Group Visualization with [i]Group Explorer[/i] - Features: From Images to Interactivity

Nathan Carter and Brad Emmons

A picture may be worth a thousand words, but an interactive experience is far superior. Group Explorer provides several tools for manipulating the objects viewed, asking questions about them, and experimenting with them.


  • Cayley diagrams reside in 3-space, and the mouse can be used to rotate any Cayley diagram in real-time on screen. We can therefore examine it from any angle and get a clearer understanding of its shape and meaning.
  • Many options for viewing Cayley diagrams are available, including labeling of nodes, selection of generators, representation of elements, styles and sizes of arrows and nodes.
  • Specific features of a group such as cosets and subgroups can be highlighted for clarity.
  • The structure of the diagram can be altered according to the user's preferences, including choosing the overall shape, adjusting the priority of the generators, and focusing on a subgroup.
  • Information can be calculated about the group and reported visually in the Cayley diagram -- e.g., information about orbits, subgroups, cosets, conjugacy classes, order classes, normalizers, or centralizers.
  • Several views of the same group can be shown simultaneously, and Group Explorer keeps them synchronized. This allows you to treat the Cayley diagram as a map of the group and use it to navigate other representations (e.g., an object with symmetries, as we will see on the next page).

In later pages, we discuss how these features combine to provide a useful tool for learning about groups, both inside and outside the classroom. To learn how to use some of these features, refer either to those pages or to the Group Explorer User Manual online (Carter, 2004).