Talk:Group (mathematics)

Closure

Although it is important to mention closure, there are a few things that disturb me about the way the definition of group is currently written. What is an operation, before the closure axiom is imposed? A function from G × G to some unspecified set? Not only is this a little vague, but it also contradicts the binary operation page it links to. Also, technically speaking, the sentence defining group is wrong, because it ends before any of the axioms are imposed.

I would propose the following, which is slightly longer, but more explicit about the role of closure, which really should be separate from the group axioms. This also breaks the definition into more manageable chunks: first understand what a binary operation is, and then understand the definition of group. Also, this would bring this page more in line with other Wikipedia pages, such as ring. Finally, there are many modern textbooks at all levels that present the definition along these lines (e.g., Artin, Lang, ...); I would add such references.

A binary operation ⋅ on a set G is a rule for combining any pair a, b of elements of G to form another element of G, denoted a ⋅ b. (The property "for all a, b in G, the value a ⋅ b belongs to the same set G" is called closure; it must be checked if it is not known initially.)

A group is a set G equipped with a binary operation ⋅ satisfying the following three additional requirements, known as the group axioms:

Associativity

For all a, b, c in G, one has (a ⋅ b) ⋅ c = a ⋅ (b ⋅ c).

Identity element

There exists an element e in G such that, for every a in G, the equations e ⋅ a = a and a ⋅ e = a hold. Such an element is unique (see below), and thus one speaks of the identity element.

Inverse element

For each a in G, there exists an element b in G such that a ⋅ b = e and b ⋅ a = e, where e is the identity element. For each a, the b is unique (see below) and it is commonly denoted a⁻¹.

^ b: Formally, a binary operation on G is a function G × G → G.

I would welcome advice about which defined terms should be bold and which should be italicized; I'm not sure what the convention is.

Ebony Jackson (talk) 02:49, 16 December 2020 (UTC)

I essentially agree, and I have edited the article accordingly. By the way, I have copy-edited the whole section for clarification and for using a simpler wording that is also more common in mathematics.

About "closure": the term is normally used for the restriction of a binary operation to a subset. Using it as it was done is thus an error. I guess that editors were confused by the usual definition of a subgroup as a nonempty subset on which the group operation and the inverse operation are closed. Using this definition, it is a theorem that a subgroup is a group, and that the groups axioms are thus satisfied. D.Lazard (talk) 10:44, 16 December 2020 (UTC)

The above wording of the last two axioms combines an axiom (one sentence) with consequent properties (e.g. uniqueness of the identity element) that is not part of the axiom. It would be good if this separation was made clearer to the reader, since the current presentation does not adequately distinguish for the reader who is not already familiar with the exact axioms. The parts that do not form part of the axiom could be moved to under the listed axioms, for example, or preceded by "This implies that ...". —Quondum 11:33, 4 May 2021 (UTC)

I'm very tempted to add Closure as one of the four group axioms, as it's already one of the "abelian group axioms". Technically the only difference is the commutativity of the operation, so it doesn't make sense to list closure as an axiom of one but not another. IBugOne (talk) 14:20, 29 December 2021 (UTC)

Please don't: "closure" is a property of subsets, and there is no subset here. The fact that the result of the operation belongs to the group is a part of the definition of an operation. By the way, I have removed the use of "closure" in abelian group#Definition. D.Lazard (talk) 15:00, 29 December 2021 (UTC)

Thank you, IBugOne, for pointing out the discrepancy. I agree with D.Lazard that the best solution to the issue you raise is that closure should not be listed an axiom either for group or for abelian group. Ebony Jackson (talk) 23:01, 29 December 2021 (UTC)

Including both left and right identity and inverse is very common mistake. The existence of the left identity and inverse can be proven using the right identity and inverse and vice versa. So it is sufficient to present only one of each in the list of the axioms. Here there are some proves, for example: https://math.stackexchange.com/questions/65239/right-identity-and-right-inverse-in-a-semigroup-imply-it-is-a-group Andrewsk (talk) 00:06, 20 January 2023 (UTC)

You are right that some of the axioms could be deduced from the others, but this is not a "mistake". The standard textbooks intentionally require the identity be a two-sided identity and so on, presumably because it is more natural not to favor one side. Therefore we should leave it as is. Ebony Jackson (talk) 00:03, 23 January 2023 (UTC)

In a similar vein, I modified the leading sentence to mention that the binary operation is closed (defined on the set). Seeing as the original sentence didn't call it a "binary operation" and instead called it an "operation that combines any two elements to form a third element", I would argue that in order to make this expansion clear and precise, it's required to mention that the domains/codomain are all in the set. So therefore I modified it to "an operation that combines any two elements of the set to produce a third element of the set". Quohx (talk) 06:58, 14 March 2022 (UTC)

Featured Article

I concerned that this article no longer meets the FA criteria. The are large sections of uncited text. Can this be resolved without a formal review? --Graham Beards (talk) 11:09, 20 April 2021 (UTC)

What sections, specifically, do you think require additional citation? Ozob (talk) 04:05, 26 April 2021 (UTC)

At least every paragraph.--Graham Beards (talk) 06:46, 26 April 2021 (UTC)

So, to be clear, this is a purely mechanical and syntactic imposition, completely divorced from any understanding of the content? It would be satisfied if we found a basic textbook on group theory and tacked it on as a footnote at the end of every paragraph? You do notice that the FA requirements emphasize that citations are needed "where appropriate", with a link that points to Wikipedia:When to cite, right? See in particular the "Subject-specific common knowledge" bullet point at that link. —David Eppstein (talk) 06:50, 26 April 2021 (UTC)

I'll nominate for WP:FAR and let the community decide.--Graham Beards (talk) 07:52, 26 April 2021 (UTC)

To me this comes across as "I don't want to answer that so I'm going to do the most hostile thing I can". —David Eppstein (talk) 16:32, 26 April 2021 (UTC)

You are mistaken. I chose not to answer your rude assertion about my understanding.--Graham Beards (talk) 16:36, 26 April 2021 (UTC)

This is an article that will have many paragraphs that fall squarely under the Subject-specific common knowledge, so we will need a list of sentences that need citations. From a quick read, it seems the article has very good bones, and it shouldn't take much time to bring it up to modern FA standards. A few points of improvement

• Standard Model not mentioned in the body

This seems to have been done by someone else already. Jakob.scholbach (talk) 09:42, 30 April 2021 (UTC)

• footnote a is a bit outdated, and is used to support that group theory impacts other fields, which isn't immediately clear

I have updated it to 2020. In my understanding this is used to support that group theory is an active mathematical discpline, not how it impacts other fields. For that purpose this note is perfectly appropriate, IMO. Jakob.scholbach (talk) 10:53, 2 May 2021 (UTC)

Ah, now I see it is used a second time. Jakob.scholbach (talk) 11:04, 2 May 2021 (UTC)

I've tweaked the article so that footnote is only used once, and now we point to the "Examples and applications" section to show how group theory has applications. XOR'easter (talk) 21:02, 2 May 2021 (UTC)

• In the rightmost example below: with people reading on phones, this should be phrased differently (last example? May be too unclear)

Sorry, what is your objection here? Jakob.scholbach (talk)

this one also has been taken care of :). FemkeMilene (talk) 19:28, 30 April 2021 (UTC)

Citations:

• Research is ongoing to simplify the proof of this classification -> cited to a 2004 study

Included a more up-to-date reference. Jakob.scholbach (talk) 19:11, 2 May 2021 (UTC)

• Kernel and image of group homomorphisms and the first isomorphism theorem address this phenomenon .. cn?

I don't think this needs a citation: the kernel is the identity element if and only if a homomorphism is injective (see the subarticle). This is domain-specific standard knowledge, IMO. Jakob.scholbach (talk) 09:49, 30 April 2021 (UTC)

• I think some of the footnotes need citations: g, j, p

Done for j. Jakob.scholbach (talk) 09:42, 30 April 2021 (UTC)

Done for p. Jakob.scholbach (talk) 17:13, 30 April 2021 (UTC)

Done for g. XOR'easter (talk) 20:47, 2 May 2021 (UTC)

• The problem can be dealt with ... cn?

I am not convinced this needs a specific citation. Basically you could name any (contemporary) book on Galois theory (such as the ones we do cite above). Jakob.scholbach (talk) 09:49, 30 April 2021 (UTC)

But Galois theory isn't standard knowledge for laypeople with a keen interest in mathematics (I quote Wikipedia:When to cite: Subject-specific common knowledge: Material that someone familiar with a topic, including laypersons, recognizes as true. Example (from Processor): "In a computer, the processor is the component that executes instructions."). Can it be found in simpler sources too? FemkeMilene (talk) 16:13, 3 May 2021 (UTC)

I added a citation to a textbook chapter. XOR'easter (talk) 03:35, 4 May 2021 (UTC)

• A presentation of a group can also be used to construct the Cayley graph .. cn?

Citation added. XOR'easter (talk) 20:35, 2 May 2021 (UTC)

• The various molecules and their properties .. cn FemkeMilene (talk) 18:48, 26 April 2021 (UTC)

I added the Standard Model to an appropriate spot in the body text and rephrased the rightmost example line. XOR'easter (talk) 21:47, 26 April 2021 (UTC)

• Please add alt text to images for accessibility

Done. Jakob.scholbach (talk) 10:53, 2 May 2021 (UTC)

• There are many duplicate links. With a technical topic as this, many are defensible, but please use User:Evad37/duplinks-alt to remove the improper ones. FemkeMilene (talk) 16:50, 27 April 2021 (UTC)

Done for those where (IMO) the nuisance of the link outweighs the benefits. If you see some more that you specifically think should go, please let me know. Jakob.scholbach (talk) 09:42, 30 April 2021 (UTC)

I've removed a few more. Hope that works. FemkeMilene (talk) 19:35, 30 April 2021 (UTC)

Thanks, Femkemilene for your comments. I have addressed some of them and will work on the remainder asap. Jakob.scholbach (talk) 09:42, 30 April 2021 (UTC)

Brilliant, thanks for your swift work! FemkeMilene (talk) 16:13, 3 May 2021 (UTC)

Comments from my second read:

• The last sentence of the first paragraph of the lede is difficult to understand. I'm not sure whether splitting it in two is sufficient.

OK, I have rephrased this. Jakob.scholbach (talk) 18:50, 3 May 2021 (UTC)

I still find it too difficult, which is a disservice to the rest of the article. I'm unfortunately not great with prose, but I see two problems with "While these are familiar from many mathematical structures, such as number systems—for example, the integers endowed with the addition operation form a group—the formulation of the axioms is detached from the concrete nature of the group and its operation."

1. such as and for example is quick succession makes it more difficult to read. I think leaving out "such as number systems" works, considering that "number systems" may not be familiar to everybody reading this. I can guess what it means, but not sure.

2. the formulation of the axioms is detached from the concrete nature of the group and its operation. Not entirely sure what this is meant to say. FemkeMilene (talk) 19:06, 3 May 2021 (UTC)

Re 1. I have broken the sentence into two. I think leaving out number systems makes the lead less informative. The directly following example of the integers should convey enough implicit meaning about number systems to be OK here.

Re 2: this is meant to say that the group axioms don't make reference to the nature of the group elements, nor to "what" the group operation actually is. This is a critical piece of information. If you have a better way of saying this, let me know! Jakob.scholbach (talk) 20:51, 4 May 2021 (UTC)

David Eppstein, could I have a third opinion here? I know a lot of your work is comprehensible, so wonder whether you can simplify or assure me it does not need simplifying. FemkeMilene (talk) 20:20, 7 May 2021 (UTC)

Re 2, it might help to figure out what the concept to be conveyed here is. It could be that you can have groups in various contexts (e.g. S₃ acting on {a,b,c} or {1,2,3} are both groups) or that all isomorphic groups are the same group (e.g. S₃ acting on {a,b,c} or {1,2,3} are the same group). Or that everything that satisfies the axioms is a group (which it really seems to be saying), but that is kinda too implicit in the idea of what axioms are for to be using such abstruse language. If the latter, wouldn't "Any set and operation that satisfies the axioms is a group" be clearer? —Quondum 20:54, 7 May 2021 (UTC)

I just came across this page, and came here to say that this sentence is very confusing to me (I am mathematician, familiar with groups, group actions, group representations, etc). Whatever exactly it is supposed to mean (Jakob.scholbach's explanation above did not clarify it for me), it seems it has to be an improperly constructed sentence: according to my reading it seems to implicitly suggest that the "concrete nature of the group and its operation" (I don't know what this means) has some manner of existence prior to the "formulation" (?) of the axioms. I assume we all agree that the opposite is the case. Gumshoe2 (talk) 05:57, 15 February 2022 (UTC)

This is regarding the two sentences "The formulation of the axioms is, however, detached from the concrete nature of the group and its operation. This allows one to handle entities of very different mathematical origins in a flexible way, while retaining essential structural aspects of many mathematical objects.", right? Material in the lead is supposed to be a summary of something. I suspect this is (or should be) thought of as a summary of the 19th-century notion of a group touched on in the History section and in more detail in History of group theory, from a time when groups were thought of in some specific formulation of what their elements should be and how they would combine (permutations and composition of permutations) rather than as anything obeying an abstract system of axioms. It's saying that the axiomatic point of view was an improvement because it allowed us to apply group theory more widely in a less cumbersome way rather than having to repeatedly translate one kind of group to another kind of group or re-prove the same theorems for every different kind of group. But if that's the intention, I don't think it expresses it very clearly. —David Eppstein (talk) 06:20, 15 February 2022 (UTC)

Yes, that would make sense and would be good to communicate. It seems a little tricky to formulate clearly in an lead-appropriate way, unfortunately I don't have any good suggestion. Gumshoe2 (talk) 06:27, 15 February 2022 (UTC)

• Should the quote from Borcherds be moved down? Those technical terms have not been introduced yet

It is true that the monster simple group has not introduced there (and is hardly introduced further down), but Borcherd's description "a huge and extraordinary mathematical object" strikes me as highly appropriate for a layman to grasp a bit of the depth out there... Other than that the quote talks just about the simplicity of these axioms, which is what this § is all about. I suggest leaving it there. Jakob.scholbach (talk) 18:50, 3 May 2021 (UTC)

• group table at the right -> check throughout for statements like this that don't make sense on phones (where that image is above the text). The majority of our readers now use mobile devices.

Done. Jakob.scholbach (talk) 18:50, 3 May 2021 (UTC)

Found one more: if the hour hand is on 9 and is advanced 4 hours, it ends up on 1, as shown at the right. FemkeMilene (talk) 20:21, 7 May 2021 (UTC)

Fixed. Jakob.scholbach (talk) 13:03, 8 May 2021 (UTC)

• For example, group theory is used to show that optical transitions between certain quantum levels cannot occur simply because of the symmetry of the states involved. cn

This is one of those things that's probably stated in some form in just about any intermediate quantum-physics book. I've added an older reference that I had at hand, and I'll poke around for a more recent one that seems particularly good. XOR'easter (talk) 23:56, 3 May 2021 (UTC)

• For example, an element of the (2,3,7) triangle group acts on a triangular tiling of the hyperbolic plane by permuting the triangles, cn

Done. —David Eppstein (talk) 03:53, 4 May 2021 (UTC)

• though the double bonds reduce this to pyritohedral symmetry, cn

I removed this second half of the sentence, none of the sources I looked at mentioned that. Added a ref. Jakob.scholbach (talk) 18:07, 7 May 2021 (UTC)

• rest of these images probs also need citations (mentioned above, but XOR'Easter's (brilliant name) response directly below may have caused confusion )

Added references. I pinged WP Chemistry about the JT-effect, will add one there, too. Jakob.scholbach (talk) 18:07, 7 May 2021 (UTC)

• the category of groups: I think this entire section is quite difficult, and uses terms that an applied mathematician won't be familiar with. I don't think it falls under domain-specific common knowledge. Can it be slightly simplified (what is a category?) and cited?

I decided to trim this § down to a sentence which is now placed in the paragraph on homomorphisms. Elaborating on the notion of a category is IMO better left to articles where this has a stronger effect (e.g., for abelian groups). Jakob.scholbach (talk) 18:50, 3 May 2021 (UTC)

• footnote q needs updating

Why? The GAP small groups lists still the same number? Jakob.scholbach (talk) 18:50, 3 May 2021 (UTC)

I think I may have misinterpreted what it says. So I was thinking, does a more modern source have "The groups of order at most 3000 are known"? From your comments it seems like there is a specific collection of groups, called the GAP small groups? Can that be clarified? FemkeMilene (talk) 19:12, 3 May 2021 (UTC)

GAP is software for doing group theory. It includes implementations of all groups of small order. I'm not clear on why this footnote would need updating; once the exhaustive search was done, it's done. XOR'easter (talk) 23:49, 3 May 2021 (UTC)

Yes, the list of groups of order <= 2000 is complete (and the result is in a sense independent of who does it). GAP does not offer a list of groups of order up to 3000, as far as I have seen. Nor does any other site (according to a search I did the other day). In this sense, the citation is still up to date. Jakob.scholbach (talk) 06:54, 4 May 2021 (UTC)

The wording still implies that group 2001 is unknown. Could it be reworded as: "Up to isomorphism, there are about 49 billion groups of order below or equal to 2000", or something in that sense? FemkeMilene (talk) 20:25, 7 May 2021 (UTC)

I have reworded it slightly, but yes, in some sense it is true that groups of order 2001 (not the 2001st group though, this makes no sense) are "unknown" in the sense that there is (to the best of my knowledge) no list available listing them all. Jakob.scholbach (talk) 13:03, 8 May 2021 (UTC)

• a problem too hard to be solved in general (footnote r), needs updating?

Hm, this is a case where some problem is super-hard, and is known to be super-hard to anyone studying this. Therefore, researchers seem not to restate this too frequently again. At least I didn't find a more recent source for that. Jakob.scholbach (talk) 18:50, 3 May 2021 (UTC)

Thanks for trying. FemkeMilene (talk) 19:13, 3 May 2021 (UTC)

FemkeMilene (talk) 16:13, 3 May 2021 (UTC)

• Simons, Jack (2003) listed in specific references, but not used. FemkeMilene (talk) 20:26, 7 May 2021 (UTC)

It is referenced when talking about symmetry of Ammonia. Jakob.scholbach (talk) 13:03, 8 May 2021 (UTC)

HF

Barging in here, from the FAR, if that's okay. I've never been taught group theory, so please bear with me when I say stupid things here.

• Is having terms such as associativity in italics compliant with Wikipedia:Manual of Style/Mathematics?

MOS allows for italics for emphasizing things. Since associativity is such an important piece of this concept I believe it is worth highlighting it. Generally my feeling is that italicization is not used too excessively. Jakob.scholbach (talk) 13:03, 8 May 2021 (UTC)

• Do It is generally preferred for computing with groups and for computer-aided proofs. and It is also useful for talking of properties of the inverse operation, as needed for defining topological groups and group objects. need references?

I have moved this § down to topological groups and added a ref for the second sentence you are asking about. I will check for the first later. Jakob.scholbach (talk) 13:03, 8 May 2021 (UTC)

• "Composition is a binary operation" - pretty sure this statement doesn't need to be italicized in the text

  I think I've addressed this. —Quondum 23:11, 7 May 2021 (UTC)

• "Via Euler angles, rotation matrices are used in computer graphics" - appears to be a sentence fragment
  • This makes sense to me now; I apparently lost my ability to read briefly. Hog Farm _Talk 23:00, 7 May 2021 (UTC)

  I've made a change here; it was a little clumsy. —Quondum 23:11, 7 May 2021 (UTC)

• Can we get a more exact citation for note a?

  I clarified the wording a little; however, a citation for this claim would be appropriate. —Quondum 23:11, 7 May 2021 (UTC)

I think having 1700 scientific papers published per year in some domain is quite aptly proving that this domain is active, no? I see absolutely no problem with this claim. Jakob.scholbach (talk) 13:06, 8 May 2021 (UTC)

I am uncomfortable with this on two levels. One: a high level of interpretation and judgement is needed to (a) classify papers and (b) to translate the number into a conclusion of how "active" the field is. In short, it is WP:SYNTH of the worst kind, even if you will not get many people disagreeing with the conclusion. It invites the query from a reader: "Are you sure?" Two: the number itself needs citation. It should not be claimed out of the blue. —Quondum 13:52, 8 May 2021 (UTC)

In all respect, I think the edit you made indicates that you are not very familiar with the situation here: Math Reviews is not a journal (as you wrote), but rather a service provided by the American Math Society (one of the, if not the most prestigious national mathematical societies). It lists all mathematical papers that have been peer-reviewed, contains secondary reviews of these papers, and contains their classifications into the several areas of maths. This information is in no way a synthesis of other knowledge that has been partly assembled here and there, it is simply a number that is out there. Questioning that 1700+ papers indicates a high level of activity strikes me as being a bit off.

Finally about 2): of course we can include a link to the Mathscinet page, but I frankly don't quite see the need for that. It is (to anyone with a subscription to MSN) a trivially verifiable information. Jakob.scholbach (talk) 12:28, 9 May 2021 (UTC)

You are presumably referring to this edit. I took that it is described as a journal from the linked article Mathematical Reviews. Perhaps, since I lack the necessary familiarity, you should edit the description in both places?

And I am not questioning that it is a high level of activity; I am looking at it from the perspective of a non-mathematician reading this: How does one get a sense of what the figure means when one has no reference, other than the claim made in WP's voice? In non-technical contexts and for schoolgoers, that might seem like a tiny number or an enormous number. Also, to cite the issue and page number that provides the mentioned list would not be strange. —Quondum 16:13, 9 May 2021 (UTC)

Relatedly, although it is correct that Mathematical Reviews was a journal (of reviews), it stopped being published as a journal well before the date given in the note, and became a database, under the different name MathSciNet. I have corrected the note to reflect its name as of the referred-to date. —David Eppstein (talk) 19:42, 9 May 2021 (UTC)

I have included a link to the MSN page (again, there are no issues / page numbers, this is an electronic database). I continue to see absolutely no problem with taking a number of 1700+ papers as an indication that this branch is highly active. If that number wouldn't make it so evidently clear that it is a highly active branch, it would require us to give further references, but this is not the case here. Jakob.scholbach (talk) 20:04, 14 May 2021 (UTC)

• There's some sort of error in the citation "{{Harvard citations|nb=yes|year=2003|last1=Simons|loc=§4.2.1"

  Fixed. —Quondum 23:11, 7 May 2021 (UTC)

• Some of these books refs it would be nice to have page numbers, if possible to help with verifiability

  I analyzed a few of these:

  • General references for broad topics that do not need page numbers: Curtis 2003 (footnote 21), Weyl 1952 (footnote 50), Bishop 1993 (footnote 52), Mumford et al (footnote 63), Fulton & Harris (footnote 66), Serre 1977 (footnote 67), Rudin 1990 (footnote 68), Artin 1998 (footnote 70), Ronan 2007 (footnote 77), Husain 1966 (footnote 79)
  • Has a page number, but in the full reference not the footnote and should probably be made more consistent: Bersuker 2006 (footnote 54)

  Done. Jakob.scholbach (talk) 09:56, 12 May 2021 (UTC)

  • Needs page numbers: Welsh 1989 (footnote 62), Kurzweil & Stellmacher 2004 (footnote 74)

  Done for Kurzweil. Welsh does not mention the Mathieu group, so this might better be replaced by some other reference. Jakob.scholbach (talk) 09:56, 12 May 2021 (UTC)

  • Can probably be replaced by a better reference: Lay (footnote 64), Kuipers (reference 65)
  • Not sure whether needs pages: Shatz 1972 (footnote 81)

  Is OK, I think. Jakob.scholbach (talk) 09:56, 12 May 2021 (UTC)

  —David Eppstein (talk) 21:50, 11 May 2021 (UTC)

Well, frankly, I understood little of this, so I may just be plain wrong on my comments. Hog Farm _Talk 22:05, 7 May 2021 (UTC)

No worries, that's just alright. Jakob.scholbach (talk) 13:03, 8 May 2021 (UTC)

Short Description

@Imaginatorium: I saw you reverted my edit of the short description. My edit removed content, but it was in-line with the purpose of the short description, see WP:SHORTDES. I made a few such edits recently and there is currently a discussion over in the Project Math talk page where I elaborate on my reasoning. To summarize here: the purpose of the short description is to briefly indicate the field covered by the article, and (explicitly) not to define the subject of the article. Notable examples exhibiting a similar degree of brevity include "American baseball player" for Babe Ruth or "U.S. State" for Florida. Feel free to add to the discussion if you wish to. Whether or not you agree with me, your opinion is welcome. Donko XI (talk) 11:24, 21 January 2022 (UTC)

Donko XI changed the short description from "Algebraic structure with one binary operation" to "Algebraic structure", and this was reverted by Imaginatorium. I support Donko XI's change for the following reasons. The previous short description is slighty too long (45 characters instead of the recommended limit of 40), but this is not the main reason for supporting the change. The aim of a short description is to allow readers to decide whether they may be interested in the article without open it. So the useful information is firstly that this is mathematics (this is already disambiguated in the article title), and to say which kind of mathematical object is the subject of the article (here, algebraic structure). Adding other details could be useful only if one could, in very few words, give an accurate definition, or, at least, disambiguate from other algebraic structures. This seems impossible. Moreover, emphasizing on the number of operations is wrong (the integers form a group that has another binary operation, the multiplication), and does not allow distinguishing groups from, say, monoids.

So Donko XI's is clearly an improvement of the short description, and I'll restore it. D.Lazard (talk) 13:58, 21 January 2022 (UTC)

I can't conceive of anyone for whom "Algebraic structure" is a meaningful description who does not already know what a group is. By contrast "American baseball player" and "U.S. state" are terms widely known to the general public. I would suggest "Set with an associative, invertible operation" (45 characters). This short description would at least give a reader with a little math background a concise reminder of what a group is. We give similar hints in the short description for Ring (mathematics), Field (mathematics) and many other articles about algebraic structures. I would also call attention to the template annotated link, which appends the short description, if it exists, to a link: "Used in lists to provide an annotated link using the short description from the linked page for annotation. Useful for disambiguation and providing an idea of what the link is about, without having to hover on the link." It's another reason to eschew absolute minimalism in short descriptions.--agr (talk) 02:54, 24 January 2022 (UTC)

You could get even closer to the 40-character soft limit by dropping the article and comma: "Set with associative invertible operation". —David Eppstein (talk) 07:26, 24 January 2022 (UTC)

Character table

A major omission is any reference to character tables. These tables used extensively in chemistry: see, for example, "Chemical Applications of Group Theory", F.A. Cotton, 3rd. edn., 1990. Petergans (talk) 08:47, 15 March 2022 (UTC)

Indentation

Why are all the main section titles double indented ==title==? They should be single indented as the menu only shows 3 levels of indentation. Currently ====items==== are present in the article, but are not shown on the menu. This will require all indents to be changed in the text. Petergans (talk) 10:48, 26 March 2022 (UTC)

See Help:Section#Creation and numbering of sections. D.Lazard (talk) 11:46, 26 March 2022 (UTC)÷

The sections of level 4 do not appear in the table of content because of the limit parameter in the template {{TOClimit|limit=3}} that appears at the end of the lead. This is a choice for having a table of content that is not too large. This choice may be discussed, but the table of content is already very large. D.Lazard (talk) 12:03, 26 March 2022 (UTC)

Restructuring article?

The edits to this featured article on mathematics have been reverted. That was due partially to the misuse of indentation, see WP:CIR; but also changes to content must be supported by reliable sources, with inline citations. Wish-lists/prayers like {{Cotton&Wilkinson}} are of no use; instead the text book "Advanced Inorganic Chemistry. A Comprehensive Text by Cotton F.A., Wilkinson G. (3rd edition)" can be found and read. If this is to be comprehensible as an article on mathematics, there should be some attempt to reconcile the terminology of physical chemistry with the standard language of theoretical physics and mathematics. In the case of the section "Symmetry"—a brief overview of a general topic—there has so far been no consensus to create separate brand new sections. Here they were sometimes done by copy-pasting content from the section on "Symmetry"; deleting the content, cited to Conway, Thurston et al, or to Weyl, was unhelpful; similarly for the citation to Graham Ellis.

As far as groups are concerned, representation theory and character theory are often first encountered in undergraduate courses on finite groups and angular momentum in quantum mechanics (see e.g. the treatment by Jean-Pierre Serre). Separate new sections at the moment seem to be WP:UNDUE, with no WP:consensus. It unbalances the article. The new image without citations is unhelpful.

The edits today to the article are a combination of vandalism, incompetence and POV pushing: why delete references to physicists or Hermann Weyl; why delete images from the section on "Symmetry"; why favour chemistry above physics? Here are diffs of recent problematic edits, including today's. [1][2][3][4][5][6][7] Mathsci (talk) 14:12, 26 March 2022 (UTC)

I think this is overly harsh. You were right to revert the changes, but that's because we should be conservative with FAs. But there were not CIR-level problems with the changes proposed. If this was not FA quality, I'd say this is what we should expect from the BRD cycle. Additionally, I think there is a problem with the article that I rasied during the FA process that it does not make enough of the applications outside mathematics. While the concept might fundamentally be a mathematical one, its most exciting applications lie in chemistry and physics and the article should not assume that the interest of the reader primarily comes from mathematics. — Charles Stewart (talk) 14:19, 26 March 2022 (UTC)

It was in an unacceptable state, given the last diff. In physics, the group-theoretic approach to quantum mechanics and representation theory can be traced back to Weyl, Heisenberg, Schrödinger, Wigner, von Neumann, M.H. Stone, Dirac, Bargmann and Harish-Chandra (cf Wiener's 1933 Cambridge book or Mackey's Chicago and Oxford lecture notes). Specific examples of character tables are undue here, compared to the character formulas of Frobenius, Schur and Weyl (which have been widely applied in theoretical physics and mathematics). Charles Stewart is completely correct that the section can be improved, but that should be done in an incremental way. Space group is encyclopedic and explained clearly on the tables in mathematics, physics and chemistry (230 cases); mathematically, Point groups in three dimensions#Finite isometry groups covers the 32 crystallographic point groups. It describes the crystallographic restriction theorem from a mathematical standpoint; and is explained in standard text books on chemistry & group theory (e.g. "Chemical Applications of Group Theory", F. Albert Cotton). Mathsci (talk) 16:43, 26 March 2022 (UTC)

• I'll note that, independently of Petergans's motives for changing the history section, we have failed to have any women in that section in the the maths FA where there would be least tokenism in avoiding that failing. Noether's contributions are as worthy of mention as anyone in the last three sentences of the penultimate paragraph, which currently reads: "As of the 20th century, groups gained wide recognition by the pioneering work of Ferdinand Georg Frobenius and William Burnside, who worked on representation theory of finite groups, Richard Brauer's modular representation theory and Issai Schur's papers.[21] The theory of Lie groups, and more generally locally compact groups was studied by Hermann Weyl, Élie Cartan and many others.[22] Its algebraic counterpart, the theory of algebraic groups, was first shaped by Claude Chevalley (from the late 1930s) and later by the work of Armand Borel and Jacques Tits.[23]". — Charles Stewart (talk) 18:47, 26 March 2022 (UTC)

• Ah, invariant theory. The finite generation of invariants of finite groups goes back to Felix Klein ("Lectures on the Icosahedron"), David Hilbert and Emmy Noether (1916). Her short, elementary and constructive proof is presented in Weyl's "The Classical Groups, Their Invariants and Representations" (Pages 275–276, 2nd edition). I don't believe it can be found on wikipedia. OTOH R.e.b. gave Hilbert's non-constructive proof using the averaging or Reynolds operator. Mathsci (talk) 20:42, 26 March 2022 (UTC)

The reversion is very disappointing. The uses of group theory in chemistry are extensive and were properly documented with references to relevant books. Symmetry in molecules is an essential part of the undergraduate curriculum in chemistry. For example chirality cannot be taught without reference to symmetry operations. The designations of many point groups are illustrated at Molecular symmetry#Common point groups, which is why the original diagrams were removed. The example of vibrations in methane illustrated the importance of group theory in relation to spectroscopy. For these reasons, I split the original section, without changing anything in the general part, and amplifying the chemical applications, albeit very briefly. The reversion should be undone so that the new material can be properly discussed, if needed. Petergans (talk) 15:13, 26 March 2022 (UTC)

• While it's natural to be disappointed when changes you've put substantial work into are rejected, my impression is that you lack experience of FA-quality editing. That's OK - little of post-high-school science is FA quality on WP - and I think the impulse behind your changes are OK, but you need to accept that getting agreement to changes to the article will be harder than you are used to. If you still think that you want to invest the time in achieving structural changes to the article, I recommend you put in some time and familiarise yourself with the changes that were made to the article over the last year, which has seen quite a big change in the degree of conformance with the style guide due to the push to get the article to FA level. — Charles Stewart (talk) 18:23, 26 March 2022 (UTC)

I agree with all of Charles Stewart's comments. Since article is FA, before making such edits it is good to discuss on talk page first. Gumshoe2 (talk) 15:25, 26 March 2022 (UTC)

OK, so be it. This means that, for people like me, the article is sub-standard and should never have been promoted to FA. I've checked with a number of chemistry texts (University level) and they all have something about symmetry; most include or discuss applications that depend on the use of point group character tables. The applications don't belong in the same place as the theory (as is the case at present). For me, that means that this discussion is now closed. Petergans (talk) 20:11, 26 March 2022 (UTC)

Groups as categories

I feel that the "category" point of view is missing : a group $G$ can be seen as a category $A_{G}$ with 1 objeect (call it $a$) where elements $g\in G$ corresponds to isomorphisms $f_{g}:a\to a$, and so that composition goes well. The reason why I didn't do the changes myself is that I don't know where to put it, or if it could only be a redirection to the (quite scarce) examples from Category, in which case I would try and extend these. GLenPLonk (talk) 14:47, 1 November 2022 (UTC)

Good idea! I tried to implement your suggestion, by adding it to the discussion of groupoids in the Generalizations section. Ebony Jackson (talk) 18:42, 1 November 2022 (UTC)

Identity and also inverse elements must be part of set

Note to 100.36.106.199 who removed (2 days ago) my words "the set contains an identity element" and returned to the previous wording "an identity element exists": The point is that it is not sufficient for an identity element to exist; it must be part of the set or else the set does not constitute a group.

Consider the first example: the integers under addition. If we consider the set without the identity element zero: ..., -3, -2, -1, +1, +2, +3, +4, ... then we have a set which is NOT a group. Zero still exists but it has to be included in the group.

As for requiring parallelism in wording for identity element and inverse elements, I actually agree that the wording should be parallel. So I will now make it parallel by adding that the inverse elements also must be part of the group (although you said you hoped not). Again for the integers under addition: the set 0, +1, +2, +3, +4, ... is NOT a group without the negative integers. The fact that they exist is not sufficient. Dirac66 (talk) 02:01, 10 July 2023 (UTC)

Your example is incoherent: the object you have presented is not a set with an operation on it (because what is -1 + 1?). Assuming you had not made this error, you would be wrong that an identity exists: the operation is defined (only) on the set, things outside the set cannot be combined using the operation with things in the set and so in particular they cannot be an identity or an inverse. --100.36.106.199 (talk) 13:49, 10 July 2023 (UTC)

I mean, it is true that students first learning abstract algebra suffer from the confusion that you are expressing here. But I think it is instructive that the first time you made the change, you did not even notice that the same argument applies to inverses as to the identity. That's because the meaning is not actually ambiguous or otherwise problematic. --100.36.106.199 (talk) 13:52, 10 July 2023 (UTC)

Having said all that: the revised wording seems fine. --100.36.106.199 (talk) 13:54, 10 July 2023 (UTC)

Undefined terms and notational elements

The statements about injective homomorphisms use several notational elements that have not been introduced previously and that will not be intuitive to a general reader: ${\stackrel {\sim }{\to }}$, $\hookrightarrow$, and $\ker \phi$.

The latter also appears in the Presentations section, along with reference to the free group

What is the fundamental group of a plane minus a point?

"The fundamental group of a plane minus a point (bold) consists of loops around the missing point. This group is isomorphic to the integers." I know very little about groups that I didn't learn from this page... but... the integers are a set, not a group, right? So "isomorphic to the integers" is a vague way of saying "isomorphic to some group that has the integers as the underyling set"? — Preceding unsigned comment added by 2404:4408:6A6E:7000:E48B:A59:8E82:2FCF (talk) 08:21, 3 October 2023 (UTC)

In this case, the relevant group is the integers under the addition operation. –jacobolus (t) 17:24, 3 October 2023 (UTC)

The given quote could be considered incorrect: the loop space consists of loops and the fundamental group crucially consists of equivalence classes of loops. The main text avoids this by saying that "elements of the fundamental group are represented by loops" which is perfectly correct but maybe overly evasive or obscure for most readers. Also, the loops don't have to go around the missing point - they just have to avoid it.

There's also the problem that the blue and orange curves in the image don't show two elements of the fundamental group: they show two different (free homotopy classes of) maps from the circle into the space. A loop representing the fundamental group has (although usually only implicitly) a fixed base point, and these two loops obviously have no common base point. So the picture is not quite illustrative of the fundamental group, even though any reader already familiar with the concepts can easily see what it's trying to communicate.

Being fully precise would obviously not be desirable in the context of the page, but perhaps a talented writer could find a way to rephrase the paragraph and image/image caption in a way that remains concise and readable but is also fully accurate. (I'm not talented enough.) Maybe it would help to move the paragraph to its own subsection "algebraic topology" or "fundamental group". Gumshoe2 (talk) 19:31, 3 October 2023 (UTC)

Looking at this picture, I agree it's weird. We probably instead want something like the pictures in Winding number § Intuitive description. –jacobolus (t) 19:37, 3 October 2023 (UTC)

@Gumshoe2 I tried rewriting the explanation here. Is that any clearer? –jacobolus (t) 22:13, 3 October 2023 (UTC)