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El. knyga: Theory of Fusion Systems: An Algebraic Approach

(University of Oxford)
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"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"--

Provided by publisher.

Recenzijos

"Craven's efforts to collect together in one work all of the main achievements stem from an excellent intention, and the book establishes a unified terminology for fusion systems, limiting the spread of conflicting definitions." Nadia P. Mazza, Mathematical Reviews "This very welcome book originates from a workshop on fusion systems held at the University of Birmingham in 2007 and is one of the first two texts on the subject, appearing at approximately the same time... The book serves as an excellent introduction to the subject and a guide to up to the cutting edget of current research." Charles W. Eaton, Mathematical Reviews

Daugiau informacijos

The first book to deal comprehensively with the theory of fusion systems.
Preface ix
PART I MOTIVATION
1(90)
1 Fusion in finite groups
3(23)
1.1 Control of fusion
4(4)
1.2 Normal p-complements
8(4)
1.3 Alperin's fusion theorem
12(4)
1.4 The focal subgroup theorem
16(4)
1.5 Fusion systems
20(4)
Exercises
24(2)
2 Fusion in representation theory
26(29)
2.1 Blocks of finite groups
27(7)
2.2 The Brauer morphism and relative traces
34(4)
2.3 Brauer pairs
38(3)
2.4 Defect groups and the first main theorem
41(6)
2.5 Fusion systems of blocks
47(7)
Exercises
54(1)
3 Fusion in topology
55(36)
3.1 Simplicial sets
56(10)
3.2 Classifying spaces
66(8)
3.3 Simplicial and cosimplicial objects
74(4)
3.4 Bousfield-Kan completions
78(5)
3.5 The centric linking systems of groups
83(3)
3.6 Constrained fusion systems
86(3)
Exercises
89(2)
PART II THE THEORY
91(267)
4 Fusion systems
93(41)
4.1 Saturated fusion systems
94(7)
4.2 Normalizing and centralizing
101(4)
4.3 The equivalent definitions
105(3)
4.4 Local subsystems
108(9)
4.5 Centric and radical subgroups
117(4)
4.6 Alperin's fusion theorem
121(6)
4.7 Weak and strong closure
127(4)
Exercises
131(3)
5 Weakly normal subsystems, quotients, and morphisms
134(54)
5.1 Morphisms of fusion systems
135(6)
5.2 The isomorphism theorems
141(7)
5.3 Normal subgroups
148(2)
5.4 Weakly normal subsystems
150(10)
5.5 Correspondences for quotients
160(11)
5.6 Simple fusion systems
171(10)
5.7 Soluble fusion systems
181(5)
Exercises
186(2)
6 Proving saturation
188(27)
6.1 The surjectivity property
189(4)
6.2 Reduction to centric subgroups
193(8)
6.3 Invariant maps
201(4)
6.4 Weakly normal maps
205(7)
Exercises
212(3)
7 Control in fusion systems
215(55)
7.1 Resistance
216(5)
7.2 Glauberman functors
221(6)
7.3 The ZJ-theorems
227(5)
7.4 Normal p-complement theorems
232(4)
7.5 The hyperfocal and residual subsystems
236(16)
7.6 Bisets
252(8)
7.7 The transfer
260(7)
Exercises
267(3)
8 Local theory of fusion systems
270(47)
8.1 Normal subsystems
271(4)
8.2 Weakly normal and normal subsystems
275(5)
8.3 Intersections of subsystems
280(7)
8.4 Constraint and normal subsystems
287(8)
8.5 Central products
295(4)
8.6 The generalized Fitting subsystem
299(9)
8.7 L-balance
308(6)
Exercises
314(3)
9 Exotic fusion systems
317(41)
9.1 Extraspecial p-groups
318(8)
9.2 The Solomon fusion system
326(4)
9.3 Blocks of finite groups
330(5)
9.4 Block exotic fusion systems
335(8)
9.5 Abstract centric linking systems
343(6)
9.6 Higher limits and centric linking systems
349(7)
Exercises
356(2)
References 358(6)
Index of notation 364(2)
Index 366
David A. Craven is a Junior Research Fellow in the Mathematical Institute at the University of Oxford.
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