Syllabus for Mathematics 305: Abstract Algebra I.
Instructor: Andres Caicedo.
Time: MWF 8:40-9:30 am. (Sorry.)
Place: MG 120.
Office Hours: MF 11-12. See here for contact information.
Text: “Adventures in Group Theory. Rubik’s Cube, Merlin’s Machine, and Other Mathematical Toys”, 2nd edn. By David Joyner. The Johns Hopkins University Press (2002). ISBN-10:0801869471. ISBN-13: 978-0801869471. Errata.
I will provide additional handouts and references as needed.
It may be a good idea to get a Rubik’s cube, as many examples we will see may be easier to understand with a cube in front of you. There are several online cube solvers (I particularly like this one), and they may be used as well, but I still recommend you get a physical copy.
The book presents many examples using the mathematics software SAGE. SAGE, developed by William Stein, is open source and may be freely downloaded. Consider installing it in your own computers so you can practice on your own. SAGE is very powerful and you will probably find it useful not just for this course.
(It was recently proved that Rubik’s cube can be solved in 20 moves or less, and 19 moves do not suffice in general.)
Contents: The usual syllabus for this course lists
Introduction to abstract algebraic systems – their motivation, definitions, and basic properties. Primary emphasis is on group theory (permutation and cyclic groups, subgroups, homomorphism, quotient groups), followed by a brief survey of rings, integral domains, and fields.
Joyner’s textbook emphasizes group theory through permutation representations. The theory is illustrated by several permutation games. Other natural examples of groups come from geometric considerations. We will see many additional examples.
(An interesting example of groups arising from geometric considerations are the plane symmetry groups, which one can see nicely illustrated in La Alhambra. I visited Granada in 2005 and have uploaded to Google+ some pictures from the trip, where you can see further examples.)
Prerequisites: 187 (Discrete and foundational mathematics). Knowledge of 301 (Linear algebra) will be useful, though I will review the matrix theory we will need.
Grading: There will not be exams. Instead, the grade will be determined based on homework.
I will frequently assign problems (many will come directly from the book) and provide deadlines. Some of these problems are routine, others are more challenging, a few may give you extra credit points due to their difficulty. Although collaboration is allowed, each student should write their own solutions. If a group of students collaborate in a problem, they should indicate so at the beginning of their solutions. Also, if additional references are consulted, they should be listed as well. It may happen that while reading a different book you see a solution for a homework problem. This is fine, as long as it is not done intentionally, and I trust your honesty in this regard. For some problems, I may specify that no collaboration is allowed.
No problems will be accepted past their deadline, and deadlines are non-negotiable.
I will pay particular attention not only to the correctness of the arguments, but also to how the arguments are presented. Your final grade will be determined based on the total score you accumulate through the term.
It may be that you do not see how to completely solve a problem, but you see how to solve it, if you could prove an intermediate result. If so, indicate this clearly, as it may result in partial credit. On the other hand, the fact that you write something does not mean you will get partial credit.
In addition, you will be assigned a project (to work in groups of two or at most three), to be turned in at the latest by the scheduled time of the final exam. This will constitute 20% or your total grade.
Attendance to lecture is not required but highly recommended.
As the term progresses, I will be getting pickier on how you write your solutions. Introduce and describe all your notation. Use words as necessary; strings of equations and implications do not suffice. You may lose points even if you have found the correct answer to a problem but it is not written appropriately. Do not turn in your scratch work, I expect to see the final product. I am not requiring that you typeset (or LaTeX) your solutions, but I expect to be able to read them without any difficulty. Additional remarks are encouraged; for example, if a problem asks you to prove a result and you find a proof of a stronger statement, this may result in additional extra credit points.Your final project must be typeset; I encourage you to consult with me through the semester in terms of how it looks and its contents.
Once your total score is determined, I will then grade on a linear scale:
- If your final score is 90% or higher, you receive an A.
- If it is between 80 and 89%, you receive a B.
- If it is between 70 and 79%, you receive a C.
- If it is between 60 and 69% you receive a D.
- If it is 59% or lower, you receive an F.
- There may be a small curve up if the distribution of scores warrants this. Plus and minus grades might be used for grades near the top or bottom of a grade range.
I will use this website to post any additional information, and encourage you to use the comments feature. If you leave a comment, please use your full name, which will simplify my life filtering spam out.