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I vaguely remember seeing something like a "picture" of various groups a while back. It was as if the elements of the group were each associated with a point and many points had segments connecting them, but not all were connected.

Does anyone know what I am talking about? If so, would you care to take the time to explain the basics (or point me to a resource, if you think google won't help)?


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up vote 5 down vote accepted

Another visualization of a group on a graph is the cycle graph. These are much less regular than a Cayley graph, so might better match your "not all points were connected." In fact, Cayley, Schreier coset, and cycle graphs are all connected, but they get less regular down the list.

The cycle graph has vertices the elements of the group, and for each maximal cyclic subgroup, a generator x is chosen, and edges are drawn between xi and xi+1. It is not entirely clear to me if every cycle graph of a group is isomorphic, but they are for small groups at least. Here are a few pictures from wikipedia: the elementary abelian group of order 16, the dihedral group of order 12, and the modular (but not Dedekind) group of order 16.

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Jack, this is exactly what I had in mind. I scoured wikipedia for a while after the initial answers to this question, with (for some reason!) no luck. Thanks again to you and the others. – The Chaz 2.0 Mar 10 '11 at 20:24

See the Cayley graph and Schreier coset graphs.

I had much enjoyment playing with such graphs when I was an MIT undergrad. In fact I recall that I submitted a very interesting one to prove a hairy identity on a problem set in a knot theory class taught by George Whitehead. He found it quite interesting (and amusing). Apparently no one had done if before so simply and vividly using a Cayley graph. It was not an easy task to impress Prof. Whitehead. Such is the beauty that some of these geometric techniques bring to the austere algebra. They are well worth your study.

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Dub: jinx. I think we posted at the same time. haha. – user02138 Mar 10 '11 at 4:52
Thanks, guys!.. – The Chaz 2.0 Mar 10 '11 at 4:55
@use: Indeed, you posted only one minute after I. No doubt neural waves travel quicker in Cambridge! (we're very close). – Bill Dubuque Mar 10 '11 at 4:56
Does this kind of representation have a distinct name: ? – Uticensis Mar 14 '11 at 3:22
@Billare: I haven't had a chance to look closely at that, but it appears to be some type of Petrie polygon, e.g. see Coxeter and Moser, Generators and relations for discrete groups. – Bill Dubuque Mar 14 '11 at 3:37

I believe you are referring to the Cayley Graph of a group.

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