Talk:Abelian group

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Finitely generated abelian groups

I have added a section with this title, with a Template:Tl template to the eponym article. However, the contents are not the same, as the new section is essentially about the computational issue. Normally it should have been added first to Finitely generated abelian group. But, as these computational aspects are very important, it seems better to introduce them first in the article that most users read first. D.Lazard (talk) 19:12, 2 March 2018 (UTC)

Incidentally that other article is full of links to nonexistent references. Here is what it looks like when I view it:

The history and credit for the fundamental theorem is complicated by the fact that it was proven when group theory was not well-established, and thus early forms, while essentially the modern result and proof, are often stated for a specific case. Briefly, an early form of the finite case was proven in (Gauss 1801), the finite case was proven in (Kronecker 1870), and stated in group-theoretic terms in (Frobenius & Stickelberger 1878). The finitely presented case is solved by Smith normal form, and hence frequently credited to (Smith 1861),[3] though the finitely generated case is sometimes instead credited to (Poincaré 1900); details follow. Harv error: link from #CITEREFGauss1801 doesn't point to any citation. Harv error: link from #CITEREFKronecker1870 doesn't point to any citation. Harv error: link from #CITEREFFrobeniusStickelberger1878 doesn't point to any citation. Harv error: link from #CITEREFSmith1861 doesn't point to any citation. Harv error: link from #CITEREFPoincaré1900 doesn't point to any citation.

The fundamental theorem for finite abelian groups was proven by Leopold Kronecker in (Kronecker 1870), using a group-theoretic proof,[4] though without stating it in group-theoretic terms;[5] a modern presentation of Kronecker's proof is given in (Stillwell 2012), 5.2.2 Kronecker's Theorem, 176–177. This generalized an earlier result of Carl Friedrich Gauss from Disquisitiones Arithmeticae (1801), which classified quadratic forms; Kronecker cited this result of Gauss's. The theorem was stated and proved in the language of groups by Ferdinand Georg Frobenius and Ludwig Stickelberger in 1878.[6][7] Another group-theoretic formulation was given by Kronecker's student Eugen Netto in 1882.[8][9] Harv error: link from #CITEREFKronecker1870 doesn't point to any citation. Harv error: link from #CITEREFStillwell2012 doesn't point to any citation.

The fundamental theorem for finitely presented abelian groups was proven by Henry John Stephen Smith in (Smith 1861),[3] as integer matrices correspond to finite presentations of abelian groups (this generalizes to finitely presented modules over a principal ideal domain), and Smith normal form corresponds to classifying finitely presented abelian groups. There is the additional technicality of showing that a finitely presented abelian group is in fact finitely generated, so Smith's classification is not a complete proof for finitely generated abelian groups. Harv error: link from #CITEREFSmith1861 doesn't point to any citation.

The fundamental theorem for finitely generated abelian groups was proven by Henri Poincaré in (Poincaré 1900), using a matrix proof (which generalizes to principal ideal domains). This was done in the context of computing the homology of a complex, specifically the Betti number and torsion coefficients of a dimension of the complex, where the Betti number corresponds to the rank of the free part, and the torsion coefficients correspond to the torsion part.[4] Harv error: link from #CITEREFPoincaré1900 doesn't point to any citation.

Kronecker's proof was generalized to finitely generated abelian groups by Emmy Noether in (Noether 1926).[4] Harv error: link from #CITEREFNoether1926 doesn't point to any citation.

—David Eppstein (talk) 19:34, 2 March 2018 (UTC)

Abstract algebra v. mathematics in the lead

I think it is more appropriate for the lead to begin, "In abstract algebra . . ." instead of "In mathematics . . ." Yes, abelian groups are encountered in almost all parts of mathematics, but the lead should indicate specifically to which area of mathematics the concept belongs.—Anita5192 (talk) 16:36, 4 April 2020 (UTC)

I disagree. The specific area to which the concept belongs, is indicated by the categorization. The fist words are there for indicating to the reader whether the article is the one for which he/she is serching. It is there for indicating that the article is not about a specific group of people. Giving a too restricted field can be confusing for some people: a student of physics learning crystallography or a computer scientist learning cryptography will certainly encounter the phrase "abelian group". If they want more information and read "In abstract algebra", they can think that the article is not for them, as "abstract" is not for them, and it is possible that they do not know well the subdivisions of mathematics. "In algebra" is also not convenient, as it may confuse someone who come here after reading that a space of differentiable functions is an abelian group under addition. So, per WP:LEAST, "In mathematics" is the best choice. D.Lazard (talk) 17:21, 4 April 2020 (UTC)

Preciseness of writing: "applied" preferable over "written"?

The current version of the text contains:

"In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written."

Would it not be more precise to change "written" into "applied" or "carried out"? I think it is the application, and not the way we choose to write things down, that really matters. (Think of function composition: (g ∘ f )(x) = g(f(x)), where we write f after g but apply f before g.)Redav (talk) 14:01, 26 November 2020 (UTC)

I just looked this up in six algebra textbooks. They all define "commutative" (implicitly) in terms of the order in which the operands are written. The article linked in the lead, commutative, also defines the term this way.—Anita5192 (talk) 17:17, 26 November 2020 (UTC)

To me it was so obvious that "applied" trumps "written". But now that you mention sources (which I cannot check, for you did not mention them), I have found a few ones too:

1. "used" [in https://www.merriam-webster.com/dictionary/commutative];
2. "operating" [in https://www.dictionary.com/browse/commutative];
3. "taken" [in https://mathworld.wolfram.com/Commutative.html combined with e.g. https://mathworld.wolfram.com/Multiplication.html];
4. "multiplied" [in https://books.google.nl/books?id=YruifIx88AQC&pg=PA4&redir_esc=y#v=onepage&q=commutative&f=false, page 6].

All of these point to "applied" or "applying" rather than "written". But rather than counting references (that may or may not be flawed), would not conveying the meaning of commutativity correctly deserve priority here?Redav (talk) 16:12, 29 November 2020 (UTC)

Here, "preciseness" (I would say "accuracy") of mathematics is more important than preciseness of English. None of your sources is reliable here as they are either English dictionaries, or wikis written by many authors, mostly anonymous, or even history books. Here, "applied" would be incorrect, because it is the operation that is applied to the operands. "Taken" and "considered" could be used in this sentenceas as well as "written". Another correct formulation would be to replace "does not depend on the order in which they are written" by "does not depend on their order". My preferred formulation is the last one, but, IMO, it is not useful to spent time for changing the article. D.Lazard (talk) 17:19, 29 November 2020 (UTC)

grothendieck group?

this is an important group.

i first thought "this should be in the sidebar" but anticipated a reversion on the basis that it's a kind of abelian group.

so the next question was: is there a mention of the grothendieck group on the abelian group page? there is not.

i was hoping someone who is familiar with group theory could add a blurb here somewhere.

my opinion is that a mention (subsection or something) on this page would make its sidebar absence justified.

thoughts?— Preceding unsigned comment added by 198.53.108.48 (talk) 00:52, 9 July 2021 (UTC)

Added to Template:Alink. No reason for adding a specific section, as the Grothendieck group is not useful for the study of abelian groups. D.Lazard (talk) 08:11, 9 July 2021 (UTC)

Shelah's Recent Work on Torsion-free Abelian Groups

Should we discuss Shelah's relatively recent proof that the family of countable Torsion-free Abelian Groups is Borel complete? Does that belong in this article, or is that too specific a topic? CessnaMan1989 (talk) 16:09, 12 September 2021 (UTC)

Do you have a reliable source? D.Lazard (talk) 16:19, 12 September 2021 (UTC)

Yes, I have both a primary source(a journal article of the derivations) and several secondary sources including a Quanta Magazine article. I think the Quanta Magazine article explains the work in the clearest terms for encyclopedic purposes. Would the Quanta Magazine article be an acceptable source to cite? CessnaMan1989 (talk) 15:52, 13 September 2021 (UTC)

My experience with Quanta is that they oversimplify to the point of unintelligibility. I don't think they can be used as a reliable source for technical material in mathematics. For the general significance of a result, with the technical parts sourced elsewhere, maybe. —David Eppstein (talk) 18:22, 13 September 2021 (UTC)

I want to dunk on Quanta too. I emerge dumber, every time I read it. It reports that someone found something that is interesting, followed by an idea-salad that conveys negative information. What's "negative information"? I don't know. It's a cognitive hazard. 67.198.37.16 (talk) 16:18, 26 May 2025 (UTC)

Grammatical disimprovement

Hi again dear Template:Ping. The sentence below Template:Quote is grammatically wrong. In the article, the "Abelian group" expression, when used as a concept, is written as singular. I think the concept "Abelian group" is a singular concept not plural. Why did you revert my edition? Please inspect these queries:

1. https://en.wikipedia.org/w/index.php?title=Special:Search&limit=500&offset=0&ns0=1&search=%22is+named+after%22&advancedSearch-current={%22fields%22:{%22phrase%22:%22\%22is%20named%20after\%22%22}}
2. https://en.wikipedia.org/w/index.php?title=Special:Search&limit=500&offset=0&ns0=1&search=%22are+named+after%22&advancedSearch-current={%22fields%22:{%22phrase%22:%22\%22is%20named%20after\%22%22}}

Thanks, Hooman Mallahzadeh (talk) 03:12, 7 April 2022 (UTC)

I didn't study enough grammar to know why, but "Abelian group is named after ... Abel" is wrong. You might say "An Abelian group..." or "The Abelian group..." but that suggests some Abelian groups were not named after Abel. There is nothing wrong with the common English expression that "Abelian groups are named...". Johnuniq (talk) 03:25, 7 April 2022 (UTC)

It's quite simple, really. Template:Ping Your mistake is thinking that a "concept" can be singular or plural: these are purely grammatical terms, and whether a *word* is singular or plural is determined by grammar, not by "whether there are two or more of them". This is why, for example, in "ten stone", the noun 'stone' is singular, because it is the singular form, even though its semantics ("concept") refers to more than one of the unit. (Lots of people don't understand this, though.) Imaginatorium (talk) 03:44, 7 April 2022 (UTC)

@Imaginatorium@Johnuniq Is the sentence below grammatically better?

Template:Quote

Or this sentence

Template:Quote

Because the subject of the word "the" is obvious, no need for the word "Abelian". Although in my opinion the second sentence (i.e., singular with "is") is still better. Hooman Mallahzadeh (talk) 04:16, 7 April 2022 (UTC)

"The group is named after" is absolutely incorrect. That wording implies that there is only one possible abelian group. "These groups are" is not wrong, so much, as it is deliberately obfuscatory (you are taking as many words as before to say things more indirectly and make the reader think about what "these groups" refers to rather than just saying it). —David Eppstein (talk) 05:18, 7 April 2022 (UTC)

Abelian Groups and Conjugacy Classes

The Conjugacy class article states a fact about abelian groups, namely that each conjugacy class of an abelian group is a singleton. I think it would be useful to add a mention of this fact in this article under the 'Properties' section Maximilien Tirard (talk) 18:28, 16 October 2022 (UTC)

Leading image

File:A composition of translations that leave a wallpaper unchanged.svg

All  translations  leaving  invariant  a  wallpaper make  up
a commutative group for the binary operation called “function composition”.

I propose to insert at the very beginning the present SVG image and its caption:  a classical example of abelian group.
  Arthur Baelde (talk) 13:17, 3 March 2023 (UTC)

What a awful image. Please, no. D.Lazard (talk) 13:30, 3 March 2023 (UTC)

Maybe a bad set of colors of your screen…
  Arthur Baelde (talk) 13:56, 3 March 2023 (UTC)

I very much doubt this meets the color contrast demands of MOS:CONTRAST. —David Eppstein (talk) 06:22, 4 March 2023 (UTC)

No, the colors are fine, but the image is awful from the aesthetic point of view, as well af from a communication point of view. In such an article, an imag illue must be an illustrative aid to understanding. This is far to be the case, as it is more difficult to decipher the message carried by the image than to understand the text of the article. So, adding this image would definitively a disimprovement of this article. D.Lazard (talk) 16:55, 3 March 2023 (UTC)

Here is an image that is more aesthetic and more easier to understand. Also, the caption avoids the pedantry of your caption (at least five technical terms that readers are not supposed to know). D.Lazard (talk) 17:22, 3 March 2023 (UTC)

We cannot use a fair-use image for this purpose. It fails WP:NFCC#1: a free equivalent could easily be found or created. —David Eppstein (talk) 22:11, 3 March 2023 (UTC)

---

File:A composition of translations that leave a wallpaper unchanged.svg

Two successive translations can be commuted without changing their composition:  a translation again,  represented along a diagonal
of  parallelogram.   All  translations  leaving  unchanged  a wallpaper  make  up
an  abelian  group  for  the  binary operation  called  “function  composition”.

I delayed my reply because of my work in progress,  in SVG.  Proposed for the very beginning of article,  this image is repeated here in a new version,  more contrasted,  because I agree with David Eppstein.  The caption also is improved.

Your comments said nothing about displacements,  composition and commutativity.  On a background very pronounced like this wallpaper previously shown here,  letters that name points or arrows that represent displacements would not be visible clearly,  notably because of its black strokes.

In advance I thank you to comment or criticize this new image or its caption.  Please,  be precise.
  Arthur Baelde (talk) 11:29, 20 April 2023 (UTC)

Everything that I have said above applies equally to the new version. I do not seen any difference between the images (only in the captions. More, the image contains formulas (discouraged by the manual of style), whose meaning is not clearly defined (in particular, two different fonts are used for the same T and U. In stead of trying to invent the wheel again, you should better to search for existing images in Wikipedia Commons. For example this one:

File:Lattice in R2.svg

The translations mapping a blue point to any other blue point form an abelian group isomorphic to Z²

This is much clearer than your image, and this does not need to try a proof inside the image (this is always a bad idea). D.Lazard (talk) 17:14, 21 April 2023 (UTC)

Again you show me an image with no translation represented.  Please,  click on the first,  and then the second of the following links,  and compare my two image versions.

• 6 October 2022
• 5 March 2023

Thank  you  for  your  goodwill.
  Arthur Baelde (talk) 10:19, 22 April 2023 (UTC)

I do not see any difference between the two versions. In any case, none of the issues is fixed in the new version. You should read WP:How to create charts for Wikipedia articles#Design: your image does not respect any of the "friendly" items that are applicable, but follows almost all "unfriendly" items. It is what I meant above when saying that your image is awful.

Here are the mean issues of the image(s):

• The main purpose of the image seems to illustrate the commutativity of addition of translation vectors. This is not the subject of this article.
• The annotations (arrows and formulas) hide the symmetries of the wall paper
• The formulas in the images are not there for labelling some part of the image. So, they do not belong to the image.
• Labels in the image are not used elsewehre. So, they are unneeded.
• Two different faces are used for the same translations (italic and blachboard bold italic). Very confusing.
• The use of blackboard bold for translations and function composition does not follow any standard convention.
• The huge arrows hide the wall paper and make the image more confusing.

In summary, this (these) image(s) cannot be used for anything suseful for readers.

Also, by using the phrase "translation represented", you probably mean "translation represented by a vector". This is a undue supposition on the reader's background, since translations have been defined and named a long time before the invention of vectors. For many people, even at very low level, a translation consists in moving the paper sheet without rotating it. No vector needed. D.Lazard (talk) 16:29, 22 April 2023 (UTC)

Thus for you,  your “awful” that qualified immediately my image was a metaphor,  of which the meaning had escaped David Eppstein and me.  And your “awful” would be now explicit,  through your stuffing of so‑called explanations.  Please,  realize your arbitrary opposition to certain images could become very visible.  Please,  be constructive now.
  Arthur Baelde (talk) 11:54, 25 April 2023 (UTC)

I agree with the criticism of Lazard, which was perfectly constructive: it was focused on the image and provided specific reasons why it is a bad choice for this article. —David Eppstein (talk) 15:34, 25 April 2023 (UTC)

Agree w/D.Lazard and David Eppstein. 67.198.37.16 (talk) 16:25, 26 May 2025 (UTC)