Characters of finite groups. Volume 1 (Q1684483)

According to \textit{C. W. Curtis} [Pioneers of representation theory: Frobenius, Burnside, Schur, and Brauer. Providence, RI: American Mathematical Society; London: London Mathematical Society (1999; Zbl 0939.01007)], the year 1897 was the centennial of two events in the history of mathematics: The publication of the first paper on representation of finite groups by Frobenius and the appearance of the first treatise in English of the theory of finite groups by Burnside. Burnside soon developed his own approach to representations of finite groups and Frobenius together with Burnside explored the new subject and its application to finite group theory. They were soon joined with Schur and a few years later by Brauer, who are called the pioneers of representation theory of finite groups. The importance of this theory appear in proving fundamental properties of finite groups, such as the existence of a Frobenius kernel in Frobenius groups which is proved using representation theory and still there is no group theoretrical proof for it. This theory is still developing and many new concepts are emerging. Character theory over the complex field is associated to representation theory and proving group theoretical properties uses characters, rather than representations. The aim of the book under review is to place character theory and its applications to finite groups within the reach of graduate students with a modest knowledge of group theory and linear algebra. Although the books by \textit{I. M. Isaacs} [Character theory of finite groups. New York-San Francisco-London: Academic Press, a subsidiary of Harcourt Brace Jovanovich, Publishers. (1976; Zbl 0337.20005)], \textit{B. Huppert} [Character theory of finite groups. Berlin: Walter de Gruyter (1998; Zbl 0932.20007)] and \textit{W. Feit} [The representation theory of finite groups. Amsterdam - New York - Oxford: North-Holland Publishing Company. (1982; Zbl 0493.20007)] focus on character theory with applications, but the present book covers new and up-date concepts and brings many problems to work on. The first edition of the book was published by \textit{Ya. G. Berkovich} and \textit{E. M. Zhmud'} [Characters of finite groups. Part 1. Transl. from the orig. Russian manuscript by P. Shumyatsky and V. Zobina. Transl. ed. by D. Louvish. Providence, RI: American Mathematical Society (1998; Zbl 0934.20008)]. But the 2nd edition is prepared by the previous authors together with L. S. Kazarin. The book has eleven chapters and covers all areas of interest in character theory over the complex field. In the preface of the book, the title of contents of Volume 1 and 2 are given and it seems that in Chapter XXVII of the second volume the authors discuss modular character theory. Exercises are an important part of the book and are of varying degree of difficulty. A list of open problems is given at the end of Volume 2, which is due to appear shortly. In my opinion, the book is highly recommended to every researcher in character theory of finite groups. For Volume 2 see [Zbl 1426.20003].