Markus Schmidmeier

Florida Atlantic University

## Spring 2018

Modern Algebra

Welcome to my Modern Algebra course MAS 4301! We meet Tuesdays and Thursdays, 12:30 - 1:50 p.m. in PS 109.

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 relevantis 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 topicsCharles 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.

Further ReadingFree 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!"

TutoringThe Math Learning Center in GS 211 provides drop-in tutoring M-R 9-6, F 9-4, U 1-5. For questions, please e-mail mlc@fau.edu.

Credit

Every week there will be homework assignments.Homework assignments:

Every Thursday we will have a quiz of about 25 minutes in which homework problems are likely to come up. The twelve best quizzes count forQuizzes:60%of the grade.

The final exam on Thursday, April 26, 2018, 10:30 - 1:00, is comprehensive. It will count forFinal Exam:40%of your grade.Here you can find some general information about my courses.

Contact Me

Tuesdays 9:30 - 11:30, Fridays 2:00 - 4:00.Office hours:

Course Web Page:http://math.fau.edu/markus/courses/algebra18.html

markus@math.fau.edu.E-mail:

Last modified:by Markus Schmidmeier