The “Getting Started” section of the help pages presents an introduction to visualizing group theory with Group Explorer. After reading these pages, you’re encouraged to either start playing with the software on your own or use the Tutorials to continue the guided tour.
Group theory is a subject that has historically had very few pictures, and when it did (such as wallpaper groups or polyhedra) they served only to exemplify a few select groups, and indirectly. Thus it was possible for a student to have a year or more of abstract algebra without ever picturing the subject in his or her head! This is a loss for any group theory student, and is a particular roadblock to visual learners.
Since about 2004, there have been a growing number of resources for learning group theory in a more visual way. Group Explorer is one of them. It provides interactive visualizations for group theory and is designed to be an aid for building intuition and understanding for students as they learn. But it may also offer some new intuitions for those seasoned in algebra as well. A brief overview of its features are given below.
These are a common introduction point for new students into binary operations and groups. Group Explorer allows users to view them, highlight them, compare them, draw homomorphisms between them, take quotients of them, and more.
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These are a less common but more potent group visualization tool and are the flagship intuition-building device used in Group Explorer. Cayley diagrams expose the structure of a group and the relationships between its elements and generators. These, too, can be viewed, highlighted, connected by homomorphisms, etc.
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Symmetry objects are very common and cycle graphs are very rare, but both are also integrated in Group Explorer.
These help pages allow the student to be guided by the software on their explorations and investigations in group theory. Group Explorer’s group library is the perfect place to start for building conjectures or finding counterexamples.
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