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Sporadic groups. (English) Zbl 0804.20011
Cambridge Tracts in Mathematics. 104. Cambridge: Cambridge University Press. xii, 314 p. (1994).
This monograph is probably the first selfcontained treatment of the finite, sporadic simple groups. Its main goals are presented in three sections. In part I general methods are developed which apply to the general study of sporadic groups and which are used in the other two parts. Part II gives existence proofs for sporadic groups. Part III gives uniqueness proofs for sporadic groups. This book uses mostly “Finite group theory” by the author [Camb. Univ. Press, 1986; Zbl 0583.20001] as reference and can be read otherwise without referring to other literature.
The 26 sporadic groups are investigated with varying emphasis. The 3- transposition groups (Fischer Groups) are only handled in a marginal way, the author plans a separate volume on this subject. The groups \(ON\), \(Ly\), \(J_ 1\), \(J_ 3\), \(J_ 4\), \(Ru\) are not treated seriously at all.
In part I the author studies graphs associated with rank 3-permutation groups, and introduces the notions of geometries in the sense of Tits. He develops as an important group theoretic tool the theory of large extraspecial 2-groups (\(Q\) is a large extraspecial 2-group in \(G\) if \(Q\) is an extraspecial 2-group with \(C_ G(Q) = Z(Q)\), \(Q\trianglelefteq C_ G(Z(Q))\), and \(\text{ И}_ G(Q,2') = 1\)). A section on algebras and forms gives the foundation for the later construction of the Griess Algebra. A somewhat more special section treats symplectic 2-loops. Here the Parker Loop (used in the construction of the Monster) is put into a general context. The last section (chap. 5) of part I gives a very readable exposition of the discovery of the sporadic groups and a discussion of existence and uniqueness proofs.
This chapter is recommended even for readers with just a minor interest in group theory. In this section the author also outlines his further approach in parts II and III. He provides tables, which explain for which groups existence or uniqueness proofs are given.
The existence part II proceeds in the following way: In a first section objects are defined whose automorphism groups in turn lead to the existence of sporadic groups. In a second section these automorphism groups and their subgroups are studied in detail which in the end leads to the existence of 20 sporadic groups. The objects being studies are Steiner systems and the Golay code for the Mathieu groups, the Leech lattice for the Conway groups, and the Griess algebra for the Monster. The sections on Steiner systems and the Leech lattice can be read independently while the other sections are more difficult and of a technical nature.
In part III first a method due to Y. Segev and the author is developed. It translates the question whether a group satisfying a certain group theoretic hypothesis into a question on graphs: For this purpose the author introduces a theory of coverings of graphs and complexes. A graph or a complex is called simply connected if it admits no proper coverings. Then the author develops a machinery which associates to the group in question a certain graph. Uniqueness then means that the graph is simply connected. Again this section is of general nature while the following ones are more technical. The first application of the above method is the uniqueness proof of \(M_{24}\), \(He\), \(L_ 5(2)\). A uniqueness proof for \(U_ 4(3)\) lays ground for uniqueness proofs for \(Co_ 1\), \(Suz\), and \(J_ 2\). The book closes with the derivation of the normalizers of subgroups of prime order in \(M_{24}\), \(J_ 2\), \(Co_ 1\), \(He\), and \(Suz\).
Each chapter of the book ends with some demanding exercises. Many parts of the book contain modifications by the author of existing proofs or completely new proofs. The author puts his main emphasis rather on the development and illustration of methods used in these investigations than on the mere collection and discussion of facts. Written by a leading group theorist this book is of utmost importance for anyone who works actively on sporadic groups or who has to deal with their applications. However, part I and sections of parts II and III will also give a larger readership an excellent introduction to methods and ideas of “sporadic” group theory.

20D08 Simple groups: sporadic groups
20-02 Research exposition (monographs, survey articles) pertaining to group theory