Most branches of mathematics use images to help communicate their content. Calculus teachers show animations of shrinking rectangles under a curve to demonstrate Riemann sums. Topology instructors draw epsilon balls on a surface because the images serve to illuminate the matter under discussion. Although it would be possible to communicate the material using only the medium of mathematical symbols, such a method is universally recognized as inferior -- it communicates far less intuition about the subject matter. Why, then, do we not use images in our introductory abstract algebra courses? This is an area where the common pedagogy is ripe for improvement.
We designed Group Explorer as part of discussions stemming from a Group Theory Visualization course Douglas Hofstadter offered at Indiana University in the fall of 2002; Doug was also part of the design team. The course demonstrated the potent intuition-building value of illustrations such as Cayley diagrams, multiplication tables, and objects of symmetry. Group Explorer was designed to automate the creation of such diagrams, and allow users to interact with and manipulate them in meaningful ways.
One of us (NC) began the development in 2003 as part of work toward an MS in Computer Science at Indiana University, and the first version was released in the summer of 2003. Later versions followed, and this article surveys version 1.5.8. (Version 2.0 has many more features but is in beta testing at the time of writing.) You are encouraged to browse the Group Explorer Web site; all versions of the software are free to download and use, and later versions are open source.
Several popular abstract algebra texts treat visual topics such as Cayley diagrams and multiplication tables, but not to their full potential. Gallian (2001) and Fraleigh (2002) are rare in that they mention Cayley diagrams, but they do not do very much in the way of visualization, and Gallian focuses on their graph theoretic properties rather than using them as a visualization tool. Grossman & Magnus (1975) spend quite a bit of time on visual group theory, but their text is not designed for a typical introductory algebra course. Hibbard & Maycock (2002) include many enlightening articles in their recent volume on abstract algebra pedagogy, and much of it involves software, e.g., Finite Group Behavior (Webb & Keppelman, 2004) and AbstractAlgebra (Hibbard & Levasseur, 2002), but very useful visualization techniques, such as Cayley diagrams, remain unaddressed. Group Explorer has a very different focus from software packages such as GAP (2004) and Magma (2005), which are designed to be powerful computational languages. Group Explorer was designed to show groups to teachers and students.
In this article, we give a tour of the most important features of Group Explorer and use it to demonstrate a foundational concept in group theory. This demonstration is suitable for standalone reading by teacher or student or presentation by an instructor in a classroom.