INTRODUCTION

Four years ago, some members of the present audience were also gathered in Montreal, for the 1982 meeting “Finite Groups: Coming of Age” organized by John McKay of Concordia University. At that time, I presented a survey lecture [Sm1] on the comparatively new and rapidly developing area of “groups and geometries”. By now the area has come to be a more established branch of modern mathematics. It seems appropriate to make today's lecture in effect a sequel to that 1982 lecture – to describe further progress on the main problems then open and indicate important new directions which have opened up since that time.

It will be convenient to follow again the general outline of the earlier lecture, namely:

1) Motivation and applications.

2) Background on geometries and diagrams.

3) “Sporadic” geometries.

4) Properties and characterizations.

Of course, events have rendered parts of this organization a little outdated, but the parallel treatment should help emphasize developments since 1982.

MOTIVATION AND APPLICATIONS

At the earlier meeting, I presented “groups and geometries” as an area whose more widespread development had begun in the latter days (late 70's) of the classification of finite simple groups. In overview, that massive result shows that a non–abelian finite simple group must be one of:

Gemoetric Methods in Group Theory

1. (a) an alternating group;

2. (b) a group of Lie type, defined qver a finite field;

3. (c) one of 26 “sporadic” groups – not contained in the families (a) (b).