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 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.
The class will consist of three hours of face-to-face lecture per week.
Abstract Algebra: A First Course, 2nd Edition by Dan Saracino. 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 computed as follows.
The grade will be based on a total homework score, two exams during the term, and a final exam.
The homework score will be worth 30%.
If your final exam score is less than or equal to the average of your Exam 1 and Exam 2 scores, then each of Exam 1 and 2 counts 20%, and the final exam counts 30%.
If your final exam score is greater than the average of your Exam 1 and Exam 2 scores, then your final exam will count 35% (instead of 30%) and each of Exams 1 and 2 will count 17.5% (instead of 20%).
Remarks about grading:
Note that attendance in lecture is not required as part of your grade. However it is expected that you will be attending all lectures, since that is the primary way information is conveyed in the course. In particular no slides or notes are available, but we will follow the presentation in the textbook for the most part.
At the end of the term, some of your homework scores will be dropped (at least 1). This way, if you don’t do as well on an assignment because of extra work in another course or for any reason, it shouldn’t affect your final homework grade. Extensions for HW are not granted except for documented reasons, as described in the Make-Up Exam section above.
After being determined by the above algorithm, the total score will be truncated down to the nearest integer less than or equal to the total score. (Truncation is not the same as rounding. For example, a score of 89.75 will be truncated to 89, not rounded to 90. Truncation instead of rounding is used to determine grades according to a policy adopted by the department.) The letter grade will then be determined by the following scale:
| A | A– | B+ | B | B– | C+ | C | C– | D+ | D | F |
|---|---|---|---|---|---|---|---|---|---|---|
| 90 | 87 | 83 | 79 | 75 | 71 | 67 | 63 | 59 | 55 | <55 |
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).
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).
This course will have two evening midterm exams and a final exam.
Exam I: Wednesday, 9 Oct, 7–9 pm. Location: LGRCA301 (this is the third floor of the Lederle Lowrise Building, not the tower).
Exam II: Wednesday, 13 Nov, 7–9 pm. Location: LGRCA301 (this is the third floor of the Lederle Lowrise Building, not the tower).
Final exam: Tuesday 17 Dec, 8–10 am. Location: LGRT 143.
Sections covered on an exam, as well as other exam policies, will be announced in class some time before the exam date. Some time in the lecture before each exam will be devoted to review.
See below for the academic honesty statement about exams.
Re-taking of exams is not allowed in this course: once an exam has been taken it cannot be retaken or made up.
You are expected to take all exams, including the final exam, during their scheduled times. All students should check your travel plans and exam schedules of your courses carefully. If you have any schedule conflicts, you may log on the Spire page, go to “Student Home” and then to “Evening Exam Conflict”. This will allow you to fill out a conflict form and submit it. Then the registrar will email your instructor who needs to provide a makeup exam. Please do so at least two weeks before the exam.
Which case and where is the official support document for the make-up request?
(1) if you have an exam (or a class) schedule conflicts with the regular exam, you should log on the Spire page, go to “Student Home” and then to “Evening Exam Conflict”. This will allow you to fill out a conflict form and submit it. Then the registrar will email your instructor who needs to provide a makeup exam.
(2) if you have a university trip for university business during the regular exam date, like an athletic competition or academic conference etc., you should ask your supervisor or your coach to write an explanation letter including his/her phone number to your instructor as the official written document. Your instructor may verify the event by phone call.
(3) if you have a religious observance on a regular exam date and can NOT take the exam, you should write an explanation letter yourself and attach the invitation letter or relevant information as the official document.
(4) if you have a medical reason and can not take the regular exam, you should ask a medical professional’s statement including his/her phone number which indicates that you were unable for medical reason to take the scheduled exam. If the medical professional’s statement is not given before the exam, your instructor may refuse your make-up request.
Problem sets will be assigned on the main course page. The method of submitting them will be determined after the grader has been chosen. 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 other words, your solutions must be submitted 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. The course is proof-based. 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 during class and office hours, 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.
The University of Massachusetts Amherst is committed to making reasonable, effective and appropriate accommodations to meet the needs of students with disabilities and help create a barrier-free campus. If you have a disability and require accommodations, please register with Disability Services (161 Whitmore Administration building; phone 413–545–0892), meet with an Access Coordinator and send an accommodation letter to your faculty. Information on services and materials for registering are also available on the website www.umass.edu/disability.
Special accommodation request: new disability students should be certified by Umass disability service center (DSC) at first, then you should ask DSC to send the special accommodation documents to your instructor at least two weeks before exam 1 or exam 2, after that you may contact DSC,
Trisha Link
Exam Proctoring Coordinator
examsaccess@admin.umass.edu
413-545-0892
169A Whitmore
directly to schedule an appointment, DSC will notify you when and where to take the exam in a few days before the exam . For the documented disability students, DSC will notify you when you should schedule the final exam in DSC.
The University of Massachusetts Amherst is committed to providing an equal educational opportunity for all students. If you have a documented physical, psychological, or learning disability on file with Disability Services (DS), you may be eligible for reasonable academic accommodations to help you succeed in this course. If you have a documented disability that requires an accommodation, please notify me within the first two weeks of the semester so that we may make appropriate arrangements.
Since the integrity of the academic enterprise of any institution of higher education requires honesty in scholarship and research, academic honesty is required of all students at the University of Massachusetts Amherst. Academic dishonesty is prohibited in all programs of the University. Academic dishonesty includes but is not limited to: cheating, fabrication, plagiarism, and facilitating dishonesty. Appropriate sanctions may be imposed on any student who has committed an act of academic dishonesty. Instructors should take reasonable steps to address academic misconduct. Any person who has reason to believe that a student has committed academic dishonesty should bring such information to the attention of the appropriate course instructor as soon as possible. Instances of academic dishonesty not related to a specific course should be brought to the attention of the appropriate department Head or Chair. Since students are expected to be familiar with this policy and the commonly accepted standards of academic integrity, ignorance of such standards is not normally sufficient evidence of lack of intent. For more information see the website of Dean of Students Office.
Expectations for our course as as follows:
For homework, as discussed above, you will be allowed to work with other students collaboratively. In fact, I encourage you to form study groups to work together. However, it is your responsibility to make sure that you are learning the material. You also must submit your own work, indicating if you collaborated with any other students. The written work you submit must be original. The use of online or other sources for solutions, or the use of generative AI tools such as ChatGPT will be considered a violation of academic honesty.
All exams will be done in-person and are closed book: no supplemental materials such as the textbook or notes may be used. Devices such as phones, watches, calculators, and computers, and not allowed to be used during exams. Use of any unauthorized resources will be considered a violation of academic honesty and will be handled accordingly.