Math 4GR3 - Groups and Rings
(Winter 2024)
Further topics in group theory and ring theory. Topics include: direct
products, Fundamental Theorem of Finite Abelian Groups, Sylow Theorems,
free groups, group presentations, fields and integral domains, special
integral domains (Euclidean, principal ideal, unique factorization),
fields of fractions of integral domains, polynomial rings in many variables,
and additional topics at the discretion of the instructor (e.g., Groebner
bases, algebraic coding theory).
Three lectures; one term
Prerequisite(s): MATH 3E03 or 3GR3
| |||||||||||||
Course Information
Instructor: Adam Van Tuyl
Office: Hamilton Hall 419
Office Hours: Monday and Thursdays 2:30-3:30
Email: vantuyl@math.mcmaster.ca
Place and Time:
Class: Consult A2L or MOSAIC for class location and times
Textbook:
-
Abstract Algebra: Theory and Applications
(2022 Edition)
by Tom Judson. This is a FREE online textbook. Hard copies can also be purchsed if you desire. Here is a link to purchase the textbook. Here is an online version. Note that we will be using the 2022 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: April 11, 2024)
Below is a summary of what we did in class, plus any relevant news and/or information.
- April 11, 2024 We had an exam review in class on Wednesday (our last class). I will have office on April 19, the day before the final exam. I will post hours later.
- April 4, 2024 Yesterday was the poster session - I hope you enjoyed it! Just a reminder that there is no class on Monday because of the eclipse.
- April 3, 2024 We finished Section 21.3 on the the compass and straightedge constructions. In particular, we proved that it was impossible to square a circle, among other things. Note that Homework 5, question 3 is missing a word. It should say ``If r and s are nonzero integers,...''.
- April 1, 2024 We start our final topic for the course: compass and straightedge constructions. Make sure you send me a copy of your poster.
- March 28, 2024 We discussed the algebraic closure of a field and splitting fields. Homework 4 was also returned.
- March 27, 2024 We looked at the connection between linear algebra and field extensions.
- March 25, 2024 We continued our discussion of 21.1 by looking at algebraic extensions.
- March 21, 2024 We jumped to Chapter 21 to start our discussion of fields extensions. Homework 4 is due on Friday, and don't forget to work on your poster.
- March 20, 2024 We finished Chapter 18 with a discussion of the properties of Euclidean domains.
- March 18, 2024 Today we looked at the properties of PIDs.
- March 14, 2024 Happy Pi Day! We looked at UFDs today.
- March 13, 2024 We started Chapter 18 on integral domains. In today's class we saw how to construct a field from any domain.
- March 11, 2024 We finished Chapter 17 by showing that all ideals in F[x] are principal. I posted the solutions to Homework 3 on A2L.
- March 7, 2024 We discussed irreducible polynomials (Section 17.3).
- March 6, 2024 Class was cancelled today.
- March 4, 2024 We proved the division algorithm for polynomial rings (Section 17.2).
- February 29, 2024 Today we looked at polynomial rings and the division algorithm for polynomial rings (see Section 17.1).
- February 28, 2024 We completed our review of rings (Chapter 16).
- February 26, 2024 Welcome back from reading break! In today's class we did a review of rings and ideals.
- February 15, 2024 We had the midterm. The midterm has been graded and has been returned via Crowdmark. We will start reviewing rings after the break.
- February 14, 2024 We reviewed some material on groups.
- February 13, 2024 Because our midterm is on Thursday, I will move my hours to Wednesday from 2:30-3:30 for this week only.
- February 12, 2024 We looked at Chapter 12 and some examples of groups that arise in linear algebra. I also posted the solutions to homework 2 on A2L.
- February 8, 2024 Today we looked at some applications of the Sylow Theorm. The midterm next week Thursday will cover up to today's material. I have also added homework 4 and 5 to the web.
- February 7, 2024 In today's class we worked through the proof of the Third Sylow Theorem.
- February 5, 2024 Today we worked through the proof of the Section Sylow Theorem. I also added an alternative version of the theorem to A2L.
- February 1, 2024 I gave the proof of the First Sylow Theorem (Section 15.1). For another proof see this proof, which does not use the class equation.
- January 31, 2024 We looked at the last section of Chapter 14 on Burnside's equation.
- January 29, 2024 We finished our discussion of group actions and the class equation. As an in-class project, we computed the class equation of the quaterions.
- January 25, 2024 Today I introduced the class equation of a group (see Sections 14.1 and 14.2).
- January 24, 2024 We started our discussion of group actions. See Section 14.1.
- January 22, 2024 Today I gave a proof of the Jordan-Holder Theorem (Section 13.2)
- January 18, 2024 We start a discusion of composition series and the Jordan-Holder Theorem (see Section 13.2).
- January 17, 2024 I proved the main two technical lemmas that were not proved in yestrday's class. For one of my proofs, I used a proof different than the one in Judson's book. This finishes the material of Section 13.1.
- January 15, 2024 I explained the main steps of the proof of the Fundamental Theorem of Finite Abelian Groups (including introducing p-groups). We proved everything except two technical lemmas which we will prove on Wednesday.
- January 11, 2024 I did a quick review on homomorphism and isomorphisms of groups. I then introduced the statement of the Fundamental Theorem of Finite Abelian Groups. My office hours were added to the webpage. Make sure you join the Discord group (I'll send a link in an email).
- January 10, 2024 I did a review of equivalence relations, and then reviewed normals subgroups and quotient groups.
- January 8, 2024 Our first day of class. I did a review of basic group theory, including a proof of Lagrange's Theorem. Look at Chapters 1-3.
- December 11, 2024 I started to set up the website for Winter 2024.
| Math 4GR3 Schedule | ||
|---|---|---|
| Week 1 (Jan 8-12) | ||
| Homework: start Assignment 1 (due Jan. 26) | ||
| Lecture | Topic | Reference |
| Lecture 1 | Group Theory Review I: Lagrange's Theorem |
3.1, 3.2, 3.3, 4.1, 6.1, 6.2 |
| Lecture 2 | Group Theory Review II: Equivalence Relations and Quotient Groups |
1.2, 10.1 |
| Lecture 3 | Fundamental Theorem of Finite Abelian Groups I | 13.1 |
| Week 2 (Jan 15-19) | ||
| Homework: work on Assignment 1 (Due Jan 26) | ||
| Lecture 4 | Fundamental Theorem of Finite Abelian Groups II: Direct Sums |
9.2, 13.1 |
| Lecture 5 | Fundamental Theorem of Finite Abelian Groups III: Technical Lemmas |
13.1 |
| Lecture 6 | Composition Series | 13.2 |
| Week 3 (Jan 22-26) | ||
| Homework: Finish Assignment 1 (Due Jan 26) | ||
| Lecture 7 | Jordan-Holder Theorem | 13.2 |
| Lecture 8 | Group Actions | 14.1 |
| Lecture 9 | Group Actions and the class equation |
14.1, 14.2 |
| Week 4 (Jan 29-Feb 2) | ||
| Homework:
Start Assignment 2 (Due Feb. 9) Poster topic due (Due Feb. 9) |
||
| Lecture 10 | Class equations: applications and a worked out example |
14.2 |
| Lecture 11 | Counting and Burnside's Equation | 14.3 |
| Lecture 12 | First Sylow Theorem | 15.1 |
| Week 5 (Feb 5-9) | ||
| Homework: Submit
Assignment 2 (Due Feb. 9) Poster topic due (Due Feb. 9) Start studying for Midterm (Feb. 15) Here is a Review Sheet |
||
| Lecture 13 | Second Sylow Theorem | 15.1 |
| Lecture 14 | Third Sylow Theorem | 15.1 |
| Lecture 15 | Applications of the Sylow Theorems | 15.2 |
| Week 6 (Feb 12-16) | ||
| Homework:
Study for the Midterm (Feb. 15) Here is a Review Sheet |
||
| Lecture 16 | Group Theory and Linear Algebra | 12.1 |
| Lecture 17 | Group Theory Review | |
| Lecture 18 |
Midterm | |
| Week 7 (Feb 19 - 23) | ||
| Reading Week (No Classes) | ||
| Week 8 (Feb 26 -March 1) | ||
| Homework: Work on Assignment 3 (Due March 8) | ||
| Lecture 19 | Review of Rings I | 16.1-16.2 |
| Lecture 20 | Review of Rings II | 16.3-16.4 |
| Lecture 21 | Polynomial Rings | 17.1 |
| Week 9 (March 4-8) | ||
| Homework: Submit Assignment 3 (Due March 8) | ||
| Lecture 22 | Division Algorithm in F[x] | 17.2 |
| Class Cancelled | ||
| Lecture 23 | Irreducible polynomials in F[x] | 17.3 |
| Week 10 (March 11-15) | ||
| Homework: Start on Assignment 4 (Due March 22) | ||
| Lecture 24 |
Ideals in F[x] | 17.3 |
| Lecture 25 | Fields from Domains | 18.1 |
| Lecture 26 | UFDs | 18.2 |
| Week 11 (March 18-22) | ||
| Homework: Submit Assignment 4 (Due March 22) | ||
| Lecture 27 | PIDs | 18.2 |
| Lecture 28 | Euclidean Domains | 18.2 |
| Lecture 29 |
Field Extensions | 21.1 |
| Week 12 (March 25-29) | ||
| Homework: Start on
Assignment 5 (Due April 8) Finish your poster (make sure to leave time to have it printed) |
||
| Lecture 30 | Algebraic Extensions | 21.1 |
| Lecture 31 | Algebraic Extensions and Linear Algebra | 21.1 |
| Lecture 32 | Algebraic Closure and Splitting Fields | 21.1-21.2 |
| Week 13 (April 1-5) | ||
| Homework:
Work on
Assignment 5 (Due April 8) Poster presentations this week |
||
| Lecture 33 | Geometric Constructions I | 21.3 |
| Lecture 34 | Geometric Constructions II | 21.3 |
| Lecture 35 | Poster Presentations | |
| Week 14 (April 8-10) | ||
|
Homework: Work on
Assignment 5 (Due April 8) Start studying for final exam (April 20) | ||
| Lecture 36 | Review Bonus: Solor Eclipse! |
Lecture 37 | Buffer/Review |
| Final Exam (April 20) | ||
|
Study for
Final Exam (April 20) | ||
Return to TOP
Homework
There will be five homework assignments. Assignments must conform to the guidelines in the course outline. Your homework will be graded as follows:
- 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.
- The remaining questions will be graded for completion (1pt each)
- Assignment 1 (Due: Jan. 26)
- Assignment 2 (Due: Feb. 9)
- Assignment 3 (Due: March 8)
- Assignment 4 (Due: March 22)
- Assignment 5 (Due: April 8)
-
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.
Return to TOP
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:
- The Sage Math Quick Reference for Abstract Algebra, one page summary sheet of the important commands for abstract algebra.
- 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
Return to TOP
Project: Creating a Poster
As part of this course, in a group of 2 to 4 students, you will create a poster on a topic in abstract algebra. At the end of the semester, your group will also provide a short presentation about your poster.
Please use the following
- Poster Project Form (link to be added) to submit your poster topic and group (before Feb 2, 2024).
- Project Information (summary of the assignment)
- Grading Rubric (how the posters will be graded)
Possible topic ideas (with some links):
- A historical overview of a particular theorem or mathematician who did work in abstract algebra. See Biographies of Mathematicians with over 1000 mathematicians listed.
- Present an open problem in abstract algebra. See Wiki Page of unsolved math problems to get started.
- Present a topic in abstract algebra not covered in classes. Possible topics include: Algebraic Coding Theory, Cryptography, Cyclotomic Extensions, Groebner Bases, Hall's Theorem, Monster Groups, Noetherian and Artinian Rings, Representation Theorem, Rubik's Cube, Semi-Direct Products, Tiling. (This is not a complete list -- you are encouraged to find your own topic! Another place to look is the Wiki List of Abstract Algebra Topics.)
- How to Prepare a Poster Information provided by SIAM (Society of Industrial and Applied Mathematics) on how to prepare a poster for a conference.
- Poster Resources A website at Bates College; it contains some information about PowerPoint.
- Examples of Posters A webpage that contains a number of good examples of a mathematical posters.
- Introduction to LaTeX If you want to install LaTeX on your own computer, this is probably the place to start.
- Overleaf If you don't want to install LaTeX on your own computer, you can use a cloudversion. You can get an account for free.
- LaTeX in 30 minutes Overleaf's half-hour tutorial to LaTeX.
- Posters in LaTeX Overleaf's introduction to posters in LaTeX.
Handouts
All class handouts are available as PDF files.
Course Information
Course handout from first day of class
You can also download a DOCX version
of the file.
Midterm Review Sheet
Handout describing the midterm.
Exam Review Sheet
Handout describing final exam.
Return to TOP
Grading Scheme
I will calculate your mark using two different weightings. Your final mark will be the higher of the two weights.
Weighting 1
25% = Assignments (5 x 5%)
20% = Midterms (1 x 20%)
15% = Poster (1 x 15%)
40% = Final Exam
Weighting 2
25% = Assignments (5 x 5%)
0% = Midterm (1 x 0%)
15% = Poster (1 x 15%)
60% = Final Exam
Return to TOP
Important Dates
Jan 8, 2024
Second semester classes begin
Feb. 15, 2024
Midterm
Feb. 19-25, 2024
Winter break (no classes)
March 15, 2024
Last day for cancelling courses without
failure by default
March 29-30, 2024
Easter break (no classes)
April 3-4, 2024
Poster presentations
April 10, 2024
First semester classes end
April 12-25, 2024
Final Exams
Return to TOP