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 charac­ teristic. 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.

Inhalt

VII. Elements of General Representation Theory.- § 1. Extension of the Ground-Field.- § 2. Splitting Fields.- § 3. The Number of Irreducible Modular Representations.- § 4. Induced Modules.- § 5. The Number of Indecomposable KG-Modules.- § 6. Indecomposable and Absolutely Indecomposable Modules.- § 7. Relative Projective and Relative Injective Modules.- § 8. The Dual Module.- § 9. Representations of Normal Subgroups.- §10. One-Sided Decompositions of the Group-Ring.- §11. Frobenius Algebras and Symmetric Algebras.- §12. Two-Sided Decompositions of Algebras.- §13. Blocks of p-Constrained Groups.- §14. Kernels of Blocks.- §15. p-Chief Factors of p-Soluble Groups.- §16. Green's Indecomposability Theorem.- Notes on Chapter VII.- VIII. Linear Methods in Nilpotent Groups.- § 1. Central Series with Elementary Abelian Factors.- § 2. Jennings' Theorem.- § 3. Transitive Linear Groups.- § 4. Some Number-Theoretical Lemmas.- § 5. Lemmas on 2-Groups.- § 6. Commutators and Bilinear Mappings.- § 7. Suzuki 2-Groups.- § 8. Lie Algebras.- § 9. The Lie Ring Method and an Application.- §10. Regular Automorphisms.- §11. The Lower Central Series of Free Groups.- §12. Remarks on the Burnside Problem.- §13. Automorphisms of p-Groups.- Notes on Chapter VIII.- IX. Linear Methods and Soluble Groups.- §1. Introduction.- §2. Hall and Higman's Theorem B.- §3. The Exceptional Case.- §4. Reduction Theorems for Burnside's Problem.- §5. Other Consequences of Theorem B.- §6. Fixed Point Free Automorphism Groups.- §7. p-Stability.- §8. Soluble Groups with One Class of Involutions.- Notes on Chapter IX.- Index of Names.