Coclass is a powerful invariant that reveals patterns in the intricate landscape of finite p -groups. This monograph offers a gentle introduction to coclass theory, presenting key developments from the past two decades within a unified framework and discussing open conjectures. Alongside a systematic exposition of established results, the book presents new findings and some corrections to the literature. Special emphasis is placed on coclass graphs and their remarkable periodic structures. Intended for readers with a graduate-level background in group theory and algebra, the book is accessible to newcomers, including graduate students, while also serving as a reliable reference for researchers.

Provides a comprehensive introduction to coclass theory Gives an overview of the major results with a focus on coclass graphs and periodic patterns Presents the main open conjectures and provides background needed to begin research in the area

Autorentext

Heiko Dietrich received his PhD from the TU Braunschweig in 2009. After postdoctoral positions in New Zealand and in Italy, he joined Monash University in Melbourne, where he is now a professor of mathematics. His research lies in computational algebra, especially group theory and Lie theory, with a strong focus on algorithmic and structural aspects of finite groups. He has published over sixty research papers and two software packages. His work includes seven publications and a PhD thesis dedicated to coclass theory. Bettina Eick received her PhD from the RWTH Aachen in 1996. After research stays in England, Australia and USA, she habilitated at the University of Kassel in 2001 and joined the TU Braunschweig as a professor later that year. Her research covers a wide range of topics in group theory and computational algebra. She has authored over ninety research papers and several software packages. She is one of the founding authors of the SmallGroups library. Coclass theory remains one of her most active research areas today.

Inhalt

1 Introduction to coclass graphs.- 2 Constructive methods and visualisations.- 3 Number theory and group constructions.- 4 Matrix groups and uniseriality.- 5 Cohomology theory and extensions.- 6 The number of infinite paths in G¹ (p,r).- 7 Periodicity I.- 8 The skeletons of a coclass tree.- 9 Groups of maximal class.- 10 Towards Periodicity II.- References.- Index.- List of Symbols.