MAT 4233 — Modern Abstract Algebra Fall 2026 · UT San Antonio
MAT 4233 Modern Abstract Algebra
Basic properties and examples of semigroups, monoids, and groups, detailed study of permutation, dihedral, and congruence groups, cyclic groups, normal subgroups, quotient groups, homomorphism, isomorphism theorems, direct products of groups, rings and fields and their basic properties, ideals, polynomial rings.
MAT 3233 Modern Algebra or equivalent
3
Traditional in-person
| Duration | Wed 19 Aug – Thu 3 Dec |
| Campus | Main Campus |
| Location | MS 2.02.12 |
| Time | Tue Thu 2:30 PM–3:45 PM |
By the end of this course you should be able to:
Eduardo Dueñez, Associate Professor of Mathematics.
| Department | Mathematics |
| Office | FLN 4.01.11 |
| Student hours | Mon 5:30PM – 6:30PM & Tue 1:30PM – 2:30PM |
| eduardo.duenez@utsa.edu | |
| Homepage | https://supernumero.us/about |
Email is the preferred method of communication.
| Component | Weight | |
|---|---|---|
| Q&A sessions | 40% | 4 sessions, 10% each; none dropped |
| Problem sets | 15% | 4 sets, 3.75% each; graded for completeness |
| Midterm Exam 1 | 20% | Thu 1 Oct in class |
| Midterm Exam 2 | 25% | Thu 3 Dec in class |
Four class meetings throughout the semester (roughly every other Thursday except when adjacent to an exam) will be Q&A sessions in which each student will be asked to explain and defend their solution to a homework problem submitted as part of a written assignment. You will only be asked questions related to an assigned problem and about the solution you submitted.
Each Q&A session is scored 0–3:
A correct answer you cannot explain is worth less than an explainable honest failure.
Note: A low Q&A average caps the course grade, whatever the other components come to:
(An average below 1.0 is of course also below 1.5; in that case the ‘D’ cap is the one that applies.)
Four sets, each with five or six theoretical problems. Maximum six pages.
Computational (SageMath) material is supplementary and entirely optional: it is never part of a problem set or a Q&A session, and it is not graded.
Each item will be scored only based on the degree of completion (not on correctness): 2 for a nominally complete solution, 1 for a clearly partial one, 0 for nothing submitted or for obvious filler with no substance.
Note: Any submission containing non-human-readable content, stray characters, or artifacts revealing non-proofread AI output scores 0 for that item.
AI assistance is allowed in preparing work for submission, as are any solution keys, collaboration, and internet sources. You are ultimately responsible for your submissions; any part of your work that has clearly not been proofread (e.g., includes non-human readable characters or LLM artifacts) automatically earns a zero grade. Include one line noting what resources you used in your submission.
This policy is by design, and it reflects how the course is graded: 85% of the grade comes from Q&A sessions and in-class exams where only what is truly understood and can be explained counts. Assignments submitted but not written/understood by you result in failing grades in Q&A sessions and exams. Use AI to learn faster —not to prepare and turn in papers you cannot defend, blocking you from passing exams as well.
All audiovisual (or audio only, or video only) recordings of lectures are explicitly forbidden; all uses of AI agents or apps to capture, process, or analyze any lecture contents are strictly forbidden. The only limited exceptions are explicitly allowed recordings per official memorandum of the office of Student Disability Services. Under no circumstances is any processing or analyzing of such recordings allowed. Any sharing of confidential course materials with individuals who are not enrolled students in the course is a violation of FERPA privacy laws.
Both exams are traditional in-class (written and closed book). Exams are based primarily on the assigned problem sets —expect some context and phrasing to differ from the homework, plus a minority of other unseen problems. Memorizing solutions will not result in good exam grades; preparing by understanding the assigned problems will make the exams straightforward and earn high Q&A grades as well.
No extensions on problem sets. A single missed Q&A session in the semester may be made up during office hours within seven days (no documentation is required). No Q&A scores are dropped.
| Grade | Range | Grade | Range |
|---|---|---|---|
| A | [90%, 100%] | A- | [85%, 90%) |
| B+ | [80%, 85%) | B | [76%, 80%) |
| B- | [72%, 76%) | C+ | [69%, 72%) |
| C | [65%, 69%) | C- | [60%, 65%) |
| D | [50%, 60%) | F | [0%, 50%) |
Expect an average of about 9 hours per week, and no fewer than 7.
This is a writing-intensive course. Standard professional standards for technical writing (mathematical writing) apply. The style is mathematical prose written for a human reader, An otherwise “correct” argument presented as a disorganized or unintelligible pile of symbols is not a solution will be returned without grading and earn no credit. For detailed requirements and required reading refer to the Writing Mathematics page.
Each student should be intimately familiar with the contents of any work submitted for grading. The instructor has the right to request adequate verbal explanation of methodology and content for any submitted work. Credit will not be awarded when such an explanation is requested but not satisfactorily provided.
| Date | Topic |
|---|---|
| Thu 20 Aug | Lecture — Course launch. The integers: division algorithm, gcd, Bezout |
| Tue 25 Aug | Lecture — Congruence and the ring Zn |
| Thu 27 Aug | Lecture — Binary operations; Cayley tables; isomorphic binary structures |
| Tue 1 Sep | Lecture — Plane isometries and finite symmetry groups: cyclic and dihedral |
| Thu 3 Sep | Lecture — Groups: axioms and first properties |
| Tue 8 Sep | Q&A (Due: Set 1) |
| Thu 10 Sep | Lecture — Subgroups; examples |
| Tue 15 Sep | Lecture — Cyclic groups; the order of an element; generators |
| Thu 17 Sep | Lecture — Generating sets |
| Tue 22 Sep | Lecture — Permutation groups; cycle notation; Cayley’s theorem |
| Thu 24 Sep | Lecture — Alternating groups; the sign of a permutation; A4 and the tetrahedron |
| Tue 29 Sep | Q&A (Due: Set 2) |
| Thu 1 Oct | Exam — Midterm Exam 1 — Sets 1 and 2 |
| Tue 6 Oct | Lecture — Cosets; Lagrange’s theorem; index |
| Thu 8 Oct | Lecture — Homomorphisms: kernel, image, and first properties |
| Thu 15 Oct | Lecture — Normal subgroups; quotient groups |
| Tue 20 Oct | Lecture — The fundamental homomorphism theorem; groups of small order |
| Thu 22 Oct | Lecture — Rings and subrings; units and zero divisors |
| Tue 27 Oct | Q&A (Due: Set 3) |
| Thu 29 Oct | Lecture — Integral domains and fields; Euler’s and Fermat’s theorems |
| Tue 3 Nov | Lecture — Ring homomorphisms, ideals and quotient rings; fundamental homomorphism theorem |
| Thu 5 Nov | Lecture — Ideal theory: principal, prime and maximal ideals |
| Tue 10 Nov | Lecture — Irreducibles and primes; unique factorization domains; principal ideal domains |
| Thu 12 Nov | Lecture — Euclidean domains; the Euclidean algorithm in Z and in k[x] |
| Tue 17 Nov | Lecture — Multiplicative norms; the Gaussian integers Z[i]; Fermat’s Two Squares theorem |
| Thu 19 Nov | Q&A (Due: Set 4) |
| Tue 24 Nov | Lecture — Sums of three and four squares; Z[sqrt(-5)] and the failure of unique factorization |
| Tue 1 Dec | Review — Review — the exam pool |
| Thu 3 Dec | Exam — Midterm Exam 2 — Sets 3 and 4 |
Readings for each meeting are on the Weekly Schedule (also available as a printable PDF).
The ombudsperson is an advocate who investigates complaints about a specific section or instructor and attempts to resolve them through mediation. Comments or complaints sent to this person will be brought to the attention of department leadership, who will follow up. If you have questions about the logistics of your class, please contact the instructor.
Contact Ilse Rosales Martinez at ilse.rosalesmartinez@utsa.edu.
As the instructor of this course, I may record meetings and lessons. You are expected to follow appropriate University policies and maintain the security of passwords used to access recorded lectures. Recordings may not be published, reproduced, or shared with those not in the class. If the instructor or a UTSA office plans any other uses for the recordings, consent of the students identifiable in the recordings is required before such use unless an exception is allowed by law.
The syllabus is subject to change at the instructor’s discretion. Any changes or corrections to the course materials, assignment dates, or other updates will be communicated to students ahead of time. You are responsible for checking Canvas for corrections or updates to the syllabus.