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This book provides a comprehensive coverage of the theory of conjugacy in finite classical groups. Given such a classical group G, the three fundamental problems considered are the following: to list a representative for each conjugacy class of G; to describe the centralizer of each representative, by giving its group structure and a generating set; and to solve the conjugacy problem in G namely, given two elements of G, establish whether they are conjugate, and if so, find a conjugating element. The book presents comprehensive theoretical solutions to all three problems, and uses these solutions to formulate practical algorithms. In parallel to the theoretical work, implementations of these algorithms have been developed in Magma. These form a critical component of various general algorithms in computational group theory for example, computing character tables and solving conjugacy problems in arbitrary finite groups.
List of contents
- 1. Introduction and Background.- 2. General and Special Linear Groups.- 3. Preliminaries on Classical Groups.- 4. Unipotent Classes in Good Characteristic.- 5. Unipotent Classes in Bad Characteristic.- 6. Semisimple Classes.- 7. General Conjugacy Classes.
About the author
Giovanni De Franceschi currently works as a software developer in Verona. He completed a PhD in mathematics at the University of Auckland in 2018; some of this work formed part of his PhD thesis.
Martin W. Liebeck has been a professor at Imperial College London since 1991, and was Head of Pure Mathematics there between 1997 and 2024. He has published about 170 articles and books on finite and algebraic groups, representation theory, probabilistic group theory, and applications of these topics to areas such as model theory in logic, Markov theory in probability, combinatorics, and the design of algorithms in computational algebra. Liebeck is a Fellow of the American Mathematical Society, and was the recipient of the London Mathematical Society’s Polya Prize in 2020.
Eamonn A. O'Brien is a professor at the University of Auckland since 2006. His research interests are in group theory and computational algebra, with a particular focus on the development and implementation of effective algorithms and their application to solving related challenging problems. He has published about 100 research papers and is a coauthor of the "Handbook of Computational Group Theory." Many of his research outputs are incorporated into the leading computational algebra systems GAP and Magma. Elected a Fellow of the Royal Society of New Zealand in 2009 and awarded its 2020 Hector Medal, he is a recipient of a 2024 Humboldt Foundation Research Award.
Summary
This book provides a comprehensive coverage of the theory of conjugacy in finite classical groups. Given such a classical group G, the three fundamental problems considered are the following: to list a representative for each conjugacy class of G; to describe the centralizer of each representative, by giving its group structure and a generating set; and to solve the conjugacy problem in G—namely, given two elements of G, establish whether they are conjugate, and if so, find a conjugating element. The book presents comprehensive theoretical solutions to all three problems, and uses these solutions to formulate practical algorithms. In parallel to the theoretical work, implementations of these algorithms have been developed in Magma. These form a critical component of various general algorithms in computational group theory—for example, computing character tables and solving conjugacy problems in arbitrary finite groups.
