Math 3GR3 - Abstract Algebra
(Fall 2019)

This course is an introduction to groups and rings with an emphasis on examples.

News Homework SAGE Quizzes Handouts Grading Scheme Schedule Policies

Course Information

Instructor: Adam Van Tuyl

Office: Hamilton Hall 419
Office Hours: M 2:30-3:20PM, Th 10:30-11:20AM
Email: vantuyl@math.mcmaster.ca

TA: Lindsay White

Office Hours: Tuesday 10:30-11:20 (4th floor hall near Hamilton Hall 407)
Email: whitela3@mcmaster.ca

Place and Time:

Class: Monday, Wednesday, Thursday 1:30-2:20 in Hamilton Hall 109
Tutorial: Tuesday 11:30-12:20 in Hamilton Hall 109

Textbook:

• Abstract Algebra: Theory and Applications (2018 Edition)
  by Tom Judson. This is a FREE online textbook. Hard copies can also be found in the bookstore. Here is an online version. Note that we will be using the 2018 edition.
• Mathematical Writing (Optional)
  by Franco Vivaldi. This book is optional, and will help you write proofs. An electronic copy is FREE through the McMaster library.

News (Last Updated: December 18, 2019)

See here for the

• Lecture Notes

Also, Linsday our TA has made her notes available via a dropbox folder:

• Tutorial Notes

Below is a summary of what we did in class, plus any relevant news and/or information.

• Dec 18, 2019 The course is now officially over! Your grades have been submitted. I hope you enjoyed the class, and have a good Christmas break! -Adam VT
• Dec 4, 2019 I summarized the course, discussed future directions, and we did a number of true-false questions.
• Dec 2, 2019 We did some review questions on rings.
• Nov 28, 2019 I proved the division algorithm for polynomials.
• Nov 27, 2019 Today we were introduced to polynomial rings (Section 17.1).
• Nov 25, 2019 I introduced prime and maximal ideals (see Section 16.4).
• Nov 21, 2019 I spent the lecture giving more examples of ideals and the First Isomorphism Theorem.
• Nov 20, 2019 I proved the First Isomorphism Theorem for rings.
• Nov 18, 2019 I returned the second midterm today. We continued our discussion of Chapter 16 with a discussion on ring homomorphisms and group homomorphisms.
• Nov 14, 2019 Today was the second midterm.
• Nov 13, 2019 Today we were introduced to ideals. We saw some examples, and were introduced to the ideal test. See Section 16.2 for more.
• Nov 11, 2019 We looked at some of the properties of integral domains and fields.
• Nov 7, 2019 We started learning about rings (see Chapter 16.1).
• Nov 6, 2019 We finished our discussion of Chapter 11; in particular, I discused the second and third isomorphism theorems.
• Nov 4, 2019 I gave a proof for the first isomorphism theorem.
• Oct 31, 2019 We looked at Section 11.1 on homomorphisms. We looked at some of their properties, and we saw some examples. We also were introduced to the kernel.
• Oct 30, 2019 We finished our discussion of normal subgroups and we also looked at simple groups (Section 10.2).
• Oct 28, 2019 We looked at Section 10.1 on normal subgroups.
• Oct 24, 2019 We finished Chapter 9 with a discussion on internal direct products (Section 9.2). I also started on Chapter 10.1.
• Oct 23, 2019 We covered the rest of Section 9.1 by giving a proof of Cayley's Theorem. I also discussed external direct products.
• Oct 21, 2019 Welcome back from the break. The midterm was handed back. We also started Chapter 9 on isomorphisms.
• Oct 17, 2019 The first midterm has been graded. It will be handed back after the break.
• Oct 10, 2019 Today was the first midterm. The midterm should be graded by the end of the break! See you in a week and half! Please note -- for the homework, your mark is out of 20, not 40. So don't panic when you see your mark from Crowdmark.
• Oct. 8, 2019 We finished Chapter 6 by discussing Fermat's Little Theorem, Euler's Theorem, and some consequences of Lagrange's Theorem.
• Oct. 7, 2019 We looked at Section 6.1 on cosets and Lagrange's Theorem.
• Oct. 3, 2019 We looked at Section 5.2 on the dihedral group.
• Oct. 2, 2019 We finished Section 5.1 by showing the every permutation is either the product of an even number of transpositions or odd number of transpositions.
• Sept. 30, 2019 We started Chapter 5 b describing perumations and cycle notation.
• Sept. 26, 2019 We finished Chapter 4 by discussing the group of n-th roots of unity in the complex plane.
• Sept. 25, 2019 We started Chapter 4 on cyclic groups.
• Sept. 23, 2019 I presented the subgroup test.
• Sept. 19, 2019 We looked at some properties all groups must share. We also started our discussion of subgroups (see Sections 3.2 and 3.3).
• Sept. 18, 2019 Today I formally defined groups and gave many examples of groups.
• Sept. 16, 2019 Today I gave some examples of groups; next class will contain the formal defintion. See Section 3.1.
• Sept. 12, 2019 We proved the Fundamental Theorem of Arithmetic. I also gave some tips on writing proofs. We have now finished Chapter 2. Quiz 1 has been graded and handed back.
• Sept. 11, 2019 We went over the Euclidean Algorithm.
• Sept. 9, 2019 I reviewed induction (Section 2.1) and I introduced the Division Algorithm.
• Sept. 6, 2019 I made some minor updates to the web page, including my office hours and the TA information.
• Sept. 5, 2019 I introduced equivalence relations (see Sections 1.1 and 1.2 of the text).
• Sept. 3, 2019 The first day of class. I did a review of sets.
• Aug. 19, 2019 Finalized a number of parts of the website, and added the course outline.
• Aug. 15, 2019 More material was added to the website.
• July 12, 2019 I started to set up the website for Fall 2019.

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Homework

There will be five homework assignments. Assignments must conform to the guidelines in the course outline. Your homework will be graded as follows:

1. For every assignment, 3 or 4 questions will graded in detail (e.g., you are required to write complete mathematical proofs). These questions will be graded out of 5 pts using the rubric described in the course handout.
2. The remaining questions will be graded for completion (1pt each)

Assignments are posted below. Assignments will be submitted via Crowdmark. You will receive an email to your McMaster account that you will use to upload your assignment. All assignments due by 11:59PM on the due date.

• Assignment 0 (Due: Sept. 13)
• Assignment 1 (Due: Sept. 20)
• Assignment 2 (Due: Oct. 4)
• Assignment 3 (Due: Oct. 25)
• Assignment 4 (Due: Nov. 8)
• Assignment 5 (Due: Nov. 29)

For more information on writing proofs, the following notes may help:

• Guide to writing proofs
  A handout on writing proofs I wrote for algebra in 2010.
• Some remarks on writing mathematical proofs
  A nice short guide to writing mathematics by John M. Lee.

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SAGE

As part of this course, the assignments will make use of SAGE, an open-source mathematics program. For the assignments, I will ask you to work through the relevant SAGE tutorials found the book's webpage. You can download your own version of SAGE, or you can try to use CoCalc. Alternatively, you can use the following shell to test your code. This shell allows you to do small calculations.

The following links may help:

• For tutorials, more information on SAGE, and even a free book on SAGE and mathematics, see George Bard's SAGE website
• SAGE's documentation for group theory

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Quizzes

There will be five quizzes. Each quiz will be about 20 minutes, and will be held during tutorial. The dates of the quizzes are

• Quiz 1 - September 10 (Chapter 1 of Vivaldi's book, Chapter 1 of Judson, my notes on proof writing)
• Quiz 2 - September 24 (Chapter 3 of Judson)
• Quiz 3 - October 22 (Chatper 5 and 6 of Judson)
• Quiz 4 - November 5 (Chapter 10 of Judson)
• Quiz 5 - November 19 November 26 (Chapter 16 of Judson)

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Handouts

All class handouts are available as PDF files.

Course Information
Course handout from first day of class

Midterm 1 Review Sheet
Handout describing first midterm.

Midterm 2 Review Sheet
Handout describing second midterm.

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Grading Scheme

I will calculate your mark using two different weightings. Your final mark will be the higher of the two weights.

Weighting 1
20% = Assignments (5 x 4%)
10% = Quizzes (5 x 2%)
30% = Midterms (2 x 15%)
40% = Final Exam

Weighting 2
20% = Assignments (5 x 4%)
10% = Quizzes (5 x 2%)
15% = Maximum of Midterm 1 and 2
55% = Final Exam

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Important Dates

Sept. 3, 2019
First semester classes begin

Oct. 10, 2019
Midterm 1

Oct. 14-18, 2019
Fall break (no classes)

Nov. 8, 2019
Last day for cancelling courses without failure by default

Nov. 14, 2019
Midterm 2

Dec. 4, 2019
First semester classes end

Dec. 6-19, 2019
Final Exams

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Class Polices

1. Policy on Academic Ethics. You are expected to exhibit honesty and use ethical behaviour in all aspects of the learning process. Academic credentials you earn are rooted in principles of honesty and academic integrity.

Academic dishonesty is to knowingly act or fail to act in a way that results or could result in unearned academic credit or advantage. This behaviour can result in serious consequences, e.g. the grade of zero on an assignment, loss of credit with a notation on the transcript (notation reads: Grade of F assigned for academic dishonesty), and/or suspension or expulsion from the university.

It is your responsibility to understand what constitutes academic dishonesty. For information on the various types of academic dishonesty please refer to the Academic Integrity Policy, located at: http://www.mcmaster.ca/academicintegrity/

The following illustrates only three forms of academic dishonesty:

• plagiarism, e.g. the submission of work that is not one's own or for which other credit has been obtained.
• improper collaboration in group work,
• copying or using unauthorized aids in tests and examinations.

2. Academic Accommodation of Students with Disabilities. Students with disabilities who require academic accommodation must contact Student Accessibility Services (SAS) to make arrangements with a Program Coordinator. Student Accessibility Services can be contacted by phone 905-525-9140 ext. 28652 or e-mail sas@mcmaster.ca. For further information, consult McMaster University's Academic Accommodation of Students with Disabilities policy.

3. Requests for Relief for Missed Academic Term Work. If you have missed work, it is your responsibility to take action. If you are absent from the university for medical and non-medical (personal) situations, lasting fewer than 3 days, you may report your absence, once per term, without documentation, using the McMaster Student Absence Form (MSAF). See Requests for Relief for Missed Academic Term Work

Absences for a longer duration or for other reasons must be reported to your Faculty/Program office, with documentation, and relief from term work may not necessarily be granted. In Math 3GR3, the percentages of the missed work will be transferred to the final examination. Please note that the MSAF may not be used for term work worth 25% or more, nor can it be used for the final examination.

4. Academic Accommodation for Religious, Indigenous or Spiritual Observances (RISO). Students requiring academic accommodation based on religious, indigenous or spiritual observances should follow the procedures set out in the RISO policy. Students requiring a RISO accommodation should submit their request to their Faculty Office normally within 10 working days of the beginning of term in which they anticipate a need for accommodation or to the Registrar's Office prior to their examinations. Students should also contact their instructors as soon as possible to make alternative arrangements for classes, assignments, and tests.

5. Important Message. The instructor and university reserve the right to modify elements of the course during the term. The University reserves the right to change the dates and deadlines for any or all courses in extreme circumstances (e.g., severe weather, labour disruptions, etc.). Changes will be communicated through regular McMaster communication channels, such as McMaster Daily News, A2L and/or McMaster email. If either type of modification becomes necessary, reasonable notice and communication with the students will be given with explanation and the opportunity to comment on changes. It is the responsibility of the student to check their McMaster email and course websites weekly during the term and to note any changes.

6. On-line Statement for Courses Requiring Online Access or Work. In this course we will be using SAGE (which can be accessed online) and Crowdmark. Students should be aware that, when they access the electronic components of this course, private information such as first and last names, user names for the McMaster e-mail accounts, and program affiliation may become apparent to all other students in the same course. The available information is dependent on the technology used. Continuation in this course will be deemed consent to this disclosure. If you have any questions or concerns about such disclosure please discuss this with the course instructor.

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