# Permutation Groups

Cambridge University Press, Feb 4, 1999 - Mathematics - 220 pages
Permutation groups are one of the oldest topics in algebra. Their study has recently been revolutionized by new developments, particularly the Classification of Finite Simple Groups, but also relations with logic and combinatorics, and importantly, computer algebra systems have been introduced that can deal with large permutation groups. This text summarizes these developments, including an introduction to relevant computer algebra systems, sketch proofs of major theorems, and many examples of applying the Classification of Finite Simple Groups. It is aimed at beginning graduate students and experts in other areas, and grew from a short course at the EIDMA institute in Eindhoven.

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### Contents

 General theory 1 12 Actions and Gspaces 2 13 Orbits and transitivity 3 14 Transitive groups and coset spaces 4 15 Sylows Theorem 6 16 Regular groups 7 17 Groups with regular normal subgroups 9 18 Multiple transitivity 10
 44 Some basic groups 104 45 The ONanScott Theorem 105 46 Maximal subgroups of Sₙ 107 47 The finite simple groups 108 Multiplytransitive groups 110 Degrees of primitive groups 111 Orders of primitive groups 112 The length of Sₙ 117

 19 Primitivity 11 111 Orbitals 13 112 Sharp ktransitivity 15 113 The SchreierSims algorithm 17 114 Jerrums filter 19 115 The length of Sₙ 20 116 At the keyboard 21 Cycles and parity 25 118 Exercises 27 Representation theory 35 22 Centraliser algebra 36 23 The OrbitCounting Lemma 37 24 Character theory 40 25 The permutation character 42 26 The diagonal group 44 27 FrobeniusSchur index 45 28 Parkers Lemma 47 29 Characters of abelian groups 51 210 Characters of the symmetric group 53 Möbius inversion 56 212 Exercises 57 Coherent configurations 63 32 Algebraic theory 66 33 Association schemes 68 34 Algebra of association schemes 73 Strongly regular graphs 76 36 The HoffmanSingleton graph 79 37 Automorphisms 83 38 Valency bounds 85 39 Distancetransitive graphs 87 310 Multiplicity bounds 90 311 Duality 92 312 Wielandts Theorem 94 313 Exercises 95 The ONanScott Theorem 99 42 Precursors 100 43 Product action and basic groups 102
 Distancetransitive graphs 118 413 Bases 120 414 Geometric groups and IBIS groups 123 Matroids 125 416 Exercises 127 Oligomorphic groups 131 52 Oligomorphic groups 135 54 Automorphism groups and topology 138 55 Countably categorical structures 139 56 Homogeneous structures 141 57 Cycle index 143 58 A graded algebra 145 59 Monotonicity 147 510 Settransitive groups 148 511 Growth rates 150 512 On complementation and switching 153 Cycle index 157 514 Exercises 160 Miscellanea 165 62 Neumanns Lemma 167 63 Cofinitary permutation groups 168 64 Theorems of Blichfeldt and Maillet 170 65 Cycleclosed permutation groups 173 67 The OrbitCounting Lemma revisited 176 68 Jordan groups 179 69 Orbits on moieties 182 610 Exercises 183 Tables 187 71 Simple groups of Lie type 188 72 Sporadic simple groups 192 73 Affine 2transitive groups 194 74 Almost simple 2transitive groups 196 75 Exercises 198 Bibliography 199 Index 213 Copyright