The book under review, Theory of Finite Simple Groups II, is the second and last volume of Michler’s exposition of his new approach to the study and classification of finite simple groups. The first volume appeared in 2006.
The “complete” classification of the finite simple groups was announced in 1981 and, quoting Gorenstein, “involved the combined efforts of several hundred mathematicians from around the world over a period of 30 years, the full proof covered something between 5000 and 10000 journal pages, spread over 300 to 500 individual papers.” Given the enormity and complexity of the proof, many mathematicians have voiced their skepticism about its reliability. Gorenstein himself added “a cautionary word about the meaning of ‘proof’ in the present context; for it seems beyond human capacity to present a closely reasoned several hundred page argument with absolute accuracy.” Moreover, as Michler points out in his second volume, “Gorenstein could not anticipate that several mathematicians involved in the original classification project have still not published their manuscripts or notes in which they announced deep results.” However, these ‘deep results’ have been quoted in several important articles by other authors. Furthermore, it is not even clear that a classification of all finite simple groups is possible, and Brauer warned that “it is not even impossible that no classification exists.” Following Brauer’s suspicions, Michler writes that, together with the first volume, “this book provides some evidence that it might be possible that there are some infinite sequences of non-isomorphic unknown simple groups with strictly increasing orders.”
In 1979, Gorenstein wrote: “There you have the 26 beautiful enigmatic sporadic simple groups […]. Arising out of so many unrelated contexts, is it yet possible that there is a single, coherent explanation of their existence? If so, it will require some new vision, seemingly beyond the capabilities of the present generation, to discover it.” In this second volume, Michler answers Gorenstein’s question affirmatively, by obtaining a uniform existence proofs for all the (known) sporadic simple groups. The solution is achieved via an algorithm due to Michler, which constructs centralizers of 2-central involutions of certain finite simple groups from indecomposable subgroups of the general linear groups GL(n, Z/2Z). Each chapter is dedicated to the existence proof of one of the sporadic simple groups, i.e. the simple groups of Dickson, Conway, Fischer, Janko, Tits, McLaughlin, Rudvalis, Lyons, Suzuki and O’Nan. The book ends with a chapter on further remarks on the classification problems, where Michler lays out a program to either prove that there are only 26 sporadic simple groups, or use his algorithm in a systematic way to find new ones.
As it was the case with the first volume, the reader with some experience using the computer packages MAGMA and GAP will particularly benefit from this book because of the numerous algorithms and pieces of code included in the text and the DVD included with the book.
Álvaro Lozano-Robledo is Assistant Professor of Mathematics and Associate Director of the Quantitative Learning Center at the University of Connecticut.