Finite Groups II

by B. Huppert, N. Blackburn

Overview

17):~t? L It CIFDr- ! wei! unsre Weisheit Einfalt ist, From "Lohengrin", Richard Wagner At the time of the appearance of the first volume of this work in 1967, the tempestuous development of finite group theory had already made it virtually impossible to give a complete presentation of the subject in one treatise. The present volume and its successor have therefore the more modest aim of giving descriptions of the recent development of certain important parts of the subject, and even in these parts no attempt at completeness has been made. Chapter VII deals with the representation theory of finite groups in arbitrary fields with particular attention to those of non-zero characteristic. That part of modular representation theory which is essentially the block theory of complex characters has not been included, as there are already monographs on this subject and others will shortly appear. Instead, we have restricted ourselves to such results as can be obtained by purely module-theoretical means.

Product Details

ISBN-13:	9783642679964
Publisher:	Springer Berlin Heidelberg
Publication date:	12/23/2011
Series:	Grundlehren der mathematischen Wissenschaften , #242
Edition description:	Softcover reprint of the original 1st ed. 1982
Pages:	534
Product dimensions:	6.10(w) x 9.25(h) x 0.04(d)

Table of Contents

Chapter VII Elements of General Representation Theory 1

1 Extension of the Ground-Field 4

2 Splitting Fields 27

3 The Number of Irreducible Modular Representations 32

4 Induced Modules 44

5 The Number of Indecomposable KG-Modules 63

6 Indecomposable and Absolutely Indecomposable Modules 71

7 Relative Projective and Relative Injective Modules 81

8 The Dual Module 97

9 Representations of Normal Subgroups 123

10 One-Sided Decompositions of the Group-Ring 147

11 Frobenius Algebras and Symmetric Algebras 165

12 Two-Sided Decompositions of Algebras 174

13 Blocks of p-Constrained Groups 184

14 Kernels of Blocks 189

15 p-Chief Factors of p-Soluble Groups 203

16 Green's Indecomposability Theorem 223

Notes on Chapter VII 237

Chapter VIII Linear Methods in Nilpotent Groups 238

1 Central Series with Elementary Abelian Factors 239

2 Jennings' Theorem 252

3 Transitive Linear Groups 266

4 Some Number-Theoretical Lemmas 270

5 Lemmas on 2-Groups 275

6 Commutators and Bilinear Mappings 286

7 Suzuki 2-Groups 299

8 Lie Algebras 316

9 The Lie Ring Method and an Application 326

10 Regular Automorphisms 349

11 The Lower Central Series of Free Groups 366

12 Remarks on the Burnside Problem 385

13 Automorphisms of p-Groups 396

Notes on Chapter VIII 404

Chapter IX Linear Methods and Soluble Groups 405

1 Introduction 407

2 Hall and Higman's Theorem B 419

3 The Exceptional Case 429

4 Reduction Theorems for Burnside's Problem 449

5 Other Consequences of Theorem B 464

6 Fixed Point Free Automorphism Groups 476

7 p-Stability 492

8 Soluble Groups with One Class of Involutions 503

Notes on Chapter IX 514

Bibliography 515

Index of Names 525

Index 527