Vectors and matrices -- Generalizations -- The principal axis transformation -- The elements of quantum mechanics -- Perturbation theory -- Transformation theory and the bases for the statistical interpretation of quantum mechanics -- Abstract group theory -- Invariant subgroups -- The general theory of representations -- Continuous groups -- Representations and eigenfunctions -- The algebra of representation theory -- The symmetric group -- The rotation groups -- The three-dimensional pure rotation group -- The representations of the direct product -- Selection rules and the splitting of spectral lines -- Partial determination of eigenfunctions from their transformation properties -- Electron spin -- The total angular momentum quantum number -- The fine structure of spectral lines -- Selection and intensity rules with spin -- Racah coefficients -- The building-up principle -- Time inversion -- Physical interpretation and classical limits of representation coefficients, three- and six-j symbols.