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The geometry of the classical groups. (English) Zbl 0767.20001
Sigma Series in Pure Mathematics. 9. Berlin: Heldermann Verlag. v, 229 p. (1992).

The book under review is the latest contribution to a list of standard references in the field of classical groups like the books due to E. Artin [Geometric algebra (1957; Zbl 0077.021)] and J. Dieudonné [La géométrie des groupes classiques (1971); see also 1955; Zbl 0067.261)] the lecture notes of C. Chevalley [The algebraic theory of spinors (1954; Zbl 0057.259)] and B. Huppert [Univ. Illinois at Chicago Circle 1968/69], J. Tits’ monograph [Buildings of spherical type and BN-pairs (1974; Zbl 0295.20047)] and the quite recent books due to K. Brown [Buildings (1989; Zbl 0715.20017)] and M. Ronan [Lectures on buildings (1989; Zbl 0694.51001)]. Here are the chapter headings: 1) Groups acting on sets, 2) Affine geometry, 3) Projective geometry, 4) The general and special linear groups, 5) BN-pairs and buildings, 6) The 7-point plane and the group ${A}_{7}$, 7) Polar geometry, 8) Symplectic groups, 9) BN-pairs, diagrams and geometries, 10) Unitary groups, 11) Orthogonal groups, 12) The Klein correspondence.

First of all the author describes the underlying geometries proving the most basic results (among others: Fundamental theorem of projective geometry, Witt’s Theorem). Then he describes the classical groups proving simplicity results based on Iwasawa’s theorem (using the results of Tamagawa on Siegel transforms in case of the orthogonal groups). Finally he determines orders. Up to this point there is nothing new. All of this can also be found in Artin’s book or Dieudonné’s book. Moreover Artin’s book contains much more information on the geometry and Dieudonné’s book is much more general, as the fields in the book under review are, when it comes to the point, usually finite.

But the book under review provides additional results, which we cannot find in the books just mentioned. The author proves all exceptional isomorphisms between classical groups and between classical groups and alternating groups. These proofs are geometric in nature. For example for proving $PSp\left(4,3\right)\cong PSU\left(4,2\right)$ he constructs the $PSp\left(4,3\right)$-quadrangle out of the $PSU\left(4,2\right)$-quadrangle. Furthermore he gives a description of the groups $Sz\left(q\right)$ including a proof of simplicity. It was a little bit strange to learn that $Sz\left(2\right)$ in this book will not be Frobenius of order 20 but ${S}_{5}$.

The second aim of the book is to point out the relation between classical geometries and groups and the modern notation of buildings and BN-pairs. For this purpose he proves some simple basic facts about buildings and describes the relation between the classical geometries, the building and the groups (in terms of parabolics and Weyl groups). Of course much more of this can be found in the books of Brown or Ronan. But the real value of the book under review is to connect geometry, groups and the modern language of buildings and BN-pairs.

The book under review contains less than the standard books on classical groups, it contains less than the recent books on buildings, but it covers enough material one has to know when entering the field. The only serious point missing is some treatment of automorphisms of the classical groups. The author combines both areas in a way which was missing in the literature until now. It is therefore highly recommended to students beginning to work with classical groups and who want to get some knowledge about the interaction between groups, classical geometries, buildings, BN-pairs and modern treatments like diagram geometries. Having acquainted the knowledge of this book the reader can study more advanced literature. For this purpose the author ends every chapter with a list for further reading.

The book is carefully written. But with students as a possible readership in mind a more detailed treatment would have been better. As the author says in the introduction the book should be accessible to students with a background in linear algebra and a little group theory. I doubt that an average student with such a background will be able to follow chapter 11 or 12. With this restriction the book is recommended for all who like to get a first view on what is going on in classical groups, buildings and BN-pairs. The book fills a gap in the existing literature.

##### MSC:
 20-01 Textbooks (group theory) 20E42 Groups with a $BN$-pair; buildings 20G15 Linear algebraic groups over arbitrary fields 20G40 Linear algebraic groups over finite fields 51E24 Buildings and the geometry of diagrams