How Can One Visualize a Group?

[gottfried]

Does anyone have any intuitive idea of how to visualize a group. The closest thing I know of in terms of a group visualization tool is a Cayley graph. I was wondering if anybody knows of a better method to visualise a group? And slightly different question what is the use of Cayley graph?

 

[Stephen Tashi]

Does anyone have any intuitive idea of how to visualize a group.

To me, there is only one "fool proof" way to think about a group and it doesn't involve a picture. Think of a set S and all the 1-to-1 functions of S onto itself. The "multiplication" of functions f and g is defined by composition of functions. You can define fg to be the function f(g(x)) (or vice versa, I suppose - anyway, it's a good aid to remember that group multiplication need not be commutative.)

In my opinion, the visual ways of representing groups are more useful to people who already have a good intuition about very elementary group theory because they involve using concepts like a set of generators for the group that are one step above the elementary ideas.

I'm guessing that you are thinking about finite groups. I recall seeing some black and white pictures in an old book that showed 3D models of interesting shapes with figures drawn on them. Some of the figures looked like "stream lines" in a physics book. I think these were models for visualizing infinite continuous groups. Anybody know about that?

 

[jbunniii]

I have not read this book, but its title is intriguing: Visual Group Theory