## Sporadic GroupsSporadic Groups is the first step in a programme to provide a uniform, self-contained treatment of the foundational material on the sporadic finite simple groups. The classification of the finite simple groups is one of the premier achievements of modern mathematics. The classification demonstrates that each finite simple group is either a finite analogue of a simple Lie group or one of 26 pathological sporadic groups. Sporadic Groups provides for the first time a self-contained treatment of the foundations of the theory of sporadic groups accessible to mathematicians with a basic background in finite groups such as in the author's text Finite Group Theory. Introductory material useful for studying the sporadics, such as a discussion of large extraspecial 2-subgroups and Tits' coset geometries, opens the book. A construction of the Mathieu groups as the automorphism groups of Steiner systems follows. The Golay and Todd modules, and the 2-local geometry for M24 are discussed. This is followed by the standard construction of Conway of the Leech lattice and the Conway group. The Monster is constructed as the automorphism group of the Griess algebra using some of the best features of the approaches of Griess, Conway, and Tits, plus a few new wrinkles. Researchers in finite group theory will find this text invaluable. The subjects treated will interest combinatorists, number theorists, and conformal field theorists. |

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### Contents

PART | 1 |

Algebras Codes and Forms | 35 |

Symplectic 2Loops | 46 |

The Discovery Existence and Uniqueness | 65 |

PART II | 77 |

The Geometry and Structure of M24 06 | 96 |

The Conway Groups and the Leech Lattice | 108 |

g Subgroups of 0 | 124 |

Subgroups of Groups of Monster Type | 172 |

The Geometry of Amalgams | 194 |

The Uniqueness of Groups of Type M24 | 212 |

The Group C43 | 241 |

Groups of Conway Suzuki and HallJanko Type | 250 |

Subgroups of Prime Order in Five | 293 |

Symbols | 304 |

311 | |

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### Common terms and phrases

2-group abelian assume bijection bilinear form centralizes CG(X CG(z Chapter classes of involutions collinearity graph completing the proof conjugate construction contains Conway groups cycle defined element of order equivalence established Exercise exists extraspecial 2-subgroup finite group fixes Further G is transitive geometry GL(V Griess algebra group G group homomorphism group of type Gx,y hence holds hypothesis induces inverts involutions irreducible isomorphism kernel large extraspecial subgroup Leech lattice Lemma Let G Mathieu groups monomial Monster morphism Moufang NG(A NG(B NG(P NM(A notation octad orbits orthogonal space particular permutation prove quadratic form quasisimple remains to show representation root 4-involutions Section sextet Similarly simple group simplicial complex split extension sporadic groups stabilizer Steiner system subgroup of G subgroups of order subset subspace Sylow Sylow 2-subgroup symmetric symplectic 2-loop Theorem transvection trilinear type J2 uniqueness system vector weakly closed