MAT 334 Introduction to Abstract Algebra

Undergraduate course, Mathematics Department, Utica College, 2021

Welcome to the home page of Dr. Xiao Xiao’s Introduction to Abstract Algebra course at Utica College. You can find all the information for this course on this page. Due to the COVID-19 pandemic, this course will be entirely online for the Spring 2021 semester. Please find the Zoom link in Google Classroom.

Important Dates

  • Add/Drop deadline: 2/5/21
  • Withdraw deadline: 4/12/21
  • SOOT: 4/30/21 - 5/7/21
  • There will be no spring break.
  • Mental health day: Thursday March 4, Friday April 2.
  • Final exam: May. 10 - May. 14

Instructor Information

  • Instructor: Prof. Xiao Xiao
  • Email:
  • Office: White Hall 255
  • Virtual Student Support Hours: book your appointment below.

An Important Course Policy

I pride myself on having a good environment for working and learning. It is very important to me that we all treat each other with care and respect, in equal measure. I know that I ask students to take risks in class almost every day, and this can be challenging for many. I ask that you help me keep our classroom a supportive place for each of the people in it. Each of us deserves the space to bring our full, authentic selves to class and be comfortable. (Borrowed from T.J. Hitchman.)

General Course Information and Policies

  • Course name: MAT 334 Introduction to Abstract Algebra
  • Course credit hours: 3 credit
  • Course Prerequisite: MAT 305 or permission of instructor.
  • Class time and location: Tuesdays and Thursdays 10:00am-11:15am online.
  • Textbook: We will not use a textbook, but rather a task-sequence adopted for inquiry-based learning. The task-sequence is written by me, adapted from notes of Margaret Morrow and David Clark, and it will be available on Google Classroom. You are expected to work out all the tasks as the semester progresses.
  • Course description: Introduction to Abstract Algebra covers basic group theory. We will discuss the following concepts in this course: groups, subgroups, abelian groups, normal subgroups, product groups, quotient groups, and group isomorphisms. Standard examples such as cyclic groups, dihedral groups, permutation groups and classical theorems such as Lagrange’s Theorem will be discussed.
  • Course learning objectives: In accordance to the learning goals of the Department of Mathematics of Utica College, students will demonstrate their proficiency of these abilities:
    • Reading and analyzing mathematical proofs.
    • Writing mathematical proofs.
    • Communicating mathematics in written form.
  • Class organization: This course will likely be different from any other math course you have taken before. As an instructor, I will not be lecturing most of the time although I love lecturing very much. Scientific research shows that most people do not learn mathematics by listening, instead, they learn by doing it! I am sure you have said to yourself before “It looked so easy when the professor was doing it, but now I am confused when I have to do it by myself.” Why? Because the knowledge belongs to your professor and does not belong to you. You do not learn the knowledge simply by hearing it once or twice from somebody else. In order for you to have a more thorough understanding of the knowledge, we will use a pedagocial practice called inquiry-based learning. Most of the time during the class, students will be working in groups and presenting solutions that they have produced by themselves and not by other people or textbooks. Attendance is mandatory. Attending class meetings will have a direct impact on your learning as well as your grade. If you miss class for any reason, you are responsible for getting the information from a classmate, and checking the course web page for any handouts and assignments.

  • You should not look to resources outside the context of this course for help. That is, you should not be consulting the web, other texts, other faculty, or students outside of our course. On the other hand, you may use each other, the course notes, me, and your own intuition. See the Homework section for more details on when and how you can use each other in class.

  • Regular attendance is expected and is vital to success in this course, but you will not explicitly be graded on attendance. Yet, repeated absences may impact your presentation grade.

  • Makeup policy: You can only make up an exam if all three conditions are met:
    • You have a legitimate reason (as determined by me) with documented proof. Visit of emergency rooms due to urgent health conditions is an example of legitimate reason. Attending non-academic events, such as someone’s wedding is an example of non-legitimate reason.
    • You have informed me well in advanced.
    • You can only make up an exam after the scheduled date.

Intellectual Property

  • My lecture and course materials, including powerpoint presentations, tests, outlines, and similar materials, are protected by U.S. copyright law and by Utica College policy. I am the exclusive owner of the copyright in those materials I create. You may take notes and make copies of course materials for your own use. You may also share those materials with another student who is registered and enrolled in this course.
  • You may not reproduce, distribute or display (post/upload) lecture notes or recordings or course materials in any other way — whether or not a fee is charged — without my express written consent. You also may not allow others to do so. If you do so, you may be subject to student conduct proceedings under the Utica College Student Code of Conduct
  • Similarly, you own the copyright in your original work. If I am interested in posting your solution on the course web site, I will ask for your permission.

Homework and Portfolio

  • Homework will be assigned each class period. Students are expected to complete (or try their best to complete) them before coming to the next class period. All assignments should be carefully, clearly, and cleanly written. Among other things, this means your work should include proper grammar, punctuation and spelling. You will almost always write a draft before you write down the final argument, so do yourself a favor and get in the habit of differentiating your scratch work from your submitted assignment. You can work with each other for the homework and you can also come to see me for hints if you want. But you must write up your solution/proof by yourself. When you write up your homework, I strongly suggest that you use the LaTeX system. There is a very good reason for doing this, which I shall explain later in the Portfolio section.

  • You will submit your draft work at Google classroom before 12:00pm on the day before we meet in class. For example, a homework assigned on Tuesday will be discussed on Thursday, so you should submit your draft work by Wednesday 12:00pm. I will review your draft and give suggestions and comments within 5 hours (namely before 5pm of the same day) of your submission. You can read the comments and make necessary changes afterwards but before you come to class on the next day. Each homework will generally consist of proving theorems or solving exercises from the task-sequence. The main purpose of submitting your homework one day before the class is for you to get feedback so you can be more prepared for the class in the next day. The majority of the class period will be devoted to students presenting a subset (maybe all) of the proofs of the theorems that you have worked on that homework.

  • After each class, you should revise and improve your write-up for every problem of your homework to finally include it in your portfolio. Every exercise, proposition, lemma, theorem and corollary that we encounter is to be included in your portfolio. Each entry in the portfolio is intended to be complete and polished. Do not include scratch work. If you have written your homework using LaTeX, it would be much easier for your to revise it and further include it in the portfolio.

  • Each of us will develop our own mathematical voice in this class. Not every solution will look the same. The form of the portfolio will be standardized. Each student will receive an electronic LaTeX template that you can use to create your own portfolio. Begin each write-up with the statement of an exercise, a proposition, a lemma, a theorem or a corollary, followed by your solution or proof. Some write-ups will be two lines long, others may be several pages. If you have done a perfect job in your homework, then you can just insert the LaTeX code into your portfolio. Though in most cases, you will have to improve your original work.

  • The portfolio will be checked three times: Tuesday, March 9, Thursday, April 8, and Thursday, May 13, 2021, at the final exam.

  • Because you will have already know whether your solutions or proofs are correct or not by discussing them during presentations, portfolios will be graded mostly on completeness and clarity. Clear and complete portfolios will earn a full mark, all others will be asked to resubmit within a week. Keep your portfolio current as you work, it will be too much effort to get it all organized and collated the night before it is due.

  • At the end of the semester, portfolios with three full marks will earn the full 30% possible. Two full marks will earn 20%, one full mark will earn 10% and no full marks will earn 0%. In the end, you will walk away with an organized and complete collection of your work on which you can look back with pride.


  • During the class, everyone (including the presenter) is expected to turn on their webcam as facial expressions is an important way to communicate and it is also a polite thing to do. Please make every effort to do so. If there are particular reasons that you would not be able to use webcam on a regular basis, please let me know in advance so we can discuss possible solutions.
  • Though the atmosphere in this class should be informal and friendly, what we do in the class is serious business. In particular, the presentations made by students are to be taken very seriously since they spearhead the work of the class.
  • The problems chosen for presentations will come from the homework. After a student has presented a proof that the class agrees is sufficient, I will often call upon another student in the audience to recap what has happened in the proof and to emphasize the salient points.
  • In order to make presentations go smoothly, presenters need to write out the proof in detail and go over the major ideas and transitions, so that they can make the proof clear to others.
  • The purpose of presentations is not just prove to me that the presenter has done the problem. The most important part is to make the ideas of the proof clear to the other students.
  • Presenters need to write in complete sentences, using proper English and mathematical grammar. Here are some suggestions on how to write a proper proof.
  • Fellow students are allowed to ask questions at any point and it is the responsibility of the presenter to answer those questions to the best of their ability.
  • When a presenter is stuck with a question from the audience or from the instructor or when the presented proof is not generally accepted by the audience, the presenter has the following choices:
    • They can think on the fly and try to solve the problem although this should not take more than a few minutes as to save class time.
    • They can yield the time to other presenters and think about the questions. After all the presentations have been completed, they can come back and address the questions.
    • They can go home and think about the questions and present the same problem again in the next class. In this case, the presenter must make an appointment and meet with the instructor before the next class.
  • Presentations will be graded using the rubric below.
2Completely correct and clear proof or solution.
1Proof has technical flaws, some unclear language, or is lacking some details. Or it is completed in more than one class.
0Minimal progress has been made.

However, you should not let the rubrics deter you from presenting if you have an idea about a proof that you would like to present, but you are worried that your proof is incomplete or you are not confident your proof is correct. You can only gain points and you will be rewarded for being courageous and sharing your creative ideas! Yet, you should not come to the board to present unless you have spent time thinking about the problem and have something meaningful to contribute.

  • In each class, a sorted class list produced by a computer program will be shown before presentations. Students whose ranks are high in the list have higher priority to choose problems. The sorted list is not produced randomly. It takes three factors into consideration:
    • The number of past presentations: the more one presented in the past, the lower one is on the list.
    • The quality of past presentations: the better one presented in the past, the lower one is on the list.
    • Recentness of past presentations: the more recently one has presented, the lower one is on the list.
  • If it is a student’s turn but they are not ready to present any problem on that day, then this is called a “pass”. A student uses a “pass” in one class must make an appointment and meet with the instructor before the next class. No one shall “pass” for two consecutive classes.

  • In order to receive a passing grade on the presentation portion of your grade, you must present at least twice times prior to each exam (2 midterms and 1 final) for a total of at least six times during the semester. Your grade on your presentations, as well as your level of interaction during student presentations, are worth 20% of your overall grade.


There will be two midterm exams and one final exam. Each exam is worth 15% of your overall grade. All exams will be oral. Details will be explained in class. Exams tentatively scheduled for Tuesday, March 9, Thursday, April 8, and Tuesday, May 11, 2021. Make-up exams will only be given under extreme circumstances, as judged by me. In general, it will be best to communicate conflicts ahead of time.


About every two weeks, you will be asked to write a journal on various topics. Details will be given in Google Classroom when they are assigned.


We will be using OverLeaf to edit and compile LaTeX files. OverLeaf is a free online program so you don’t have to pay anything to use it but you need to have an internet connection. I will run a special session on how to use LaTeX before the class starts. If you have experiences with LaTeX, then you do not have to attend the training session.


Your final grade will be determined by the scores of your presentations, portfolio, journals, and exams.

Midterm Exam 115%
Midterm Exam 215%
Final Exam15%

The correspondences between percentage and letter grades are explained in the following table:

PercentageLetter Grade

Academic Integrity

I have zero tolerance on dishonesty. Any forms of dishonesty such as copying homework or cheating on quizzes and examinations, will result in zero credit for that particular assignment, and will be reported to the Academic Standards Committee. The highest penalty a student can receive is “F for cheating” for the course. There might be additional sanctions by the Academic Standards Committee such as dismissal from the college. See Utica College official page for Academic Honesty for more details.

Special Accommodation

The stuff just below is the University approved language, and is a bit… ``legalese’’. The point is, if you need accommodations to succeed in this course, talk to me and we can make sure you get what you need. And the social environment of this course is important to me, too. Let’s work together to make a welcoming and affirming space for everyone.

Any student who has need of special accommodations in this class due to a documented disability should speak with me as soon as possible, preferably within the first two weeks of class. You should also contact Judy Borner, Director of Learning Services in the Academic Support Services Center (315-792-3032 or ) in order to determine eligibility for services and to receive an accommodation letter. We will work with you to help you in your efforts to master the course content in an effective and appropriate way. See Utica College official page for Office of Learning Services.


It is the students’ responsibility to keep informed of all announcements, syllabus adjustments, or policy changes during the semester via this web page or via school emails. The author of this syllabus reserves the right to change it with notice at any time during the semester.