Markus Schmidmeier
Mathematical Sciences
Florida Atlantic University

# Modern Algebra

Fall 2011

Welcome to my Modern Algebra course MAS 4301! We meet Mondays and Wednesdays 5:30 - 6:50 p.m. in PS 112.

Why Abstract Algebra?

As scientists, when we capture facts, we never try to reproduce them in full, but only that side which is important or relevant in a particular context. This process of selecting what is relevant is the very essence of abstraction.

We therefore will not study, say, the integers as one subject, the complex numbers as another, and matrices as a third subject. Rather, particular aspects will be isolated, put in axiomatic form, and studied without reference to any specific objects. The other side is that each aspect is shared by many traditional systems.

One of the most basic and ubiquitous structures in mathematics is the concept of groups, of which we will study both theoretical aspects and many applications: As you know, the integers form a group, but also the functions on the real line, the letters in a code word, the symmetries in a wallpaper pattern...

Textbook and preliminary topics

Charles C. Pinter, A book of abstract algebra, 2nd edition, Dover, ISBN-13: 978-0-486-47417-5. I plan to cover in particular the following sections.

 Groups 2: Operations; 3: Definition of groups; 4: Elementary properties; Subgroups 5: Cayley diagrams, introduction to coding theory; 10: working with group elements Examples 6: Functions, finite state machines; 7&8: permutations; 11: cyclic groups; Comparing groups & New groups 9: Isomorphisms in mathematics, isomorphic and non-isomorphic groups; 14: Homomorphisms, kernel and range; 15: Quotient group construction; Symmetry Groups describing symmetry, discussion of fries groups and wallpaper patterns; Abelian groups The decomposition theorem for finite abelian groups, see 16.

Course Objectives

• Discover mathematical structures in applications.
• Establish results about abstract mathematical structures, working from the axioms.
• Present in writing sequences of logical steps which demonstrate the validity of mathematical statements.
• Contribute to classroom discussion, in particular by presenting the solution of an algebra problem.

Free online course at the Harvard University Extension School: My former student Larry sent me the link, writing: "Oh, and here is a link to the Harvard lecture series on Abstract Algebra. I might not have survived Dr. X's Modern Algebra class without it!"

Credit

Homework assignments:   Every week there will be homework assignments.

Quizzes:   Every Wednesday we will have a quiz of about 15 minutes in which homework problems are likely to come up. The ten best quizzes count for 40% of the grade. The last quiz is take-home: Click here!

Presentation:   One 10-minute presentation, say about a solution of a suitable problem in our textbook, will count for 10% of your grade.

Midterm Exam:   The midterm exam, on October 3, will count for 20% of your grade. The material is based on class discussion and homework problems.

Final Exam:   The final exam on December 7 (probably 4:00 - 6:30 p.m.) is comprehensive.  It will count for  30% of your grade.

Here you can find some general information about my courses.

Contact Me

Office hours:  MW, 2-4 p.m. in SE 230, or after class.

Course Web Page:   http://www.math.fau.edu/schmidme/algebra11.html

E-mail:   markus@math.fau.edu.