Sporadic Groups

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Cambridge University Press, Mar 25, 1994 - Mathematics - 314 pages
Sporadic 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
Index
311

The Griess Algebra and the Monster
142

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