## Permutation GroupsPermutation 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 |

213 | |

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### Common terms and phrases

2-transitive groups action of G acts transitively alternating group association scheme automorphism group centraliser algebra CFSG character of G coherent configuration commutative conjugacy classes conjugate contains coset countable cycle index defined disjoint distance-regular graphs distance-transitive graphs eigenvalues element of G example finite simple groups fixed points fixed-point-free element fn(G follows Frobenius function g 6 G G acts G is transitive G-orbits G-space geometric group given group G imprimitive induced infinite intersection irreducible characters irredundant base isomorphic Jordan set Lemma Let G linear matrix matroid Moore graph multiplication n-sets n-tuples non-identity element non-trivial number of orbits oligomorphic group orbital graph orbits of G permutation character polynomial primitive groups primitive permutation group proof Prove rank regular normal subgroup relation representation Show stabiliser strongly regular graph structure subgroup of G subset subspaces symmetric group Theorem theory transitive permutation group vertex set vertices wreath product