INTRODUCTION

These notes represent the lectures which I gave in St. Andrews under the title “Burnside groups of exponent 4”. Some details which I had to suppress in the talks as time grew short have survived in these notes. Apart from that and some improvements mentioned below, they correspond pretty well to the ground which I tried to cover.

The idea of the course was to survey the present state of knowledge about groups with exponent four, since several outstanding problems have been settled in the last five years - but I wished to provide some insight into these developments by giving some guidance to the history of these problems. I hoped also to show that this is an interesting and now accessible branch of group theory and so I have given perhaps more detail than usual at certain points to help clarify new ideas for the non-expert. The course is self-contained up to a point; complete proofs of many results are given - on the other hand I have not tried to compress long proofs but I have mentioned instead the ideas involved. I have given full references and attributions - correctly, I hope.

The plan, corresponding to the five talks, is as follows. §1 gives a brief resume of work on the Burnside problem and then proves those (few) general results on exponent four groups which were known before 1960.