Explorations for Abstract Algebra

Janet Beery, University of Redlands

 

MATH 341, Abstract Algebra, is a one-semester course introducing students to groups, rings, and fields, with emphasis on the structure and classification of groups. [Jump to Explorations]

 

The text I have used most recently is Contemporary Abstract Algebra (Fifth Edition), by Joseph A. Gallian.  Newer editions of this text seem to have changed very little from the 5th edition.

 

Prerequisites:  MATH 201 or 204 (Discrete Math), and MATH 241 (Linear Algebra)

 

Course objectives:  - To understand several important concepts in abstract algebra, including group, ring, field, vector space, homomorphism, isomorphism, substructure, and quotient structure, and to apply these concepts to such real world problems as describing molecular symmetry and detecting and correcting errors in coded information;

-   more generally, to gain experience with the process of abstraction in mathematics through the study of algebraic structure;

-   to improve your ability to think logically and abstractly; 

-   to improve your ability to prove mathematical theorems; and

-   to improve your ability to communicate mathematics, both orally and in writing.

 

Despite all this talk of abstraction, the course is very concrete.  Students spend much time (and energy!) exploring examples in order to help them understand abstract concepts and to give them practice in the process of abstraction.  We begin this exploration on the very first day of class, when students begin the first two activities listed below, Symmetry Groups and Group Tables of Symmetry Groups.  By the second class session, students are proposing properties that a composition table of a set of rigid motions of a regular polygon must have and, more generally, that the Cayley table for a group must have (see the third activity, More Group Tables).

 

Ideally, such exploration would be repeated for each new concept.  In particular, it is important for students to explore examples of groups with both additive and multiplicative operations in order to understand cosets and factor groups.

 

Helping students structure their proof-writing with outlines like those given in activities 6 and 7 has its advantages and disadvantages.  One disadvantage is that students are unlikely to come up with different proofs of a theorem that they then could be prompted to share with one another.  Different approaches could be outlined either on paper or in class (e.g. What if a student started her proof like this Ö?  How could she finish it?).  I donít believe Iíve ever tried giving different outlines for a proof of the same theorem to different students. 

 

Explorations:

1.  Symmetry Groups

2.  Group Tables of Symmetry Groups

3.  More Group Tables

4.  Groups of Orders 3 and 4

5.  Not a Group!

6.  Proofs of Theorems about Groups

7.  Subgroup Proofs

8.  Groups of Order 4

 

Posted 2/2010





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