This book is one of the most famous monographs devoted to the application of group theory to quantum mechanics. The theory of groups is set out taking into account its use in physical applications, with the greatest attention paid to the symmetric group, the group of rotations and the most important section for applications - the theory of representations. Before proceeding to the applications, the author summarizes the main provisions and the apparatus of quantum mechanics and the theory of atomic spectra. The general theory developed in the book is applied to atomic spectra in a form that allows it to be used for a wider range of nuclear spectrum problems, field theory and elementary particles, and so on. p. In this regard, such issues as the properties of vector coupling coefficients and cancer coefficients, as well as the circulation of time. are presented. The book is intended for researchers and graduate students of physicists, especially theoretical physicists working in the field of atomic and nuclear spectroscopy, the study of the structure of molecules, solid state physics, as well as mathematicians interested in the physical applications of group theory.