Perhaps simple is not the best term for some groups that tend to be very complicated. But thanks to the Jordan-Hölder theorem, all finite groups can be built from extensions of finite simple groups and so the importance of these not-so-simple groups becomes apparent.
Looking at any table of finite simple groups of order less than a given number, one notices that most of them are abelian. But abelian simple groups are indeed simple: By Lagrange’s theorem all finite abelian simple groups are cyclic of prime order. Thus, the game becomes interesting only when one looks at the non abelian simple groups.
As with the whole history of group theory, the first examples of non-abelian finite simple groups were studied by Galois, who certainly knew that the alternating groups An are simple for n greater than four and also studied the linear groups PSL2(p) for primes p ≥ 5. In his systematization of group theory in 1870, C. Jordan, amongst other things, constructed the simple groups PSLn(p) for all n ≥ 2 and primes p ≥ 5. And in 1861–1873 Mathiew constructed the five simple groups that bear his name, the first examples of “sporadic” simple groups, so named by Burnside, because they do not belong to any infinite families.
In a paper published in 1892, Hölder wrote that “It would be of the greatest interest if a survey of all (finite) simple groups could be known.” This might have been the birth of the program whose main goal was to classify all finite simple groups, and as the title of Hölder’s paper implies, he determined all finite simple groups of order up to 200, by using the only tool available then, Sylow’s theorems. Hölder’s list was extended little by little until Frobenius and Burnside introduced character theory to the arsenal. Then it could be proved, for example, that all finite groups whose order is the product of the powers of two primes are not simple.
At the beginning of the 20th century, Dickson generalized Jordan’s construction of linear groups to arbitrary finite fields, following an analogy with Killing’s classification of (complex) Lie algebras, including some exceptional cases. A more general and systematic construction was given by Chevalley in the 1950s that also included analogues of the exceptional groups of Lie type. This work was completed by Steinberg, Tits, Herzig, Ree and Suzuki, including now the remaining finite groups of Lie type.
By mid 20th century, R. Brauer, in his talk at the 1954 ICM in Amsterdam, called the attention of the group theory community to the importance and urgency of the classification of all finite simple groups, emphasizing the new methods now available (many due to him) and laying out a strategy to achieve this goal. This call to arms felt into fertile ground and many young mathematicians were attracted to the task. The first groundbreaking result came at the beginning of the 1960s, when Feit and Thompson proved the celebrated odd-order theorem: All finite groups of odd order are solvable, or equivalently, every finite non abelian simple group has even order. An immediate consequence is that every finite non abelian simple group has an element of order two, an involution, and as the author reminds us in his introduction, somehow the idea that using this involution one could achieve Hölder and Brauer’s dream and have a classification of all finite simple groups started to look not so far fetched.
As with many other dreams, this turned out not to be so easy to achieve. A first sign was Janko’s construction, in the mid 1960s, of the simple groups now named after him. One could almost hear the “Who ordered that?” question that Physicists asked in a similar context, when news of a new elementary particle crushed their early dreams of having a complete classification of all elementary particles. In the years that followed, more “sporadic” simple groups were found to exist, culminating with Griess’s construction of Fischer’s Monster group at the beginning of the 1980s.
At about that time, group-theorists announced, very prematurely as it would turn out, the proof of the classification theorem for finite simple groups, a proof that has somehow just been completed with the publication of Aschbacher and Smith's two volume The Classification of Quasithin Groups (AMS, 2004). There are now two ongoing projects to provide all the details of the proof of this theorem. A “revisionist project” led by Gorenstein, Lyons and Solomon, with six volumes already published (AMS, 1994–2005). Their work, not yet completed, focuses on giving a complete revised and simplified proof of the classification theorem following the original ideas. The second such project, led by Meierfrankenfeld, Stellmacher and Stroth, focuses on giving a different proof using the so-called amalgam method.
The book under review has as its main goal to give an introductory overview of the construction and main properties of all finite simple groups. The first chapter of the book starts with the formulation of the classification theorem, sketching some of the history of these groups as outlined above. It includes some remarks about the length and reliability of the proof, and lists the prerequisites for reading the book (they are remarkably few, making the book accessible to a non specialist in group theory and graduate students). The book starts properly in Chapter 2, studying the alternating groups and proving their simplicity for n≥ 5. As is be the case with all simple groups studied in the book, in addition to the definition, construction and description of the groups, the author studies some of their main properties, for example, computes their automorphism groups (in the case of the symmetric groups, he also singles out the special case of the outer automorphism of S6), covering groups of both symmetric and alternating groups, and includes a proof of the O’Nan-Scott theorem that classifies the maximal subgroups of the alternating An and symmetric groups Sn. The chapter ends with a review of Coxeter and Weyl groups.
Chapter 3 is devoted to define and prove the main properties of the six families of classical simple groups (linear, unitary, symplectic, and the 3 families of orthogonal groups) over finite fields. The chapter starts by computing the orders of the linear groups, proving their simplicity using Iwasawa’s lemma (stated and proved on page 45) and then obtaining some of their subgroups, outer automorphisms and covering groups. Next, the author reviews some facts about bilinear, sesquilinear and quadratic forms over finite fields of odd characteristic, and then applies these results to study the symplectic groups, prove their simplicity, and study some of their subgroups, automorphisms and covers. The same is done for the unitary groups, and lastly for the more difficult orthogonal groups, first in odd characteristic and then for characteristic two.
Chapter 4 studies the ten families of exceptional simple groups of Lie type using the approach via octonions and Jordan algebras, an approach that allows the author to obtain some of their main properties in a reasonable way.
Chapter 5, the last chapter of the book, is devoted to the 26 sporadic simple groups. This chapter contains a mixture of complete proofs, proofs that are only sketched, and some calls to the literature, since it is unreasonable to expect at this time a complete detailed constructions and proofs of the main facts about all these groups in a reasonable number of pages. For example, the five Mathieu groups are treated using the classical Steiner systems and some more recent binary codes. All main properties of these groups are given complete proofs. Next come the three Conway groups, whose construction is given using the Leech lattice, and the groups of McLaughlin and Higman-Sims viewed as subgroups of two of the Conway groups. Here the treatment is more or less complete. Next, the three Fischer groups are described in terms of their actions on some large graphs. By studying a sort of “double cover” of the Golay code, the Parker loop, the author can give a description of the Monster group, the baby Monster and the remaining monstruous groups (the Thompson, Harada-Norton and Held groups) and their relations. The last section of chapter five is devoted to the remaining six sporadic simple groups, the so-called pariahs because they don’t seem to fit into a reasonable pattern or be related to the other 20 sporadic groups. It seems that, for now, these pariahs are too bizarre to be tamed.
Before the publication of the book under review, there were few books that attempted to give an overview of all finite simple groups, the oldest perhaps the well-known Atlas of Finite Groups (Oxford, 1986) of Conway et al, a set of character tables and information about most finite simple groups, now accessible through the libraries of some computer programs, e.g., GAP. It is no coincidence that the author of the book under review is one of the coauthors of the Atlas. (Amusingly enough, in my local library the Atlas is shelved with the oversized books in the Geography section, with books of similar size filled with beautiful pictures and maps of earthly places. Certainly the “pictures and maps” of the Atlas of Finite Groups are also beautiful, but “earthly” I couldn’t say.)
This book is the first one that attempts to give a systematic treatment of all finite simple groups, using the more recent and efficient constructions, allowing the reader to get a sense that this is not a rarified field, and that calculations with these groups can be done. The author succeeds in making this important but difficult area of mathematics readily accessible to a large sector of the mathematical community, and for this we should be grateful.
Felipe Zaldivar is Professor of Mathematics at the Universidad Autonoma Metropolitana-I, in Mexico City. His e-mail address is email@example.com.