Syllabus

… il y aura, j’espère, des gens qui trouveront leur profit à déchiffrer tout ce gâchis.

… there will be, I hope, some people who will find it to their advantage to decipher all this mess. —Évariste Galois (1832)

Wherever groups disclosed themselves, … simplicity crystallized out of comparative chaos. —Eric Temple Bell (1938)

Overview

Content

This course is an introduction to “structural” thinking in mathematics. To acquaint ourselves with this mode of thinking, we’ll focus our attention on group theory, a branch of mathematics that studies symmetries. The idea of a group has nebulous beginnings, but it condensed significantly in the late 1700s and early 1800s (in the work of mathematicians such as Niels Abel, Évariste Galois, and Arthur Cayley). The theory has developed significantly since, and the past century has also witnessed a proliferation of applications outside of mathematics, including in physics, chemistry, biology, and computer science.

Topics we plan to cover include: groups, subgroups, cosets, Lagrange’s theorem, normal subgroups, factor groups, direct products, homomorphisms, isomorphisms, permutation groups, Cayley’s theorem, and the first isomorphism theorem.

Prerequisites

The course assumes that you have some prior exposure to proof-writing, naive set theory, and some basic arithmetic (eg, Math 109). Occasionally, some exposure to matrices (eg, Math 18) will also be useful. If you are unsure about whether this course is suitable for you, feel free to reach out!

Course Materials

The primary textbook for the course is Gallian’s Contemporary Abstract Algebra (9th edition). I encourage you to use any other resources you find helpful.

I’ve also set up a Zulip server with a dedicated stream for our class. I’ll use this to make announcements, and I encourage you to use it to talk to each other about anything class-related (asking math questions, asking logistical questions, setting up study sessions, etc). Zulip is open-source software that you can use either in a browser or by installing an app on any platform. You can get an invite link from Canvas (if you don’t have access to Canvas, just email me!).

Course Structure

Philosophy

In the long run, more important than learning any particular piece of math is learning how to learn math independently: in pedagogical jargon, that’s “how to be a self-regulated learner of math.” Research shows that the following three things are key aspects of training to be a self-regulated learner:

1. Active reading. Reading math is very different from other kinds of reading. You can’t read a math textbook the same way you’d read a novel for pleasure if you’re hoping to get anything out of it. You have to stop constantly as you’re reading math. You have to try to work out examples yourself, instead of just reading through them. You have to doodle pictures to make sure you have some kind of a picture in your head of what’s going on. You have to try to formulate precise questions about things you don’t understand.

2. Reciprocal teaching. Talking to your peers about math is incredibly important. If you don’t understand a particular concept and ask your peers, you’re much more likely to get an explanation that you actually find helpful. If you think you do understand a particular concept and help a peer who’s struggling, you’ll almost certainly find that the process of explaining the concept to your peer will solidify your own understanding of it.

3. Metacognition. A key part of learning how to learn is reflecting on your learning and taking the time to ask yourself questions about your learning. What parts of your study habits are working well for you? What parts aren’t working and how can you change these parts? What kind of a mindset do you have towards math, and what can you be doing to help cultivate a growth mindset in yourself?

Evidence-based methods for accomplishing all three of these are built into course structure. It would benefit you to bear these three things in mind as you go through the course.

Class

This class will be mostly “flipped.” You’ll be expected to read and make a preliminary attempt to understand material on your own before coming to class. I’ll typically start class off with a brief summary of your latest reading (not a substitute for having done the reading!), and we’ll spend most of our time working on problems together.

Assignments

There will be a few different types of assignments in the course:

• Reading Assignments: These involve reading some part of the textbook, attempting some exercises, and formulating a question about the reading. One of these will be due before most class sessions.

• Reflection Assignments: These ask you to reflect on various aspects of your relationship with mathematics, your mathematical learning, etc. You’ll have about 1 of these due every week.

All assignments are due at 8am on the day indicated on the schedule. Deadlines are strict and late assignments will not be accepted. Note that the 8am deadline will most likely mean that you have to submit the assignment the night before!

Exams

There will be three exams: two midterms and a final. The midterms will take place during class. The final will take place in our allocated time of finals week. You will be allowed to use one handwritten sheet of notes. No other resources will be allowed. There will be no make-up exams.

After the midterms (but not the final), you’ll have a chance to earn some points back by doing corrections.

Assessment

Scores will be calculated as follows.

Further details:

• Reading Assignments. This is split up evenly into two components:

  • Reading Questions. You get 1 point for each day that you submit a reading question as a part of your reading assignments, up to a maximum of \(n-3\) where \(n\) is the number of days that a reading is assigned. In other words, you don’t need to submit every day to get full credit (and there’s no extra credit for doing so).

  • Comprehension Checks. You get 1 point for each CC problem you attempt from the reading assignments, up to a maximum of \(n-12\), where \(n\) is the total number of CC problems that are assigned. In other words, you don’t need to submit attempts to every CC problem every day to get full credit (and there’s no extra credit for doing so).

• Reflection Assignments. Half of this score is allocated to the “Weekly Reflections.” The remaining half is split evenly among other reflection assignments (eg, the “Mathematical Autobiography”).

• Midterms. Each midterm has equal weight. If your final exam score is higher than your lowest midterm score, it will replace that score. In particular, this policy automatically comes into effect if you miss a midterm.

• Final Exam. You must take the final exam to pass the class.

Your score will then be converted to a letter grade using the following cutoffs:

At the end of the quarter, I may lower some of these cutoffs. I will not increase them.

Accommodations

If you experience any disability-related barriers to your learning, please contact the Office for Students with Disabilities right away to have them provide a current Authorization for Accommodation (AFA) by the end of week 2. We might be unable to accommodate late requests. Note also that, if you have accommodations that compel you to take an exam at a different time than the rest of the class, you may be given a different exam of equivalent difficulty.

Integrity

Act with integrity. You’ll learn more math that way, and you’ll be practicing good habits for ethical decisions that you’ll have to make for the rest of your life. If you have a question about whether something class-related is integrous, just ask first. Academic integrity violations have to be reported to university authorities and result in serious consequences, ranging from failing the course to expulsion from the university. None of that is any fun for anyone, so please, just don’t do it.

Diversity

The pursuit of knowledge thrives in diverse and inclusive environments. I expect that all of us, myself included, will work towards making our class a welcoming space for everyone, no matter how we might identify on race, ethnicity, nationality, socioeconomic background, gender, sexual orientation, ability, age, and the many other dimensions of identity. I also encourage you to reach out if there are ways to have our space be more comfortable for you. If anyone says anything in class that makes you feel unwelcome, please let me know.