The predictive power of mathematics in quantum phenomena is one of the great intellectual successes of the 20th century. This textbook, aimed at undergraduate or graduate level students (depending on the college or university), concentrates on how to make predictions about the numbers of each kind of basic state of a quantum system from only two ingredients: the symmetry and the linear model of quantum mechanics. This method, involving the mathematical area of representation theory or group theory, combines three core mathematical subjects, namely, linear algebra, analysis and abstract algebra. Wide applications of this method occur in crystallography, atomic structure, classification of manifolds with symmetry, and other areas.