$$\def\Fix{\mathrm{Fix}} \def\D{\mathsf{D}} \def\C{\mathsf{C}} \def\D{\mathsf{D}} \def\GL{\mathsf{GL}} \def\OO{\mathsf{O}} \def\SO{\mathsf{SO}} \def\RR{\mathbb{R}} \def\ZZ{\mathbb{Z}} \def\Aut{\mathrm{Aut}} \def\xx{\mathbf{x}}$$
jam - wiki

## Summary - lecture by lecture

Lectures in green have not happened yet

Chapter 1

1. What is symmetry? Transformations and groups. And a reminder about permutations.

2. Group actions. Orbits and stabilizers.

Do Problems 1.1 - 1.4

3. More on orbits and stabilizers

4. Orbit-Stabilizer theorem

Do Problems 1.6, 1.9, 1.11, 1.12

5. Free, transitive and effective actions. Actions of a group on itself.

6. Action on set of left cosets $$G/H$$ as model for any transitive action

You can now do all problems from Chapter 1, except for 19, 20 and 21, which require Sec 1.6 (which we haven't covered).

Chapter 2

7. Euclidean transformations, $$\OO(n)$$ and $$\SO(n)$$; Seitz symbol

8. Cyclic and dihedral subgroups of $$\OO(2)$$

Do problems 2.1-2.7 & 2.9

9. The finite subgroups of $$\OO(2)$$. Euclidean transformations in the plane: rotations, reflections and glide-reflections.

10. Euclidean transformations and classification of triangles.

Can now do all problems of chapter 2

Chapter 3

11. Lattices in $$\RR^2$$ (the 5 types) and their symmetries.

12. Symmetries of the 5 types of lattice continued;

Do problems 3.1-3.5

13. Wallpaper groups and overview of their classification.

14. Wallpaper groups continued.

Can do all problems from Chapter 3

Chapter 4

15. Symmetric problems and invariant functions

16. Critical points of invariant functions and Fixed point subspaces

Can do problems 4.1 to 4.4 (in new version)

17. Principle of symmetric criticality

18. Bifurcations and symmetry breaking;

19. Example in 3D: tetrahedral group

Chapter 5

20. Symmetry in differential equations

21. Conservation of symmetry and coupled cell systems

Can do problems from Chapter 5

22. Revision discussion