## 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

II | 1 |

III | 2 |

IV | 3 |

V | 4 |

VI | 6 |

VII | 7 |

VIII | 9 |

IX | 10 |

XLVIII | 104 |

XLIX | 105 |

L | 107 |

LI | 108 |

LII | 110 |

LIII | 111 |

LIV | 112 |

LV | 117 |

X | 11 |

XII | 13 |

XIII | 15 |

XIV | 17 |

XV | 19 |

XVI | 20 |

XVII | 21 |

XVIII | 25 |

XIX | 27 |

XX | 35 |

XXI | 36 |

XXII | 37 |

XXIII | 40 |

XXIV | 42 |

XXV | 44 |

XXVI | 45 |

XXVII | 47 |

XXVIII | 51 |

XXIX | 53 |

XXX | 56 |

XXXI | 57 |

XXXII | 63 |

XXXIII | 66 |

XXXIV | 68 |

XXXV | 73 |

XXXVI | 76 |

XXXVII | 79 |

XXXVIII | 83 |

XXXIX | 85 |

XL | 87 |

XLI | 90 |

XLII | 92 |

XLIII | 94 |

XLIV | 95 |

XLV | 99 |

XLVI | 100 |

XLVII | 102 |

LVI | 118 |

LVII | 120 |

LVIII | 123 |

LIX | 125 |

LX | 127 |

LXI | 131 |

LXII | 135 |

LXIV | 138 |

LXV | 139 |

LXVI | 141 |

LXVII | 143 |

LXVIII | 145 |

LXIX | 147 |

LXX | 148 |

LXXI | 150 |

LXXII | 153 |

LXXIII | 157 |

LXXIV | 160 |

LXXV | 165 |

LXXVI | 167 |

LXXVII | 168 |

LXXVIII | 170 |

LXXIX | 173 |

LXXXI | 176 |

LXXXII | 179 |

LXXXIII | 182 |

LXXXIV | 183 |

LXXXV | 187 |

LXXXVI | 188 |

LXXXVII | 192 |

LXXXVIII | 194 |

LXXXIX | 196 |

XC | 198 |

XCI | 199 |

213 | |

### Common terms and phrases

2-transitive groups action of G acts transitively alternating group association scheme automorphism group basis algebra 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 group 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 transitive permutation group vertex set vertices wreath product