Scott / Gross Proceedings of the Conference on Finite Groups

1. Auflage 2014
ISBN: 978-1-4832-6108-9
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 580 Seiten, Web PDF

Proceedings of the Conference on Finite Groups provides information pertinent to the fundamental aspects of finite group theory. This book presents the problem of characterizing simple groups in terms of the local structure of a group. Organized into five parts encompassing 43 chapters, this book begins with an overview of the characterization of the Chevalley groups over fields of odd order and indicates the role of this characterization in the theory of component type groups. This text then examines the structure as well as the representations of specific simple groups. Other chapters consider the general theory of representations and characters of finite groups. This book discusses as well permutation groups and the connection between group theory and geometry. The final chapter deals with finite solvable groups as well as the theory of formations. This book is a valuable resource for mathematicians, graduate students, and research workers.

Autoren/Hrsg.

Scott, William R.

Gross, Fletcher

Inhaltsverzeichnis

1;Front Cover;1
2;Proceedings of the Conference on Finite Groups;4
3;Copyright Page;5
4;Table of Contents;6
5;List of Contributors;10
6;Preface;14
7;PART I: LOCAL STRUCTURE;16
7.1;CHAPTER 1. A CHARACTERIZATION OF CERTAIN CHEVALLEY GROUPS AND
ITS APPLICATION TO COMPONENT TYPE GROUPS;18
7.1.1;THEOREM 1;18
7.1.2;COROLLARY 2;18
7.1.3;COROLLARY 3;19
7.1.4;MAIN PROBLEM;21
7.1.5;THEOREM 4;22
7.1.6;THEOREM 5;23
7.1.7;REFERENCES;25
7.2;CHAPTER 2. FINITE GROUPS OF ALTERNATING TYPE;28
7.2.1;THEOREM;29
7.2.2;COROLLARY;30
7.2.3;UNBALANCED GROUP CONJECTURE;30
7.2.4;THEOREM;31
7.2.5;COROLLARY;32
7.2.6;THEOREM ;33
7.2.7;THEOREM;33
7.2.8;REFERENCES;36
7.3;CHAPTER 3. FINITE GROUPS OF 2-LOCAL 3-RANK AT MOST
1;40
7.3.1;THEOREM A;45
7.3.2;THEOREM B;46
7.3.3;THEOREM C;47
7.3.4;CONJECTURE;49
7.4;CHAPTER 4. FINITE SIMPLE GROUPS OF CHARACTERISTIC
2,3-TYPE;52
7.4.1;1. INTRODUCTION;52
7.4.2;2· BACKGROUND;52
7.4.3;3. STATEMENT OF MAIN THEOREMS;54
7.4.4;4. OUTLINE OF THE PROOF;56
7.4.5;5. CONCLUDING REMARKS;58
7.4.6;REFERENCES;60
7.5;CHAPTER 5. 3-STRUCTURE IN FINITE SIMPLE GROUPS;62
7.5.1;PROBLEM 1;64
7.5.2;PROBLEM 2;65
7.5.3;PROPOSITION;67
7.5.4;THEOREM;70
7.5.5;REFERENCES;76
7.6;CHAPTER 6. FINITE GROUPS OF
PSL(2,q)–TYPE;78
7.6.1;PROBLEM;78
7.6.2;B
(G)-CONJECTURE;79
7.6.3;PROPOSITION;79
7.6.4;REFERENCES;93
7.7;CHAPTER 7. SOME CHARACTERIZATIONS BY CENTRALIZERS OF ELEMENTS OF ORDER THREE;94
7.8;CHAPTER 8. A CHARACTERIZATION OF PSP4(3m) BY THE CENTRALIZER OF AN ELEMENT OF ORDER THREE;100
7.8.1;CONJECTURE;100
7.8.2;1. THE NONSIMPLE CASE;101
7.8.3;REFERENCES;116
7.9;CHAPTER 9. CHARACTERIZATION OF 3D4 (q3), q = 2n BY ITS SYLOW
2–SUBGROUP;118
7.9.1;THEOREM A.;118
7.9.2;THEOREM B;119
7.9.3;OUTLINE OF PROOF OF THEOREM B;119
7.10;CHAPTER 10. SIGNALIZER FUNCTORS;122
7.11;CHAPTER 11. STRONGLY CLOSED 2-SUBGROUPS OF FINITE GROUPS;124
8;PART II: THE KNOWN SIMPLE GROUPS;126
8.1;CHAPTER 12. THE STRUCTURE OF THE "MONSTER" SIMPLE GROUP;128
8.1.1;THEOREM 1;129
8.1.2;THEOREM 2;130
8.1.3;THEOREM 3;130
8.1.4;CONJECTURE;130
8.1.5;LEMMA;130
8.1.6;PROPOSITION.;131
8.1.7;LEMMA.;132
8.1.8;REFERENCES;133
8.2;CHAPTER 13. ON THE SIMPLE GROUP F OF
ORDER;134
8.2.1;1. THE STRUCTURES OF
HS, AUT (HS) , HS AND AUT (HS);136
8.2.2;2, FUSION-SIMPLE GROUPS HAVING
AUT (HS) OR HS AS THE CENTRALIZER OF AN INVOLUTION;147
8.2.3;3. THE ORDER OF G;179
8.2.4;4. THE CONJUGACY CLASSES AND THE CHARACTER TABLES OF SOME LOCAL SUBGROUPS;181
8.2.5;5. THE CONJUGACY CLASSES OF G;191
8.2.6;6, THE EXISTENCE OF THE CHARACTER OF DEGREE
133;199
8.2.7;7. THE CHARACTER TABLE OF G;203
8.2.8;REFERENCES;211
8.3;CHAPTER 14. A MONOMIAL CHARACTER OF FISCHER'S BABY MONSTER;292
8.3.1;1. MONOMIAL SPACES AND WEIGHTS;293
8.3.2;2. FISCHER'S BABY MONSTER;295
8.3.3;REFERENCES;298
8.4;CHAPTER 15. ON THE IRREDUCIBLE CHARACTERS OF A SIMPLE GROUP OF
ORDER;300
8.4.1;REFERENCES;314
8.5;CHAPTER 16. A SETTING FOR THE LEECH
LATTICE;316
8.6;CHAPTER 17. THE SUBMODULE STRUCTURE OF WEYL
MODULES FOR GROUPS OF TYPE A1;318
8.6.1;1. THE COMPOSITION FACTORS OF VM;320
8.6.2;2. THE STRUCTURE GRAPH OF A WEYL MODULE;321
8.6.3;REFERENCES;326
8.7;CHAPTER 18. ON THE 1-COHOMOLOGY OF FINITE GROUPS OF LIE TYPE;328
8.7.1;1.
INTRODUCTION;328
8.7.2;2. DESCRIPTION OF THE GROUPS;328
8.7.3;3. MODULES AND LOWER BOUNDS;330
8.7.4;4. UPPER BOUNDS;330
8.7.5;5. DETERMINATION OF H](Gs ,
V ) , V MINIMAL,;336
8.7.6;REFERENCES;343
8.8;CHAPTER 19. FIELD AUTOMORPHISMS AND MAXIMAL SUBGROUPS OF FINITE CHEVALLEY GROUPS;344
8.8.1;THEOREM 1;344
8.8.2;THEOREM 2;344
8.8.3;COROLLARY.;345
8.9;CHAPTER 20 .ON THE DEGREES OF CERTAIN CHARACTERS OF CHEVALLEY GROUPS;346
8.9.1;THEOREM 1;348
8.9.2;THEOREM 2;349
8.9.3;THEOREM 3;350
8.9.4;THEOREM 4;350
8.9.5;REFERENCES;351
9;PART III: REPRESENTATIONS;354
9.1;CHAPTER 21. THE MAIN PROBLEM OF BLOCK THEORY;356
9.1.1;1. INTRODUCTION;356
9.1.2;2.
SIZE OF A BLOCK;358
9.1.3;3. MCKAY'S CONJECTURE;360
9.1.4;4. GLAUBERMAN'S THEOREM [2];362
9.1.5;5. THE GENERAL LINEAR GROUP;367
9.1.6;REFERENCES;371
9.2;CHAPTER 22. ON PROJECTIVE REPRESENTATIONS OF FINITE WREATH PRODUCTS;372
9.2.1;1. INTRODUCTION;372
9.2.2;2. DEFINITIONS;373
9.2.3;3. CONSTRUCTION;373
9.2.4;4. PROJECTIVELY MONOMIAL GROUPS.;377
9.2.5;REFERENCES;378
9.3;CHAPTER 23. THEOREMS RELATING FINITE GROUPS AND DIVISION ALGEBRAS;380
9.3.1;GENERAL
CASE;385
9.3.2;BRAUER-WITT
THEOREM;386
9.3.3;LOCALIZATION;388
9.3.4;REDUCTION THEOREM;389
9.3.5;A GENERAL THEOREM;390
9.3.6;RECIPROCITY;394
9.3.7;CONCLUDING REMARK;396
9.3.8;REFERENCES;397
9.4;CHAPTER 24. EXCEPTIONAL CHARACTERS OF FINITE GROUPS WITH A FROBENIUS SUBGROUP;400
9.4.1;HYPOTHESIS;400
9.4.2;THEOREM 1;401
9.4.3;THEOREM 2;402
9.4.4;REFERENCES;403
9.5;CHAPTER 25. SIMPLE GROUPS WITH A CYCLIC SYLOW SUBGROUP;404
9.5.1;THEOREM 1;404
9.5.2;THEOREM 2;405
9.5.3;THEOREM 3;405
9.5.4;THEOREM 4;406
9.5.5;LEMMA 5;407
9.5.6;REFERENCES;410
9.6;CHAPTER 26. ON FINITE LINEAR GROUPS IN DIMENSION AT MOST 10;412
9.6.1;1. INTRODUCTION;412
9.6.2;2. STATEMENT OF RESULTS;413
9.6.3;3. OUTLINE OF PROOFS;418
9.6.4;REFERENCES;421
9.7;CHAPTER 27. NONEXISTENCE OF A FINITE GROUP WITH A SPECIFIED 7-BLOCK;424
9.7.1;1. INTRODUCTION;424
9.7.2;2. GENERAL METHODS USED;425
9.7.3;3. THE SPECIFIC BLOCK;427
9.7.4;REFERENCES;439
9.8;CHAPTER 28. LINEAR GROUPS CONTAINING AN ELEMENT WITH AN EIGENSPACE OF CODIMENSION TWO;440
9.8.1;REFERENCES;444
9.9;CHAPTER 29.
ON FINITE COMPLEX LINEAR GROUPS OF DEGREE;446
9.9.1;INTRODUCTION;446
9.9.2;HYPOTHES IS 1;446
9.9.3;THEOREM;446
9.9.4;CONJECTURE;447
9.9.5;AN ATTEMPT TO PROVE THE CONJECTURE;447
9.9.6;HYPOTHESIS 2;447
9.9.7;NOTATION;450
9.9.8;HYPOTHESIS 3;455
9.9.9;SOME SPECIAL CASES;458
9.9.10;REFERENCES;459
9.10;CHAPTER 30. SYLOW AUTOMIZERS OF ODD ORDER OR AN APPLICATION OF COHERENCE;460
9.11;CHAPTER 31. ON GROUPS OF CENTRAL TYPE;466
9.11.1;THEOREM;467
9.11.2;COROLLARY;467
10;PART IV: PERMUTATION GROUPS;468
10.1;CHAPTER 32. TWO-TRANSITIVE EXTENSIONS OF SOME GROUPS;470
10.1.1;THEOREM;470
10.2;CHAPTER 33. THE NON-EXISTENCE OF RANK-3 TRANSITIVE EXTENSIONS OF THE HIGMAN-SIMS SIMPLE GROUP;472
10.2.1;2. PRELIMINARY;473
10.2.2;3. PROOF OF THEOREM;476
10.2.3;REFERENCES;484
10.3;CHAPTER 33. ON THE n,2n PROBLEM OF MICHAEL FRIED;486
10.4;CHAPTER 34. BLOCK DESIGNS FROM FROBENIUS GROUPS AND PLANAR NEAR-RINGS;488
10.4.1;1. BASIC PROPERTIES OF PLANAR NEAR-RINGS AND THEIR
CONNECTION WITH FROBENIUS GROUPS;489
10.4.2;2. BALANCED INCOMPLETE BLOCK DESIGNS (BIBD) FROM FROBENIUS GROUPS;496
10.4.3;3. EXAMPLES;505
10.4.4;4. FUTURE DEVELOPMENTS;511
10.4.5;REFERENCES;515
10.5;CHAPTER 35. GEOMETRY OF GROUPS OF LIE TYPE;518
10.5.1;1. NOTATION AND DEFINITIONS;518
10.5.2;2. EXAMPLES;520
10.5.3;3. SOME LIE INCIDENCE STRUCTURES, POLAR AND PREPOLAR SPACES;521
10.5.4;4. SKETCH OF THE PROOF;525
10.5.5;REFERENCES;527
10.6;CHAPTER 36. LOCALLY DUAL AFFINE GEOMETRIES;528
10.6.1;CONJECTURE;529
10.6.2;REFERENCES;533
11;PART V: SOLVABLE GROUPS;534
11.1;CHAPTER 37. THE HUGHES PROBLEM AND GENERALIZATIONS;536
11.1.1;REFERENCES;542
11.2;CHAPTER 38. A NORMALIZER CONDITION ON FINITE P-GROUPS;544
11.3;CHAPTER 39. PRODUCTS OF FORMATIONS;546
11.3.1;LEMMA 1;547
11.3.2;PROPOSITION 1;547
11.3.3;PROPOSITION 2;548
11.3.4;LEMMA 2;549
11.3.5;LEMMA 3;549
11.3.6;PROPOSITION 3;550
11.3.7;REFERENCES;551
11.4;CHAPTER 40. p-N GROUPS AND p-SATURATED FORMATIONS;552
11.4.1;LEMMA 1;552
11.4.2;THEOREM A;554
11.4.3;THEOREM B;554
11.4.4;THEOREM C;555
11.4.5;REFERENCES;555
11.5;CHAPTER 41. IRREDUCIBLE MODULES OF SOLVABLE GROUPS ARE ALGEBRAIC;556
11.5.1;THEOREM 1;559
11.5.2;THEOREM 2;559
11.5.3;COROLLARY 3;559
11.5.4;THEOREM 4;560
11.5.5;COROLLARY 5;560
11.5.6;THEOREM 6;561
11.5.7;LEMMA 7;565
11.5.8;LEMMA 8;565
11.5.9;THEOREM 9;565
11.5.10;LEMMA 10;566
11.5.11;REFERENCES;567
11.6;CHAPTER 42. CERTAIN FROBENIUS GROUPS ACTING FIXED-POINT-FREE ON SOLVABLE GROUPS;570
11.6.1;CONJECTURE;570
11.6.2;THEOREM 1;571
11.6.3;COROLLARY;572
11.6.4;THEOREM 2;572
11.6.5;LEMMA;574
11.6.6;THEOREM 3;574
11.6.7;REFERENCES;578
11.7;CHAPTER 43. BOUNDING THE FITTING LENGTH OF A FINITE GROUP;580