This book gives a proof of Cherlin's conjecture for finite binary primitive permutation groups. Motivated by the part of model theory concerned with Lachlan's theory of finite homogeneous relational structures, this conjecture proposes a classification of those finite primitive permutation groups that have relational complexity equal to 2. The first part gives a full introduction to Cherlin's conjecture, including all the key ideas that have been used in the literature to prove some of its special cases. The second part completes the proof by dealing with primitive permutation groups that are almost simple with socle a group of Lie type. A great deal of material concerning properties of primitive permutation groups and almost simple groups is included, and new ideas are introduced. Addressing a hot topic which cuts across the disciplines of group theory, model theory and logic, this book will be of interest to a wide range of readers. It will be particularly useful for graduate students and researchers who need to work with simple groups of Lie type.

Since the classification of finite simple groups was announced in 1980 the subject has continued to expand opening many new areas of research. This volume contains a collection of papers, both survey and research, arising from the 1990 Durham conference in which the excellent progress of the decade was surveyed and new goals considered. The material is divided into eight sections: sporadic groups; moonshine; local and geometric methods in group theory; geometries and related groups; finite and algebraic groups of Lie type; finite permutation groups; further aspects of Lie groups; related topics. The list of contributors is impressive and the subjects covered include many of the fascinating developments in group theory that have occurred in recent years. It will be an invaluable document for mathematicians working in group theory, combinatorics and geometry.

With the classification of the finite simple groups complete, much work has gone into the study of maximal subgroups of almost simple groups. In this volume the authors investigate the maximal subgroups of the finite classical groups and present research into these groups as well as proving many new results. In particular, the authors develop a unified treatment of the theory of the 'geometric subgroups' of the classical groups, introduced by Aschbacher, and they answer the questions of maximality and conjugacy and obtain the precise shapes of these groups. Both authors are experts in the field and the book will be of considerable value not only to group theorists, but also to combinatorialists and geometers interested in these techniques and results. Graduate students will find it a very readable introduction to the topic and it will bring them to the very forefront of research in group theory.

cOver the past 20 years, the theory of groups -- in particular simple groups, finite and algebraic -- has influenced a number of diverse areas of mathematics. Such areas include topics where groups have been traditionally applied, such as algebraic combinatorics, finite geometries, Galois theory and permutation groups, as well as several more recent developments. Among the latter are probabilistic and computational group theory, the theory of algebraic groups over number fields, and model theory, in each of which there has been a major recent impetus provided by simple group theory. In addition, there is still great interest in local analysis in finite groups, with substantial new input from methods of geometry and amalgams, and particular emphasis on the revision project for the classification of finite simple groups.

This important book contains 20 survey articles covering many of the above developments. It should prove invaluable for those working in the theory of groups and its applications.