## Topic outline

• This topic

### General Information

• Attendance in both lectures and tutorials is compulsory and will be recorded for the purpose of engagement monitoring, in line with the School's Student Engagement Policy.
• Tutorials will start in week 1.
• The first five weeks of this module will involve going over Peter Cameron's Group Theory Revision Notes. How much of this is review, how much is new, and how much forms a crash course in basic group theory will depend on how much group theory you have already studied. I am happy to further explain any parts of this material in the tutorials.
• There may be a lecture in week 7 please follow the announcements.
• ### Useful References

Useful texts for background reading include:

• Walter Ledermann and Alan J. Weir, Introduction to Group Theory (Second Edition), Longman, 1996.
• Peter J. Cameron, Introduction to Algebra (Second Edition), Oxford University Press, 2008.

A useful reference for group actions and permutation group theory is:

• Peter J. Cameron, Permutation Groups, L.M.S. Student Texts 45, Cambridge University Press, 1999.

A very thorough and useful reference for all aspects of group theory is:

• Derek J.S. Robinson, A Course in the Theory of Groups (Second Edition), Graduate Texts in Mathematics 80, Springer, New York, 1996.

• ### Week 1 and Week 2

The following in the notes below are NOT examinable:

• Section 1.5 (Presentations);
• from the last two paragraphs of page 12 (starting with "The group D2n has a presentation") until the end of page 15 (including the Exercise on page 15). You are still responsible, however, for the statement of the Fundamental Theorem of Abelian Groups and its application (as given in Section 2.2).
• ### Week 3 and week 4

Section 3.4 (Appendix: How many groups?) in the notes below is NOT examinable.

Supplementary reading on group actions (from the blog of Tim Gowers):

• ### Week 5 and Week 6

In the notes below, Theorem 4.4 part (b) and its proof, and the proof of the Jordan–Hölder Theorem in Appendix 5.5, are NOT examinable.

• ### Week 8 and Week 9

On Mon, Nov 5 we have a revision lecture (instead of a usual lecture), at the usual time but not at the usual place: 10–12, Queens LG3. We have no tutorial on Friday Nov 9.

In the notes below, Section 6.3 (Digression: Minimal normal subgroups) is not examinable.

• ### Week 10 and Week 11

For the purposes of the final examination, you are only responsible to know one proof of the simplicity of A(for n greater than or equal to 5), and you are only responsible to know one proof that S6 has an outer automorphism.

A description of the outer automorphism of S6  in terms of "mystic pentagons" can be found in the notes http://math.stanford.edu/~vakil/files/sixjan2308.pdf .

• ### Final Exam Information and Solutions to Coursework

• The rubric for the exam will be: "You should attempt all questions. Marks awarded are shown next to the questions". Calculators will not be permitted in the exam.
• Below you can find hints and brief solutions to the different coursework, which will hopefully be of help when preparing for the exam. You should only look at the solutions after you have tried to solve the problems on your own, as a way of checking your answer.
• Sample solutions are not provided for past Group Theory exam papers, since it is best to work out the solutions for yourself and to understand the concepts involved. See also your lecture notes for help. Once you have worked out a solution to an exam question, you should think about how to check it, as this is a useful skill for the final exam. If you have any questions about a solution that you have worked out and tried to check, I will be very happy to answer these questions. Please email me for an appointment.