Jordan-Jordan:

1. Is motivation for studying orbits and stabilizers, and groups in general. We viewed symmetries of graphs (lecture 1) as our motivation instead.

2. Chapter 2 is all revision from Proofs and Problem Solving. We covered aspects in lecture 3 and lecture 16.

3. Chapter 3 is all revision from linear algebra.

4.1 is notation that we used throughout.

4.2 Operations: lecture 2

4.3 Definition of a Group: lecture 2.

4.4 Examples of Groups: mainly lecture 2, but also throughout the course.

4.5 Consequences of the axioms: lecture 4.

4.6 Direct products: lecture 5.

5.1 Subgroups: lecture 5

5.2 Examples of Subgroups: mainly lecture 5, but also throughout the course.

5.3 Groups of Symmetries. We did not cover exactly 5.3; we covered symmetries of graphs and Platonic solids instead. See lectures 1, 2 and 3.

6.1-6.3 Cyclic groups, subgroups, order of elements: all lecture 6 and 7.

6.4-6.5 Orders of products. We did not cover this in lectures, but similar problems were on the Exercise Sheets.

6.6-6.7 Subgroups and Products of Cyclic Groups: lecture 7.

7.1 Definition of a Group Action: lecture 11.

7.2 -7.3 Orbits and Stabilizers: lecture 13.

7.4 Permutations from Group Actions: this is linked to Chapter 9.3, and was the main theorem in lecture 12.

7.5 The alternating group: lecture 16.

8. Chapter 8 is mainly revision of Proofs and Problem Solving. We covered aspects at the end of lecture 7, and beginning of lecture 8.

9.1 Homomorphisms and Isomorphisms: lecture 10.

9.2 Properties of group homomorphisms: lecture 10.

9.3 Homomorphisms from group actions: this is linked to Chapter 7.4, and was the main theorem in lecture 12.

9.4 Cayley's Theorem: lecture 12.

9.5 All cyclic groups of fixed order are isomorphic: lecture 7.

10.1-10.2 Left cosets: lecture 8.

10.3-10.5 Lagrange's Theorem and Applications: lectures 8 and 9.

10.6 Right cosets: lecture 9.

11.1 Orbit-Stabilizer: lecture 14.

11.2-11.3 Fixed subsets and Polya counting: lecture 15.

12. Chapter 12 is Polya counting examples: lecture 15.

13.1-13.2 Conjugates and Conjugacy Classes: lecture 17.

13.3 Conjugacy in S_n: lecture 18.

13.4-13.6 Centres and Centralizers: lecture 17.

14.1 is just an example of a group action.

14.2 Cauchy's Theorem: lecture 14.

14.3 Direct Products: lecture 11.

15.1 Kernels of Homomorphisms: lecture 10.

15.2 Kernels of Actions: lecture 12.

15.3 Conjugates of a Subgroup: we did not formally cover this in lectures, but it is very similar to aspects of lecture 18 and 19.

15.4 Normal Subgroups: lecture 18 and 19.

15.5 Normal Subgroups and conjugacy classes: lecture 18.

15.6 is Problem 5.27 on the exercise sheets.

16. I gave an overview of factor groups in lecture 20. None of Chapter 16 is examinable.