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pgroups of maximal class: Periodic structures in the graph associated with pgroups of maximal class,Used
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The classification of finite pgroups up to isomorphism is an intricate problem in group theory. In particular, a classification by order seems to be hopeless in general. In 1980, LeedhamGreen and Newman suggested to classify pgroups by coclass. This has led to a new research area which delivered a deep insight into the structure of pgroups. New approaches and results concerning a classification by coclass have been developed since 1999, and it is known that the 2groups of fixed coclass can be classified. A fundamental tool in this approach is the investigation of the coclass graph associated with these groups. For odd primes, a similar classification is still open. As an important special case, the author Heiko Dietrich investigates the pgroups of coclass 1, that is, the pgroups of maximal class. These are the pgroups of order p to the power of n with nilpotency class n1. He gives a survey on the known structure of these groups and he proved two types of periodic patterns in the associated coclass graph. These patterns are reflected in the structure of the groups and the achieved results strongly support the conjecture that the pgroups of maximal class can be classified.
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