Title The Theory of Fusion Systems : an Algebraic Approach

Published Cambridge : Cambridge University Press, 2011

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Description 1 online resource (386 pages) Series Cambridge Studies in Advanced Mathematics, 131 Cambridge Studies in Advanced Mathematics, 131 Contents Cover; THE THEORY OF FUSION SYSTEMS; CAMBRIDGE STUDIES IN ADVANCED MATHEMATICS; Title; Copyright; Contents; Preface; Part I: Motivation; 1 Fusion in finite groups; 1.1 Control of fusion; 1.2 Normal p-complements; 1.3 Alperin's fusion theorem; 1.4 The focal subgroup theorem; 1.5 Fusion systems; Exercises; 2 Fusion in representation theory; 2.1 Blocks of finite groups; 2.2 The Brauer morphism and relative traces; 2.3 Brauer pairs; 2.4 Defect groups and the?rst main theorem; 2.5 Fusion systems of blocks; Exercises; 3 Fusion in topology; 3.1 Simplicial sets; 3.2 Classifying spaces 3.3 Simplicial and cosimplicial objects3.4 Bousfield-Kan completions; 3.5 The centric linking systems of groups; 3.6 Constrained fusion systems; Exercises; Part II: The theory; 4 Fusion systems; 4.1 Saturated fusion systems; 4.2 Normalizing and centralizing; 4.3 The equivalent definitions; 4.4 Local subsystems; 4.5 Centric and radical subgroups; 4.6 Alperin's fusion theorem; 4.7 Weak and strong closure; Exercises; 5 Weakly normal subsystems, quotients, and morphisms; 5.1 Morphisms of fusion systems; 5.3 Normal subgroups; 5.4 Weakly normal subsystems; 5.2 The isomorphism theorems 5.5 Correspondences for quotients5.6 Simple fusion systems; 5.7 Soluble fusion systems; Exercises; 6 Proving saturation; 6.1 The surjectivity property; 6.2 Reduction to centric subgroups; 6.3 Invariant maps; 6.4 Weakly normal maps; Exercises; 7 Control in fusion systems; 7.1 Resistance; 7.2 Glauberman functors; 7.3 The ZJ-theorems; 7.4 Normal p-complement theorems; 7.5 The hyperfocal and residual subsystems; 7.6 Bisets; 7.7 The transfer; Exercises; 8 Local theory of fusion systems; 8.1 Normal subsystems; 8.2 Weakly normal and normal subsystems; 8.3 Intersections of subsystems 8.4 Constraint and normal subsystems8.5 Central products; 8.6 The generalized Fitting subsystem; 8.7 L-balance; Exercises; 9 Exotic fusion systems; 9.1 Extraspecial p-groups; 9.2 The Solomon fusion system; 9.3 Blocks of finite groups; 9.4 Block exotic fusion systems; 9.5 Abstract centric linking systems; 9.6 Higher limits and centric linking systems; Exercises; References; Index of notation; Index Summary The first book to deal comprehensively with the theory of fusion systems Fusion systems are a recent development in finite group theory and sit at the intersection of algebra and topology. This book is the first to deal comprehensively with this new and expanding field, taking the reader from the basics of the theory right to the state of the art. Three motivational chapters, indicating the interaction of fusion and fusion systems in group theory, representation theory and topology are followed by six chapters that explore the theory of fusion systems themselves. Starting with the basic definitions, the topics covered include: weakly normal and normal subsystems; morphisms and quotients; saturation theorems; results about control of fusion; and the local theory of fusion systems. At the end there is also a discussion of exotic fusion systems. Designed for use as a text and reference work, this book is suitable for graduate students and experts alike Bibliography Includes bibliographical references (pages 358-363) and indexes Notes English Print version record Subject Finite groups. Representations of algebras. Algebraic topology. MATHEMATICS -- Algebra -- General. Topología algebraica Grupos finitos Algebraic topology Finite groups Representations of algebras Endliche Gruppe Algebraische Topologie Form Electronic book ISBN 9781139092128 113909212X 9781139093156 1139093150 1139091239 9781139091237 9780511794506 0511794509 1107221420 9781107221420 1139092634 9781139092630 9786613686374 6613686379 1139090321 9781139090322 9781107471931 1107471931

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