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E-grāmata: Biset Functors for Finite Groups

  • Formāts: PDF+DRM
  • Sērija : Lecture Notes in Mathematics 1990
  • Izdošanas datums: 10-Mar-2010
  • Izdevniecība: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
  • Valoda: eng
  • ISBN-13: 9783642112973
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Biset Functors for Finite GroupsPalielināt
  • Formāts: PDF+DRM
  • Sērija : Lecture Notes in Mathematics 1990
  • Izdošanas datums: 10-Mar-2010
  • Izdevniecība: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
  • Valoda: eng
  • ISBN-13: 9783642112973
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This volume exposes the theory of biset functors for finite groups, which yields a unified framework for operations of induction, restriction, inflation, deflation and transport by isomorphism. The first part recalls the basics on biset categories and biset functors. The second part is concerned with the Burnside functor and the functor of complex characters, together with semisimplicity issues and an overview of Green biset functors. The last part is devoted to biset functors defined over p-groups for a fixed prime number p. This includes the structure of the functor of rational representations and rational p-biset functors. The last two chapters expose three applications of biset functors to long-standing open problems, in particular the structure of the Dade group of an arbitrary finite p-group.This book is intended both to students and researchers, as it gives a didactic exposition of the basics and a rewriting of advanced results in the area, with some new ideas and proofs.

Recenzijas

From the reviews:

The theory of biset functors was developed to give a unified construction of the elementary operations of restriction, induction, inflation, deflation and transport by isomorphism. This volume gives an excellent introduction to the topic, which until now was only available in a series of research papers. The book can be read by graduate students with a good knowledge of algebra, but will also be a valuable source for researchers. (Robert Hartmann, Mathematical Reviews, Issue 2011 d)

1 Examples 1
1.1 Representation Groups
1
1.2 Other Examples
6
1.3 Biset Functors
9
1.4 Historical Notes
10
1.5 About This Book
10
Part I General Properties
2 G-Sets and (H, G)-Bisets
15
2.1 Left G-Sets and Right G-Sets
15
2.2 Operations on G-Sets
17
2.3 Bisets
18
2.4 Burnside Groups
28
2.5 Burnside Rings
31
3 Biset Functors
41
3.1 The Biset Category of Finite Groups
41
3.2 Biset Functors
43
3.3 Restriction to Subcategories
47
4 Simple Functors
53
4.1 Admissible Subcategories
53
4.2 Restriction and Induction
56
4.3 The Case of an Admissible Subcategory
58
4.4 Examples
69
Part II Biset Functors on Replete Subcategories
5 The Burnside Functor
75
5.1 The Burnside Functor
75
5.2 Effect of Biset Operations on Idempotents
76
5.3 Properties of the mG,N's
79
5.4 Subfunctors in Coprime Characteristic
83
5.5 Application: Some Simple Biset Functors
89
5.6 Examples
91
6 Endomorphism Algebras
97
6.1 Simple Modules and Radical
97
6.2 Idempotents
104
6.3 Faithful Elements
109
6.4 More Idempotents
111
6.5 The Case of Invertible Group Order
114
7 The Functor CRc
121
7.1 Definition
121
7.2 Decomposition
124
7.3 Semisimplicity
130
7.4 The Simple Summands of CRc
133
8 Tensor Product and Internal Hom
135
8.1 Bisets and Direct Products
135
8.2 The Yoneda-Dress Construction
136
8.3 Internal Hom for Biset Functors
139
8.4 Tensor Product of Biset Functors
140
8.5 Green Biset Functors
147
8.6 More on A-Modules
151
Part III p-Biset Functors
9 Rational Representations of p-Groups
155
9.1 The Functor of Rational Representations
155
9.2 The Ritter-Segal Theorem Revisited
156
9.3 Groups of Normal p-Rank 1
158
9.4 The Roquette Theorem Revisited
162
9.5 Characterization of Genetic Subgroups
165
9.6 Genetic Bases
170
9.7 Genetic Bases and Subgroups
178
10 p-Biset Functors
183
10.1 Rational p-Biset Functors
183
10.2 Subfunctors of Rational p-Biset Functors
186
10.3 The Subfunctors of RQ
193
10.4 The Subfunctors of R*Q
199
10.5 The Subfunctors of kRQ
202
10.6 A Characterization
206
10.7 Yoneda-Dress Construction
209
11 Applications
215
11.1 The Kernel of the Linearization Morphism
215
11.2 Units of Burnside Rings
223
12 The Dade Group
241
12.1 Permutation Modules and Algebras
242
12.2 Endo-Permutation Modules
243
12.3 Dade P-Algebras
248
12.4 Bisets and Permutation Modules
253
12.5 Bisets and Permutation Algebras
263
12.6 Relative Syzygies
272
12.7 A Short Exact Sequence of Biset Functors
275
12.8 Borel-Smith Functions
277
12.9 The Dade Group up to Relative Syzygies
283
12.10 The Torsion Subgroup of the Dade Group
291
References 293
Index 297
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