# Talk:Fundamental theorem of cyclic groups

WikiProject Mathematics (Rated Start-class, Mid-importance)
This article is within the scope of WikiProject Mathematics, a collaborative effort to improve the coverage of Mathematics on Wikipedia. If you would like to participate, please visit the project page, where you can join the discussion and see a list of open tasks.
Mathematics rating:
 Start Class
 Mid Importance
Field: Algebra

## Justification to add new proof, using Bézout's Identity

As requested by Arthur Rubin, my justifications for adding this new proof to teh article are:

• While it is equivalent to teh first, it is smaller and lighter to read.
• It emphasizes teh that $g^a, g^b \in H \Leftrightarrow g^{gcd(a, b)} \in H$

Also, it could be improved and made shorter, without extra complexity.

--AYGHOR (talk) 09:36, 25 May 2012 (UTC)

## Merger proposal

There was a body of opinion at Wikipedia:Articles for deletion/Fundamental theorem of cyclic groups that this article should be merged into Cyclic group: I don't think it can quite be described as the consensus, though. My point was that while this is clearly a name for the theorem, there seems no reason to believe that it is the name. There is essentially only one book in Google Books using this name (the alternative name "Characterization Theorem for Cyclic Groups" mentioned above scores two) and 12 hits for the phrase on Google Scholar. Compare with 37,700 hits on Google Books for "Fundamental theorem of algebra". Deltahedron (talk) 18:07, 30 September 2012 (UTC)

My opinion is that there is enough material on this specific theorem to warrant having a separate article on it. I don't feel that very strongly, though, and I don't much care whether we keep the present article title or change it to something else, such as Subgroups of cyclic groups. —David Eppstein (talk) 18:38, 1 October 2012 (UTC)
Why not? The theorem itself is too trivial to deserve an article on its own right. Incnis Mrsi (talk) 19:00, 1 October 2012 (UTC)
It's nontrivial enough to get a page of coverage in several books. In any case, triviality is not the correct criterion, rather it should be notability: is this theorem the subject of nontrivial coverage in multiple reliable sources. I think it clearly is. We have the sources in the article, and even the AfD nomination statement agreed that "this is a standard homework problem" which to me indicates that it is well-known and therefore notable. —David Eppstein (talk) 00:59, 2 October 2012 (UTC)
I'm definitely in favor of a merge. Everything here is already in the Cyclic group article except for the Generalization section. That section is pretty clearly unreferenced original research. I'm not sure whether the theorem really gets a page of coverage, I can't think of any important consequences of the theorem, everything just comes down to the fact that cyclic groups aren't very complicated. Dingo1729 (talk) 17:28, 8 October 2012 (UTC)
I've removed the unreferenced section (which wouldn't belong in a merged article) but I've also removed the merge tags, and instead used the {{main}} template to link here from the cyclic group article. The reason is that I think this article now covers the subject in more detail than the coverage of the same subject in the cyclic group article and I worry that expanding the coverage there would unbalance the article. —David Eppstein (talk) 03:17, 24 November 2013 (UTC)

## Infinite Case

Of the three references for the theorem only [1] uses the name "Fundamental Theorem". Only reference [2] includes the part "For the infinite cyclic group, every subgroup is infinite cyclic, of finite index, and for each integer i there is exactly one subgroup of index i". I know that it's both true and simple to prove. It may even be included in other nearby propositions or theorems. But it doesn't seem right to include it from a reference that doesn't include the name of the theorem. I've removed the sentence. Dingo1729 (talk) 03:44, 2 October 2012 (UTC)

I think WP:NOTDICT is relevant. Our article should cover the mathematics, not the name. And the relevant mathematics includes infinite groups, whether or not it's in the source with the name. Perhaps the material you removed should not be in the lead, but it should certainly not have been eliminated from the article altogether. —David Eppstein (talk) 04:01, 2 October 2012 (UTC)
I agree with the general point about covering mathematics rather than a name. Unfortunately, this article is a bit of a straitjacket because there seems to be only one author who uses this name. We can't just re-write stuff the way he should have written it. It's a bit of a misnomer too, because it really isn't fundamental in the sense of having lots of other stuff built up from it. It's just facts about finite cyclic groups. We could avoid this problem if the material were merged into the main article. Dingo1729 (talk) 13:35, 2 October 2012 (UTC)

## Suggest move

Fundamental theorem of cyclic groupssubgroups of cyclic groups – We have lots of high-quality textbook sources covering the topic of subgroups of cyclic groups (the content of this article regardless of what it's called), so I think it is a worthy subject for an article (subsidiary to the main cyclic group article per Wikipedia:Summary style). However, I agree with the criticism above that the name "fundamental theorem of..." is not the best choice. The proposed move might ease some of the objections to this article, which as I understand it are that this is not fundamental enough to be called a fundamental theorem and that really only one source uses that name. Although most Wikipedia article titles are singular I think plural works better in this case because most of the material in the article considers all subgroups at once rather than a single subgroup in isolation. --Relisted.  — Amakuru (talk) 13:26, 2 December 2013 (UTC) David Eppstein (talk) 03:41, 24 November 2013 (UTC)

Support. Though I still think this should be shortened and merged into Cyclic groups. At some stage the amount of stuff we write obscures the basic simplicity of these groups. (That wasn't meant to mean they are all simple groups.) Dingo1729 (talk) 16:45, 24 November 2013 (UTC)

Support/merge: actually I still think the merger is a good idea. The problem remains: why do we need a "separate article" on subgroups of a cyclic group, a theorem about it or otherwise. If cyclic group gets too long, we can always split it off. It should also be pointed out that cyclic group already has a discussion on subgroups. -- Taku (talk) 16:32, 25 November 2013 (UTC)