groups. Their basic building blocks are the simplest of the finite simple groups: the cyclic groups of prime order. However, there is a huge number of possibilities for the compositions of these basic building blocks. A complete classification of all p-groups is therefore a very difficult and currently open problem. Colass theory provides a new approach towards the structural analysis of p-groups. This approach has been very successful and many deep and interesting new insights into the structure of p-groups have been obtained with it.The central goal in this project is to use the colass theory to work towards a complete classification of the p-groups maximal class. For this purpose a combination of new theoretical ideas and new algorithmic methods are presented and used. A variety of possible applications of this new approach towards a classification of p-groups of maximal class are discussed.The p-groups of maximal class are a central class of examples in coclass theory. A successful classification of the groups in this class would have an impact on the overall theory of p-groups.

群在代数中起着中心作用。它们形式化了对称性的概念，因此在许多不同的领域都有应用，如晶体学，编码理论，图论或伽罗瓦理论。有限群的分类和结构研究是代数中一个有趣的问题。所有有限群的基本构件，有限单群，已经成功地分类。这样的分类对于一般的有限群来说是一个很大的问题。素数幂阶群（p-群）在某些方面是有限单群的极端对应。它们的基本构造块是有限单群中最简单的：素数阶循环群。然而，这些基本构建块的组成有大量的可能性。因此，所有p-群的完全分类是一个非常困难的问题。Colass理论为p-群的结构分析提供了一种新的方法。这种方法已经非常成功，许多深刻的和有趣的新见解的结构的p-群已经获得了它。在这个项目的中心目标是使用colass理论的工作对一个完整的分类的p-群的最大类。为此，提出并使用了新的理论思想和新的算法方法的组合。本文讨论了极大类p-群的分类问题，极大类p-群是上类理论的一个中心类。这类群的成功分类将对p-群的整体理论产生影响。