# Talk:Abelian group

WikiProject Mathematics (Rated B-class, Top-importance)
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Field: Algebra

## Typo?

Unless I'm too tired, there's an error in the table of small groups. The subgroup list of Z24 and Z42 has been interchanged, but I don't know how to correct it...

Now I've changed it. Can someone please check it?

## Capitalization

Wikipedia articles are fairly consistent in writing "Euclidean", "Newtonian", "Eulerian", "Riemannian" with upper-case first letter. Is there any reason why, instead of "Abelian", mostly "abelian" is being used?
S.

We follow the mathematical usage; "abelian" is much more common than "Abelian". Generally, it is considered to be an honour if you have made it to an adjective and are written in lower-case. AxelBoldt

## Humor of anonymous edits

A stalwart from our lexicon of maths jokes to get us through lectures: What's purple and commutes? An abelian grape.

Just as bad is "What's yellow and equivalent to the Axiom of Choice?"

Zorn's lemon JHobson3 (talk) 11:19, 11 March 2014 (UTC)
Yet another is "What is furry and equivalent to the Axiom of Choice?"
Zorn's lemmingAnita5192 (talk) 17:56, 11 March 2014 (UTC)

So, in the unlikely event that you were wondering, that's why the recent anon made those pecular edits! Pete/Pcb21 (talk) 12:45, 9 Dec 2003 (UTC)

Could be someone who's central heating's out and has to travel on a very cold bus to work? Dysprosia 12:47, 9 Dec 2003 (UTC)

## Translation of Gruppentafel

I don't know the right word, so I'll post my addition on this talk page, so that someone can put the corrected version in the article.

A finite group can easily be checked to be abelian by creating its group table (what's "Gruppentafel" in english?): The group is abelian iff the table is symmetric along the main diagonal. --SirJective 12:10, 19 Dec 2003 (UTC)
Group table, probably. Would you like me to add this, or do you want to go ahead? I have something to add related to your potention addition... Dysprosia 12:14, 19 Dec 2003 (UTC)
I'm a bit unsure about my language being correct. After 8 years of English lessons at school I can read and understand the most, but never was good at writing things myself. So I would prefer you to add my sentence. It irritates me that I cannot find the group table in this wikipedia, nor via google... A search in MathWorld yields the term multiplication table... That seems to be the right term!
You may also want to look at Cayley diagram or cayley table as it applies specifically to groups.
See also Talk:Cantor-Bernstein-Schroeder_theorem for another addition, which hasn't inspired anyone since the end of october. --SirJective 12:41, 19 Dec 2003 (UTC)
For this, as you can see, I've avoided the problem altogether :) Dysprosia 23:45, 19 Dec 2003 (UTC)
Fine, thanks. 217.80.248.173 12:24, 20 Dec 2003 (UTC)

## lowercase notation a great honor?

Can someone please explain or justify this rather bizarre sounding remark? -Lethe | Talk 15:50, May 5, 2005 (UTC)

First, you are referring to the very first sentence on this talk page. That could have been made explicit, since I first looked at the page history, then at the article, and only later here. But I found it. :)
I think it means to say that once words become generic and widespread, they usually get to be lowercase. Like "gramophone" which I think used to be "Gramophone". Basically the point is that "abelian" is now the defacto spelling, and not "Abelian", which means this lad, called Abel, is very famous now. :) Oleg Alexandrov 16:58, 5 May 2005 (UTC)
So actually I was referring to the final section of the article where it reads:
"The abelian group is rare in being expressed with a lowercase a, rather than A."
I completely missed what that sentence was saying. I thought that somehow denoting abelian groups with a lowercase letter (i.e. let a be the infinite cyclic group) was supposed to be a great honor to the group. Which was fucked up for a whole bunch of reasons: we don't honor mathematical objects, we honor people; most people write groups with capital letters, or at least no one writes the names of ablelian groups in particular as lowercase.
But now I see that the "a" in question was the first letter of the word "abelian", and everything makes perfect sense. Thanks Oleg. -Lethe | Talk 23:02, May 5, 2005 (UTC)
I was thinking about it, and it always seemed to me that when we write abelian group (or sometimes I'll see, for example, euclidean geometry or something), it's lowercase because the word has become so common, not because we're trying to bestow even greater honor on Abel. True, it is a great honor that one of his mathematical inventions is so important and ubiquitous that we deem it a nonproper word. but the lowercase letter isn't there for the honor, it's there to indicate the commonness of the word. Or at least that's my opinion. -Lethe | Talk 22:51, May 12, 2005 (UTC)

## Groups from rings

Every commutative ring gives rise to two abelian groups in the same fashion

Shouldn't it be: "every unital commutative ring"? What if there are no units in a ring? Or is that not possible? Andres 21:45, 8 May 2005 (UTC)
Wikipedia's definition of ring doesn't allow non-unital rings. --Zundark 22:05, 8 May 2005 (UTC)

## List of Small Abelition Groups: Notation

I wanted to have clarification on notation used in the list of small abelian groups. Frequently one encounters, in the subgroup section, things like n x Sn where n is an element of the natural numbers. I have never seen this notation before. For instance, the article states that a subgroup of Klein's 4-group is 3 x Z 2. What does the 3 represent? Clearly all subgroups of Klein's 4-group contain 2 elements and hence are isomorphic to Z2. Does the 3 just mean there exist 3 such isomorphic subgroups? If so the fact there are three of them is a bit of a triviality, and I wonder if it only confuses things. -- Shawn M. O'Hare 13:07, 5 November 2005 (UTC)

Yes, that is what is meant. I do not think the number of subgroups of the same type is in general obvious and/or uninteresting.--Patrick 13:52, 5 November 2005 (UTC)
I explained the notation.--Patrick 13:59, 5 November 2005 (UTC)

## Notation for powers

in the table for notation under powers it is unclear what 'na' stands for, is it not-applicable or n * a, given that right underneath we have a^n. i am assuming the former, but i don't know with certainty so do not wish to edit the page. --201.130.133.221

It's just the notation: in additive notation we write nx rather than xn. --Zundark 08:13, 11 April 2006 (UTC)

## Identity

Do ableian groups include the identity element? --HappyCamper 23:39, 14 September 2006 (UTC)

Any group, abelian or not, has an identity element. It's a requirement for it to be a group. - grubber 00:03, 15 September 2006 (UTC)
Hm...I should have known that. Thank you. --HappyCamper 00:28, 15 September 2006 (UTC)
No worries :) Group theory can be a lot to handle some days! grubber 14:35, 15 September 2006 (UTC)

Just to make this clear, because someone could get this wrong: Every group has an identity element ("includes the identity element"). But in an abelian group that is written additively, i.e. with +, the identity element is 0. E.g. every subgroup of the group of integers (under addition) contains the identity element 0, but the even numbers form a subgroup that does not contain 1 (which is not an identity element in the sense of the group we consider). --Hans Adler (talk) 11:36, 28 March 2008 (UTC)

## Finite abelian groups

The large section on finite abelian groups is not really appropriate here. It would be more appropriate in the article on finitely generated abelian groups. But even better, I think, would be to make it a separate article. I intend to do this if there are no objections. --Zundark 09:48, 16 July 2007 (UTC)

## My favourite mathematical joke

Q: What's purple and commutes?

A: An Abelian grape. —Preceding unsigned comment added by 24.15.135.55 (talk) 19:24, 14 October 2007 (UTC)

## Abelian vs. commutative

For groups these two words are clearly synonyms. But when I read "Let G be a commutative group", then a priori I expect it to be written multiplicatively, while when I read "Let G be an abelian group" , I expect it to be written additively. Are there any sources that discuss this question? I couldn't find any, and of course I may just be wrong. --Hans Adler (talk) 11:56, 28 March 2008 (UTC)

## Direct sum/product

I removed direct sum/product from the table comparing multiplicative and additive notation, since I think it is highly misleading to regard the difference between them as merely a difference in notation. Even if $G \oplus H$ and $G \times H$ are isomorphic for abelian groups, the same does not generally hold for sums $\bigoplus G_i$ and products $\prod G_i$ taken over infinitely many groups. —Preceding unsigned comment added by 213.113.150.132 (talk) 21:05, 9 June 2008 (UTC)

Agreed on the removal, for this and other reasons. The table now is much more to the point. For operations other than on group elements, the influence of additive or multiplicative notation is much less standard (where do automorphisms go? if this group acts, how is the action denoted?), and would just pollute the table. JackSchmidt (talk) 21:46, 9 June 2008 (UTC)

At the London wikimeet today we were discussing the problem with technical pages and how many users find our articles too difficult to read. This is especially true with mathematics articles but also of other subjects, medicine springs to mind. I came up with idea of a summary box for the article, a sort of "in a nutshell" box for articles like we have with policies.The idea was thought to be a good one (among the small group there) and James F. got straight to making the template as soon as he got home from the meet. Gordo suggested Abelian group as a topic that it would be very difficult to do this for so I've had a go as an experiment. Please feel free to edit the lead section bearing in mind that it is intended for non technical readers who will probably not understand the rest of the article. It is not intended to replace the lead proper of the article. Also please do comment on the merits of the idea. Either here, or on the template talk page. Theresa Knott | The otter sank 22:14, 10 August 2008 (UTC)

Sorry Theresa, but in my opinion the template is ugly and unnecessary, and the text is more confusing than the old lead that it has replaced (the potted explanation of a group is incomplete; "the order of the elements is unimportant" is waffly and nearly meaningless; reference to matrix multiplication is unnecessary - if reader does not know what a group is, will they know what a matrix is ?). If it supposed to be "as well as" the old lead section then it is distracting and unnecessary, and if it is supposed to be "instead of" then it makes the article worse. I am all in favour of improving lead sections, but IMO introducing complicated and intrusive templates is not the way to go. (BTW, I have mentioned your request for comments at Wikipedia talk:WikiProject Mathematics) Gandalf61 (talk) 21:05, 11 August 2008 (UTC)
Actually most people with an American undergraduate degree who only took math their first couple years will be in the situation of not knowing what a group is but knowing what matrix multiplication is. I think Theresa's on to something with that observation. --C S (talk) 17:04, 12 August 2008 (UTC)
Theresa, what does "it is not intended to replace the lead proper" mean? The template has replaced the lead proper at the moment, hasn't it? I also think that's what the template page says. So as far as I can see, all the template does is to have the lead section in a bigger font and have a box around it? -- Jitse Niesen (talk) 21:58, 11 August 2008 (UTC)
There seems to be some disagreement between Theresa and Jdforrester on that point - Theresa added it as an extra, then Jdforrester removed the original lead. --Tango (talk) 22:23, 11 August 2008 (UTC)
Looks ugly to me. We should concentrate on writing good leads, not on putting funny boxes around them. --Tango (talk) 22:23, 11 August 2008 (UTC)

My intention was to add it as an extra. My concern is that too many articles are worded so technically that they put off the people we want to read them. For example many people are unfamiliar with the use of * as a notation for an operator an if the read "so that for all a and b in G, a * b = b * a" will have no idea what the article is talking about and will read no further. I think it very important to give an easy non technically worded section expressly aimed at non mathematicians who are unfamiliar with mathematical writing.

The idea of using a box is to give the reader a visual clue that the contents of the box contain an easier,less formal, text than the main article that can be read on its own, and gives the more serious reader a chance to skip this part and scroll down to to the main part of the article skipping the summary if they want. Is also gives us a visual clue that stuff inside the box needs to be as accessible to as many people as possible, and we need to take extra care to include as little jargon as possible as well as some examples.

As for the actual contents of the box. It was my first attempt at it! If you feel it isn't good enough then edit it. Personally I learned what a matrix is at 15 and didn't learn what a group was until university. But I take your point that many readers will not know what a matrix is. So come up with a better example. Theresa Knott | The otter sank 06:42, 12 August 2008 (UTC)

I oppose putting a box round leads. Many articles already contain 2 boxes at / near the top, the TOC and an infobox, and boxing the lead would distract attention from these. The example used in the trial also had rather short leads, and I hate to think how a box would look round a longer lead, e.g. Dinosaur. -- Philcha (talk) 07:37, 12 August 2008 (UTC)

The idea is not about putting a box around the lead but about putting a short summary of the article at the top.

Dinosaurs were the dominant vertebrate animals on land from the late Triassic period (about 230 million years ago) to the end of the Cretaceous period (65 million years ago), when most of them became extinct in the Cretaceous–Tertiary extinction event. They are closely related to birds and most paleontologists regard birds as the only surviving dinosaurs.

The term "dinosaur" was first coined in 1842 by Sir Richard Owen and derives from Greek δεινός (deinos) "terrible, powerful, wondrous" + σαῦρος (sauros) "lizard".

The idea it is that it is a summary of the article. Not a replacement for the lead proper (I know James sees it differently). Theresa Knott | The otter sank 07:51, 12 August 2008 (UTC)

(ec) I do see some merit to the summary as in this revision where it complements the introduction. But I see no point in this revision where the whole lead is boxed. In any case the intro text is a lot easier to read than the current revision. I do actually like the larger font size, it makes it clear that this is the minimal amount you need to read. --Salix alba (talk) 07:56, 12 August 2008 (UTC)
Theresa Knott's example can be interpreted another way - ideally the 1st para of a good lead summarises the lead. But I wouldn't bet that it works for all topics. I still wouldn't like to have yet another box, for the reasons stated above. -- Philcha (talk) 08:11, 12 August 2008 (UTC)

The discussion at the Wikimeet was that every article should have a summary as the first section. The "box" could be a style choice for the reader. I like the box, but it should be a choice, and the summary information must be there for articles of any reasonable length. Gordo (talk) 08:19, 12 August 2008 (UTC)

But isn't this what WP:LEAD already says? "The lead serves both as an introduction to the article below and as a short, independent summary of the important aspects of the article's topic." -- Jitse Niesen (talk) 10:43, 12 August 2008 (UTC)
Precisely. All this seems to be is that a few people don't think our current leads are very good. The solution is to fix them, not to add something else. --Tango (talk) 14:53, 12 August 2008 (UTC)

I don't think Theresa's idea should just be dismissed so easily. Formatting can make a big difference in readability. Yes, even putting a shiny box around something. In fact, I'm reading a fairly serious book on 4-manifolds right now that has 4 layers of text: each layer is in smaller font than the previous and goes into more details. it works surprisingly well (and a good choice of font for each level seems crucial). I probably wouldn't advise this particular changing font size approach, not because it's not good or appropriate, but in reality, to do it well would require far more effort than people are putting into articles (in general) right now.

Another more prosaic reason to allow this is if it gets lay readers interested in making such boxes, actually reading these articles and asking for input and trying to create nice little blurbs, "it's a good thing". Look at the folks interested in infoboxes, even though such boxes rarely provide useful additional info (for which I am against them). Hopefully with Theresa's "dumbing down" box, people will find they are useful. Seeing the discussion below is a bit funny with regard to the "dumbing down" issue. A well-written (but wrong) summary has metamorphisized into a correct (but impenetrable) one. Apparently "order of the elements" is meaningless to some here, although I found it makes perfect sense and partial vindication for this wording is that a layman also came up with it. Even serious texts (unreadable by those without a math Ph.D. essentially) sometimes have "dumb down" moments, where they might say "ok, pretend for now this space is compact and all such functions on this space converge" and then 50 pages later they will say "ok, it's not really compact, we have to use the Hirzebruch-Milnor-Leray spectral compactification as modified by Grothendieck to get it to be gnarly-compact. Even though it's not compact, we will be able to get a convergence result for all blah-functions on the gnarly-compactification" (ok this example is made up, but it's in line with what occurs in these cases). My point is that mathematicians sometimes forget that even serious mathematicians "lie" in order to make a good exposition. So what if the boxed material is not pinpoint accurate? As long as people have the expectation that the boxed summary is just a loose summary, it shouldn't pose a problem. It's all a matter of expectations. --C S (talk) 17:04, 12 August 2008 (UTC)

You'll note that we deliberately used 120% text inside the box, following the usual convention that the smaller the text, the harder the material. Asl the appearancve of the template could easily be tweaked to make it less obtrusive. What if the background wasn't blue? Do we actually need the surrounding box? (I think we probably do but perhaps we could try simply having the bigger text and seeing if that is enough. Does it have to go in the lead? Again i think it does but I'd be willing to give it a go at the end of the article if that is more acceptable. Theresa Knott | The otter sank 00:00, 13 August 2008 (UTC)
I think in terms of visual presentability, it is good. I think putting it at the end would be more than useless. I think the real issue here is what an "encyclopedic" article is supposed to look like. For example, one could argue that like in magazine style, we could have the first few sentences of the lede in larger text. This would impact more than the visual aspects, as I expect people will work harder on those initial sentences once they realize it is much more visible. I don't think you are just asking to have the first few sentences set off in larger font though. I think you actually want the very beginning to be an introduction to the more serious lede and it should be even more informal than a "serious encyclopedia article" is generally held to be. Correct me if I'm wrong. Unfortunately, even if math project was willing to go along with this, there is always resistance at large from people who are "serious". I'm reminded of the whole bring radical image incident. Math editors were actually fairly amenable in general to having the cartoon. But no, the "serious" editors stepped in and said no way Jose this is an encyclopedia! --C S (talk) 15:09, 13 August 2008 (UTC)
Looking through the Bring radical discussion, I made the following remark: "There is much on Wikipedia that strictly speaking, does not convey information in terms of some fact. There are materials that lend additional insight, make the learning experience more pleasurable, and work to reinforce material that is explained in another part of the article." I think Theresa' proposal ought to be judged in this light. For example, does it "make the learning experience more pleasurable?" I'm not convinced of that, but on the other hand, I have to foolishly admit that the font size in the book I mentioned I'm reading does make the whole experience much more enjoyable. --C S (talk) 15:18, 13 August 2008 (UTC)
Anything which can make the task of writing an accessible but technically thorough article should be encouraged. However, I am also not convinced of the efficacy of this particular template. If editors can make good use of it in their own style, then they are welcome to it. But it has the feel of something that is about to be "imposed from above", particularly since the template is currently editprotected for no good reason. siℓℓy rabbit (talk) 16:17, 13 August 2008 (UTC)
Yes, that seems odd. There are already three people editing the (protected) template. Is this meant to be an inclusive or exclusive proposal? --C S (talk) 16:40, 13 August 2008 (UTC)
I think it was protected for the same reason that templates are always protected. They are a target for vandals as they are transcluded to many other pages. In this case hpwever i think protection is rather premature, as we haven't had the chance to decide on the apperance. I shall unprotect it. Feel free to tweak all you want. Theresa Knott | The otter sank 06:34, 14 August 2008 (UTC)

New section to separate discussion of lead text from discussion of boxes, formats etc. above.

I was about to revert back to the old lead text again, but I see Zundark has already done that. Sorry, but there are just too many problem with the new text:

1. "A group is ... a set of elements, and an operator which have the property that when the operator acts on any two of the elements the result is also a member of the group" : Yes, as far as it goes, but many other algebraic structures have this closure property as well, and a group is much more than that - this is like saying a horse is an animal with four legs. What this sentence actually defines is called a magma.
2. "for the set of natural numbers the addition operator creates a group" : No it doesn't - where are the inverses ? The natural numbers under addition (assuming we include 0 as a natural number) form a commutative monoid.
3. Matrix multiplication is a terrible example for non-abelian groups. It is overly complex, and some groups of matrices under multiplication are abelian anyway e.g. 2x2 rotation matrices. The simplest and best example of a non-abelian group is the permutation group on 3 elements, S3.

Actually, I think the old lead is pretty good anyway. But if we are going to discuss improvements to the lead text, let's at least start from a version that is correct. Gandalf61 (talk) 09:21, 12 August 2008 (UTC)

Yes that is the problem with me ( an non mathematician) writing a lead text. The problem with having a mathematician editing it is they tend to use more abstract notation and jargon than is really necessary.Which puts of some readers. Perhaps an ideal solution would be for the mathematicians to correct the problems in the text rather than revert an attempt to make the lead more user friendly? Theresa Knott | The otter sank 09:30, 12 August 2008 (UTC)
Theresa - I won't personally be attempting to fix the problems in the proposed new lead text because I think the old lead text, although not perfect, is far better, and I see no point in spending time trying to improve a 40% solution when we already had a 90% solution there anyway. I must admit that I am struggling to see what you are trying to achieve here. The first sentence of the old lead is already an accurate and concise summary of what an abelian group is - it is a group in which the group operation is commutative. Yes, the reader has to understand what "group", "group operation" and "commutative" mean, but if they don't understand these terms then they can read the linked articles. We should not try to import a definition of a group into this article, because then we would also have to explain what a set is and what a binary operation is etc. etc. Gandalf61 (talk) 10:07, 12 August 2008 (UTC)
The first sentence of the old lead
In mathematics, an abelian group, also called a commutative group, is a group (G, * ) with the additional property that the group operation * is commutative, so that for all a and b in G, a * b = b * a.
Illustrates a lot of problems common to mathematical/technical article leads.
1. Prior knowledge - it requires the reader knows what a group is, yes they can click on the link, but this can lead to a definition chase through other links. For example Group requires the reader to click on set. So the uninformed reader have already left the page by word 12.
2. Unnecessary notation. The notation (G, * ) will be novel for reader below undergraduate level, causing further confusion. The notation is not necessary for understanding the concept so could be moved later.
3. Terms before definition. The word commutative is introduced before its meaning a * b = b * a. Switching the order to say: An abelian group satisfies the property a * b = b * a this is called commutative operation. The order subtly changes how a reader interprets the text. The current way round they panic at another unknown word, with the other order its "yes I understand a*b=b*a", "ah thats called commutative".
--Salix alba (talk) 10:41, 12 August 2008 (UTC)
I agree with your second and third points, but not the first. I don't think that the first paragraph of the article on Abelian groups should be spent discussing groups in general. If we want to repeat the definition of groups in this article, we should do so in a section Background just after the lede. But I think it's better to link to our article, in a way that makes the priority clear, as the current first sentence does. I think the untrained reader you are concerned with wouldn't be able to read a two-sentence summary of groups and come away feeling they grasp the topic, anyway.
Another option would be to rearrange the lede:
In abstract algebra, an Abelian group consists of a set and a binary operation that is associative and commutative, has an identity element, and under which each element has an inverse. An example is the set of integers together with the addition operation. Abelian groups can be equivalently defined as groups in which the operation is commutative.
That's pretty rough, but it includes the first-level definition without being too verbose. — Carl (CBM · talk) 11:47, 12 August 2008 (UTC)
You've just added 5 extra words that need to be defined in order to remove 1. What's the point of that? I really don't see the point in not assuming that someone reading about a special case of a group doesn't know what a group is. We don't define "number" in our article on 5, we just link to the article and assume people will read it if they want to know more. --Tango (talk) 14:52, 12 August 2008 (UTC)
I think there might be a lot of people who encounter an Abelian group before they encounter a group. There are 250+ articles linking to this one, people are likely to come across this page
In Algebra an Abelian group consists of a set of objects and a binary operation which satisfies a*b=b*a (commutativity) as well as the other group axioms (has an identity element, every element has an inverse, the operation is associative). An example is the set of integers together with the addition operation.
Note in the above I've linked to Algebra#Abstract_algebra which probably has the simplest definitions of the terms involved. --Salix alba (talk) 17:02, 12 August 2008 (UTC)

I support reverting to the old lead and going from there, eg with Salix alba's first sentence. Explanations can be put into the first section of the article proper, overburdening lead with details defeats its purpose. Incidentally, this article is very thin on substance, dealing as it does primarily with the classification of finite abelian groups and their automorphism groups. No history, no description of the scope of the subject... That should be the main focus of the impromptu "improvement drive", now that we've got plenty of interested editors here. Arcfrk (talk) 23:31, 12 August 2008 (UTC)

Although I am the author of the new lead, I don't particularly care for it myself. My main concern with the old lead was that it ran afoul of WP:MTAA, although I have historically been a strong believer that WP:LEAD takes precedent. If you want to revert it, go ahead; it's fine with me. I'm content to let it stay until someone else with a strong opinion steps in and either wrestles with the lead in its present form, or hits the undo button. I won't be offended either way. (Well, as long as the edit summary isn't something like: "ROTFLMAO! What idiot wrote this?!") Cheers, siℓℓy rabbit (talk) 23:41, 12 August 2008 (UTC)
This is the beauty of having a summary box. The box is complies with WP:MTAA whilst the lead section of the article complies with WP:LEAD. Why settle for either/or when we can have both? It's not as if Wikipedia is going to run out of words if we have some repetition between the summary and the lead section. Theresa Knott | The otter sank 23:53, 12 August 2008 (UTC)
As I understand it, WP:MTAA is a guideline that applies to the whole article, not just its lead. Our objective should be to write accessible articles that start with accessible leads. If an article is accessible and the lead conforms to WP:LEAD then a summary box is unnecessary. If an article isn't accessible then a summary box won't fix that problem.
This article is in fact pretty close to WP:MTAA - it is in order of increasing complexity; it gives examples; it uses active sentences; maybe it needs a picture or two. I still don't understand exactly what problems the summary box and the summary text in it were meant to fix here. Gandalf61 (talk) 08:39, 13 August 2008 (UTC)
The main problem with meeting WP:MTAA is that anyone who is not a mathematician won't be able to understand anything after the first 12 words. Ty 14:38, 13 August 2008 (UTC)
Unfortunately, it is ignorant comments like this that make the mathematicians take people like you less seriously. That also leads to a tendency to group people like you with other people who make complaints for valid reasons. --C S (talk) 14:51, 13 August 2008 (UTC)
As I understand Theresa's proposal, I believe what is being proposed is a kind of "cartoon" summary. Something that is not quite as dry and encyclopedic as a lede is expected to be (according to current norms). The language would be less precise, more looser. I think she wants something that is even less formal than math articles where people have really tried to make the lede informal and approachable. The second aspect of the proposal is to have something set off from the main text in bigger text, as visually, it could make a difference in readability for general readers. It might be good for Theresa and others for this proposal to take a look at group (mathematics) and see if the lede there already does what they want, in terms of the content (putting aside the issue of visual presentability). It has been worked on extensively. --C S (talk) 14:55, 13 August 2008 (UTC)
The current lead might suffer from trying to define the object (twice). The lead just needs to indicate to the reader why the article is interesting and what the article covers. The reader does not need to understand the object being described by the article, only how the article describes it.
One might add some language to the lead that indicates that abelian groups are one of the first objects studied in undergraduate abstract algebra (easy to cite from intro algebra textbooks TOC and preface), that they form the basis for many other important algebraic objects studied in introductory algebra such as rings and modules (first paragraph of the abelian groups chapter), and are an area of active research (MSC classification).
The article itself is basically stub class except for a few topics that are treated exhaustively. Roughly speaking, none of the content of Fuchs's Abelian Groups is covered, and this is a fairly standard textbook on the topic. It is hard to write a good lead to a stub, so it might be better to make the article not a stub first. JackSchmidt (talk) 15:13, 13 August 2008 (UTC)

## Stub fix

I added some sections to help expand this article from its current state (roughly a stub with a full article on automorphisms of finite abelian groups merged in). This is a basic history as was passed down to me through oral tradition (and basically sourceable from prefaces): In the 1960s abelian groups were understood as being torsion, torsion-free, or mixed. Torsion groups were handled locally and in some sense "finished", but torsion-free groups had a strange variety of techniques that only sort of worked, and mixed groups was mostly just a name for the "general" theory, where basic splitting theorems were developed. One of the main research problems was describing the additive groups of rings (and less so, the multiplicative groups of rings versus automorphism groups of abelian groups). In the 1970s homological methods gained widespread acceptance, and torsion products became tor functors, Baer sums became ext groups, etc. I'm not sure of exact dates (traces present in the 70s, but I think not until the 80s), but the use of modules became increasingly important, especially in torsion-free groups: most nice torsion-free abelian groups are modules over nice rings, so either implicitly or explicitly, people began studying their submodules rather than their subgroups. "Basic subgroups" are a very early version of this, but later on you get explicit descriptions of modules over valuation rings, etc. More recently (though maybe like 80s recently), set theoretic methods came into their own, primarily using the work of Shelah. Basically there were still lots of unanswered questions, despite quite a lot of research, and suddenly the reason for this was made clear: most of the questions were independent of ZFC. There's some reasonably neat stuff along these lines in constructive mathematics too, and the old notions of purity and flatness became very important there.

At any rate, after the 4 sections (torsion, torsion-free, mixed, additive rings) are expanded, it might be wise to add sections for:

• module theoretic methods
• homological methods
• set theoretic methods

The intersection of the three is a fairly interesting 21st century topic often called set-theoretic homological algebra (Eklof, Goebel, Trlifaj, Enochs are common authors). Also of course, post 1960s results in the previous four sections should be added once the 1960s material is in prose form.

In a completely different direction, I'll mention again my opinion that describing the topic in terms of math curriculum is important. Abelian groups are not just a mathematical object, but also a social construct used to create a sense of common culture during math education. Similarly, they are a social construct used to organize communities of researchers and create a sense of common purpose. The former could easily be sourced, and would help the lead. The latter is sufficiently handled by adding the three new sections. JackSchmidt (talk) 16:20, 13 August 2008 (UTC)

## Capitalization: rare?

Is "abelian" rare (or even unique) among eponymous adjectives in being spelt with a lowercase letter? The source quoted in the article says only

more commonly known as "abelian groups" (the lack of capitalization being a tacit acknowledgement of the degree to which his name has been institutionalized).

Shreevatsa (talk) 18:21, 19 February 2009 (UTC)

It's quite common for eponymous adjectives to be written in lowercase in mathematics, when the adjectives in question have become standard terminology (e.g., boolean, or noetherian). The only difference with abelian is how frequently this is done (e.g., I think abelian is the only one that is written in lowercase in the MSC). --Zundark (talk) 18:41, 19 February 2009 (UTC)
I don't buy that argument per se; but English is not consistent about eponymous adjectives (unlike I thought), according to reputable sources like the Chicago Manual of Style. I can't check it myself, but I trust this quote in the Wikipedia entry on eponyms. So abelian is no stranger than, for instance, mendelian. --Blaisorblade (talk) 15:12, 24 October 2013 (UTC)

## Should be capitalized

For what its worth ... my grad school abstract algebra professor, a distinguished group theorist, never failed to emphasize that Abelian is always capitalized in honor of Abel. One vote here for capitalization.

189.222.214.1 (talk) 20:19, 11 April 2013 (UTC)

As long as you are referring to voting, my college professor claimed that the fact that gaussian is not capitalized actually honors Gauss more than capitalizing, because it acknowledges that anything gaussian is so well-known and accepted that it has become a household term. I believe that this applies to any concept named after a famous person, e.g. abelian, with the caveat that it depends upon how well-known the concept is. My algebra textbooks have abelian in all lower case. One vote here against capitalization. — Anita5192 (talk) 18:52, 24 October 2013 (UTC)
I have heard this before, roughly in the form of: it is a really great accolade when something to which a name has been applied starts being spelled using lower-case. It is, inevitably a one-way progression: eventually these all become lower case with use and not the reverse; it is only a matter of how rapidly in each case. I support lower-case "abelian", since it has been already used extensively this way, and it is definitely not a sign of disregard or disrespect. — Quondum 17:04, 25 October 2013 (UTC)