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.

Preface	p. ix
Motivation	p. 1
Fusion in finite groups	p. 3
Control of fusion	p. 4
Normal p -complements	p. 8
Alperin's fusion theorem	p. 12
The focal subgroup theorem	p. 16
Fusion systems	p. 20
Exercises	p. 24
Fusion in representation theory	p. 26
Blocks of finite groups	p. 27
The Brauer morphism and relative traces	p. 34
Brauer pairs	p. 38
Defect groups and the first main theorem	p. 41
Fusion systems of blocks	p. 47
Exerdises	p. 54
Fusion in topology	p. 55
Simplicial sets	p. 56
Classifying spaces	p. 66
Simplicial and cosimplicial objects	p. 74
Bousfield-Kan completions	p. 78
The centric linking systems of groups	p. 83
Constrained fusion systems	p. 86
Exercises	p. 89
The Theory	p. 91
Fusion systems	p. 93
Saturated fusion systems	p. 94
Normalizing and centralizing	p. 101
The equivalent definitions	p. 105
Local subsystems	p. 108
	p. 117
Alperin's fusion theorem	p. 121
Weak and strong closure	p. 127
Exercises	p. 131
Weakly normal subsystems, quotients, and morphisms	p. 134
Morphisms of fusion systems	p. 135
The isomorphism theorems	p. 141
Normal subgroups	p. 148
Weakly normal subsystems	p. 150
Correspondences for quotients	p. 160
Simple fusion 'systems	p. 171
Soluble fusion systems	p. 181
Exercises	p. 186
Proving saturation	p. 188
The surjectivity property	p. 189
Reduction to centric subgroups	p. 193
Invariant maps	p. 201
Weakly normal maps	p. 205
Exercises	p. 212
Control in fusion systems	p. 215
Resistance	p. 216
Glauberman functors	p. 221
The ZJ-theorems	p. 227
Normal P-complement theorems	p. 232
The hyperfocal and residual subsystems	p. 236
Bisets	p. 252
The transfer	p. 260
Exercises	p. 267
Local theory of fusion systems	p. 270
Normal subsystems	p. 271
Weakly normal and normal subsystems	p. 275
Intersections of subsystems	p. 280
Constraint and normal subsystems	p. 287
Central products	p. 295
The generalized Fitting subsystem	p. 299
L-balance	p. 308
Exercises	p. 314
Exotic fusion systems	p. 317
Extraspecial p-groups	p. 318
The Solomon fusion system	p. 326
Blocks of finite groups	p. 330
Block exotic fusion systems	p. 335
Abstract centric linking systems	p. 343
Higher limits and centric linking systems	p. 349
Exercises	p. 356
References	p. 358
Index of notation	p. 364
Index	p. 366
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The Theory of Fusion Systems

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