Increasing Accessibility of Examples in Abstract AlgebraUsing 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

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.

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

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!

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