Sporadic Groups [Paperback]

The first self-contained treatment of the foundations of the theory of sporadic groups.Takes a first step in a program to provide a uniform self-contained treatment of the foundational material on the sporadic finite simple groups.Takes a first step in a program to provide a uniform self-contained treatment of the foundational material on the sporadic finite simple 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. The existence treatment finishes with an application of the theory of large extraspecial subgroups to produce the twenty sporadics involved in the Monster. The Aschbacher-Segev approach addresses the uniqueness of the sporadics via coverings of graphs and simplicial complexes. The basics of this approach are developed and used to establish the uniqueness of five of the sporadics.Preface; 1. Preliminary results; 2. 2-Structure in finite groups; 3. Algebras, codes and forms; 4. Symplectic 2-loops; 5. The discovery, existence, and uniqueness of the sporadics; 6. The Mathieu groups, their Steiner systems, and the Golay code; 7. The geometry and structure of M24; 8. The Conway groups and the Leech lattice; 9. Subgroups of .0; 10. The Griess algebra and the Monster; 11. Sul3%