The origin of this book was a seminar held at All Souls College, Oxford, in the Spring of 1989. The aim of the seminar was to work through Michel Lazard's paper Groupes analytiques p-adiques [L], at least far enough to understand the proof of ‘Lubotzky's linearity criterion’ (Lubotzky 1988). In fact, Lubotzky's proof combined Lazard's characterisation of p-adic analytic groups with some recent results of Lubotzky and Mann (1987b) on ‘powerful’ pro-p groups. We found that by reversing the historical order of development, and starting with powerful pro-p groups, we could reconstruct most of the group-theoretic consequences of Lazard's theory without having to introduce any ‘analytic’ machinery. This was a comforting insight for us (as group theorists), and gave us the confidence to go on and develop what we hope is a fairly straightforward account of the theory of p-adic analytic groups.

The first edition was divided (like Gaul) into three parts. Parts I and II were essentially linear in structure. The point of view in Part I was group-theoretic; in Part II, more machinery was introduced, such as normed algebras and formal power series. Between Parts I and II was an Interlude (Chapter 6): this consisted of a series of more or less independent digressions, describing applications of the results to various aspects of group theory.