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Finite simple groups. An introduction to their classification. (English) Zbl 0483.20008

The University Series in Mathematics. New York-London: Plenum Press. x, 333 p. $ 29.50 (1982).
This book wants to outline basic ideas behind the classification of finite simple groups. It is an expanded version of the author’s survey article [Bull. Am. Math. Soc., New Ser. 1, 43–199 (1979; Zbl 0414.20009)]. As the proof of the classification theorem consists of many thousand pages clearly this book can not contain any substantial parts of the actual proof itself. Only little arguments which illustrate special techniques are given. The aim of this book is rather to explain the fundamental methods involved in the classification theorem to the following audience (we quote from the introduction): ”Although this book is clearly aimed at a mathematical audience, portions of it should be of interest to physicists, crystallographers, and other theoretical scientists...”
The author promises additional volumes following the one under review. There it shall be shown how the concepts developed in this volume interact in the proof of the classification theorem. This book wants to give a discussion of the ingredients of the proof of the classification theorem. Chapter 1 of this book gives an outline of the proof of the classification of finite simple groups. It is shown why the proof splits into the following three parts:
A. The classification of nonconnected simple groups.
B. The classification of simple groups of component type.
C. The classification of simple groups \(G\) with \(e(G)\le 2\).
D. The classification of simple groups \(G\) with \(e(G)\le 3\).
Here for instance the effect of Aschbacher’s “proper 2-generated core theorem” and Gorenstein-Harada’s “sectional 2-rank 4”-paper on the subdivision of the classification program is discussed. Also consequences of the classification of finite simple groups are investigated (Schreier conjecture, generation of simple groups by two elements, etc.).
Chapter 2 gives a thorough discussion of the known (now all) simple groups. The section for the finite groups of Lie type includes topics as the classification of the complex finite-dimensional simple Lie algebras, Cartan subalgebras, root subspaces, Chevalley’s integral basis theorem, Dynkin diagrams, the construction of finite Chevalley groups, the twisted variations of Steinberg, Suzuki and Ree, the \((3,N)\)-pair structure. In different sections of this chapter the evidence for the discovery of simple sporadic groups, their construction and uniqueness are explained. Among others the following topics are illuminated: Janko’s first group and the Janko-Thompson (partial) classification of the Ree groups, Fischer’s work on groups generated by 3-transpositions, the Fischer-Griess monster and the Norton algebra, the Leech lattice and the Conway groups, rank 3-extensions, and computer constructions of simple groups.
Chapter 3 deals with recognition theorems, i.e. theorems of the kind: If an abstract simple group \(G\) resembles a known finite simple group \(G^+\) with \(e(G)\le 2\) ”close enough” then \(G\sim G^+\). Again we only can give a few headlines which shall indicate the contents of this chapter: Tits work on groups with \((B,N)\)-pair of rank \(\ge 3\) and the work of Hering, Fong, Kantor and Seitz on groups with a split \((3,N)\)-pair of rank \(\le 2\), the classification of Zassenhaus groups, O’Nan’s identification of \(U_3(q)\) and Bombieri’s identification of \({}^2G_2(3^n)\), and identifications via centralizers of involutions.
Chapter 4 with the title “general techniques of local analysis” is the longest and dominant part of the book. This chapter deals with the main tools which are used in the proof of the classification theorem. It includes among others a broad discussion of the following topics: Bender’s strongly embedded subgroup theorem and Aschbacher’s work on groups with proper 2-generated core, signalizer functor theorems and their connection with “k-balance” and “L-balance”, fusion theorems as Glauberman’s \(Z^*\)-theorem, the notion of the Thompson subgroup, the \(ZJ\)-theorem and Thompson’s work on quadratic pairs, the Bender method (on maximal subgroups \(M\) with \(\vert \pi(F^*(M)\ge 2)\), Goldschmidt’s work on groups which possess a strongly closed abelian 2-group. Incidently departing from his usual practice (in order to exemplify the Bender method), the author gives a complete proof of the \(p^aq^b\)-theorem of Burnside.
Further important subjects covered by the author are: the product fusion theorem of Goldschmidt, Timmesfeld’s work on root involutions and his classification of simple groups containing a nontrivial elementary abelian (TI)-subgroup which is weakly closed in its centralizer, various factorization results among them Thompson’s and Glauberman’s triple factorization theorems. Towards the end of this chapter we find a thorough discussion of theorems which deal with “failure of factorization”. Here we find Cooperstein’s and Mason’s classification of “F-pairs”, Aschbacher’s classification of \(K\)-groups \(X\) where \(F^*(X)\) is a 2-group, which do not have a \((Z,J)\)-factorization. The longest part of this chapter is devoted to pushing up results, i.e. the work of Baumann, Glauberman, and Niles, Aschbacher’s notion of \(\chi\)-blocks. In the final paragraph properties of \(K\)-groups needed in the classification proof are collected.
This book is written in the same vivid style as the classic textbook on group theory of the same author [”Finite groups.” New York etc.: Harper & Row (1968; Zbl 0185.05701)j. The author knows splendidly how to convey his enthusiasm on his subject to the reader. (Here it should be remembered that this enthusiasm was always a driving force for the participants in the classification program of finite simple groups.) It shows that it can be a bargain to have to dispense of proofs of theorems. General concepts become more clear as one is not occupied by technicalities. However the problem the book deals with is an extremely difficult one. Thus only the first three chapters may be easy to digest by the audience the author refers to (see the above quotation). Chapter 4 really requires a deeper understanding of group theory and can only the fully appreciated when the theorems discussed there are applied - i.e. when the author finishes the successors of this book.
For anyone seriously interested in the classification of finite simple groups this book is a must to read.

MSC:

20D05 Finite simple groups and their classification
20-02 Research exposition (monographs, survey articles) pertaining to group theory
20D08 Simple groups: sporadic groups
20D06 Simple groups: alternating groups and groups of Lie type