This is an introduction to abstract algebra with a focus on group theory. Topics include basics of group theory, subgroups, homomorphisms, normal subgroups, cosets, quotient groups, cyclic groups, and abelian groups.
Prof. Paul Gunnells, gunnells at math dot umass dot edu. The best way to contact me is by email. Please don’t leave a message on my office phone; I almost never listen to messages there.
Abstract Algebra: A First Course, 2nd Edition by Dan Saracino. Here is a link to the publisher’s site, which gives a table of contents of the book. The goal is to cover roughly the first half of the book, up to the topic of rings.
Please be sure to read the textbook to supplement the lectures. Ideally you will have read the relevant material before it is covered in lecture. Also, not every topic covered in the problems will be explicitly lectured on. See below for more advice about taking the course.
The grading for the course will be as follows. There will be a final exam worth 40%, and two exams during the semester, each worth 20%. The remaining 20% will be based on homework exercises.
The grading scale will be the traditional decile scale (if changed, it will only be changed in your favor, i.e. an A will never be higher than 93, etc.):
A: 100–93, A-: 92–90, B+: 89–87, B: 86–83, B-: 82–80, C+: 79–77, C: 76–73, C-: 70–72, D+: 69–67, D: 60–66, F: below 60.
The final will be cumulative, with some emphasis placed on topics covered after the second exam.
The date and time of the final exam have been scheduled by the university. The final will only be given at that time, and not at any other time for any reason, with the exception of the reasons outlined in the academic regulations (see below for more information). In particular, adjust your travel plans accordingly; planning to leave for vacation before the final exam is a bad idea.
The University has a byzantine final examination policy for resolving conflicts. The details are contained in the academic regulations specifically Section X.C. Please read it carefully and make sure that you have no final exam conflicts when the schedule becomes available. It is your responsibility to understand and follow this policy (note that part of the process is getting proof of a conflict from the Registrar’s office, since no faculty member can parse the text of the academic regulations).
The dates of the exams during the semester are the following:
These exam dates do not conflict with any religious observances, as determined by the
2018 NYC Alternate Side Parking Rules Suspension Calendar,
which is the most complete list of holidays I know.
Please be aware of these exam dates and record them. Exams will not be given at any other time. Sections covered on an exam, as well as other exam policies, will be announced in class some time before the exam date. The lecture before each exam will be a review.
Make-up exams will only be given in the case of family or medical emergency. Both situations will require a note from your advisor, and the latter will require a note from your physician. No make-up exams will be given for any other reason.
Problem sets will be assigned on the main course page and will be collected in-class. Late problem sets will not be accepted for any reason, and will simply be marked late and returned ungraded. At the end of the term, a few problem set grades will be dropped, so missing one or two problem set submissions shouldn’t affect your grade. Only selected problems (randomly chosen by me) will be graded.
I encourage you to form study groups and to work on the problem sets together. In fact you will learn a lot more about the material through discussing it with your fellow students. However, there are a few guidelines to follow:
Remember that ultimately you’ll be taking exams by yourself, so if you choose to work with others, make sure that you’re understanding what’s going on.
If you do work with other students, you are responsible for writing up the problems yourself in your own words. In other words, your solution must be sunmitted separately; there are no group homework submissions allowed.
If you do work with your students on a problem set, you must indicate this on the first page of your HW submission. Please include the names of the students you worked with. This alerts the grader that some of your solutions may closely resemble other students and that there is nothing amiss.
Successful completion of the problem sets is essential to help you monitor your progress in the course. The homework problems will be very similar to problems that appear on exams. Please don’t postpone working on the problems; try to take a look at them shortly after the material is covered in class.
Math 411 is traditionally regarded by students as a very difficult and abstract course. Unlike calculus courses and other lower-division courses (such as Math 235 and Math 331), where one learns algorithms to do computations that get immediately applied to homework problems, the emphasis is on development through definitions, examples, theorems, and proofs. A much higher level of active involvement with the material is necessary to succeed.
One good thing about abstract algebra is that it is a lot more concrete than one might expect. The textbook has many examples and sample computations with groups. Theorems are illustrated with examples and counterexamples. Free high quality software, such as GAP and SAGE, is available for you to experiment with to supplement your understanding. There will be a lot of information to digest in this course. Don’t be suprised if you have to invest significant time outside of class to make progress!
I try to answer as many questions as possible during lecture. If you have a question, don’t be afraid to ask. Chances are other students also have the same question. I also usually stick around a few minutes after class to answer quick questions (such as questions about parts of the lecture, a homework problem you’ve tried, etc.). Most students find this to be a good way to clear up confusion.
Outside of class, the best way to get help is through my office hours. Sometimes only a little bit of consultation is all that’s needed to deal with difficulties. One thing to remember is that you will get much more out of office hours if you make a serious effort to do the problem on your own first.
Although I like to get a lot of questions from students, it is not possible to answer mathematical questions by email. Please don’t be offended if you ask me a mathematical question by email and I don’t respond. I’ve found in the past that trying to discuss mathematics by email rarely helps anyone, and usually only causes more confusion. It’s much more effective to ask me such questions during class or office hours.