the advantages of group theory in physics were not recognized till 1925 when it was applied for formal study of theoretical foundations of quantum mechanics, atomic structures and spectra by, to name a few, H A Bethe, E P Wigner, etc. It has now become indispensable in several branches of physics and physical chemistry. Dr. Joshi develops the mathematics of group theory and then goes on to present its applications to quantum mechanics, crystallography, and solid state physics. For proper comprehension of representation theory, he has covered thoroughly such diverse but relevant topics as Hilbert spaces, function spaces, operators, and direct sum and product of matrices. He often proceeds from the particular to the general so that the beginning student does not have an impression that group theory is merely a branch of abstract mathematics. Various concepts have been explained consistently by the use of the C4v. Besides, it contains an improved and more general proof of the Schurs first lemma and an interpretation of the orthogonality theorem in the language of vector spaces (Chapter 3).Throughout the text the author gives attention to details and avoids complicated notation.