Twelve Sporadic Groups

1. Front Matter

   Pages I-4

   PDF

2. Background from General Group Theory

   • Robert L. Griess Jr.

   Pages 5-10

3. Assumed Results about Particular Groups

   • Robert L. Griess Jr.

   Pages 11-23

4. Codes

   • Robert L. Griess Jr.

   Pages 24-29

5. The Hexacode

   • Robert L. Griess Jr.

   Pages 30-35

6. The Golay Code

   • Robert L. Griess Jr.

   Pages 36-53

7. Subgroups of M ₂₄

   • Robert L. Griess Jr.

   Pages 54-75

8. The Ternary Golay Code and 2·M ₁₂

   • Robert L. Griess Jr.

   Pages 76-87

9. Lattices

   • Robert L. Griess Jr.

   Pages 88-94

10. The Leech Lattice and Conway Groups

    • Robert L. Griess Jr.

    Pages 95-103

11. Subgroups of the Conway Groups; the Simple Groups of Higman-Sims, McLaughlin, Hall-Janko and Suzuki; Local Subgroups; Conjugacy Classes

    • Robert L. Griess Jr.

    Pages 104-145

12. Generation Three of the Happy Family and the Pariahs

    • Robert L. Griess Jr.

    Pages 146-153

13. Back Matter

    Pages 154-169

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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.

• Group theory
• Lattice
• algebra
• infinite families
• simple finite groups
• sporadic groups

• Department of Mathematics, University of Michigan, Ann Arbor, USA

  Robert L. Griess

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