Increasing Accessibility of Examples
in Abstract Algebra
Using Computer-based Projects

A Joint Project by:

Peter Blanchard (Miami University Hamilton)
& Judy Holdener (Kenyon College)

GAP Projects

In studying abstract algebra, the process of experimentation, conjecture, and proof is strongly inhibited by a lack of data. While it is true that a good textbook will contain many well-known examples, those examples are usually introduced in the context of a single specific topic. Exploring an example in more depth or in a different context typically requires a prohibitive amount of computation.


What follows are six computer-based projects designed to enhance student exploration and understanding by making examples, data, and computations more accessible to students. The projects were used as a supplement to a first-semester Abstract Algebra course. They rely on the software package GAP (Groups, Algorithms, and Programming), a freely distributed program designed to handle large computations within and relating to groups.

GAP Primer --
A supplemental resource and reading assignment for the class.

Project 1. -- An Introduction to GAP
Illustrates the basics of GAP in the context of the group of rotations of a cube; Assumes no prior knowledge of GAP

pdf file: An Introduction to GAP

Project 2. -- Subgroups Generated by Subsets
Discusses subgroups generated by a subset from two viewpoints: the "top-down" approach using intersections, and the "bottom-up" approach using group closure

pdf file: Subgroups Generated by Subsets

Project 3. -- Exploring Rubik's Cube with GAP
Investigates the transformation group of Rubik's cube.

pdf file: Exploring Rubiks Cube

Project 4. -- Conjugation in Permutation Groups
Explores the relationship between the cycle structure of a permutation and cycle structure of its conjugate; Revisits permutations of the Rubik's cube.

pdf file: Conjugation in Permutation Groups

Project 5. -- Exploring Normal Subgroups and Quotient Groups.

pdf file: Exploring Normal Subgroups

Project 6. -- The Number of Groups of a Given Order
Explores the number of possible group structures for any given order; the class will need hints and encouragement on the last problem!

pdf file: Number of Groups of a Given Order

Questions or Comments?

E-mail: holdenerj@kenyon.edu or blanchpf@muohio.edu

via an

Enhancing Learning through Technology with Collaboration Grant

Many thanks to

Scott Siddall

for his support

on this

project.

Back to the Kenyon Homepage Back to the Math Homepage Back to JAH's Homepage Last Modified: August 19, 2005