1 Preliminaries -- 1.1 The Concept of Groups -- 1.2 Price Index in Economics -- 1.3 The Realization of Groups -- 1.4 Representation of Groups -- 1.5 Equivalence of Representations -- 1.6 Reducibility of Representations -- 1.7 Complete Reducibility -- 1.8 Basic Conclusions -- 1.9 Representations of Special Finite Groups -- 1.10 Kronecker Products -- 1.11 Unitary Representations -- Problems -- 2 Linear Operators with Symmetries -- 2.1 Schurโ{128}{153}s Lemma -- 2.2 Symmetry of a Matrix -- 2.3 The Fundamental Theorem -- Problems -- 3 Symmetry Adapted Basis Functions -- 3.1 Illustration by Dihedral Groups -- 3.2 Application in Quantum Physics -- 3.3 Application to Finite Element Method -- 3.4 Perturbed Problems with Symmetry -- 3.5 Fast Fourier Transform on Finite Groups -- 4 Continuous Groups And Representations -- 4.1 Continuous Matrix Groups -- 4.2 Relationship Between Some Groups -- 4.3 Constructing Representations -- 4.4 Clebsch-Gordan Coefficients -- 4.5 The Lorentz group and SL(2,C) -- Problems -- 5 Symmetry Ad. Vectors, Characters -- 5.1 Orthogonality of Representations -- 5.2 Algorithm for Symmetry Adapted Bases -- 5.3 Applications -- 5.4 Similarity Classes of Groups -- 5.5 Characters -- 5.6 Representation Theory of Finite Groups -- 5.7 Extension to Compact Lie Groups -- Problems -- 6 Various Topics of Application -- 6.1 Bifurcation and A New Technique -- 6.2 A Diffusion Model in Probability Theory -- Problems -- 7 Lie Algebras -- 7.1 Infinitesimal Operator and Exponential Map -- 7.2 Lie Algebra of a Continuous Group -- 7.3 Representation of Lie Algebras -- 7.4 Representations of SU(2) and SO(3) -- 7.5 Examples from Quantum Mechanics -- Problems -- 8 Applications to Solid State Physics -- 8.1 Lattices -- 8.2 Point Groups and Representations -- 8.3 The 32 Crystal Classes -- 8.4 Symmetries and the Ritz Method -- 8.5 Examples of Applications -- 8.6 Crystallographic Space Groups -- Problems -- 9 Unitary and Orthogonal Groups -- 9.1 The Groups U(n) and SU(n) -- 9.2 The Special Orthogonal Group SO(n) -- 9.3 Subspaces of Representations of SU(3) -- A -- Answers to Selected Problems