1. Introduction
Part 1. Recollections
Chapter 1. Linear representations of finite groups
Chapter 2. Rings and algebras
Part 2. Introduction and Göbel’s bound
Chapter 3. Rings of polynomial invariants
Chapter 4. Permutation representations
Application: Decay of a spinless particle
Application: Counting weighted graphs
Part 3. The first fundamental theorem of invariant theory and Noether’s bound
Chapter 5. Construction of invariants
Chapter 6. Noether’s bound
Chapter 7. Some families of invariants
Application: Production of fibre composites
Application: Gaussian quadrature
Part 4. Noether’s theorems
Chapter 8. Modules
Chapter 9. Integral dependence and the Krull relations
Chapter 10. Noether’s theorems
Application: Self-dual codes
Part 5. Advanced counting methods and the Shephard-Todd-Chevalley theorem
Chapter 11. Poincaré series
Chapter 12. Systems of parameters
Chapter 13. Pseudoreflection representations
Application: Counting partitions
Appendix A. Rational invariants