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We have been experimenting with using computer software in

almost all of our undergraduate mathematics courses. In particular,

I have focused on using GAP in our beginning course in abstract

algebra. As a preface to explaining what we have settled on,

let me begin by describing our beginning our algebra course.

All mathematics majors are required to take one term of abstract

algebra. One reason for this is that it is one of only two

courses in which primary emphasis is placed on developing

"mathematical maturity." That is to say, we have

regarded our beginning undergraduate algebra course as

a theoretical course where students are expected to learn

to write proofs precisely and accurately, to think and reason

logically, and to gain an appreciation for generalization and

abstraction.

The course introduces groups, rings, and fields, with lots of

examples, and covers the standard theorems (e.g., Lagrange's Theorem,

Fundamental Homomorphism Theorem for groups and rings, Fundamental

Theorem of Field Theory). The content is a slightly abridged

version of the first 25 chapters of Joseph A. Gallian's

``Contemporary Abstract Algebra'', Third Edition, 1994,

D. C. Heath and Company, Lexingon, Massachusetts.

Such a full agenda leaves little time for extended computer

projects. Nevertheless, with careful planning we

have supplemented this theoretical approach with

four computer projects, using GAP (for groups) and MAPLE (for

rings and fields). About one class period is devoted to each

project. These examples show the value of the

computer in pedagogy, understanding, research, and application.

In the first project the computer is used to solve various sliding

block puzzles. Specifically, students specify sets of generators

and the computer calculates their subgroups. Students might wonder

how the computer is able to do this, and it could lead,

although not in this course, to further study of generators and

relations, and the Todd-Coxeter algorithm.

The second project helps students understand the Fundamental Theorem

of Finite Abelian Groups. Specifically, for arbitrary $n$, students

use the computer to help them identify the group of units

modulo $n$ as a direct sum of cyclic groups of prime power order,

and to construct its cycle graph.

In the third project, students ask questions such as ``What

proportion of the elements of a group have property $P$?''

The computer grinds out data and calculates the proportion for

any desired group. This, in turn, leads to conjectures and

possible proofs.

The fourth project uses the computer to calculate

``encoding'' polynomials for multiple error correcting BCH codes.

The computer is also used to detect and correct

errors in received messages. Among other things, this project

reinforces the importance of finite field algebra.

These projects show students the scope and power of computer

software, and hopefully motivates them to want to continue

their studies in mathematics and algebra in particular.

I am currently preparing versions of these projects for

distribution by e-mail (in plain TEX). If interested in

getting a copy by e-mail, my address is

larson@stolaf.edu

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