Groups of prime-power order and coclass theory

群在代数中起着核心作用。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，如结晶学、编码论、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。对所有有限群的基本构成块--有限单群进行了成功的分类。这样的分类对于一般的有限群来说是一个广泛公开的问题。素数幂阶群(p-群)在某些方面是有限单群的极端对应。它们的基本构件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且目前尚未解决的问题。Colass理论为p-群的结构分析提供了一种新的途径。这一方法已经非常成功，并获得了许多关于p-群结构的深刻而有趣的新见解。本项目的中心目标是利用科拉斯理论来努力实现对p-群极大类的完全分类。为此，提出并使用了新的理论思想和新的算法方法相结合。讨论了这种新方法在极大类的p-群分类中的各种可能的应用。极大类的p-群是余类理论中的一类中心实例。对这一类中的群进行成功的分类将对p-群的整体理论产生影响。

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.