# Mathematics

## Math 344: Abstract Algebra

Spring 2001, 4 credits

I.   INTRODUCTORY INFORMATION

INSTRUCTOR:     Dr. Larry Krajewski
Office: MC 526
Phone:  (608) 796-3658 (office)
(608) 782-1648 (home)[before 10 p.m.]
Office hours: 10 MWF & by appointment
e-mail: llkrajewski@mail.Viterbo.Edu

PREREQUISITES:  C or better in Math 260

TEXTBOOK:  An Introduction to Abstract Algebra, A. Richard Mitchell and Roger W. Mitchell, Brooks/Cole, 1971 (permission to copy granted by publisher)

II.    GOALS

To help the student see the beauty, elegance and usefulness of abstract mathematical structures.

To investigate mathematical structures that unify the observed patterns and properties that are shared by several diverse examples.

To relate properties derived logically from the defining characteristics of a mathematical structure to properties of specific examples of the structure.

To explore the processes involved in building new structures from given structures and investigate properties and uses of such structures.

To continue to develop the problem solving and proof making ability of the students.

III.    CONTENT

Review of set theory
Groups
Binary operations
Groups
Subgroups
Cyclic groups and Generating Subsets
Cosets
Isomorphisms
Morphisms
Kernels
Normal Subgroups and Quotient Groups
Isomorphism Theorems
Automorphisms
Symmetric Groups
Cayley's Theorem

Rings
Definition
Subrings
Ideals
Quotient Rings
Ring Morphisms and Isomorphisms

IV.  Evaluation

Each Monday you will be given a problem set which is due the following Monday. These will be graded on how clearly you explain your solution, not just the final answer. Try to write so that the average student in the class could easily follow what you are doing without having seen the problem before.

I encourage you to work together on exercises and in_class activities; but the problem sets should represent your own work. You may discuss the problems with me but not anyone else. As per Viterbo's academic honesty policy, each Problem Set assignment must have the following written at the end and must have your signature. "I have read and understand the policies of Viterbo College regarding academic honesty. I assert this represents my own work and adheres to college policies." Your signature is saying that the statement is true so please be sure you understand the policies. If you are unsure about anything, please talk to me about it. To save yourself writing simply write "Pledged" followed by your signature instead of the entire statement on problem sets after the first one.

90%_100%        A                       Problem Sets (13)    80%
80%_  89%       B
70%_  79%       C                       Final Exam               20%
60%_  69%       D

V. Philosophy
Some of you may have had mathematics courses that were based on the transmission, or absorption, view of teaching and learning. In this view, students passively "absorb" mathematical structures invented by others and recorded in texts or known by authoritative adults. Teaching consists of transmitting sets of established facts, skills, and concepts to students. I do not accept this view. I am a constructivist. Constructivists believe that knowledge is actively created or invented by the person, not passively received from the environment. No one true reality exists, only individual interpretations of the world. These interpretations are shaped by experience and social interactions. Thus, learning mathematics should be thought of as a process, of adapting to and organizing one's quantitative world, not discovering preexisting ideas imposed by others.

Consequently, I have three goals when I teach. The first is to help you develop mathematical structures that are more complex, abstract, and powerful than the ones you currently possess so that you will be capable of solving a wide variety of meaningful problems. The second is to help you become autonomous and self-motivated in your mathematical activities. You will not "get" mathematics from me but from your own explorations, thinking, reflecting, and participation in
discussions. As independent students you will see your responsibility is to make sense of, and communicate about, mathematics. Hopefully you will see mathematics as an open-ended, creative activity and not a rigid collection of recipes. And the last is to help you become a skeptical student who looks for evidence, example, counterexample and proof, not simply because school exercises demand it, but because of an internalized compulsion to know and to understand,

VI.  BIBLIOGRAPHY
Asiala, Mark et al, Development of Students' Understanding of Cosets, Normality,
and Quotient Groups, Journal of Mathematical Behavior, 16(3), 241-309.

Brown, Anne, et al, Learning Binary Operations, Groups, and Subgroups, Journal
of Mathematical Behavior 16(3), 187-239.

Clark, Julie M. et al, An Investigation of Students' Understanding of Abstract
Algebra, Journal of Mathematical Behavior 16(3), 181-185.

Cupilari, Antonella, The Nuts and Bolts of Proofs, Wadsworth, 1989.

Dinkines, Flora, Abstract Mathematical Systems, Appleton-Century-Crofts, 1964.

Fraleigh, John, A First Course in Abstract Algebra, Addison-Wesley, 1989.

Gallian, Joseph, Contemporary Abstract Algebra, D.C. Heath, 1990.

Hungerford, Thomas, Abstract Algebra, Saunders, 1990.

Solow, Daniel, How to Read and Do Proofs, John Wiley & Sons, 1990..

VII.    AMERICANS WITH DISABILITIES ACT

"If you are a person with a disability and require any auxiliary aids, services or other
accommodations for this class, please see me and Wayne Wojciechowski, The Americans
With Disabilities Act Coordinator (MC 320_796- 3085) within ten days to discuss your
accommodation needs."