## 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.

3. More on orbits and stabilizers.

4. Orbit-Stabilizer theorem. Free, transitive and effective actions.

5. Actions of a group on itself.

6. Action on set of left cosets \(G/H\) as model for any transitive action. Burnside type of action.

#### Chapter 2

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

8. Elements of \(O(2)\) are reflections or rotations. The subgroup \(C_n\)

9. Dihedral subgroups of \(\OO(2)\). The finite subgroups of \(\OO(2)\). Euclidean transformations in the plane: rotations, . . .

10. . . . reflections and glide-reflections. Euclidean transformations and classification of triangles.

11a. Finish classification of triangles. – can now do all problems of Chapter 2

#### Chapter 3

11b. Lattices

12. The 5 types of lattice in \(\RR^2\);

13. Symmetry groups of lattices. Can do problems 3.1 to 3.12

14. Wallpaper groups. Can do all problems of Chapter 3

#### Chapter 4

15. Symmetric problems and invariant functions can do problem 4.1

16. Critical points of invariant functions and Fixed point subspaces

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

#### Chapter 6

22. Symmetries of periodic orbits

23. Revision discussion