Math 3230-001 COURSE INFORMATION
Fall 2009
Course Description
This course is an introduction to the study of algebraic structures. Topics covered include binary operations, abstract groups, cyclic and permutation groups, homomorphisms, normal subgroups, quotient groups, and a brief introduction to rings and fields.
Prerequisites
Math 2710 (Transition to Advanced Mathematics). It is recommended that you also have taken Math 2210Q (Applied Linear Algebra).
Textbook
Abstract Algebra, A First Course, by Dan Saracino, Waveland Press, Inc., 1st or 2nd edition
Syllabus
We will try to cover most of the book (we will definitely skip sections 15 and 21). Also, be aware that I will not necessarily go through the material in the order in which it appears in the book. I will at times present material differently than in the text as well as add additional material to the lectures.
Topical Outline
(not in order of presentation)
1. Preliminaries: Integers (Divisibility, Fundamental
Theorem of Arithmetic), Modular
Arithmetic, Induction, Equivalence Relations, Mappings
2. Groups: Binary Operations, Groups, Subgroups, Cyclic
Groups, Isomorphisms,
Cosets, Lagrange’s Theorem, Homomorphisms, Normal Subgroups,
Quotient Groups (Factor Groups)
3. Permutations: Symmetric and Dihedral Groups
4. Rings: Definition, Basic Properties, Subrings, Ideals and
Quotient Rings, Integral Domains, Fields
Grades
As I'm sure you already know, mathematics is not a spectator's sport. This is especially true in Abstract Algebra where the emphasis is on deductive reasoning (proofs). Therefore, assignments are a vital and integral part of this course and so you should be rewarded accordingly for doing them conscientiously. Class attendance and participation are also important. Finally I cannot emphasize enough the importance of reading the book carefully as we go along and of also reviewing your class notes. I strongly suggest that you take some time every night to go over the definitions we used that day -- it will make your life a lot easier. It is also important to read the sections of the text we will be going over before the class meets.
The grading will incorporate all this as follows:
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Class Participation, Quizzes, Homework: |
30% |
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Exam 1: |
20% |
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Exam 2: |
20% |
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Final Exam: |
30% |
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Here is a tentative schedule for when the exams will be given.
Quizzes
There will be in class quizzes which will emphasize definitions and results seen in class and the homework. A missed quiz is assigned a grade of zero. I WILL DROP YOUR LOWEST QUIZ GRADE. There will also be some take home quizzes. These will involve proofs that require more time than can be done quickly in class. They will be done individually without consultation with anyone else. You can come to me for guidance if you are stuck.
Tests/Final
The first exam will probably cover the material from sections 0-7, and Exam 2 will cover the material through section 14. The final exam will be comprehensive. THERE WILL BE NO MAKE-UP TESTS. If an exam is missed because of an emergency (and there are very few reasons for a true emergency!) the weight of that exam will be incorporated with the other grades. You can link to the tentative schedule for the exams here or from the course homepage, click on link for important dates.
Homework and Study Groups
To assist you in mastering the material, the class will break up into study groups of two to three students. Each week after the first, homework will be assigned at the Tuesday and Thursday classes for submission at the next week's Tuesday meeting. Try to work every assigned question, but for each study group, a good practice is to rotate definite responsibilities for each assigned question among the members. For each problem, one designated person should be able to work the problem, and explain it to the others. The responsible person may obtain assistance from me or anyone else willing to provide help, such as students who have completed the course, graduate students, and other instructors. Prior to Tuesday's class, the group meets to go over the homework and prepare it for submission by the group. At that meeting, the experts explain the solution of any problems that other members were unable to complete. In this way, everyone gets a reliable and understandable explanation of all the challenging problems.
Be Advised, everyone is responsible for understanding (and possibly reproducing) all the material in the homework.
It would be a mistake to skip homework, because no skill (in mathematics, foreign language, athletics, and so on) can be learned by passive involvement, but only by regular practice. Moreover, many skills are learned over time, so do not expect to understand everything perfectly right away. You should find your understanding of basic topics improving gradually from one week to the next.
Proofs on homework should not be simply a string of logical and mathematical symbols, but include complete sentences in English. Proof should be written in an essay style explaining the reasons why the claim is true. These reasons should be clearly elucidated and conform to standard proof techniques learned in Math 213.
Each group member receives the same grade for that submission, which should represent the collective work of all members. If someone does not contribute to a submission, the remaining group members can omit his or her name from the group's paper. You are free to change study groups at any time; always inform the other group members beforehand. If there are problems, please come speak with me about them.
Rational: Numerous studies have found value in working together to master challenging mathematical material. I profited greatly as a student from working with my classmates, and once you form the habit of batting around mathematical ideas you should find you not only have more success but also actually start to enjoy the dynamic flow of ideas! If you get stuck, you don't have to suffer in isolation. On the contrary, you have a ready source of support from your group partners. I am ready and willing (and should be able!) to provide help if you and your group mates can't generate any ideas for attacking assigned problems.
