MA 423-01: Modern Algebra I
Fall 2018
Section 1
Course Description:
Bulletin description: Elementary notions in groups, Fundamental Theorem of Finitely Generated Groups, permutation groups, quotient groups, isomorphism theorems, and applications of transformation groups.
More accurate description: Monoids, groups, subgroups, quotient groups, isomorphism and the Isomorphism Theorem, permutation groups, Cayley’s Theorem.
Prerequisite: MAT 326 (Linear Algebra I) and MAT 340 (Discrete Mathematics). These prerequisites constitute necessary preparation for the course; if you lack either you are at a tremendous disadvantage.
Schedule: (very tentative, what on account of my rethinking how I do this class)
| Week | Sections |
| 1 | Introduction; Sets and Relations |
| 2 | Division; Linear Algebra |
| 3 | “Real,” “Imaginary,” and “Complex” numbers; catch-up |
| 4 | From integers and monomials to monoids; Isomorphism |
| 5 | Direct Products; Absorption and the Ascending Chain Condition |
| 6 | Catch-up; Groups |
| 7 | The symmetries of a triangle; Cyclic groups and the order of an element |
| 8 | The roots of unity; Catch-up |
| 9 | Test #1; Subgroups |
| 10 | Cosets; Lagrange’s Theorem; Quotient groups |
| 11 | “Clockwork” groups; Catch-up |
| 12 | Homomorphisms; Consequences of isomorphism |
| 13 | The Isomorphism Theorem; Automorphisms and groups of automorphisms |
| 14 | Permutations; Cycle notation |
| 15 | Dihedral Groups; Cayley’s Theorem |
| 16 | Catch-up |
| 17 | Final Exam |
Grading: The semester grade will be determined by a weighted average, according to the weights listed below.
| Tests (there will be three) |
60% of total |
| Homework (problems may be graded randomly) |
30% of total |
| Quizzes (generally definitions, sometimes reading) |
10% of total |
Graduate students enrolled in MAT 523 must also perform an independent research project. They must find an article in College Math Journal, Math Magazine, or a similar scholarly journal, read it, and write a brief summary of the article. The article must be related to the concepts of abstract algebra that we study in this class: namely, monoids, groups, isomorphisms, permutations. Please check with me that the article is appropriate before getting too far into this. This counts as a test grade.
The Major Field Test: In 2010, the mathematics department voted to make the Major Field Test in mathematics a component of MAT 423. You will receive a grade for this test. We will discuss the details later in the semester.
Homework:
I generally collect the homework for grading,
but in order to return it to you in timely fashion,
I might skip part of the assignment.
Nevertheless, it is important for you to understand every problem on the
assignment, so be sure to review the graded problems, try to fix what was
wrong, and ask me about any problem that wasn't graded, and which you did
not understand.
Late Assignments: Any assignment turned in past the specified due date and time will receive a grade lowered by ten percent for each school day late.
Makeup work: I do not give makeup tests/quizzes/etc. without an excused absence. If you must miss class, then you must also produce documentation of the reason for your absence. If you were sick, you can show me the receipt from the hospital or doctor; if you had a sports event, you can show me the schedule; if someone died, you can show me an obituary notice; if the tire on your car blew out, you can show me the receipt from the mechanic.
A word about definitions: There is no intelligible conversation when the participants have different notions of what the words mean. The most fundamental part of algebra — of mathematics itself, really — lies in the meanings of the words. If you don’t know what the words mean, you cannot reason about them, let alone about the ideas that spring from them. Your first task before attacking any problem should be to ensure you know what the new terms mean. If you don’t know, there is no point in wasting time on the problem: go through the notes, review the definitions, think about what they mean and how they are used in the text. Only then should you attempt the problem.
A word about homework: Many math majors see the purpose of homework as a “verification” that they have learned the material that was presented in class, or as “practice”. It’s a disgraceful fact that most undergraduate courses are taught that way: students are rarely challenged on the assignments, and even the “practice” given them is so easy as to infantilize them. This naturally flies in the face of what a university education should be about, which is to develop your mind and your reasoning skills.
Algebra is different. In order to answer some of the difficult questions at the heart of algebra, mathematicians found they had to organize certain patterns and properties into structures. Thus, the very nature of the course requires a “detour” into deep theory, and thus into proofs. It is more or less impossible to assign a good proof question where you mimic a technique learned in class (unless the class is about proofs itself, a là Discrete Mathematics).
Thus, the homework is neither verification nor practice, but exploration. It is an extension of the class lectures. It is not unusual for a homework assignment of five to ten questions to take several hours or even more. It is vitally important that you struggle with the problems, consult other students and the professor, and generally consider finding the solutions more important than pretty much anything else. If this notion offends you, then frankly you’ve chosen the wrong major.
A word about tests: Tests will consist of problems that you have not seen in class or in the homework. You should be able to solve them based on what you learned in the course. A good study guide for your tests will consist of (1) reviewing the homework and making sure you understand it, and (2) studying additional problems in each chapter.
Tutoring and study groups: I encourage you to work together on homework assignments, to look at each other’s solutions, and to explain answers to each other. This is not the same thing as copying each other’s homework.
Relevant wisdom from Calvin and Hobbes:
Image copyright Bill Watterson.
Usage is for educational purposes, and falls
under the principle of “fair use.”
Clicking on the image will take the reader to the source.
Use a pencil. Work that is written in pen will receive a 10% deduction.
Mobile phones: Turn your phone off or set it to vibrate once class begins, and to leave it there. If you absolutely must use the phone, please step out of class, take care of business, then return. If you use the phone in the class, I will ask you to leave. If you do not comply with this request, you will forfeit the next test. If you use the phone during a test, even as a calculator, you will forfeit the test.
Important dates:
Statement on academic integrity: All students at the University of Southern Mississippi are expected to demonstrate the highest levels of academic integrity in all that they do. Forms of academic dishonesty include (but are not limited to):
Engaging in any of these behaviors or supporting others who do so will result in academic penalties and/or other sanctions. If a faculty member determines that a student has violated our Academic Integrity Policy, sanctions ranging from resubmission of work to course failure may occur, including the possibility of receiving a grade of “XF” for the course, which will be on the student’s transcript with the notation “Failure due to academic misconduct.” For more details, please see the University’s Academic Integrity Policy. Note that repeated acts of academic misconduct will lead to expulsion from the University.
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ADA Syllabus Statement If a student believes that they have a disability which is covered by the Americans with Disabilities Act (ADA) and makes them eligible to receive classroom or housing accommodations, they should contact the Office for Disability Accommodations (ODA) for information regarding the registration process. Disabilities covered by the ADA may include but are not limited to ADHD, learning disabilities, psychiatric disabilities, physical disabilities, chronic health disorders, temporary illnesses or injuries and pregnancies. Students should contact ODA if they are not certain whether their documented medical condition qualifies for ODA services. Students are only required to disclose their disability to the Office for Disability Accommodations. All information submitted to ODA by the student is held with strict confidentiality.
The University of Southern Mississippi |