Twelve Sporadic Groups
The finite simple groups are basic objects in algebra since many questions about general finite groups can be reduced to questions about the simple groups. Finite simple groups occur naturally in certain infinite families, but not so for all of them: the exceptions are called sporadic groups, a term used in the classic book of Burnside [Bur] to refer to the five Mathieu groups. There are twenty six sporadic groups, not definitively organized by any simple theme. The largest of these is the monster, the simple group of Fischer and Griess, and twenty of the sporadic groups are involved in the monster as subquotients. These twenty constitute the Happy Family, and they occur naturally in three generations. In this book, we treat the twelve sporadics in the first two generations. I like these twelve simple groups very much, so have chosen an exposition to appreciate their beauty, linger on details and develop unifying themes in their structure theory. Most of our book is accessible to someone with a basic graduate course in abstract algebra and a little experience with group theory, especially with permu tation groups and matrix groups. In fact, this book has been used as the basis for second-year graduate courses.
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action algebra Aut(TQ basis bilinear form binary Golay code central extension classes of elements CM CM commutes conjugacy classes conjugate contradiction coordinates coset covering group cycle shape define Definition denotes dodecad element of order Exercise field automorphism finite group finite simple groups fixed point follows frame Frattini argument fus ind fus Golay code group of order hexacode word homomorphism implies ind fus ind inner product integer involution irreducible isomorphic kernel labeling Leech lattice Lemma Let G linear Mathieu group matrix maximal subgroups module Niemeier lattice nonsplit nontrivial nonzero normal subgroup Notation octad orbit of length p-group permutation matrix permutation representation point stabilizer Proof prove rank self-orthogonal sextet Show span sporadic groups standard Steiner system sublattice submodule subset subspace Suppose Sylow ternary Golay code Theorem theory transitive triangles of type trivial unique vector space vectors of type weight whence
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