Francis Buekenhout and Arjeh M. Cohen wroteDiagram Geometry — Related to Classical Groups and Buildings with the goal of providing a self-contained introduction to diagram geometry. They do an effective job of accomplishing this goal. The authors assume the reader has a background on the basics of group theory, but all other necessary material is introduced. Anyone who has taken some graduate courses in group theory should be able to understand this theory after reading the book. It contains several chapters of introduction to the field before touching on some of the treatment of specific geometric spaces.
Even though this book does contain exercises it would not make a great text book. The chapters are very complete but it would be difficult to cover many sections in a course due to the many topics contained in each chapter. Additionally the text does not contain the kind of additional description and motivation that one would expect from a course text. For this reason the book would make an excellent reference and a good guide for someone interested in doing self study to work in the field. In addition to some exercises at the end of each section the authors offer extra details about the history and various other topics that could be used to spur further research. I enjoyed these insights as they provide the reader with a glimpse into the original motivation for various concepts. As one begins studying in a new field, these insights can be very valuable as they help the development make more sense.
As someone who has been exposed to the classical groups from a more algebraic point of view, I found this text gave a great background to understanding the geometric interpretation of these groups.
Ellen Ziliak is an Assistant Professor of mathematics at Benedictine University in Lisle IL. Her training is in computational group theory, particularly using geometric properties of groups. More recently she has become interested in ways to introduce undergraduate students to research in abstract algebra through applications.