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Full Description
The Leech lattice Λ, the Conway group ∙O, and the Monster group M are immensely famous structures. They each grow out of the Mathieu group M24 and its underlying combinatorial structure, and play an important role in various branches of mathematics and in theoretical physics. Written by an expert in the field, this book provides a new generation of mathematicians with the intimate knowledge of M24 needed to understand these beautiful objects, and many others. It starts by exploring Steiner systems, before introducing the Miracle Octad Generator (MOG) as a device for working with the Steiner system S(5,8,24). Emphasizing how theoretical and computational approaches complement one another, the author describes how familiarity with M24 leads to the concept of 'symmetric generation' of groups. The final chapter brings together the various strands of the book to produce a nested chain of groups culminating in the largest Conway simple group Co1.
Contents
1. Introduction; 2. Steiner systems; 3. The Miracle Octad Generator; 4. The binary Golay code; 5. Uniqueness of the Steiner system S(5,8,24) and the group M24; 6. The hexacode; 7. Elements of the Mathieu group M24; 8. The maximal subgroups of M24; 9. The Mathieu group M12; 10. The Leech lattice Λ; 11. The Conway group ·O; 12. Permutation actions of M24; 13. Natural generators of the Mathieu groups; 14. Symmetric Generation using M24; 15. The Thompson chain of subgroups of Co1; Appendix. Magma code for 7*36 : A9 ↦ Co1; References; Index.
