This book is based on a special course that the author delivered to the Faculty of Mechanics and Mathematics at Moscow University in the academic years 1971/72 and 1972/73. It presents a new and improved version of the method of investigating groups with an identical relation of the form [lowercase italic]x[lowercase italic superscript]n = 1 evolved by P. S. Novikov and the author for solving Burnside's problem on periodic groups, first published in a joint paper. In the interval since the Russian edition was published, the method described has found new applications.

This volume contains papers which are based primarily on talks given at an inter national conference on Algorithmic Problems in Groups and Semigroups held at the University of Nebraska-Lincoln from May ll-May 16, 1998. The conference coincided with the Centennial Celebration of the Department of Mathematics and Statistics at the University of Nebraska-Lincoln on the occasion of the one hun dredth anniversary of the granting of the first Ph.D. by the department. Funding was provided by the US National Science Foundation, the Department of Math ematics and Statistics, and the College of Arts and Sciences at the University of Nebraska-Lincoln, through the College's focus program in Discrete, Experimental and Applied Mathematics. The purpose of the conference was to bring together researchers with interests in algorithmic problems in group theory, semigroup theory and computer science. A particularly useful feature of this conference was that it provided a framework for exchange of ideas between the research communities in semigroup theory and group theory, and several of the papers collected here reflect this interac tion of ideas. The papers collected in this volume represent a cross section of some of the results and ideas that were discussed in the conference. They reflect a synthesis of overlapping ideas and techniques stimulated by problems concerning finite monoids, finitely presented mono ids, finitely presented groups and free groups.

William Burnside was one of the three most important algebraists who were involved in the transformation of group theory from its nineteenth-century origins to a deep twentieth-century subject. Building on work of earlier mathematicians, they were able to develop sophisticated tools for solving difficult problems. All of Burnside's papers are reproduced here, organized chronologically and with a detailed bibliography. Walter Feit has contributed a foreword, and a collection of introductory essays are included to provide a commentary on Burnside's work and set it in perspective along with a modern biography that draws on archive material.

"An excellent up-to-date introduction to the theory of groups. It is general yet comprehensive, covering various branches of group theory. The 15 chapters contain the following main topics: free groups and presentations, free products, decompositions, Abelian groups, finite permutation groups, representations of groups, finite and infinite soluble groups, group extensions, generalizations of nilpotent and soluble groups, finiteness properties." —-ACTA SCIENTIARUM MATHEMATICARUM

Authoritative collection of surveys and papers that will be indispensable to all research workers in the area.

During the past forty years, a new trend in the theory of associative algebras, Lie algebras, and their representations has formed under the influence of mathematical logic and universal algebra, namely, the theory of varieties and identities of associative algebras, Lie algebras, and their representations. The last twenty years have seen the creation of the method of 2-words and *a-functions, which allowed a number of problems in the theory of groups, rings, Lie algebras, and their representations to be solved in a unified way. The possibilities of this method are far from exhausted. This book sums up the applications of the method of 2-words and *a-functions in the theory of varieties and gives a systematic exposition of contemporary achievements in the theory of identities of algebras and their representations closely related to this method. The aim is to make these topics accessible to a wider group of mathematicians.