## MTH714U/MTHM024 - Group Theory - 2016/17

This module provides an introduction to advanced group theory. The aim is to explore the theory of finite groups by studying important examples in detail, such as simple groups. In particular, the projective special linear groups over small fields provide a rich vein of interesting cases on which to hang the general theory.

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## Assessment Overview

• 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 2. Exceptionally in week 2, the tutorial will take place in Scape 1.02 (instead of M203).
• 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 will be a lecture and a tutorial in week 7.
• ### 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

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.

• ### Week 4 and Week 5

Theorem 4.4 part (b) and its proof in the notes below are NOT examinable.

• ### Week 6 and Week 7

Please note that there will be both a Group Theory lecture and a tutorial in week 7 (at the usual times in the usual place). I will also be holding my Office Hour in week 7 (at the usual time and place).

Section 6.3 (Digression: Minimal normal subgroups) in the notes below is NOT examinable.

Iwasawa's Lemma (Theorem 6.9 in the notes below) will be stated and proved at the beginning of week 8, before starting on Notes Section 7.

• ### Week 8 and Week 9

Iwasawa's Lemma (Theorem 6.9 in Notes Section 6) will be stated and proved at the beginning of week 8, before starting on Notes Section 7.

For the purposes of the final examination, you are only responsible to know one proof of the simplicity of Afor n greater than or equal to 5, and you are only responsible for one proof that S6 has an automorphism that is not an inner automorphism.

• ### Week 10, Week 11 and Week 12

There will be no Group Theory tutorial in week 11 (that is, no tutorial on December 7th).

The proof of Theorem 8.2 in the notes below is NOT examinable.

• ### Revision and Final Exam Information

• There will be one Group Theory revision lecture during Revision Week. It will take place on Monday 24 April 2017, 15.00-16.00, in room W316 in the Queens' Building.
• 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 during an Office Hour.