Simon Fraser University
Engaging the World

Course Outlines

Fall 2025 - MATH 817 G100

Groups and Rings (4)

Class Number: 6024

Delivery Method: In Person

Overview

  • Course Times + Location:

    Sep 3 – Dec 2, 2025: Tue, Thu, 10:30 a.m.–12:20 p.m.
    Burnaby

Description

CALENDAR DESCRIPTION:

A survey of graduate group and/or ring theory. Possible topics include generators and relations, composition series, Sylow theory, permutation groups, abelian groups, p-groups, nilpotent and solvable groups, aspects of simple groups, representation theory, group algebras, chain conditions, Jacobson radical, Chevalley-Jacobson density theorem, Wedderburn-Artin theorems.

COURSE DETAILS:

This course presupposes basic familiarity with groups and rings at the level of e.g. MATH 340 and 341. It will provide an introduction to more advanced topics in abstract algebra, with an emphasis on the study of how groups and rings can transform other objects. In the case of groups, the natural notion is that of a group representation, which is simply a vector space endowed with the action of a group. We will study representations of finite groups, where we will see how they can be completely classified (at least in principle). In the case of rings, the natural notion is that of a module, which is a generaliziation of the notion of a vector space over a field. In fact, we will be particularly interested in finding situations in which many of the nice properties of vector spaces (existence of bases, complete classification) still hold. The concepts and techniques occuring in the this course appear in many areas of mathematics, including algebraic and differential geometry, number theory, representation theory, mathematical physics, and topology.

Topics covered will include:
1. Multilinear algebra
2. Representations of finite groups
3. Basics on modules
4. Chain conditions and Hilbert basis theorem
5. Modules over PIDS
6. Basics on category theory
7. Representations of finite-dimensional algebras

Grading

  • Assignments & Participation 60%
  • Take Home Final Exam 40%

NOTES:

Five to ten questions will be assigned each week Thursday and due by the beginning of class on Thursday of the following week, to be dropped off in class. Absolutely no late homework will be accepted; one homework assignment may be missed. A correct solution to a homework problem will include a restatement of the problem and only consist of complete sentences. I will mark the assignments for completion. Additionally, you will be required to present solutions to 3 problems in class over the course of the semester. The final exam will be a comprehensive take-home exam.

Materials

MATERIALS + SUPPLIES:

There is no required textbook for the course. Each Thursday, I will post a list of topics (and relevant references) with which I expect you to be familiar in preparation for the next week’s lectures. It is recommended that you have access to at least one of the following texts:
  • Algebra, Serg Lang
  • Abstract Algebra, David Dummit and Richard Foote

RECOMMENDED READING:

Algebra, Serg Lang

Abstract Algebra, David Dummit and Richard Foote

REQUIRED READING NOTES:

Your personalized Course Material list, including digital and physical textbooks, are available through the SFU Bookstore website by simply entering your Computing ID at: shop.sfu.ca/course-materials/my-personalized-course-materials.

Graduate Studies Notes:

Important dates and deadlines for graduate students are found here: http://www.sfu.ca/dean-gradstudies/current/important_dates/guidelines.html. The deadline to drop a course with a 100% refund is the end of week 2. The deadline to drop with no notation on your transcript is the end of week 3.

Registrar Notes:

ACADEMIC INTEGRITY: YOUR WORK, YOUR SUCCESS

At SFU, you are expected to act honestly and responsibly in all your academic work. Cheating, plagiarism, or any other form of academic dishonesty harms your own learning, undermines the efforts of your classmates who pursue their studies honestly, and goes against the core values of the university.

To learn more about the academic disciplinary process and relevant academic supports, visit: 


RELIGIOUS ACCOMMODATION

Students with a faith background who may need accommodations during the term are encouraged to assess their needs as soon as possible and review the Multifaith religious accommodations website. The page outlines ways they begin working toward an accommodation and ensure solutions can be reached in a timely fashion.