- Volume
-
2 : gw ISBN 9783540106326
Description
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.
Table of Contents
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.
- Volume
-
3 : gw ISBN 9783540106333
Description
Und dann erst kommt der "Ab -ge - sa. ng\' da. /3 der nidlt kurz und nicht zu la. ng, From "Die Meistersinger von Nurnberg", Richard Wagner This final volume is concerned with some of the developments of the subject in the 1960's. In attempting to determine the simple groups, the first step was to settle the conjecture of Burnside that groups of odd order are soluble. The proof that this conjecture was correct is much too long and complicated for presentation in this text, but a number of ideas in the early stages of it led to a local theory of finite groups, so me aspects of which are discussed in Chapter X. Much of this discussion is a con- tinuation of the theory of the transfer (see Chapter IV), but we also introduce the generalized Fitting subgroup, which played a basic role in characterization theorems, that is, in descriptions of specific groups in terms of group-theoretical properties alone. One of the earliest and most important such characterizations was given for Zassenhaus groups; this is presented in Chapter XI. Characterizations in terms of the centralizer of an involution are of particular importance in view of the theorem of Brauer and Fowler.
In Chapter XII, one such theorem is given, in which the Mathieu group 9J'l1l and PSL(3, 3) are characterized.
Table of Contents
X. Local Finite Group Theory.- 1. Elementary Lemmas.- 2. Groups of Order Divisible by at Most Two Primes.- 3. The J-Subgroup.- 4. Conjugate p-Subgroups.- 5. Characteristic p-Functors.- 6. Transfer Theorems.- 7. Maximal p-Factor Groups.- 8. Glauberman's K-Subgroups.- 9. Further Properties of J, ZJ and K.- 10. The Product Theorem for J.- 11. Fixed Point Free Automorphism Groups.- 12. Local Methods and Cohomology.- 13. The Generalized Fitting Subgroup.- 14. The Generalized p?-Core.- 15. Applications of the Generalized Fitting Subgroup.- 16. Signalizer Functors and a Transitivity Theorem.- Notes on Chapter X.- XI. Zassenhaus Groups.- 1. Elementary Theory of Zassenhaus Groups.- 2. Sharply Triply Transitive Permutation Groups.- 3. The Suzuki Groups.- 4. Exceptional Characters.- 5. Characters of Zassenhaus Groups.- 6. Feit's Theorem.- 7. Non-Regular Normal Subgroups of Multiply Transitive Permutation Groups.- 8. Real Characters.- 9. Zassenhaus Groups of Even Degree.- 10. Zassenhaus Groups of Odd Degree and a Characterization of PGL(2, 2f).- 11. The Characterization of the Suzuki Groups.- 12. Order Formulae.- 13. Survey of Ree Groups.- Notes on Chapter XI.- XII. Multiply Transitive Permutation Groups.- 1. The Mathieu Groups.- 2. Transitive Extensions of Groups of Suzuki Type.- 3. Sharply Multiply Transitive Permutation Groups.- 4. On the Existence of 6- and 7-Fold Transitive Permutation Groups.- 5. A Characterization of M11 and PSL(3, 3).- 6. Multiply Homogeneous Groups.- 7. Doubly Transitive Soluble Permutation Groups.- 8. A Characterization of SL(2, 5).- 9. Sharply Doubly Transitive Permutation Groups.- 10. Permutation Groups of Prime Degree.- Notes on Chapter XII.- Index of Names.
- Volume
-
2 : pbk ISBN 9783642679964
Description
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.
Table of Contents
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.
- Volume
-
3 : pbk ISBN 9783642679995
Description
Und dann erst kommt der "Ab -ge - sa. ng\' da. /3 der nidlt kurz und nicht zu la. ng, From "Die Meistersinger von Nurnberg", Richard Wagner This final volume is concerned with some of the developments of the subject in the 1960's. In attempting to determine the simple groups, the first step was to settle the conjecture of Burnside that groups of odd order are soluble. The proof that this conjecture was correct is much too long and complicated for presentation in this text, but a number of ideas in the early stages of it led to a local theory of finite groups, so me aspects of which are discussed in Chapter X. Much of this discussion is a con- tinuation of the theory of the transfer (see Chapter IV), but we also introduce the generalized Fitting subgroup, which played a basic role in characterization theorems, that is, in descriptions of specific groups in terms of group-theoretical properties alone. One of the earliest and most important such characterizations was given for Zassenhaus groups; this is presented in Chapter XI. Characterizations in terms of the centralizer of an involution are of particular importance in view of the theorem of Brauer and Fowler.
In Chapter XII, one such theorem is given, in which the Mathieu group 9J'l1l and PSL(3, 3) are characterized.
Table of Contents
X. Local Finite Group Theory.- 1. Elementary Lemmas.- 2. Groups of Order Divisible by at Most Two Primes.- 3. The J-Subgroup.- 4. Conjugate p-Subgroups.- 5. Characteristic p-Functors.- 6. Transfer Theorems.- 7. Maximal p-Factor Groups.- 8. Glauberman's K-Subgroups.- 9. Further Properties of J, ZJ and K.- 10. The Product Theorem for J.- 11. Fixed Point Free Automorphism Groups.- 12. Local Methods and Cohomology.- 13. The Generalized Fitting Subgroup.- 14. The Generalized p?-Core.- 15. Applications of the Generalized Fitting Subgroup.- 16. Signalizer Functors and a Transitivity Theorem.- Notes on Chapter X.- XI. Zassenhaus Groups.- 1. Elementary Theory of Zassenhaus Groups.- 2. Sharply Triply Transitive Permutation Groups.- 3. The Suzuki Groups.- 4. Exceptional Characters.- 5. Characters of Zassenhaus Groups.- 6. Feit's Theorem.- 7. Non-Regular Normal Subgroups of Multiply Transitive Permutation Groups.- 8. Real Characters.- 9. Zassenhaus Groups of Even Degree.- 10. Zassenhaus Groups of Odd Degree and a Characterization of PGL(2, 2f).- 11. The Characterization of the Suzuki Groups.- 12. Order Formulae.- 13. Survey of Ree Groups.- Notes on Chapter XI.- XII. Multiply Transitive Permutation Groups.- 1. The Mathieu Groups.- 2. Transitive Extensions of Groups of Suzuki Type.- 3. Sharply Multiply Transitive Permutation Groups.- 4. On the Existence of 6- and 7-Fold Transitive Permutation Groups.- 5. A Characterization of M11 and PSL(3, 3).- 6. Multiply Homogeneous Groups.- 7. Doubly Transitive Soluble Permutation Groups.- 8. A Characterization of SL(2, 5).- 9. Sharply Doubly Transitive Permutation Groups.- 10. Permutation Groups of Prime Degree.- Notes on Chapter XII.- Index of Names.
by "Nielsen BookData"