Introduction to Group Explorer
Why are there no pictures in group theory?
In topology, geometry, statistics, and analysis, mathematicians draw
pictures. In computer science, data structures and algorithms have
diagrams or flowcharts.
But in group theory, pictures rarely (if ever)
appear. As a result, many students leave group theory knowing facts,
but neither appreciating them, nor truly being able to assign deep meaning
to the theorems they know.
Coupling traditional theorem-proof methods with graphical interaction
with groups would breathe new intuitive life into group theory.
How does one "graphically interact with a group?"
Every group has at least one visual representation, called
a Cayley diagram. Nearly all groups have many Cayley diagrams.
A Cayley
diagram shows the interaction among those generators. The entire structure of a group can be seen in one
glance, as in this diagram of the tetrahedral group A4.
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A truncated tetrahedron Cayley diagram for the
tetrahedral group A4. |
This diagram has two generators, one of order two shown in
bright blue lines, and another of order three shown in red arrows.
There are other attractive, symmetric Cayley diagrams for A4,
but of course all have the same twelve elements.
How can this help students learn group theory?
Though the above diagram may simply seem pretty, imagine
augmenting it with the following features that we've put in Group
Explorer.
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View the object with a fully interactive 3D interface,
to look at it from any side. |
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Label the elements of the group in any of a variety
of representations. |
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Highlight any of a long list of properties and
parts of the group (subgroups, cosets, conjugacy classes, etc.). |
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Alter the structure of the diagram based on various
group theoretic properties you wish to investigate. |
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Navigate the Cayley diagram using the group's
generators, and watch as symmetric 3D objects respond to the group
actions. |
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View as many Cayley diagrams or symmetry objects
simultaneously as you like, while also browsing and editing
multiplication tables, and more. |
Not convinced? Examine our Lessons and Assignments page
for specific ideas for getting your abstract algebra students visualizing
groups.
Tell me more...
If you're interested in Group Explorer, please see
the How to Use page or any of
the Tutorials.
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