Before reading this article, you may not have heard much about Cayley diagrams. More common techniques for visualizing groups include multiplication tables (or "Cayley tables") and three-dimensional objects of symmetry. Group Explorer also has the capability to use each of these visualization techniques, and we discuss each briefly here. Group Explorer also comes with a "Permutation View," a "History View," and a "Group Navigator," but they are less commonly used, and so we do not address them here. For more information, refer to Carter (2004).
Multiplication tables are probably the most common means for visualizing a group -- most introductory group theory textbooks discuss them. They have the advantage of fully displaying the group operation, and thereby exhibiting some of the patterns in that operation. But this comes at the expense of not giving a clear picture of the set of elements of the group, except redundantly (on each axis, and throughout the table). Figures 4 through 6 show the multiplication tables for the same three groups whose Cayley diagrams appeared in Figures 1 through 3; the Cayley diagrams are repeated for ease of comparison. The background color of a cell in the table is determined by the element in that cell, to more readily illustrate patterns in the group operation.
Figure 4. Multiplication table and Cayley diagram for Z3
Figure 5. Multiplication table and Cayley diagram for S3
Figure 6. Multiplication table and Cayley diagram for S4
In Group Explorer you may explore the tables interactively just as you can the Cayley diagrams. You can choose the order of the elements along the axes and the generators and their priority, you can focus on subgroups, cosets, and you can do numerous computations about the group in question.
Furthermore, because Group Explorer can display many views of a group simultaneously and keep them synchronized, if an element is highlighted in a Cayley diagram view, then that same element will be highlighted in the multiplication table view. Figure 7 shows a screen shot of Group Explorer with several different windows available for the group Z9. These views are available for most groups of small order.
Figure 7. Group Explorer viewing Z9 using several different visualization media
Now that we have introduced Group Explorer a bit, let us see how these images (and interacting with them) can be used to help us learn about groups.