Around Burnside

Perhaps it is not inappropriate for me to begin with the comment that this book has been an interesting challenge to the translator. It is most unusual, in a text of this type, in that the style is racy, with many literary allusions and witticisms: not the easiest to translate, but a source of inspiration to continue through material that could daunt by its combinatorial complexity. Moreover, there have been many changes to the text during the translating period, reflecting the ferment that the subject of the restricted Burnside problem is passing through at present. I concur with Professor Kostrikin's "Note in Proof', where he describes the book as fortunate. I would put it slightly differently: its appearance has surely been partly instrumental in inspiring much endeavour, including such things as the paper of A. I. Adian and A. A. Razborov producing the first published recursive upper bound for the order of the universal finite group B(d,p) of prime exponent (the English version contains a different treatment of this result, due to E. I. Zel'manov); M. R. Vaughan-Lee's new approach to the subject; and finally, the crowning achievement of Zel'manov in establishing RBP for all prime-power exponents, thereby (via the classification theorem for finite simple groups and Hall-Higman) settling it for all exponents. The book is encyclopaedic in its coverage of facts and problems on RBP, and will continue to have an important influence in the area.

Table of Contents:
1 Introduction.- § 1. Historical Survey.- § 2. Engel Lie Algebras: Definitions and Examples.- § 3. The Locally Nilpotent Radical.- § 4. Basic Conventions. Elementary Combinatorics.- § 5. The Method of Sandwiches.- § 6. Filtrations in Lie Algebras.- § 7. Main results and Structure of Proofs.- § 8. Commentary.- 2 The Descent to Sandwiches.- § 1. Descent to nil-elements of index 3.- § 2. Descent to Thin Sandwiches (General Case).- § 3. Descent to Thin Sandwiches (the Case p > n).- § 4. Descent from C2* to C(p-3)/2* 41.- § 5. Commentary.- 3 Local Analysis on Thin Sandwiches.- § 1. A First Footbridge Between Thin and Thick Sandwiches.- § 2. A Second Footbridge Between Thin and Thick Sandwiches.- § 3. Two Necessary Lemmas.- § 4. Sandwich Algebras.- § 5. Commentary.- 4 Proof of the Main Theorem.- § 1. Pairs of Thin Sandwiches.- § 2. Thick Pairs of Thin Sandwiches.- § 3. Completion of the Proof of the Main Theorem.- § 4. Commentary.- 5 Evolution of the Method of Sandwiches.- § 1. Rehabilitation of False Sandwiches.- § 2. A Geodesic Connecting Theorems 3.4.1 and 1.7.4.- § 3. The Local Nilpotency of Sandwich Algebras.- § 4. The Sandwich Radical and its Applications.- § 5. Commentary.- 6 The Problem of Global Nilpotency.- § 1. The Nilpotency Class of a Lie Algebra with En.- § 2. Combination of Solubility and the Engel Condition En.- § 3. Insolubility for n Close to p.- § 4. Global Nilpotency for p > n.- § 5. Commentary.- 7 Finite p-Groups and Lie Algebras.- § 1. Fundamental Relations Between Groups and Lie Algebras.- § 2. The Ideal of Relations (a general Survey).- § 3. Multilinear Relations: Proofs of Theorems 2.3 and 2.4.- § 4. Commentary.- Appendix I An Effective Version of the Proof of Theorem 1.7.4 in Terms of Recursive Functions (due toE. I. Zel’manov).- Appendix II A Short Biography of William Burnside (after A.R. Forsyth [52]).- Epilogue.- Notes Added in Proof.- References.- Author Index.- Notation.