Linear algebraic groups are affine varieties (the algebraic part of the name) over a field of arbitrary characteristic, which can be realized as groups of matrices (the linear part of the name). Most of the classification and structure of such groups was obtained by C. Chevalley, J. Tits, A. Borel, R. Steinberg and others by the mid twentieth century.
The first exposition of this theory was in the celebrated mimeographed notes of the Séminaire C. Chevalley (1956–1958), now republished as Volume 3 of Chevalley’s Collected Works as Classification des Groupes Algébriques Semi-simples (Springer, 2005), edited by P. Cartier. Amazingly enough, this book is still quite readable and amply rewarding.
The second exposition of the theory of linear algebraic groups is by one the masters, A. Borel’s Linear Algebraic Groups (First Edition, Benjamin, 1969, Second Edition, Springer, 1991). A well-written graduate textbook, Linear Algebraic Groups by J. Humphreys (Springer, 1975) followed Borel and Chevalley’s lead, and then came another fine exposition, T. Springer’s Linear Algebraic Groups (Birkhäuser, First Edition 1981, Second Edition 1998, Reprinted in 2009).
A byproduct of Chevalley’s approach to the construction of (simple) linear algebraic groups over an arbitrary (algebraically closed) field was its generalization to arbitrary base fields, both for the classical groups and, less obviously but also important, to the exceptional groups. A particularly interesting case is when the base field is a finite field. It was again Chevalley in “Sur certains groupes simples”, Tôhoku Math. J. (2) 7 (1955), 14–66, who systematically studied these groups as analogues of Lie groups over a finite field. He constructed them, gave their orders, described their Bruhat decomposition and proved their simplicity. A fine, exhaustive exposition of all of these is given in the two books by R. W. Carter, Simple Groups of Lie Type (Wiley, 1972) and Finite Groups of Lie Type (Wiley, 1985). This was a major development, almost half a century after the pioneering work of C. Jordan and L. Dickson, who studied linear, orthogonal, symplectic and some exceptional groups over finite fields.
The book under review has as its main goal to give a systematic exposition of the subgroup structure of the finite groups of Lie type based on the general properties of linear algebraic groups. In order to do this, the authors first develop the basic theory of linear algebraic groups, assuming that the reader is familiar with the elements of algebraic geometry. The exposition is economical: sometimes the arguments are just sketched or the reader is referred to one of the classical books mentioned above. Consider, for example, the basic result that any connected one-dimensional linear algebraic group is isomorphic to either the additive or multiplicative group. The authors just prove that the group is commutative and either semisimple or unipotent. They then refer to Springer’s book for an algebraic proof, but they could also have referred to Borel’s text for a geometric proof.
The main goal of Part I of the book is to describe the classification of semisimple algebraic groups. As with the classification of complex semisimple Lie algebras by Cartan and Killing, this is based on a root system that does no depend on the characteristic of the base field. Part II of the book is devoted to the fine structure of semisimple algebraic groups: Bruhat decomposition, parabolic subgroups, subgroups of maximal rank, centralizers and conjugacy classes, the classical simple algebraic groups and the exceptional algebraic groups.
Part III is devoted to the finite analogues of the simple linear algebraic groups: the finite groups of Lie type. The exposition is based on Steinberg’s approach: the analogues of the linear algebraic groups over a finite field are obtained from the corresponding groups over the algebraic closure of the given field by means of Galois descent, i.e., the finite groups of Lie type arise as the fixed points of suitable variants of the Frobenius automorphism or of compositions of the Frobenius with an automorphism of the given algebraic group. Once this is done, it is put to use to find the structure of the finite groups of Lie type exhibiting a root system and a Bruhat decomposition, from which it can be computed, for example, the orders of these groups, automorphism groups, Sylow subgroups, subgroups of maximal rank or parabolic subgroups.
This is a well-written book. The economical approach allows the authors to cover many topics that are dispersed on the literature, with many examples and exercises at the end of each of the three parts. Graduate students interested on these areas will be well served by it.
Felipe Zaldivar is Professor of Mathematics at the Universidad Autonoma Metropolitana-I, in Mexico City. His e-mail address is firstname.lastname@example.org.