Handbook of Computational Group Theory

2. Auflage 2027
ISBN: 978-1-041-25869-8
Verlag: Taylor & Francis

Buch, Englisch, 568 Seiten, Format (B × H): 156 mm x 234 mm

Reihe: Discrete Mathematics and Its Applications

Praise for the Previous Edition

"This is a book I am very happy to have, both for the choice of content and the quality of exposition. Its subject is a very complete and up-to-date review of computational group theory. …All together, the book contains of a huge amount of information. …I think every mathematician will want this book on his shelf."
—Mathematics of Computation

"It will be an indispensable source for any user in this field."

—G. Kowol, in Monatshefte fur Math, 2007, Vol. 151, No. 3

Handbook of Computational Group Theory, Second Edition, offers a complete treatment of all the fundamental methods and algorithms in CGT presented at a level accessible even to advanced undergraduate students. It develops the theory of algorithms in full detail and highlights the connections between the different aspects of CGT and other areas of computer algebra. More advanced and specialized algorithms, particularly those developed more recently, are cited and summarized. While acknowledging the importance of the complexity analysis of CGT algorithms, the authors' primary focus is on algorithms that perform well in practice rather than on those with the best theoretical complexity.

Throughout the book, applications of the key topics and algorithms to areas both within and outside of mathematics demonstrate how CGT fits into the wider world of mathematics and science. The authors include detailed pseudocode for all the fundamental algorithms and provide detailed worked examples that bring the theorems and algorithms to life.

New to the Second Edition

- Detailed survey of progress on the matrix groups recognition project, which provides tools for investigating the structure of large groups of matrices over finite fields - it is now possible, for example, to identify quickly the composition and chief factors of groups of this type

- Discussion of a new and effective method of computing character tables of finite groups based on a result of Brauer

- Description of new methods for computing the structure of infinite polycyclic groups

- Summary of algorithms for studying infinite groups of matrices, including deciding finiteness and the Tits alternative

- Brief description of new fast methods for solving the word problem in certain types of word-hyperbolic groups, including small cancellation groups

- Summary of other significant developments during the past two decades, including a new system for computing in the Monster simple group, which has enabled the construction of its maximal subgroups

Zielgruppe

Postgraduate

Autoren/Hrsg.

Holt, Derek F.

Eick, Bettina

O'Brien, Eamonn A.

Fachgebiete

• Mathematik | Informatik Mathematik Mathematik Allgemein Zahlensysteme
• Mathematik | Informatik Mathematik Algebra

Inhaltsverzeichnis

1. A Historical Review of Computational Group Theory 2. Background Material 3. Representing Groups on a Computer 4. Computation in Finite Permutation Groups 5. Coset Enumeration 6. Presentations of Given Groups 7. Representation Theory and Computation in Matrix Groups 8. Computation with Polycyclic Groups 9. Computing Quotients of Finitely Presented Groups 10. Advanced Computations in Finite Groups 11. Libraries and Databases 12. Rewriting Systems and the Knuth-Bendix Completion Process 13. Finite State Automata and Automatic Groups

Über Autor(innen)

Derek Holt received his PhD at Oxford in 1974, supervised by Dr Michael Collins. Following Postdocs at Tübingen (Germany) and Oxford, he moved to Warwick in 1978. During his time at Warwick, he has had extended periods of academic leave, principally spent at Aachen (Germany) and Sydney (Australia). He received a London Mathematical Society Junior Whitehead Prize in 1981, principally for completing the classification of groups with more than one end. He has authored or co-authored about 130 research papers, and four academic books. His research area is Group Theory, including both finite and geometric, and much of his research has involved the theoretical and practical development of algorithms for computing in finite groups, and in certain classes of infinite groups. He has written a substantial volume of package code for Magma, and also two GAP packages.

Bettina Eick received her PhD in mathematics from the RWTH Aachen (Germany) in 1996, studying group theory and algorithms. Spending one year at the University of Würzburg (Germany), she was PostDoc at the University of Kassel (Germany) where she habilitated in 2001. Immediately thereafter she became Professor at the TU Braunschweig (Germany). All throughout her research career, Bettina Eick spent significant time as guest researcher at the University of Auckland (New Zealand), Queen Mary University of London (England), and at the Australian National University in Canberra (Australia). She is an author of over 90 research articles and co-author of three books. She developed several software packages in computational algebra, including the SmallGroups library. Her software is available in computer algebra systems such as GAP, Oscar and Magma.

Eamonn O'Brien is Professor of Mathematics at the University of Auckland. His research interests are in group theory, with a particular focus on the development, implementation and application of effective algorithms for their study. He is the author of about 100 research papers and two books. Many of his research outputs are incorporated into GAP and Magma. Eamonn obtained his PhD from the Australian National University in 1988. He was a Research Fellow at the ANU and a Humboldt Research Fellow at RWTH University of Aachen before joining Auckland in 1997. 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. Eamonn was managing editor of the Journal of Algebra (Computational Section) from 2018-2026.