This book offers a single source of basic facts about the structure of the finite simple groups with emphasis on a detailed description of their local subgroup structures, coverings and automorphisms. The method is by examination of the specific groups, rather than by the development of an abstract theory of simple groups. While the purpose of the book is to provide the background for the proof of the classification of the finite simple groupsdictating the choice of topicsthe subject matter is covered in such depth and detail that the book should be of interest to anyone seeking information about the structure of the finite simple groups. This volume offers a wealth of basic facts and computations. Much of the material is not readily available from any other source. In particular, the book contains the statements and proofs of the fundamental BorelTits Theorem and CurtisTits Theorem. It also contains complete information about the centralizers of semisimple involutions in groups of Lie type, as well as many other local subgroups. Readership Graduate students and research mathematicians interested in the subgroup structure of the finite simple groups of Lie type, the alternating groups and the sporadic simple groups. Reviews "This is the third volume in a series in which the authors aim to write down a complete proof of the classification of simple finite groups. This third volume concentrates entirely on various basic properties of the known finite simple groups. The volume is written in the careful, clear and thorough style we have come to expect from the authors. Quite apart from its role in the series, it contains a wealth of information about the known simple groups which is essential for use in applications of finite group theory. For this reason, it will surely stand on its own as a standard text on simple groups."  Bulletin of the London Mathematical Society "The book is carefully written and much of the material presented has uses well beyond the task at hand. There is a wealth of information in this volume, including quite a number of useful tables ... will be a valuable reference for future generations of mathematicians."  Mathematical Reviews Table of Contents  Some theory of linear algebraic groups
 The finite groups of Lie type
 Local subgroups of groups of Lie type. I
 Local subgroups of groups of Lie type. II
 The alternating groups and the twentysix sporadic groups
 Coverings and embeddings of quasisimple \(\mathcal {K}\)groups
 General properties of \(\mathcal {K}\)groups
 Background references
 Expository references
 Errata for numbers 1 and 2
 Glossary
 Index
