Algebraic Structures, Spring 2021

Robot juggling rubik cubes
Group ring
© 2021 Laure Bukh
Used with permission


Mondays, Wednesdays, Fridays 4:30


Online via Zoom


The aim of this course is to introduce algebraic structures that pervade mathematics: groups and rings. We will learn what they are, will see many examples, learn how to reason about them. Topics to be covered include permutation groups, abelian groups, cyclic groups, homomorphisms, quotient groups, group actions, group classification, rings, ring homomorphisms, ideals, integral domains, quotient rings, unique factorization domains, principal ideal domains, and fields.

The prerequisites are being comfortable with reading and writing proofs, and a little of bit of linear algebra.


Class format:

This is a remote class. The meeting ID is 932 5786 1340, which is accessible only through CMU Zoom accounts. The password was e-mailed to the registered students on 31st of January. If you add the class after this date or want to audit the class, e-mail me.

To maintain interactive and informal spirit, the students are required to keep their cameras on.

Office hours:

The office hours will be at 11:30am on Tuesdays and 8:30am on Thursdays, online via Zoom meeting ID 998 3409 8577 using the same password as for the lectures. The hours are subject to change. I am also available by appointment.

Course activities:

There will be weekly homeworks, two mid-terms and and a final. The mid-terms will take place on March 5th and April 9th. The final exam will be scheduled by the registrar.

Students are expected to fully participate in the class. Discussions during the lectures are encouraged.

Homework will count for 15% of the grade. The mid-terms will count for 22% each, whereas the final will count for 41%.


Practice is an integral to learning mathematics. You are encouraged to do as much homework as possible on your own; this way you will learn more. Though collaboration is allowed, you must write the solutions yourself. Turning in solutions that you do not understand will be treated as cheating.

Homework must be neat. Each word must be readable. Anything that you do not want to be graded must be completely crossed out. If in doubt, either re-write solution from scratch or typeset it in LaTeX. Any solution that fails to be neat will receive 0.

In order to ensure academic integrity, each week I will call on some subset of students to explain their solutions to me outside class.

The homework must be submitted via Gradescope.


Exams are open book. You may use your notes, references, and any inanimate online resources.

You may not use assistance from other people, either in-person or online. This includes e-mail, SMS, any other kind messaging, or posting on online forums/disscussion boards or similar websites.

You must take exam at any time on the exam day, between 12:01am and 11:59pm Eastern time. You may not discuss the exam with anyone (except me) until the exam day ends. This includes even comments such as ``The exam was easy/hard.''.

All exams are self-proctored. You must record yourself using Zoom (audio, video and screen), for the entire duration of the exam.

Academic integrity:

Violation of academic integrity include, but are not limited to,

Any violation will result in automatic grade of R for the class, and will be reported to the university.


This syllabus is more likely to change than a syllabus during a non-pandemic semester. I will strive to minimize disruptions, and will communicate any changes promptly via e-mail (and in class if possible).

Staying sane and healthy:

This is an advanced mathematics course. It is designed to challenge your brain with new and exciting mathematics, not to wear your body down with sleepless nights. Start the assignments early, and get good nutrition and exercise. Pace yourself, for semester is long. If you find yourself falling behind or constantly tired, talk to me.


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