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.