Book excerpt: 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.