Induced representation method makes it possible to construct representation of the entire group G on the basis of the irreducible representation of its subgroup H and the structure of the left coset decomposition of G with respect to H. The application of theorems of induced representation method, i.e. Frobenius reciprocity theorem and a set of Mackey theorems makes it possible to simplify symmetry analysis of quantum systems with subgroups of symmetry. Following problems of the theory of electron structure are considered making use of induced representation method: (a) Normal vibrations of molecules. A method based on Frobenius reciprocity makes it possible to simplify the existing method. It is shown that the analysis of vibrations can be performed by checking the orthogonality of IRs of the local subgroup and of the whole group without construction transformation matrix on the whole group. (b) MO (molecular orbitals) or SALC (symmetry-adapted linear combinations) in clusters are also analyzed by making use of Frobenius reciprocity theorem. Also the reciprocity theorem is generalized on matrix elements and transitivity of induction theorem is used to label repeating IRs in large symmetrical clusters. Hybrid orbitals Hybrid orbitalsin molecules are also considered in a framework of induced representation method. (c) A short theory of space-group IRs based on induced representations is presented. These sections are introductory for Chap. 4 . (d) The examples are presented of the application of induced representation method to compatibility relations, selection rules, and SALCs in solids, making use Mackey intertwining number theorem the difference of SALCs in symmorphic and non-symmorphic groups is established.