Talk:Cycle graph (algebra)

Could the introduction please clarify when elements a and b are joined in this graph?

Charles Matthews 16:35, 2 Apr 2005 (UTC)

If I understand correctly, we choose a generator g for each maximal cyclic subgroup, then we join g^n-1 to gⁿ for each generator g and each positive integer n. David Radcliffe 22:59, 12 March 2006 (UTC)

This desperately needs to go into the article itself. ciphergoth (talk) 17:32, 29 July 2016 (UTC)

Tetrahedral diagram

I removed the tetrahedral group diagram, because it is both more and less than a cycle diagram. If I was a newcomer wishing to learn about cycle graphs, that diagram would be pretty intimitdating. It contains WAY more information than a simple cycle graph, yet it FAILS to point out the identity element, which is necessary for a cycle graph. I think this diagram should go to an article specifically dealing with the A4 group, perhaps the alternating group article or the tetrahedral symmetry article. PAR 00:10, 8 January 2006 (UTC)

Hello Par - I think it was a mistake to remove the tetrahedral group cycle graph (not that I am biased...). Here's why. Wikipedia math entries should be ideally accesible to anyone, not just the already math-literate or math-educated. We don't need it to become another Mathworld clone. So I made that diagram with the specific intent of having the concept of a group cycle graph, along with other concepts such as symmetry group/rotation group etc, be presented visually. Many people learn best visually, and to be able to see how doing three 120degree rotations brings you back to the original position will help someone understand what a group cycle means, in a way that (a²b)²=e might not. As for not showing all features of a graph cycle (such as labelling e), well, that can be easily addressed. What do you say? Debivort 03:46, 8 January 2006 (UTC)

Hi - Regarding the diagram in the cycle graph (algebra) page, I'm not convinced, I still think its too complicated, it requires some serious study in order to understand it. Cycle graphs are so simple, I don't want to complicate things. I am not infallible, however. I have asked User:Patrick to check in on our discussion. He is a contributor to these and similar pages where cycle graphs are used (e.g. List of small groups. Can we agree to go with what he decides? -- (PAR) [excerpted from an email to user:debivort, placed here with permission]

I think a community consensus is more appropriate than either of us agreeing on some binding arbitration. I feel that cycle graphs are simple because they abbreviate lots of other information, such as what an edge indicates. In my figure, that information is given (for the particular example illustrated). I think a more appropriate person to ask might actually be a non-specialist, and they can say whether or not the image builds their understanding or detracts from it from a more objective stand point. That is, if you think it is important that non-specialists be able to understand these sorts of pages.

You'll also notice that I didn't just go put the image back up. I do think it contributes to the article, but I am not rabidly committed to having it there. It already has a decent home in symmetry group, and it looks User:Patrick has also already placed it in tetrahedral symmetry, so if it doesn't end up on cycle graph so be it. I just feel it is a good contribution for non specialists. Debivort 06:46, 8 January 2006 (UTC)

The diagram is perhaps somewhat too complicated as a first example at the top, especially for people who are less geometrically inclined, but we could put it lower, as one of the examples.--Patrick 11:45, 8 January 2006 (UTC)

I've never understood the actual need for this kind of graph. The reference to the Shanks book - what does it say? Charles Matthews 11:01, 8 January 2006 (UTC)

Responding to Charles Matthews - I think it was Richard Feynman who noted that there are word people and picture people. Some people process information more verbally, others more pictorially. Being a severe pictorial type myself, my first reaction to your question is "you're pulling my leg, right?. A cycle graph reveals a huge amount of group structure in one glimpse, while a group table is a mind-boggling mish-mash of symbols that has to be scanned over and over again for structure." But then, I remember Feynman's comment, and I change that to "because for certain people, it is a very informative way of displaying certain aspects of group structure." PAR 14:46, 8 January 2006 (UTC)

Certainly the tables are bad. But some of these diagrams (order 16 groups) mean little to me, and I taught this stuff to students for at least five years. Is this standard, or original research, I ask? And, once more, does the reference refer to it? Odd for a number theory book, really, though I could see it doing the finite abelian case. Charles Matthews 14:57, 8 January 2006 (UTC)

I don't own the Shanks reference, but if I remember correctly, it only gave one example of a cycle graph for a group. The concept is certainly not original research by me (I wish), although I did calculate many of the diagrams from scratch. See also the mathworld entry at http://mathworld.wolfram.com/CycleGraph.html. Maybe this should be mentioned as a reference as well. PAR 18:05, 8 January 2006 (UTC)

Definitely pretty standard stuff, not Original Research (sorry PAR and Patrick). That link would be nice, I'll add it. Debivort 20:55, 8 January 2006 (UTC)

I took out the proof that cycles do close; it isn't in cyclic group after all, but I don't think we need it. This is an encyclopedia, not a text book. Septentrionalis 23:42, 8 July 2006 (UTC)

Z_n^m

Groups of the form (Z_n)^m will have (n^m-1)/(n-1) n-element cycles sharing the common identity element.

Are you sure? Let Z_4^2 = <a,b: a^4=b^4=1, ba=ab>. The cycle {1, a, a^2, a^3} and the cycle {1, a.b^2, a^2, a^3.b^2} share both 1 and a^2. I think your diagram is wrong. --74.100.225.24 02:32, 15 May 2007 (UTC)

Right - Its fixed. PAR 11:48, 23 May 2007 (UTC)

Cycle graph does not give enough information to uniquely define the group

The article completely fails to mention this, and in my opinion, this is a really important detail that should be in the article (as it's easy to naively think that this is true since the first counterexample is a pair of order 16 groups, one of which is nonabelian, and hence is likely to be unfamiliar to someone that is not well read regarding finite group theory and cycle graphs). If my memory serves, this first example used to be in this article and in fact, this was where I learned that cycle graphs are not enough information on their own. I think it should have a short section at the end, with a brief mention of the fact in the introduction (with reference to where it is in the article). 76.14.232.104 (talk) 06:00, 24 August 2015 (UTC)