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!