Introduction

Summary

This book contains the proceedings of the L.M.S. Durham Symposium on Groups and Combinatorics, July 5–15, 1990, supported by the Science and Engineering Research Council of Great Britain.

The classification of finite simple groups was completed in 1980, and most of the conference was concerned with trends in group theory and related areas which have come to the fore since then. We have divided the material into eight sections, which we now outline.

Sporadic groups

During the conference, a spectacular proof of the ten-year-old Conway “Y₅₅₅ conjecture” concerning a Coxeter-type presentation of the Monster sporadic group, was achieved by Norton and Ivanov. In their articles, Norton and Ivanov give details of their proof; Ivanov obtains a new geometric characterization of the Monster, and this is used by Norton to prove the conjecture. Conway's short paper outlines the background. The article of Conway and Pritchard gives a proof of the Y₅₅₂ presentation for the Fischer group Fi₂₄. In a different vein, the papers of Aschbacher and Segev concern proofs of uniqueness results for sporadic groups. In their first article they survey their general framework for providing uniqueness results; this uses a graph-theoretical setting and some ideas with a topological flavour. Their second paper discusses in detail the uniqueness proof for the group J₄.

Moonshine

This section concerns the remarkable relationships between the Monster group and certain modular functions. Much insight into moonshine is gained from studying the Monster Lie algebra, which is described by Borcherds in his article.