An international conference ‘Groups – St Andrews 1989’ was held in the Mathematical Institute, University of St Andrews, Scotland during the period 29 July to 12 August 1989. A total of 293 people from 37 different countries registered for the conference. The initial planning for the conference began in July 1986 and in the summer of 1987 invitations were given to Professor J A Green (Warwick), Professor N D Gupta (Manitoba), Professor O H Kegel (Freiburg), Professor A Yu Ol'shanskii (Moscow) and Professor J G Thompson (Cambridge). They all accepted our invitation and gave courses at the conference of three or four lectures. We were particularly pleased that Professor Ol'shanskii was able to make his first visit to the West. The above courses formed the main part of the first week of the conference. All the above speakers have contributed articles based on these courses to the Proceedings.

In the second week of the conference there were fourteen one-hour invited survey lectures and a CAYLEY workshop with four main lectures. In addition there was a full programme of research seminars. The remaining articles in the two parts of the Proceedings arise from these invited lectures and research seminars.

The two volumes of the Proceedings of Groups – St Andrews 1989 are similar in style to ‘Groups – St Andrews 1981’ and ‘Proceedings of Groups – St Andrews 1985’ both published by Cambridge University Press in the London Mathematical Society Lecture Notes Series.

An international conference ‘Groups - St Andrews 1989’ was held in the Mathematical Institute, University of St Andrews, Scotland during the period 29 July to 12 August 1989. A total of 293 people from 37 different countries registered for the conference. The initial planning for the conference began in July 1986 and in the summer of 1987 invitations were given to Professor J A Green (Warwick), Professor N D Gupta (Manitoba), Professor O H Kegel (Freiburg), Professor A Yu Ol'shanskii (Moscow) and Professor J G Thompson (Cambridge). They all accepted our invitation and gave courses at the conference of three or four lectures. We were particularly pleased that Professor Ol'shanskii was able to make his first visit to the West. The above courses formed the main part of the first week of the conference. All the above speakers have contributed articles based on these courses to the Proceedings.

In the second week of the conference there were fourteen one-hour invited survey lectures and a CAYLEY workshop with four main lectures. In addition there was a full programme of research seminars. The remaining articles in the two parts of the Proceedings arise from these invited lectures and research seminars.

The two volumes of the Proceedings of Groups - St Andrews 1989 are similar in style to ‘Groups - St Andrews 1981’ and ‘Proceedings of Groups - St Andrews 1985’ both published by Cambridge University Press in the London Mathematical Society Lecture Notes Series.

In 1979 we held a small meeting in St Andrews at which Joachim Neubüser from RWTH Aachen spoke on Counterexamples to the class-breadth conjecture. At this time we discussed the possibility of organising a much larger group theory meeting in St Andrews in 1981. Preliminary dates were suggested to fit the German school holidays. Indeed choosing dates for all the meetings has proved an interesting task: fitting in with the end of the English academic year, the start of the American academic year, the Galway races (Galway 1993), the Open University Summer School (Bath 1997), the Open Golf Championship (St Andrews 2005). For Groups 1981 we invited main speakers whose mathematical interests were close to our own. By chance, three of the four — Joachim Neubiiser (RWTH Aachen), Seán Tobin (Galway), and Jim Wiegold (Cardiff) — had been friends from postgraduate days in Manchester. The fourth, Derek Robinson (Urbana), was originally from Montrose (visible on a good day across the Tay estuary from the Mathematical Institute in St Andrews). Despite our planning of the 1981 dates, the wedding of Prince Charles and Princess Diana was announced to take place during the period of the conference. Residences provided only packed lunches on the wedding day. However Jim Wiegold, the ‘Mathematical Prince of Wales’, provided our own star attraction! We had intended the conference to last a week but some participants wanted to stay in St Andrews for a further week.

We would like to thank Cambridge University Press for encouraging us to produce this new edition of the Proceedings of Groups St Andrews 1981. At the suggestion of Roger Astley of Cambridge University Press we have asked the four main speakers at the 1981 conference to provide brief addenda to their articles. We are delighted that they have all responded positively to this task. Three of the authors have provided their own new pages. The fourth article on ‘An elementary introduction to coset table methods in computational group theory’ has been prepared by us with our friend and collaborator George Havas after some helpful suggestions from Joachim Neubüser. We have also added a short article looking back at twenty-five years of Groups St Andrews conferences.

Although for the 1981 Proceedings we put all the references into a standard form, we have, twenty-five years later, adopted a more relaxed approach and have kept the refereeing style of the addenda as provided by the authors.

Thanks are also due to our colleague Martyn Quick for his help with the preparation of the additional material.

Even after 25 years the article by Joachim Neubiiser remains the first source to which all three of us refer those who want to find out about the use of coset tables for studying groups. Our view is confirmed by the 14 Reference Citations from 1998 to 2005 which MathSciNet reveals for this article. Here we loosely follow the structure of the original article and provide some updates on the area (oriented towards our own interests).

First we point out that two newer books include comprehensive details on coset enumeration and related topics in works which are much broader studies. They give excellent coverage of the areas addressed in this article and, further, provide much additional material. They also provide some alternative points of view and many references (as do the other materials cited here).

One of Neubüser's aims in writing his survey was to provide a unified view on coset table methods in computational group theory. He addressed the way coset table concepts were developed, implemented and used. In Derek Holt follows the same kind of approach, including a long chapter “Coset Enumeration” and a shorter one “Presentations for Given Groups”. Charles Sims in focuses on finitely presented groups and he takes a perspective significantly based on some fundamental methods from theoretical computer science, namely automata theory and formal languages.

An international conference ‘Groups - St. Andrews 1981’ was held in the Mathematical Institute, University of St. Andrews during the period 25th July to 8th August 1981. The main topics of the conference: combinatorial group theory; infinite groups; general groups, finite or infinite; computational group theory are all well-represented in the survey and research articles that form these Proceedings. Four courses each providing a five-lecture survey, given by Joachim Neubüser, Derek Robinson, Sean Tobin and Jim Wiegold have been expanded, subsequently, into articles forming the first four chapters of the volume. Many of the themes in these chapters recur in the survey and research articles which form the second part of the volume.

Methods and techniques such as homology, geometrical methods and computer implementation of algorithms are used to obtain group theoretical results. Computational methods are surveyed in several articles in particular the major survey by Joachim Neubüser and find application in papers on Burnside groups and finite simple groups. In fact Burnside groups are discussed in two rather different papers, a survey of groups of exponent four by Sean Tobin and a major contribution to the exponent five case by Marshall Hall and Charles Sims. Derek Robinson exploits the way in which cohomology groups arise in group theory to establish some splitting and near-splitting theorems. Rudolf Beyl also uses homological techniques to discuss group extensions.