# MATH2968 Algebra (advanced)

## General Information

This page contains information on the Intermediate Unit of Study MATH2968 Algebra (advanced).

This unit is offered in Semester 2.

Lecturer(s): Stephan Tillmann

For further information on Intermediate Mathematics and Statistics, refer to the Intermediate Handbook. In particular, see the MATH2968 handbook entry for further information relating to MATH2968.

You may also view the description of MATH2968 in the central units of study database.

- Credit point value: 6CP.
- Classes per week: Three lectures, one tutorial and one practice class.

Email enquiries about MATH2968 may be sent to MATH2968@maths.usyd.edu.au.

**Students: Please give your name and SID when emailing us.
Anonymous emails will not be replied to.**

## MATH2968 Information in 2016

## News

The final exam is on Saturday 12 November. Check your personal exam timetables for venues and other details!

**Consultations before the exam:** Tuesday 8 November and Thursday 10 November from 10-12 in Carslaw 710

Here is the handout that was distributed in class: Excerpt from Brieskorn (1985)

## Lecturer

Name:Stephan Tillmann

Room:Carslaw 710

Phone:9351 2005

Email:stephan.tillmann@sydney.edu.au

Consultation:Tuesdays 10-11 in Carslaw 710

## Help and advice

The best way of approaching this subject is to start immediately and to work steadily each week. In lectures, I will motivate and explain key concepts, give examples, and go through complete proofs or sketch the main ideas of proofs. You are expected to read the recommended sections or examples in the notes, which are posted below. Some concepts will not be defined rigorously in class as they are best processed at your own time, either alone or in a study group. Doing the recommended reading will also raise the long term retention of the concepts and techniques. Then begin to work on the suggested problems associated with the lecture. There will be about 2-4 problems to work on associated with each lecture. Work on the problems and get them done — with or without the help from friends, tutors or me. Only consult the solutions after you have found your own solution or sought some help; the main purpose of many of the problems is: trying to solve them will help you understand the course material better and hence help you solve other problems more easily. This is not achieved by reading and trying to understand someone else's solution.

The first source of help is the person next to you in lectures or tutorials. They are doing the same problems as you and are having similar but perhaps not the same difficulties as you do. The second is your tutor. Don't miss a tutorial and have a list of problems on which you need help. I am also more than happy to answer your questions; my office hours are given above.

Please let me know if you have any problems or comments on the subject; both with regard to the contents and how it is run.

## Notes

The current draft of the lecture notes is available below. Please let me know if you find any misprints or errors. I will maintain a change history below, which you should consult from time to time.

Many proofs are not given in the text, but rather in a separate section at the end of each chapter. This is meant to entice you to prove a result before reading its proof or seeing it in class.

The notes contain exercises for the tutorials. I will keep you up-to-date as to which problems you should work on. You can find hints or solutions to most problems in the notes. However, I strongly encourage you to only look at them after you have spent some time trying to solve them (with or without help from friends, or hints from your tutor or me). It is more important to discover your own solution than to understand someone else's solution!

Change history:

- Groups and group actions (Version 12.07.2016)
- Excerpt from Brieskorn (1985)
Below is Daniel Daners' script for MATH2961, which will be a useful reference when we expand upon the linear algebra from Semester 1.

- None.

- Linear Algebra (Lecture Notes for MATH2961)

## Practice classes

Sheets with problems for the practice classes will be posted below, and you are expected to

bring a printed copy to class.Solutions will be posted separately. If you miss a practice class,do not look at the solutions before attempting the problems!If the questions require props (such as tetrahedra), they will be provided in class.You are not expected to write down formal proofs during the practice class! I would like you to find the key ideas through discussion, and to write down the formal arguments afterwards. Collaboration on these problems is strongly encouraged! Some sheets will be longer, and finished in subsequent tutorials.

More difficult questions will be indicated * or **, where the difficulty may be conceptual or technical.

- (Thu Week 1) Practice Class 1 and here are the solutions.
- (Thu Week 2) Practice Class 2 and here are the solutions.
- (Thu Week 3) Practice Class 3 and here are the solutions.
- (Thu Week 4) Practice Class 4 and here are the solutions.
- (Thu Week 5) Practice Class 5 and here are the solutions.
- (Thu Week 6 Quiz) Here is a Practice Quiz and here are the solutions. Here is the actual Quiz and here are the solutions.
- (Mon Week 7) Practice Class 6 and here are the solutions.
- (Thu Week 7) Practice Class 7 and here are the solutions.
- (Thu Week 8) Practice Class 8a with solutions; and Practice Class 8b with solutions.
- (Thu Week 9) Practice Class 9 and here are the solutions.
- (Thu Week 10) Practice Class 10 and here are the solutions.
- (Thu Week 11) Practice Class 11 and here are the solutions.
- (Thu Week 12 Quiz) Here is a Practice Quiz and here are the solutions. Here is the actual Quiz and here are the solutions.
- (Wed Week 13) Practice Class 12 and here are the solutions.
- (Thu Week 13) Practice Class 13 and here are the solutions.

## Class times

- Lectures will be held on
**Monday, Tuesday, Wednesday at 9am in Carslaw 175**, except for the public holiday on Monday in Week 10 (3 October). - Practice classes/quizzes will usually be held on
**Thursday at 9am in Carslaw 175 in Weeks 1-13**. Sheets with problems for the practice classes are posted below. Solutions will be posted at the beginning of each week. The quizzes will be held in**Weeks 6 and 12**. - Tutorials will be held on
**Monday in Weeks 2-9 and 11-13**, at the time and place indicated on your timetable; Carslaw 452 at 13:00 and Carslaw 451 at 15:00. Problems will be taken from the notes or from the practice sheets, and additional warm-up problems may be handed out at the start of each class. There will be**no tutorial on Monday 3 October (Week 10)**due to the public holiday.

## Unit outline

This unit introduces the theory of groups, which is the basis of modern algebra. Groups provide a unifying framework for topics such as geometric symmetry, permutations, matrix arithmetic and more. Group theory is vital in many areas of mathematics (algebra, number theory, geometry, harmonic analysis, representation theory, geometric mechanics) and in areas of science such as theoretical physics and quantum chemistry.

MATH2968 is the keystone of the sequence of algebra units, marking the point at which powerful abstract theory enters and takes the subject to a more sophisticated level. It lays the foundations for the applications of algebra in other areas of mathematics and science, where groups often arise as algebraic groups, arithmetic groups, topological groups or Lie groups and not only an understanding of the abstract structure of the group is important, but also their linear representations.

The rough plan of week-by-week topics is as follows (this will change, depending on progress and inspiration):

Week 1:Definition and examples of groups and subgroups

Week 2:Cosets and Lagrange's Theorem

Week 3:Homomorphisms, normal subgroups

Week 4:Quotient groups, the isomorphism theorems

Week 5:Some key examples

Week 6:Centre, commutators, automorphisms, characteristic subgroups

Week 7:Direct and semi-direct products, abelian groups, group actions

Week 8:Sylow's theorems and applications

Week 9:Sylow's theorems and applications

(Midsemester break)

Week 10:Minimal polynomials and invariant subspaces

Week 11:Nilpotent linear transformations and Jordan Normal Form

Week 12:Action of GL(2,R) on Mat(2,R); matrix exponentionals

Week 13:Action of elements of GL(2,R) on the plane; finite subgroups of GL(2,R)

## Learning outcomes

By the end of the unit, you should be able to

- know a range of examples of groups, and their varying properties;
- understand the axioms and basic definitions of group theory;
- carry out calculations in specific groups of permutations and matrices, using general results;
- conceive and construct proofs of general results concerning elements, subgroups, homomorphisms, conjugacy, etc.;
- understand the relationship between homomorphisms and quotients, expressed in the Isomorphism Theorems;
- state and use elementary results in group theory such as Lagrange's Theorem, Cauchy's Theorems, the Sylow Theorems, and the Isomorphism Theorems, with a fair understanding of how these results are proved;
- state and use the classification of finite abelian groups, and appreciate the subtlety of the classification problem for non-abelian groups;
- state and use the Jordan Canonical Form Theorem and the Cayley-Hamilton Theorem, with some understanding of how these results are proved.

## Assessment

Your mark for MATH2968 is calculated as follows.

**Two quizzes**, each 45 minutes long and worth 15%, will be held in the 9am Thursday class:- Quiz 1 on Thursday 1 September (Week 6), will draw on material in Weeks 1-5;
- Quiz 2, on Thursday 20 October (Week 12), will draw on material in Weeks 6-11;

**Two assignments**, each worth 5%, to be handed in via Turnitin. The assignments will give practice in investigating examples and constructing proofs, and feedback should help with your mathematical writing skills and exam preparation.- Assignment 1 was due on Friday 12 August (Week 3) and here are the solutions.
- Assignment 2 was due on Thursday 22 September (Week 9) and here are the solutions.

**Final exam**, 2 hours long and worth 60%, during examination period. No notes, books, or calculators are allowed (no questions will require calculators).- Your final mark will be the maximum of (30% quizzes, 10% assignments, 60% final exam) and (15% quiz 1, 10% assignments, 75% final exam) and (15% quiz 2, 10% assignments, 75% final exam) and (10% assignments, 90% final exam) and (100% final exam).

## Reference books

The content of the unit is defined by the lectures and the above notes. Introductory group theory is covered in a multitude of sources, which you may find it helpful to consult for extra problems or alternative explanations. Most online mathematical encyclopedias contain material relevant to this unit. Be aware that conventions and notation may differ slightly from those of the lectures. The following books have been placed on reserve in the library:

*Abstract Algebra*by David S. Dummit and Richard M. Foote*Concrete Abstract Algebra*by Niels Lauritzen*An Introduction to the Theory of Groups*by Joseph J. Rotman*Indra's pearls. The vision of Felix Klein.*by David Mumford, Caroline Series and David Wright

## Past exams

Past exam papers, as well as partial solutions and a discussion of relevance of the questions to this year's syllabus, will be posted closer to the end of semester.

**Warning:** With the exception of 2013, past exams were set by different lecturers, who had different conventions and covered somewhat different material. The format of the exam in 2013 can be used as a guide for this year's exam.

The exams in 2007, 2009, and 2010 had more questions than were necessary to obtain full marks, allowing students to choose which questions to answer; this year's exam is the traditional style. When looking at the papers from 2007, 2009, and 2010, please refer to the comments made on the first page of the solutions to the 2010 paper.

I have also indicated, which questions are not relevant for this year's exam.

**Advice:** First study and revise the course material. Then attempt the questions, and don't give up too quickly! Meet up with friends, and compare your strategies (or solutions), and try to find a solution (or different solutions) together. The overall aim in this is not to know how to solve a particular problem, but to gain a better understanding of the course material, so that you can solve different problems more easily. If you still have problems solving the questions, ask me for hints. Only look at the solutions as a last resort!

2013 exam and solutions (Ignore questions 6 and 10.)

2011 exam and solutions

2010 exam and solutions with comments (Ignore questions 6 and 7(c).)

2009 exam and solutions (Ignore questions 5, 6 and 9(c).)

2008 exam and solutions (Ignore question 5.)

2007 exam (Ignore questions 6, 7 and 8(c).)

2006 exam (Ignore questions 1, 2, 5 and 6.)

2005 exam (Ignore questions 1, 2, 9, 10. For question 3(d), you would be given the definition of "transitive.")

## Timetable

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