Twelve Sporadic Groups (Springer Monographs in Mathematics) 1st Edition by Robert L. Jr. Griess (PDF)

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The 20 sporadics involved in the Monster, the largest sporadic group, constitute the Happy Family. This book is a leisurely and rigorous study of two of their three generations. The level is suitable for graduate students with little background in general finite group theory, established mathematicians and mathematical physicists.

User’s Reviews

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⭐This book is about some anomalies in group theory. It begins with exposition on the 5 Mathieu groups that were discovered in the 19th century, then goes on to 7 more groups found around 1970. In all there are 26 sporadic simple groups, not found in any of several infinite families of simple groups. They are anomalies.The first 6 chapters develop the Mathieu groups, principally through coding theory rather than strictly from permutation theory. M24 includes the other 4 as subgroups and is the full automorphism group of the binary Golay code. The latter is a 24-bit alphabet of codewords different enough that miscopies can be recognized and often corrected. This code has proved valuable in interplanetary communication.Chapter 6 gives an extensive treatment of the maximal subgroups of M24.Chapter 7 treats the automorphism group of the ternary Golay code; this group is the double cover of the Mathieu group M12.The book goes on to develop the Leech lattice and its automorphism group Conway-0. M24 is a subgroup of Conway-0.Chapter10, on some important subgroups of Conway-0, has some errors at the beginning. Perhaps Griess was testing whether the reader is on his/her toes. I had to do computations to get correct conclusions.The last chapter sketches the nature of the remaining 14 sporadic simple groups. Griess himself says it is anecdotal.Permutations in M24 are generally indicated by diagrams on a 6-by-4 grid and this is very nice.I am not so pleased with his notation for describing Leech lattice vectors. In fact the chapters on Conway groups seem less clear than those on Mathieu groups.Griess coined some terminology. He refers the Monster group and the sporadic simple groups (20) within it as the Happy Family. He calls the 6 others the Pariahs. He divided the Happy into 3 generations. The first generation comprises the 5 Mathieu groups. The second comprises the 7 Leech lattice groups.

⭐Interesting topic, wonderful to see so many exotic examples explored. Unfortunately the text is riddled with typos and errors.

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