## The Classification of the Finite Simple Groups, Issue 3This work offers a single source of basic facts about the structure of the finite simple groups with emphasis on a detailed description of their local subgroup structures, coverings, and automorphisms. The method is by examination of the specific groups, rather than by the development of an abstract theory of simple groups. While the purpose of the book is to provide the background for the proof of the classification of the finite simple groups - dictating the choice of topics - the subject matter is covered in such depth and detail that the book should be of interest to anyone seeking information about the structure of the finite simple groups. The treatment, however, is not self-contained. The authors rely on a small number of standard references whose results they extend and develop. This volume offers a wealth of basic facts and computations. Much of the material is not readily available from any other source. In particular, the book contains the statements and proofs of the fundamental Borel-Tits Theorem and Curtis-Tits Theorem. It also contains complete information about the centralizers of semisimple involutions in groups of Lie type, as well as many other local subgroups. |

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### Contents

I | 1 |

II | 2 |

III | 4 |

IV | 5 |

V | 6 |

VII | 7 |

IX | 8 |

X | 13 |

XXXI | 134 |

XXXII | 143 |

XXXIII | 153 |

XXXIV | 170 |

XXXV | 186 |

XXXVI | 204 |

XXXVII | 219 |

XXXVIII | 227 |

XI | 15 |

XII | 16 |

XIII | 17 |

XIV | 22 |

XV | 23 |

XVI | 27 |

XVII | 30 |

XVIII | 31 |

XIX | 36 |

XX | 40 |

XXI | 45 |

XXII | 53 |

XXIII | 63 |

XXIV | 67 |

XXV | 73 |

XXVI | 89 |

XXVII | 91 |

XXVIII | 95 |

XXIX | 106 |

XXX | 119 |

XXXIX | 236 |

XL | 249 |

XLI | 252 |

XLII | 288 |

XLIII | 294 |

XLIV | 302 |

XLV | 311 |

XLVI | 313 |

XLVII | 317 |

XLVIII | 324 |

XLIX | 328 |

L | 335 |

LI | 343 |

LII | 346 |

LIII | 349 |

LIV | 380 |

LV | 383 |

LVI | 385 |

LVII | 399 |

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action adjoint algebraic group asserted assume Aut(A Aut(K Borel subgroup Cc(L Chevalley Ck(x column completes the proof conjugacy class conjugate contains contradiction corresponding cyclic Definition direct product Dynkin diagram element elementary abelian factors field automorphism finite groups following conditions hold fundamental system graph automorphism graph-field automorphism group of order groups of Lie Hence high weight implies induces Inndiag(A inner automorphisms inverts involutions irreducible isometry isomorphism kernel Lemma Lie components Lie type Lie(r mapping maximal torus Moreover nontrivial normal subgroup normalizes notation orthogonal Outdiag Outdiag(A p-group parabolic subgroup permutes preimage prime Proposition prove quasisimple quasisimple group respectively root groups root subgroups root system Section semisimple short root simple groups solvable sporadic groups Steinberg endomorphism Steinberg group structure subspace Suppose Sylow 2-subgroup T-root Table Theorem trivial type 2A unipotent unique untwisted Weyl group whence Z(Ku