Reviews

Robert Steinberg spent a sabbatical at Yale University during the 1967–68 academic year and gave a course on Chevalley groups. There were two note-takers in the course (John Faulkner and Robert Wilson), and Steinberg wrote up his lectures based on the notes. These notes were then mimeographed and circulated among group theorists and others for decades. Steinberg kept a record of some minor alterations, but never published the notes officially. Finally, after his death on his 92nd birthday in 2012, a revised version was prepared (supervised by Steinberg’s nephew, Robert Snapp, a physicist at the University of Vermont) which included these changes. They also incorporated some small remarks by various mathematicians and some footnotes that had appeared in the Russian edition of the notes. The American Mathematical Society (AMS) has now republished the notes, making them readily available to a new generation of group theorists. The influence of Steinberg’s notes has been remarkable and has impacted almost everyone who does serious work in the area. Steinberg’s elegant and comprehensible writing style made his work accessible to a very wide audience. While there are more recent and advanced books in the area (see particularly Jantzen [4]), they mostly assume that one knows all the material in Steinberg’s book. There is still no substitute for Steinberg’s work. Some aspects of these notes are covered in the very nice book of Carter [2] (with much of his book influenced by Steinberg’s work). One should also consult Steinberg’s collected works [8]. The book gives a wonderful exposition of Chevalley’s seminal work [3] as well as fundamental extensions by Steinberg himself [7]. The theory of finite Chevalley groups, and more generally finite groups of Lie type, is of central importance in group theory, but also in number theory, theoretical physics, and many other areas. Recall that the classification of finite simple groups, completed in 2004, states that a nonabelian finite simple group is either one of 26 sporadic groups, an alternating group, or a finite group of Lie type (i.e., a Chevalley group or a variation due to Steinberg, Suzuki, and Ree). Thus, if one wants to understand finite groups, it is essential to understand finite groups of Lie type. Indeed, many of the properties developed by Steinberg and others (including some developed in the book) are used in the proof of the classification of finite simple groups. Given a possibly unknown simple group, one wants to identify it as a group of Lie type using various properties, such as generators and relations, centralizers of unipotent elements, involutions, etc. Let us briefly describe how the finite groups of Lie type can be obtained from algebraic groups. Let G be a simple algebraic group defined over k, the algebraic closure of a finite field. These are classified (up to isogeny) by Dynkin diagrams (just as for the case of complex Lie groups). Let σ be an endomorphism of G with a finite set of fixed points. The simplest example is when σ is a Frobenius field automorphism of k