Groups of prime-power order and coclass theory

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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.