Preface

This book is inspired by the workshop Groups, Combinatorics, Computing, held at National University of Ireland, Galway from April 11 to 16, 2011—the Fifth “de Brún Workshop” run under the auspices of Science Foundation Ireland’s Mathematics Initiative Programme. A principal theme of the workshop was interactions between group theory and combinatorics with algorithmic or computational aspects. Areas encompassed by this theme are currently the focus of intense research activity. The core part of the workshop was formed by three lecture courses. These contained a wide and unique selection of material, for the first time providing an accessible introduction to frontier research in thematic areas. It became clear that the courses should be made available to a larger audience. The book has three chapters, one per lecture course. Each chapter is selfcontained; beginning with background material including historical roots, the reader is led to the latest results and open problems. Illustrative examples, some proofs and algorithms, and extensive bibliographies are given. The first chapter, by Martin Liebeck, is an exposition of recent developments in probabilistic and asymptotic theory of finite groups, particularly finite simple groups. The first two sections are on random generation of finite groups and maximal subgroups. The next topic is representation varieties and charactertheoretic methods. Finally, diameter and growth of Cayley graphs of simple groups are considered. The chapter traces progress on fundamental conjectures which have driven the development of this subject. The second chapter is by Alice Niemeyer, Cheryl Praeger, and Ákos Seress. This chapter again has a strong probabilistic flavour. It discusses the role of estimation in the design and analysis of randomised algorithms for computing with finite groups, and approaches to estimating proportions of important element classes. Among the latter are geometric methods, the use of generating functions, and theory of Lie type groups. The chapter also surveys numerous results concerning estimation in permutation groups and finite classical groups. An application to the construction of involution centralisers, a key part of the constructive recognition of finite simple groups, is given. Connections with theoretical computer science are made.