- This topic
- 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 . 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 will be a lecture in week 7.
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).
Section 3.3 (The Orbit-Counting Lemma) and Section 3.4 (Appendix: How many groups?) in the notes below are NOT examinable, except for the determination of the groups of order 4 and those of order 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.
Please note that there will be a Group Theory lecture in week 7 (at the usual time and place).
In the notes below, Section 6.3 (Digression: Minimal normal subgroups) is not examinable.
Proofs concerning commutator subgroups and Theorem 6.9 will be covered in week 8.
Proofs concerning commutator subgroups and Theorem 6.9 in the Section 6 notes will be covered in week 8.For the purposes of the final examination, you are only responsible to know one proof of the simplicity of An for n greater than or equal to 5, and you are only responsible for one proof that S6 has an outer automorphism.
We shall prove that S6 has an outer automorphism in week 10.
- We shall prove that S6 has an outer automorphism in week 10, before moving on to the topic of linear groups.
- There will be a Group Theory revision lecture on Tuesday, 24 April 2018, 12.00-13.00 in room 2.41 in the Bancroft Building (near the library).
- The rubric for the Group Theory exam will be: "You should attempt all questions. Marks awarded are shown next to the questions.". Calculators will not be permitted in the exam.
- 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 will be important on 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.