THE CLASSIFICATION OF FINITE SIMPLE GROUPS
CONTENTS CHAPTER 1 OVERVIEW 3 INTRODUCTION TO THE SERIES 3 A THE FINITE SIMPLE GROUPS 6 1 SIMPLE GROUPS 6 2 K GROUPS 12 B THE STRUCTURE OF FINITE GROUPS 12 3 THE JORDAN H OLDER THEOREM AND SIMPLE GROUPS 12 4 THE GENERALIZED FITTING SUBGROUP AND QUASISIMPLE GROUPS 15 5 P CORES AND P COMPONENTS 19 6 THE EMBEDDING OF P CORES AND P COMPONENTS 20 7 TERMINAL AND P TERMINAL P COMPONENTS 22 8 P CONSTRAINED AND P SOLVABLE GROUPS 24 C CLASSIFYING SIMPLE GROUPS 27 9 INTERNAL ANALYSIS TARGETING LOCAL STRUCTURE 27 10 INTERNAL ANALYSIS PASSING FROM GLOBAL TO LOCAL INFORMATION 29 11 IDENTIFYING SIMPLE GROUPS 31 12 A CAPSULE SUMMARY OF THIS SERIES 35 13 THE EXISTING CLASSI CATION PROOF 38 14 SIMPLIFYING THE CLASSI CATION THEOREM 41 D THE BACKGROUND RESULTS 44 15 FOUNDATIONAL MATERIAL 44 16 THE INITIAL ASSUMPTIONS 46 17 BACKGROUND RESULTS BASIC MATERIAL 47 18 BACKGROUND RESULTS REVISED PORTIONS OF THE PROOF AND SELECTED PAPERS 48 E SKETCH OF THE SIMPLI ED PROOF 51 19 CENTRALIZERS OF SEMISIMPLE ELEMENTS 51 20 UNIQUENESS SUBGROUPS 52 21 THE SETS LP G AND GROUPS OF EVEN TYPE 53 22 GENERIC SIMPLE GROUPS AND NEIGHBORHOODS 55 23 THE MAIN CASE DIVISION 58 24 SPECIAL SIMPLE GROUPS 59 25 STAGES OF THE PROOF 61 26 GENERIC SIMPLE GROUPS 63 27 THE IDENTI CATION OF G 68 PART 1 PRELIMINARIES 1 F ADDITIONAL COMMENTS 72 28 THE LENGTH OF THE PROOF 72 29 THE K GROUP ENVIRONMENT 75 30 THE TERM G G 76 31 READING THIS SERIES 77 CHAPTER 2 OUTLINE OF PROOF 79 INTRODUCTION 79 A THE GRIDS 80 1 SOME BASIC TERMINOLOGY 80 2 THE UNIQUENESS GRID 83 3 THE CLASSI CATION GRID 83 B THE UNIQUENESS GRID 87 4 2 UNIQUENESS SUBGROUPS 87 5 GROUPS OF RESTRICTED EVEN TYPE WITH M S 1 90 6 COMPONENT PREUNIQUENESS SUBGROUPS 90 7 THE ODD UNIQUENESS THEOREM 92 8 SOME TECHNICAL DE NITIONS 93 9 GROUPS WITH A STRONGLY EMBEDDED SUBGROUP 95 10 THEOREM M S 96 11 THEOREM U 98 C THE CLASSI CATION GRID GENERIC AND SPECIAL SIMPLE GROUPS 99 12 CP GROUPS 99 13 GP GROUPS AND TP GROUPS 102 14 GROUPS OF SPECIAL ODD TYPE 103 15 GROUPS OF SPECIAL EVEN TYPE 105 16 GROUPS OF GENERIC TYPE 106 D THE CLASSI CATION GRID THE STAGES OF THE PROOF 106 17 THEOREM C1 106 18 P TERMINAL P COMPONENTS 108 19 CENTRALIZER OF INVOLUTION PATTERNS 109 20 THEOREM C2 110 21 THEOREM C3 114 22 THEOREM C4 114 23 THEOREM C5 116 24 THEOREM C6 118 25 VERTICAL NEIGHBORHOODS 118 26 THEOREM C7 120 E PRINCIPAL TECHNIQUES OF THE PROOF 122 27 FUSION 122 28 THE BENDER METHOD 123 29 SIGNALIZER FUNCTORS AND K BALANCED GROUPS 124 30 LP BALANCE THE BP PROPERTY AND PUMPUPS 127 31 THE SIGNALIZER FUNCTOR METHOD 128 32 NEAR COMPONENTS PUSHING UP AND FAILURE OF FACTORIZATION 129 33 THE AMALGAM METHOD 131 34 LOCAL ANALYSIS AT TWO PRIMES 134 35 CHARACTER THEORY AND GROUP ORDER FORMULAS 135 36 IDENTI CATION OF THE GROUPS OF LIE TYPE 137 37 PROPERTIES OF K GROUPS 138 F NOTATIONAL CONVENTIONS 139 BACKGROUND REFERENCES 140 EXPOSITORY REFERENCES 141 PART I PRELIMINARIES