Finite Simple Groups : Thirty Years of the Atlas and Beyond.

Cover -- Title page -- Contents -- Preface -- Acknowledgments -- Moonshine and the meaning of life -- References -- The Monster is fabulous -- 1. Introduction -- 2. Transposition Groups -- 3. -diagrams -- 4. Working in the Projective Plane -- 5. Fabulous Groups -- References -- Majorana representation of the Monster group -- 1. Majorana algebras -- 2. The Monster algebra and Norton-Sakuma subalgebras -- 3. Classifying Majorana representations -- References -- Letter to Donna Testerman -- Reliability and reproducibility of Atlas information -- 1. Introduction -- 2. Maximal subgroups -- 3. Character tables -- Acknowledgments -- References -- Characters and Brauer trees of the covering group of ² ₆(2) -- 1. Introduction -- 2. The character table of \boldmath3. -- 3. Determination of Brauer trees -- References -- Maximal subgroups of sporadic groups -- 1. Introduction -- 2. Methods -- 3. Historical survey -- 4. Results -- References -- Construction of the Thompson Chain of subgroups of the Conway group ⋅ and complete graphs on letters -- 1. Introduction -- 2. The Leech lattice Λ -- 3. The Miracle Octad Generator or MOG -- 4. Verification of the presentation -- 5. Conclusion -- References -- Conway's groupoid and its relatives -- 1. The first Conway groupoid ₁₃ -- 2. A more general setting for groupoids -- 3. Conway groupoids and codes -- 4. Classification results -- 5. Generation games -- References -- The subgroup structure of finite groups -- 1. Reductions for permutation groups -- 2. The finite simple groups -- 3. The alternating and symmetric groups -- 4. Groups of Lie type -- 5. Exceptional groups of Lie type -- 6. The Palfy-Pudlak Question -- 7. Lower signalizer lattices. -- 8. A question and a theorem -- References -- Some remarks on maximal subgroups of finite classical groups -- 1. Introduction.

2. Maximality of members of \SSS( ): An overview -- 3. Irreducible representations of ₁₁ -- Acknowledgement -- References -- Toward a classification of endotrivial modules -- 1. Introduction -- 2. Definitions and preliminaries -- 3. A little history -- 4. The kernel of restriction and Green correspondents -- 5. The kernel of restriction to the Sylow subgroup -- 6. Another viewpoint -- 7. Simple endotrivial modules -- 8. Acknowledgments -- References -- Some remarks on global/local conjectures -- 1. On the ( )-Conjecture -- 2. ( ) and -- 3. AWC and the Glauberman Correspondence -- Acknowledgements -- References -- Minuscule weights and Chevalley groups -- 1. Introduction -- 2. Minuscule weight modules -- 3. Weyl group action -- 4. Chevalley groups via minuscule weights -- 5. Remarks and examples -- References -- A method for building permutation representations of finitely presented groups -- 1. Introduction -- 2. The construction of mosaics -- 3. Some examples -- References -- Character ratios for finite groups of Lie type, and applications -- 1. Introduction -- 2. Previous results -- 3. A new result -- 4. Applications of Theorem 3.1 -- 5. Remarks on the proof of Theorem 3.1 -- References -- Conjugacy classes, growth and complexity -- 1. Introduction -- 2. Covering -- 3. Growth -- 4. Mixing -- 5. Interleaved products and communication complexity -- References -- Permutation groups where non-trivial elements have few fixed points -- 1. Introduction -- 2. The main results for simple groups -- 3. Current work and open questions -- Acknowledgments -- References -- Back Cover.

This volume contains the proceedings of the international conference Finite Simple Groups: Thirty Years of the Atlas and Beyond Celebrating the Atlases and Honoring John Conway, which was held from November 2-5, 2015, at Princeton University, Princeton, New Jersey. Classification of Finite Simple Groups, one of the most monumental accomplishments of modern mathematics, was announced in 1983 with the proof completed in 2004. Since then, it has opened up a new and powerful strategy to approach and resolve many previously inaccessible problems in group theory, number theory, combinatorics, coding theory, algebraic geometry, and other areas of mathematics. This strategy crucially utilizes various information about finite simple groups, part of which is catalogued in the Atlas of Finite Groups (John H. Conway et al.), and in An Atlas of Brauer Characters (Christoph Jansen et al.). It is impossible to overestimate the roles of the Atlases and the related computer algebra systems in the everyday life of researchers in many areas of contemporary mathematics. The main objective of the conference was to discuss numerous applications of the Atlases and to explore recent developments and future directions of research, with focus on the interaction between computation and theory and applications to number theory and algebraic geometry. The papers in this volume are based on talks given at the conference. They present a comprehensive survey on current research in all of these fields.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2024. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.