The first edition of this book has been the indispensable reference for researchers in the theory of pro-p groups. In this second edition the presentation has been improved and important new material has been added.
The theory of p-adic analytic pro-p groups has undergone significant development since the seminal work of Lazard in 1965. This book presents a complete and self-contained account of this theory, which has many applications in both group theory and number theory.
The first part of the book is group-theoretic. It develops the theory of pro-p groups of finite rank, starting from first principles and using elementary methods. Part II introduces p-adic analytic groups: by taking advantage of the theory developed in Part I, it is possible to define these, and derive all the main results of p-adic Lie theory, without having to develop any sophisticated analytic machinery.
Part III, consisting of material new to the second edition, takes the theory further. Among those topics dealt with are the theory of pro-p groups of finite coclass, the dimension subgroup series, and its associated graded Lie algebra. The final chapter sketches the beginnings of a theory of analytic groups over pro-p rings other than the p-adic integers.
New results obtained since the publication of the first edition have been incorporated in the text and in many new exercises.