Abstract Algebra






Dynamic Content Table


Free textbooks and other resources

FALL 2022

This course provides an introduction to algebraic structures, beginning with the study of group theory. Topics include binary operations, modular arithmetic, groups (matrix, symmetry, permutation), subgroups and Lagrange’s Theorem, homomorphisms and Cayley’s Theorem, and basic properties of rings and fields.

1) Important dates
2) Guidelines for homework
3) Practice problems for the midterm
4) What to expect in this course?
5) Final Comprehensive Review
6) Final Comprehensive Review
Lesson 0: Review of basic proof techniques
Lesson 1: Review of functions and maps
Lesson 2: Euclid’s Algorithm and consequences
Lesson 3: The Principle of Mathematical Induction
Lesson 4: The symmetries of the square and the symmetric groups
Lesson 5: Groups
Lesson 6: Subgroups and Cayley diagrams
Lesson 7: Lagrange’s theorem
Lesson 8: Group homomorphisms
Lesson 9: Normal and quotient groups
Lesson 10: The isomorphism theorems
Lesson 11: Cauchy’s theorem
Lesson 12: Direct Products
Lesson 13: Fundamental theorem of finite Abelian groups
Lesson 14: Symmetric groups
Lesson 15: Alternating groups
Lesson 16: Rings and Fields

  • Meeting days and hours: MWF (11:15-12:05)
  • Sept 7: Fall classes begin
  • Oct 10: Columbus Day (No classes)
  • Oct 26: Friday Class Schedule (Wed classes will not meet)
  • Midterm: Friday October 28th
  • Nov 11: Veteran’s Day – No classes
  • Nov 23: Thanksgiving recess (Nov 23 – Nov 28)
  • Dec 14: Last Day of Instruction – Day Classes
  • Dec 15: Reading Day
  • Final Exam: Friday, Dec 16, 11 AM-1PM

Before you attempt any homework set, make sure to:

  • Watch the relevant video lecture
  • Read the relevant part of the textbook corresponding to the lesson
  • Do not read passively; work out examples given both in the lecture and your book.

Try to work independently as much as possible, and only look at the solutions posted once you have spent a reasonable amount of time thinking about these problems and their solutions.

For this course, I intend to use a differentiated instruction style (my own version of a flipped classroom), which I believe is much more suitable to all student’s needs:

Less time in class doing low learning activities (note-taking, listening to my lectures,…) and more time focusing on intermediate to high learning activities centered around specific learning outcomes. The concept of a flipped classroom is self-explanatory and intuitive: lower-level learning activities, such as note-taking and watching lectures, are conducted at home, and a significant portion of intermediate to higher learning activities occurs within the confine of the classroom under the instructor’s guidance and supervision. All lectures are recorded and organized by the instructor and provided in the format of videos on this blog. Additionally, students are required to watch preassigned online videos made available on the blog. Also, students are instructed to take careful notes just as they would in class. In these settings, students fully control the pace and rhythm of their exposure to the concepts of interest. They can pause and rewind as often as necessary to digest the introduced ideas and concepts. These video lectures are, however, not substitutes for reading. As such, students are also instructed to supplement the videos with additional reading (taken from an assigned companion textbook). I also recommend that students read their books before and after watching my lessons.

In the second phase of this teaching format, I will prepare a set of in-class activities (a list of problems that, in my estimation, is relevant to the topics at hand) made available to students to be completed in the setting of a classroom. For these activities, students will be asked to work either individually or in small groups, and a procedure will be put in place to provide an opportunity for every student to present solutions to the in-class activities. Additionally, throughout these presentations, students receive a wealth of feedback in the form of questions, comments, and remarks by both instructors and peers to improve their solutions and presentations.

This modified version of a flipped classroom is weekly organized as follows:

(1) Notes portfolio: A list of video lectures is made available on the blog, and students are expected to interact with this content at their convenience and pace. Students are to maintain and organize all notes taken during the semester in a way that allows them, at the end of the semester, to upload a single document containing all notes taken during the course of the semester to receive credit.

(2) Proof portfolio for in-class activities: Before each class meeting, students are instructed to prepare by attempting the in-class activities (a list of selected problems made available) associated with the topic at hand. These in-class activities are also posted on the course blog, and each written solution serves as an entry to students’ proof portfolio: an essential component of the grade in the course.

(3) Homework: A set of homework to submit on a predetermined platform (Gradescope or Teams) and to be graded by the instructor are assigned to students.

The importance of homework as a learning tool: Mathematical concepts are best acquired and mastered through iteration. The first time a student is introduced to a new idea, the student is encouraged to take time to understand every aspect of this new concept. This often requires reading definitions, propositions, and theorems introduced in video lectures several times. Students are also encouraged to revisit examples presented to them by the instructor and those accessible in the assigned companion textbook. Multiple exposures are often crucial to success in mathematics. In the second step, students are also encouraged to put notes, books, and other similar media aside and to spend time unpacking materials by working out small examples independently. Thirdly, having grasped a new concept, students are provided an opportunity to exploit the acquired tools now at their disposal to solve assigned problems. Solving a problem in mathematics often requires that students understand the specific obstruction(s) embedded within the formulation of the problem. Indeed, every math problem has a hidden obstacle that may or may not be visible (this is a function of the difficulty level.) Unveiling this obstruction is a journey that students are encouraged to take individually or in small groups. Wrestling with math problems also offers an opportunity to exercise creativity in applying the tools acquired to the particulars of each assigned problem. This often requires multiple attempts, and each iteration reinforces the integration of the newly acquired concepts into students’ general knowledge of mathematics. However, should the ratio of time spent in solving a problem to available time be too high (as judged by each student), students are encouraged to seek help by either discussing assigned problems with a peer, a tutor, or the instructor. For instance, office hours offer an excellent opportunity to have in-depth discussions and build relationships with the instructor. In summary, assigning homework plays a multi-faceted role:

  • Practice mastering the content being delivered
  • Expansion in the creative dimension of their cognitive abilities
  • Engagement with peers and instructor in meaningful ways.
  • Midterm: 1/4 of your score
  • Proof Portfolio: 1/4 of your score
  • Note portfolio: 1/4 of your score
  • Final exam: 1/4 of your score

At 8:40, the result in the bracket should be [(2l)^3+3(2l)^2+3(2l)+1].
However, it doesn’t affect the answer at all (Correction by Zenan)

In this lecture we study the following concepts: functions, domain, codomain, range, injective, surjective, bijective, inverse, identity, image, preimage, composition of functions, associativity and commutativity of composition

Corrections: At 0:50, f(g) should be f(x)

In this lesson, we study the following concepts: Well-Ordering Principle, Euclid’s Algorithm, GCD, prime numbers, relatively prime numbers and related concepts

Some obscurity worth clarifying: In the proof of the existence of the Greatest Common Divisor of a,b, I would like to clarify that at around 21:14, one should pick a nonzero element x \neq 0 and then prove that its opposite -x also belongs to A.

In this lesson, we study the Principle of Mathematical Induction used to prove statements of the type: for all natural numbers n, is true

Review of matrix multiplication


Only watch the lectures below once you how to multiply matrices


Correction: At 1:41, it should be these phenomena not this phenomena. Phenomenon is a noun that means an observable fact or event in philosophy, and more commonly something remarkable or unusual outside the world of philosophy. Phenomenon is the only acceptable singular form. Phenomena is its plural, In certain instances, phenomenons can be used as a plural.


Correction by Justin Carpender at 10:23 A child cannot be a parent of its parent would show that it is not symmetrical whereas reflexive substantiated by the following justification: a child cannot be the parent of itself.

Since the concept of quotient groups can be a bit tricky to grasp when you are first introduced to it, as a starting point let us take a birdseye view approach before we dive more into the specific and technical aspect of the notion.

Having being exposed in a slightly informal way to the concept of quotient groups, we can now undertake a much more detailed treatment of the topic.