The following is an archived discussion of a featured article nomination. Please do not modify it. Subsequent comments should be made on the article's talk page or in Wikipedia talk:Featured article candidates. No further edits should be made to this page.

The article was promoted by User:SandyGeorgia 00:40, 17 September 2008 [1].

Group (mathematics)[edit]

Nominator(s): Jakob.scholbach (talk)


I'd like to nominate group (mathematics), a core topic in mathematics, for FAC. This article has passed GA review, has had a peer review and has been significantly improved expanded since. The three main contributors (in terms of numbers of edits) are myself, JackSchmidt and TimothyRias. Jakob.scholbach (talk) 12:24, 1 September 2008 (UTC)[reply]

Comment Wow. This is such a superior introduction to the subject over anything I've seen in a textbook. Quick comments on "the 6th roots of unity": Per WP:MOSNUM#Typography, and not contradicted in WP:MSM, don't superscript the "th". (That is: if you want to superscript the th, then please argue that at WP:MOSNUM or WP:MSM, but personally, I agree with the guideline.) Most readers will have no idea how to pronounce ζ or what it is; write out "zeta" and tell them that it's just a variable often used for roots. I think it would also be a good idea to explain that "the 6th roots of unity" are the 6 complex numbers that give 1 when raised to the 6th power. - Dan Dank55 (send/receive) 14:27, 1 September 2008 (UTC)[reply]

Comments

Otherwise sources look okay. Links checked out with the link checker tool. (I'm scared that I have some of these books on my shelves. No, I did not buy them myself, they are my father's and my husband's books.. but still...) Ealdgyth - Talk 14:36, 1 September 2008 (UTC)[reply]
This seems very strange to me too. The references being cited are the printed journal articles; JSTOR links are convenience links for those who can use them - it is preferable to have a link which some readers can use than no links at all. Septentrionalis PMAnderson 03:27, 4 September 2008 (UTC)[reply]

Comments by Dabomb87 (talk · contribs):

Dabomb87 (talk) 14:43, 1 September 2008 (UTC)[reply]

Comment while I think the article is really good, it could really help having references in each of the introductory sections paragraphs. For example the definition, the first paragraph after the intro has no ref (although it is a definition...) Nergaal (talk) 17:29, 1 September 2008 (UTC)[reply]

Question What is the audience for this article? Are non-mathy people like myself, people who have not taken calculus since high school, supposed to be able to understand it? If so, I would be happy to perform my "what does the layperson get out of this article" test on it. :) Awadewit (talk) 18:12, 1 September 2008 (UTC)[reply]

  • I will perform the test and see what happens. I will get back to you later today. Awadewit (talk) 18:39, 1 September 2008 (UTC)[reply]

Comments

1) I would leave "technical" out from the second sentence in the lead - these conditions are or course critical for the concept.

2) I find the two expanded examples (Z,+) and D4 to stretch the limits of encyclopedic style. Detailed worked-out examples come close to being at odds with WP:NOTTEXTBOOK. More specifically, the verbal explanation following a+(b+c)=(a+b)+c etc. is likely to be unnecessary for anyone reading the article (taught by 3rd grade?).

Butting in again: I am generally happy to let the editors make the call on length of proofs, which worked-out examples will help, etc., but I sometimes ask that they move those proofs or examples to wikibooks. If desired, the relevant section on wikibooks can be linked directly to and from the relevant section of the article, to make it easier to get back and forth. - Dan Dank55 (send/receive) 21:10, 3 September 2008 (UTC)[reply]

3) In Definition, it could make sense to spell out that a binary operation is just a function mapping an ordered pair (x,y) of elements of G to an element of G. Can keep the wikilink, of course.

4) In the history section the last paragraph would be better if it did not list a few selected (if important) developments, but conveyed the general importance of groups within mathematics as well as its liveliness as an autonomous subject in maths.

5) The history section should elaborate on the emergence of the concept of group as independent of its constituent elements, a development that took place in late 19th century. The abstraction of a "group" from a transformation group is in a sense the essence of group theory and its applications.

6) The section "Simple consequences of the group axioms" seems incoherent. The first paragraphs and subsections match the section title (while including proof of the uniqueness of identity may be overkill, could be just stated to be a consequence), while the two last subsection are not really connected.

7) "Elementary group theory" is presented as if a clearly demarcated discipline. However, as I have seen it used is prgamatically to denote what ever early lecture courses or first chapters in textbooks cover.

8) Basic concepts:

9) Examples introduces (Z,+) again. No need to have same example twice (three times, as it is also in the lead)

10) Discussion of the multiplicative groups of (finite) prime fields should make it clearer up front that (i) non-zero integers modulo a prime are considered with (ii) multiplication as the operation. Now risk of confusion with the additive group Z/(p).

11) Topological groups should be mentioned if Lie groups and algebraic groups are.

12) Galois groups section should make it clear that what is defined is the Galois group of a polynomial. Should elaborate to include the Galois group of a Galois extension. Stca74 (talk) 20:21, 1 September 2008 (UTC)[reply]

General reply to Stca74: thanks for your comments. The very first sentence states

"This article covers only some of the basic notions related to groups. Further ways of studying groups are treated in Group theory."

Your ideas seem partly to be oriented to a more trained audience. As explained to Awadewit above, we tried to keep things as down-to-earth as possible, obviously without getting to blah-blah. While loving to talk about more sophisticated stuff such as universal properties, I do think it important to keep this current orientation. With a total limit in mind, we have to - sadly or not - exclude many topics which would be nice. You will have noticed that there is another article, group theory which is both deeper and more conceptual in scope. That said, I disagree with you in a number of points, but am surely willing to find a good consensus. I numbered your points for easier reference.

I did notice the first sentence and do appreciate the intention of keeping this article down to earth in style. In fact, my comments were made essentially from that viewpoint. As an example, if the target audience is supposed to need the amount of guidance given in the explicit examples, then motivating concepts such as quotient groups should also be critical (8 above), and this calls for a quick (and not too technical) discussion of equivalence relations. The universal property of a quotient group, when expressed directly and without general concepts should only help to make quotients motivated and accessible, in my view at least. Stca74 (talk) 04:22, 2 September 2008 (UTC)[reply]
All right. Then we are on the same track. Jakob.scholbach (talk) 17:53, 2 September 2008 (UTC)[reply]

1) OK Jakob.scholbach (talk) 21:13, 1 September 2008 (UTC)[reply]

2) Interestingly, I see the "illustration" section as the one which is actually the key step helping to understand an interested layman in understanding the topic. You are right, that the writing style there could be found in a textbook, but reducing down the slowliness there would create a less understandable article in favor of a quick treatment, which is IMO not the intention encyclopedias goal. The meaning of a+(b+c)=(a+b)+c may be clear to most readers, but was explained in order to stress the common properties of Z and D4 and any group in general. I would not like to suppress this. Jakob.scholbach (talk) 21:13, 1 September 2008 (UTC)[reply]

See additional comments below. Stca74 (talk)

3) The need of ordered pairs is spelled out pretty clearly (?) in the definitions section ("The order in which the group operation is carried out can be significant. In other words, the result of combining element a with element b need not yield the same result as combining element b with element a; the equation a • b = b • a may not always be true.") Jakob.scholbach (talk) 21:13, 1 September 2008 (UTC)[reply]

Yes, the importance of order comes through in the text already. My point was rather to avoid leaving something as easy to explain in a few words as binary operation behind a wikilink.Stca74 (talk) 04:22, 2 September 2008 (UTC)[reply]
Well, right at the start it says "an operation "•" that combines any two elements a and b to form another element denoted a • b". Jakob.scholbach (talk) 17:48, 2 September 2008 (UTC)[reply]

4) OK, that's right. I have now tried to do better. Satisfying? (From a glance at the references mentioned there, history of groups is itself a vast topic). Jakob.scholbach (talk) 18:00, 4 September 2008 (UTC)[reply]

I very much like the rewrite. Having just tried very hard to write a summary of the history of groups, I can say Jakob has succeeded in condensing a very rich history into short, but interesting prose giving a feel for some of the important, people, topics, and movements of group theory. I think there is much more continuity now (not just straight to Gorenstein from von Dyck), so that the claim that group theory is currently active is much easier to believe. Actually, previous to Jakob's nice work, it might even have only said Gorenstein's legacy was still active, not group theory in general! At any rate, very nice work. JackSchmidt (talk) 21:15, 4 September 2008 (UTC)[reply]
History section good now. Stca74 (talk) 13:47, 7 September 2008 (UTC)[reply]

5) I'm not sure I completely understand what you mean by "its constituents". Please re-explain to me. But, the 19th century period is covered to some extent, right? Jakob.scholbach (talk) 18:00, 4 September 2008 (UTC)[reply]

Dealt with now. The von Dyck reference addresses my concern. Stca74 (talk) 13:47, 7 September 2008 (UTC)[reply]

6) Yeah, I'm not exactly happy with this either. On the other hand, I don't really know how to do better? A separate section for the two last subsections? What do you think? Jakob.scholbach (talk) 17:45, 2 September 2008 (UTC)[reply]

I made them full sections. I'm not sure this order: Definition, History, Elementary theorems, Variant definitions, Notation makes sense. All of them should come before homomorphisms; but should they be permuted among themselves? (I see how the present order came about: start with the definition and widen out in all directions; but does it work?) Septentrionalis PMAnderson 16:51, 4 September 2008 (UTC)[reply]
I'm not sure what's best. To be honest, I think giving a full section to "Notations" (as short as it is) is putting it too high in the TOC-hierarchy. The simplest solution to this problem would perhaps be completely removing the Notations section. What do you think? It's a bit odd, this one. As for Variants, this could be the very last (true content) section. Another option is to put the first paragraph of this section to "Simple consequences" and merge the second one to the Applications section. I guess I will do this if nobody opposes. Jakob.scholbach (talk) 18:07, 4 September 2008 (UTC)[reply]
OK, the content is now put at several places. I removed the notations section, which was the weakest and least essential to this article. Jakob.scholbach (talk) 18:53, 5 September 2008 (UTC)[reply]

7) OK, reworded. Jakob.scholbach (talk) 16:57, 2 September 2008 (UTC)[reply]

8) Good point. Homomorphisms are now right at the start. Hopefully better explanation for cosets and quotients. Universal properties mentioned. (Semi)direct products scrapped (I had thought about this earlier, too, but was hesitating. But a thorough explanation of the basics is more crucial). Kernels : hm. That would be a bit of a dead end now. In light of "summary style" I'd like to refer to group homomorphism. OK? Jakob.scholbach (talk) 17:45, 2 September 2008 (UTC)[reply]

Clearly improved. To further motivate quotient groups, I would expand still further on how the concepts of quotient group, normal subgroup and cosets follow from the compatibility of an equivalence relation with the group law: that the equivalence class of x*y is insensitive to which representatives x and y for their equivalence classes one picks should be the most natural thing to ask. That would also motivate why one often wants to consider two elements the same if they differ by an element in a fixed subgroup, as it is now stated. I would also consider switching the order of cosets and quotients.
As for kernels, I think they would be a very good source of simple yet non-trivial examples of subgroups. Moreover, the isomorphism of G/ker(f) with im(f) would be a good way to produce a non-trivial example of isomorphism between groups, to add some feeling for the types of results one can have with groups. I would not include a proof, just statement. Stca74 (talk) 07:50, 3 September 2008 (UTC)[reply]
I perfectly understand your idea to explain and motivate every single bit of the realm we created. However, and this is a serious problem, we must not indulge ourselves in going to far. I hope to have your agreement that we have to stick to the most essential points. I asked myself: would the addition of kernels and first isomorphism theorem be something which fundamentally exceeds the current article or adds to the general understanding. I think not. The "basic notions" section is to give a general feeling of these notions, much as the "simple consequences" should give a feeling how the most elementary steps are done. The points you are calling for should be, I believe, explained in glossary of group theory. Jakob.scholbach (talk) 18:52, 5 September 2008 (UTC)[reply]
I see this as an article-level consistency question - yes, not possible to carefully motivate everything in a long article, but now there is quite some discrepancy between how much time is spent on illustrating the axioms and how some important further concepts are introduced. See below for further comments. Stca74 (talk) 13:47, 7 September 2008 (UTC)[reply]


9) Hm. The idea is to get a "quadrangle" of (Z, +), (Z, ·), (Q, +) and (Q\{0}, ·). In view of this, i.e. to (more or less explicitly) point to rings, I think briefly mentioning (Z, +) does help, and does not hurt. Also, it is to underline that a given set may (or may not) allow several group operations, a thing which I assume not to be evident from the start. Does that make sense? Jakob.scholbach (talk) 21:13, 1 September 2008 (UTC)[reply]

Makes sense, but then could be written to discuss the above more explicitly ("The following examples show that...") Stca74 (talk)
I have written a little explanatory intro phrase. Better? Jakob.scholbach (talk) 16:57, 2 September 2008 (UTC)[reply]
Better. However, still quite a lot on rather simple examples this late in the article Stca74 (talk) 13:47, 7 September 2008 (UTC)[reply]

10) OK. Jakob.scholbach (talk) 16:57, 2 September 2008 (UTC)[reply]

11) Do you think of an additional section such as "General linear group and matrix groups"? Jakob.scholbach (talk) 21:13, 1 September 2008 (UTC)[reply]

That, and algebraic groups are also mentioned in History and Representation Theory. Rather than removing the references, I think topological groups should be mentioned - after all, they are much simpler conceptually than Lie groups.Stca74 (talk) 04:22, 2 September 2008 (UTC)[reply]
Yes, they are simpler, but AFAIK they don't have that widespread use as Lie groups. What particular application/theorem etc. do you have in mind?
I grant you're obviously not asking for the trivial remarks that all Lie groups and all topological vector spaces are topological groups. To start off the long list where just the topological group structure is in the play: Haar measure on locally compact groups, all of (abstract) Harmonic analysis, idèle groups, Galois groups of (infinite) Galois extensions, fundamental groups in algebraic geometry, etc. Stca74 (talk) 07:26, 3 September 2008 (UTC)[reply]
While I (more or less know and) like your points, I feel that there is less of a general common basis to these concepts than to with Lie groups. I tried to come up with something ("Group representations are an organizing principle in the theory of finite groups, Lie groups, algebraic groups and topological groups, especially (locally) compact groups. Via Haar measures and harmonic analysis, the latter present another case where the entanglement of a group with an additional structure, allows considerably deeper insights than the two notions considered separately.). Perhaps this does the job? I think we can't afford an additional section for topological groups (and this would, I believe, also overemphasize the topic). Jakob.scholbach (talk) 15:43, 3 September 2008 (UTC)[reply]
See below for further commentsStca74 (talk) 13:47, 7 September 2008 (UTC)[reply]
Topological groups don't need a detailed discussion, but a rough definition would probably be good. "A group combined with an apropriate topology is called a topological group." should be enough. You may want to define "appropriate" a little more, but I think people can read the article if they care (we shouldn't assume an understanding of topology in an article on groups, so it's pretty meaningless to start talking about continuity of operations). --Tango (talk) 02:16, 8 September 2008 (UTC)[reply]

12) Is "Abstract properties of Galois groups [...] associated to polynomials" not explicit enough? For G.gps of Galois extensions: I think we don't have the space here (would have to tell about field extensions, field homomorphisms). The case presented is the key case, right? Jakob.scholbach (talk) 16:57, 2 September 2008 (UTC)[reply]

My issue is with the first sentence: "Galois groups are groups of substitutions of the solutions of polynomial equations." It looks conspicuously like the definition of Galois groups, for which it is inappropriate. I do not think mentioning fields here would be an issue for anyone who has read the article this far - we've already talked in passing about differentiable manifolds, fundamental groups, error-correcting codes (as we should). Stca74 (talk) 13:47, 7 September 2008 (UTC)[reply]
I've fixed most (all?) of the WP:access violations. There still remain some MOS issues with some images on the page, but (obvious) resolutions of those lead to other MOS issues, so I'm still contemplating how to fix those. (The main issues are the left floated images at the start of the "notations", "examples" and "Lie groups" (sub)sections, which when floated right can lead to unwanted stacking behaviour for wide screen layouts.) (TimothyRias (talk) 09:21, 2 September 2008 (UTC))[reply]

Conditional Support. So far nobody seems prepared to commit themselves on this one. I've spent the best part of this morning thinking about this. It is certainly an excellent article; but there is an issue with regard to accessibility. I've looked at the 14 or so mathematics FA, (eight of which are biographies) and only 0.999... and 1-2+3-4+... are comparable, (the second one has been on the Main Page). And although it would be difficult to make these FAs and this candidate as engaging to the lay reader as Infinite monkey theorem, more of a effort should be made. My condition is a re-write of the Lead—since this is what will appear on the Main Page should it get there. In the current Lead the links do not help the lay reader, please try to introduce the subject in plain English where possible. I would start with the history paragraph, then give the subject some context in the real world and save the more esoteric definitions for the end. Graham Colm Talk 11:07, 2 September 2008 (UTC)[reply]

Comments First off, for the layperson unused to reading about mathematics and unfamiliar with groups, this article is long. I grew exhausted and had to give up before the end. I got to "Cosets" and then the article became too complicated. I skipped to "Examples and applications", but I sort of lost steam after "Rationals", so I didn't read any further. If anything after that point is really important, you might consider moving it earlier in the article.

Thanks to you and your roommate. It is true that the article too long/dense for easy reading. It may be comforting to know that the content of the "Basic concepts" section takes weeks to teach in undergrad, let alone the more advanced concepts. Jakob.scholbach (talk) 15:43, 3 September 2008 (UTC)[reply]
  • For every integer a, there is another integer b, such that a+b=b+a=0. b is called the inverse element of the integer a, which we denote -a.
  • For every integer a, there is another integer -a, such that a+ -a=-a+a=0. -a is called the inverse element of the integer a. (I like this one the best.)
I adjusted this. I chose (a variant of) your first suggestion as it better reflects the way one would check the axioms for a more difficult case. (It makes it clear to the readers that -a is used as a notation for the inverse, which in this case coincides with minus the number in the familiar sense.) (TimothyRias (talk) 08:47, 3 September 2008 (UTC))[reply]
Fixed. (Should have been D4 rather than G. (TimothyRias (talk) 08:57, 3 September 2008 (UTC))[reply]
Made this more consistent. (i.e. removed birth and death dates for Galois) (TimothyRias (talk) 10:12, 3 September 2008 (UTC))[reply]
l = l • e e is the identity element.
= l • (a • r) r is an inverse of a so e = a • r.
= (l • a) • r Use associativity to rearrange the parentheses.
= e • r l is an inverse of a so l • a = e.
= r e is the identity element.
would that make it easier to understand? (Of course coupled with the explanation in the text that this shows that l = r i.e. they are the same element.) (TimothyRias (talk) 22:22, 5 September 2008 (UTC))[reply]

Overall, I thought the article was accessible, but somewhat difficult; I definitely understood the major ideas (my roommate made me explain them back to him), but not the more advanced concepts (to be expected). I sat for awhile and read it, digesting it and rereading it. I did not understand everything the first time I read it, but I think most of the ideas are clearly stated both abstractly and with examples. I believe most readers unfamiliar with groups will struggle a bit, but that is the nature of highly abstract subjects. Nice work. I will read the rest of the article over the next few days. Awadewit (talk) 23:16, 2 September 2008 (UTC)[reply]

Support with final comments - I've read the rest of the article now and with some minor prose adjustments I am happy to say that I think it is a good introduction to the topic. I learned! I am so happy! Since I am a newbie to this topic, I cannot speak to the comprehensiveness of the article. I did note the Springer-Verlag textbooks, though. They look very reliable. :) Might I also add how eye-catching the Rubik's Cube image is at the top of the article? When I clicked on the article, I said to myself "Cool! A Rubik's Cube! I'm going to read this article!" Very effective. (Am I lame or what?) Awadewit (talk) 16:02, 6 September 2008 (UTC)[reply]


Comments - I found this article was a very good introduction; most of the concepts were explained clearly. Just a few comments:

I don't think I can fairly judge an article like this, so I'll avoid taking a position. Nousernamesleft (talk) 03:00, 5 September 2008 (UTC)[reply]

Thanks for the comments. Let me reply point by point:
  • Well the actual number of conditions can be debated. The whole old 3 or 4 axioms debate. The actual number itself is not important so saying "a few" in the lead seems accurate.
  • Not sure on this. As a non-native speaker I'm not quite sure whether it should be plural or not.
  • Good question. The short answer is because of some recent edits due to some other FAC comments (see above somewhere.) I think we could get away with moving it to the definition section. (although the abstractness of this sub section may deter some readers) I'll let Jakob decide.
    • I have now split the "Variants" into "Generalizations" (top-level), and merged the rest into a footnote (concerning mentioning or not the closure axiom) and a paragraph in the "Simple consequences of the axioms" section. That seems to be more natural. Jakob.scholbach (talk) 22:06, 5 September 2008 (UTC)[reply]
  • Well, "first" because it was the same contemporaries that later published his results. (Galois died very young).
  • Yes you might argue that root of unity already in some sense implies complex, but there are plenty of readers that would not know this so I see vey little harm in it being there.
  • I'll have a look a this.

Thanks again. (TimothyRias (talk) 08:06, 5 September 2008 (UTC))[reply]

  • Ah, okay. Two were unambiguously identified in the lead before, though I note that's been changed.
  • It's been changed.
  • It's been moved down and renamed "generalizations"; I won't argue, though I still disagree with that placement.
  • Yes, but something like "...was first rejected bu his contemporaries, then..." would make more sense in that context. Right now, there's no identified "second" to go with the first. (A rather bad explanation; sorry)
  • All right.
  • It's better now.
Nousernamesleft (talk) 22:11, 5 September 2008 (UTC)[reply]
Arb break (temporary break added by SG on request)

Further comments After a fresh complete reading of the article, I have the following comments to make. The article has improved during the FAC process, but I think there is still work to do.

Structural concerns:

  • OK, I have put the definition in the middle (as suggested by Awadewit above), so that the reader should be able to see at the same time the Z example and the def. I also trimmed down the explanations a little bit. I think calling the definition "very simple" is not appropriate for part of the readership we are aiming for. Citing Awadewit above: "I rarely look at equations." Such readers will even more rarely come across abstract symbols. So, if we want the lay reader to understand anything at all, we have to be lengthy on the illustrations. I propose not to stick too much to the WP:not a textbook dogma at this point.Jakob.scholbach (talk) 20:02, 7 September 2008 (UTC)[reply]
  • My idea of the article (and the topic itself) is roughly this: the definition is the starting point. Next level is fooling around with the symbols we just created, e.g. show that there is only one inverse. (Having the proof here is not because it is a terribly important proof, but to give the flavor of what the topic is about. Again, by the above responses, things like that are far from trivial; and could not be written down ad hoc by a newcomer). The third level is basic structural concepts like homomorphisms etc. The fourth level (has to be dealt with more detailed in group theory) is using "external" ideas such as representations, geometric actions etc. to understand groups. To structure this "ascending path" of complexity (steps 1-3) in etapes, I prefer having a separate section for "Simple consequences". Jakob.scholbach (talk) 20:02, 7 September 2008 (UTC)[reply]
Fine with that. Structure around the section now cleaner. Stca74 (talk) 17:20, 12 September 2008 (UTC)[reply]
  • OK. Abelianness was indeed defined several times. I moved this section to Examples, prioritizing cyclic (the only "fact" we had about abelian groups was the fundamental theorem on f.g. gps). This is now also in succession of F_p^x, a cyclic group, so the flow is better. Jakob.scholbach (talk) 20:02, 7 September 2008 (UTC)[reply]
  1. Topological groups: natural to make the point that introducing distance between points makes sense, and that there is a slightly more general concept of topology, and that if either is present, it makes sense to require that the groups operations are continuous.
  2. Lie groups: could be moved from the present article
  3. Group objects in a category. More "advanced", granted, but the natural place to have the main article reference to group objects is rather Group than Group Theory, which in my books is about ordinary groups, ie, group objects in (Sets). Should mention how the previously discussed examples are all group objects (in sets, topological spaces and manifolds).
  • I really think introducing such content escapes the scope of the article, both in length and depth. It's not that I don't write about the topics (actually we did have something similar at some point), but AFAIK neither group objects nor topological groups are top-priority notions related to groups (so as to deserve a separate sub[sub]section). However, I will ask the guys at WP:WPM to comment on this here. If there is a consensus toward your opinion, I agree to include more, but then we have to sacrifice some other content. Jakob.scholbach (talk) 20:02, 7 September 2008 (UTC)[reply]
I tend to agree with Stca74 on this. A section devoted to groups with additional structure should be very instructive for readers as to the one of the ways groups play a role in modern mathematics. As for the length concern, as I have said before, the article is not extremely long, (at least for a large topic such as groups too which many books have been devoted in their entirety) There is some room for expansion. As long is the whole thing stays below 100k (it is at 81k now), I think we should be OK. (TimothyRias (talk) 09:04, 8 September 2008 (UTC))[reply]
But a previous reviewer (Awadewit, I think) has said that the article is already too long, and the rule of thumb in WP:LENGTH says that an article that is larger than 60 KB "probably should be divided" for better readability. As often happens with mathematics article, we are in a cleft stick here - if we attempt to make the article accessible to a general audience then it becomes too long and the general reader becomes bored; if we keep it concise, then we are accused of being inaccessible. Gandalf61 (talk) 10:11, 8 September 2008 (UTC)[reply]

(<-)I have added some material on topological groups and a word about group objects. Jakob.scholbach (talk) 19:33, 8 September 2008 (UTC)[reply]

Details:

  • Hm. First I think the lead should not be too detailed. On the other hand, I don't see a priority of the inverse axiom over the other ones, esp. associativity: the inverse axiom as stated requires the identity axiom, and comparing loops or quasigroups with monoids, I think monoids are the thing which can be actually used for something, where as the other seem a pretty much esoteric idea(?) Also, you can associate a group to any semigroup; can one do so in absence of associativity? Jakob.scholbach (talk) 06:02, 9 September 2008 (UTC)[reply]
Funny, my reason for my proposal was essentially your argument against it: associativity is so natural (for audiences that haven't yet seen Lie algebras) that it goes unnoticed, and thus it is the invertibility that catches attention. Surely agree with you on monoids. And as formally the axioms have equal weight, happy with your point.Stca74 (talk) 17:20, 12 September 2008 (UTC)[reply]
  • I tried to emphasize the necessity to put the definitions the way they are (by calling for a general structural principle, i.e. more or less in the background stands category theory). It's right that this wasn't done so much yet. Now it is. Saying that the map G → G/N should be a group homomorphism is, IMO, a shorthand for the explanation that the equivalence relation has to be compatible with the group structure etc. Right? (Kernel and image also mentioned). Jakob.scholbach (talk) 20:02, 7 September 2008 (UTC)[reply]
  • How would you elementarily define quotient groups without having cosets at hand? AFAIK, this order of presenting things is the standard one. Jakob.scholbach (talk) 06:02, 9 September 2008 (UTC)[reply]
As follows: It often makes sense to partition a group G into disjoint subsets, considering any two elements x and y equivalent (written x~ y) if they belong to same subset of the partitioning; one calls ~ an equivalence relation on G and the sets in the partitioning equivalence classes (for the equivalence relation in question). In such a situation it is natural to ask if the group operation is compatible with the equivalence relation in the following sense: whenever x~x ' and y~y ', then xy~x'y'. If this holds, then one can define a binary operation for the set of all equivalence classes in the following manner: if X and Y are two equivalence classes, one sets XY to be the equivalence class of xy for any x in equivalence class X and y in Y - the compatibility between the group law and the equivalence relation guaranteeing that the the choice of x and y in their equivalence classes does not affect the outcome. Under these conditions the set of equivalence classes is in fact a group: its neutral element is the equivalence class of e, and the inverse of the equivalence class of x is provided by the equivalence class of x-1. This group of equivalence classes is called the quotient group (by the equivalence relation) and denoted by G/~or more often by G/H if H is the equivalence class of e. The function which maps an element x of G to its equivalence class is in fact a homomorphism from G to G/H, called the canonical projection, and its kernel is precisely H.
If ~is an equivalence relation compatible with the group law, then the equivalence class containing the neutral element e of G is in fact a subgroup H of G (it is the kernel of the canonical projection homomorphism). Moreover, it follows that then x~y precisely when x-1y belongs to H, or in other words when y belongs to the coset xH of all those elements of G that are of the form xh with h in H. In particular x ~e precisely when x is in H. In fact, it can be seen that the subgroup H satisfies further the condition that xH = Hx; such subgroups are called normal. Conversely, it can be verified that if H is any normal subgroup of G, then the equivalence relation for which x~y precisely when x-1y belongs to H, is compatible with the group law of G. Then the equivalence classes are precisely the cosets xH with x an element of G.
Obviously the previous discussion can be expanded if desired. It could be good to discuss a simple example between the two paragraphs. The above is (a simplified version) of the approach taken by, e.g., Bourbaki, and in my books fairly normal way of introducing the concepts. Stca74 (talk) 22:48, 9 September 2008 (UTC)[reply]
Let's remember that we are not writing a textbook here, and our goals should be clarity and simplicity, not abstract elegance and completeness. I think the current order - subgroups, cosets, quotient groups - is fine. Gandalf61 (talk) 08:51, 10 September 2008 (UTC)[reply]
Couldn't agree more on the first sentence. Apparently clarity is in the eye of the beholder - I understand you are saying that the up-front introduction of cosets is likely to be clearer to the intended (possibly non-mathematical) audience than the one via equivalence classes? Be it as it may, I do not see how the sketch above would be "more abstract" or "complete" than what is in the article now. Cheers, Stca74 (talk) 11:53, 10 September 2008 (UTC)[reply]
Thanks, Stca, for your draft, but I also prefer the current version. We all agree that the mathematics is the same. So its only about which fact comes when and how: you are starting with any partition (which I have trouble to motivate as such, I have to say) of the group and then come to the conclusion that it better be a partition of cosets. I was starting with a subgroup and the associated cosets and said in the end that for G -> G/N to be a group homomorphism, the given definition is the only possible one. So, the only difference is that you gain this insight at an earlier stage, at the cost of talking explicitly about equivalence relations.
By "elementarily" I meant something which avoids any additional notions (such as eqn. rel.). A quick glance at two books (Artin Algebra and Lang Algebra) reveals that Artin chooses your account, Lang starts with cosets of normal subgroups. I don't know how to resolve the "problem" w.r.t. to some termination of this FAC, but I feel it fair to say that both approaches have their merits, the one is more direct and elementary, the other one a bit more conceptual, but none of them clearly outweighs the other? Jakob.scholbach (talk) 15:10, 10 September 2008 (UTC)[reply]
Happy to go with the consensus here. Stca74 (talk) 17:20, 12 September 2008 (UTC)[reply]
Why would you say Lie groups are called continuous groups, other than the fact that they are (locally) path connected? (TimothyRias (talk) 09:08, 8 September 2008 (UTC))[reply]
Well, I don't have historical references at hand, but my understanding is that Lie introduced the term in the late 19th century in a non-technical manner to distinguish "continuous" transformation groups from discrete ones. The Montgomery-Zippin result that a finite-dimensional locally compact locally connected metrizable group is a Lie group is from the 1950's and thus much later than the introduction of the name continuous group - how much could have been conjectured in the 19th century I cannot verify right now. But in any case the present text ("Because of the manifold structure it is possible to consider continuous paths in the group. For this reason they are also referred to as continuous groups.") should be clarified. One can obviously consider continuous paths in any topological space, thus in any topological group (even if one would then get only trivial paths in some cases). One could also leave the whole remark out and just mention that historically Lie groups have been also called "continuous groups", a terminology which is less used nowadays as it is somewhat ambiguous wrt topological groups. Stca74 (talk) 12:13, 8 September 2008 (UTC)[reply]
The hard part of the Montgomery-Zippin result is the implication that any finite-dimensional locally compact locally connected metrizable group allows a compatible manifold structure. The inverse implication is fairly trivial and probably was so to Lie. (well at least at an intuitive level since at time most of the above concepts weren't really formally developed.) Note also that (a non-trivial) path in a (transformation) group is precisely what is intuitivly meant by a continuous transformation. But I agree that the phrasing in the article is somewhat awkward. Pending what we decide to do with "groups with additional structure", I will rephrase it. (TimothyRias (talk) 15:28, 8 September 2008 (UTC))[reply]
Sure, the inverse implication is completely trivial, and my point indeed was that it is unobvious and was unknown in the late 19th century that there is a topological characterisation of Lie groups. The best historical account I could check was the Historical Note to Chapters I-III of Bourbaki's Lie Groups; it discusses Lie's original papers and letters in some detail and appears to confirm my view of how the term "continuous group" was used. No reference to paths in the group. Agree with your point on paths in transformation groups as corresponding to intuition, though. Agree also with your proposal. Stca74 (talk) 16:49, 8 September 2008 (UTC)[reply]
I have personally never heard "continuous group". Do what you want with this piece of text. Jakob.scholbach (talk) 19:33, 8 September 2008 (UTC)[reply]
I have removed this phrase. Jakob.scholbach (talk) 19:50, 10 September 2008 (UTC)[reply]

Small comments support (from Randomblue (talk) 10:32, 10 September 2008 (UTC))[reply]

I created Janko group (disambiguation), left inverse (disambiguation), and hyperbolic plane (disambiguation) and linked the article to these as per WP:DAB#NAME. Ozob (talk) 22:10, 11 September 2008 (UTC)[reply]
Sorry, I don't get it. What exactly is disturbing you? Unrelated is supposed to mean a group law in the subset which has nothing to do with the group law in the bigger set. Jakob.scholbach (talk) 19:50, 10 September 2008 (UTC)[reply]
What bothers me is that you don't go straight to the point. "a subset with an unrelated group law does not qualify as a subgroup" is not the same thing as "a subset with operation a restriction of the mother group's operation is a subgroup" so you can't write "the group structure has to be respected when passing from the smaller group to the bigger one, i.e. a subset with an unrelated group law does not qualify as a subgroup". Anyway, you seem to want to say the same thing three times in a row:

->"the group structure has to be respected when passing from the smaller group to the bigger one" (an "intuitive" definition)
->"a subset with an unrelated group law does not qualify as a subgroup" (IMH useless)
->"H is called a subgroup if the restriction of • to H is a group operation on H" (the formal definition)Randomblue (talk) 20:43, 10 September 2008 (UTC)[reply]

That's perfectly right. I removed all allusions in this direction in that subsection, just at the end of quotient groups there is a little survol over the relation of subs to quotients and homomorphisms. Jakob.scholbach (talk) 15:51, 11 September 2008 (UTC)[reply]
  • Yes, this is what I thought: replace 'counterpart' with 'dual'. Randomblue (talk) 15:20, 10 September 2008 (UTC)[reply]
  • OK. I added a footnote (dual category may frighten people). Jakob.scholbach (talk) 19:50, 10 September 2008 (UTC)[reply]
Fixed one misplaced pair of hyphens. (For things like n-by-n matrix, hyphen is correct, right?) Jakob.scholbach (talk) 21:43, 11 September 2008 (UTC)[reply]
" and – if division is possible, such as in Q – fields" or "objects – be they of geometric nature, such as the introductory symmetry group of the square" or "models – imposing, say, axial symmetry on a situation will typically" or "serves – in the absence of significant gravitation – as a" Randomblue (talk) 21:55, 11 September 2008 (UTC)[reply]
Done. Jakob.scholbach (talk) 14:42, 12 September 2008 (UTC)[reply]

Support I can't say that I understood every word, but I thought it was as clear as an article of this nature is ever going to be I even made a tiny edit, jimfbleak (talk) 16:58, 11 September 2008 (UTC)[reply]

Images (temporary break added by SG at request)
The images have now been tagged by their authors. Jakob.scholbach (talk) 08:25, 13 September 2008 (UTC)[reply]

))

I preferred to remove the mention of the Grothendieck group in the Generalizations, this is defined for abelian semigroups (though there is a similar construction for semigroups in general) and seems to stray away from the main point. However, I think that all the material on group objects should be moved in this section, it's not really needed when dealing with topological groups and would fill the blank space. I also think that the example for a monoid should be the set of natural numbers with 0 and +. This is the most canonical and fundamental non-nul monoid. Cenarium Talk 22:44, 15 September 2008 (UTC)[reply]
I reinstated that sentence and did now mention N. I think the Gr.gp. is interesting insofar as it generalizes a well-known and "natural" construction, of whose generality few readers will be aware. So I think it's not off-topic. The group objects should be in one place. (As an aside: I think content matters have priority over layout questions. Layout depends on so many factors.) Rethinking the situation w.r.t. the external links, in particular the SPringer ref you mentioned, I now agree with Eubulides below that these pages hardly add any content. So I removed that. Jakob.scholbach (talk) 15:30, 16 September 2008 (UTC)[reply]
  • All right, I have replaced the link by the four concrete pages. Jakob.scholbach (talk) 09:58, 14 September 2008 (UTC)[reply]
Also, can you all have a look at the non-standard use of WP:ITALICS throughout the article text? It's unclear why they're used. Also, there are mixed citation methods (((cite book)) and ((citation)), Ealdgyth catches those, so perhaps they were added after she went through, see WP:CITE, the citation and cite xxx templates produce different citation styles and they shouldn't be mixed. In this case, cite book should be changed to citation.) SandyGeorgia (Talk) 01:09, 14 September 2008 (UTC)[reply]
I changed the cite books to citations. —David Eppstein (talk) 06:20, 14 September 2008 (UTC)[reply]
What exactly do you mean? I have removed some cases of over-emphasis here and there, but grosso modo it seemed OK to me. The types of italics we have are highlighted notions that are crucial (such as simple group), emphasis ("it does not form a group"), titles of mathematical works (Disquisitiones), and a lot of variables. The manual is not so clear whether highlighting most important notions is OK, but I believe it is (it is also standard practice in math texts, btw). For example, in the Cosets section, not highlighting left and right cosets would make the actual definition more difficult to find without reading the text. Jakob.scholbach (talk) 09:58, 14 September 2008 (UTC)[reply]
Looks better now (last time I looked I saw lists in italics, when lists are bolded). SandyGeorgia (Talk) 23:35, 14 September 2008 (UTC)[reply]
Why is closure left out of the definition of the axioms in the lead? Why is finite or infinite left out of the lead? Just curious. SandyGeorgia (Talk) 23:35, 14 September 2008 (UTC)[reply]
I don't think closure should be mentioned in the lead. Closure is not a group axiom, it's a tautology, still used nowadays because of the influence of history and traditions. Finiteness is mentioned at the end of the intro. Cenarium Talk 00:30, 15 September 2008 (UTC)[reply]
I think that the first sentence makes it clear enough that the result of the operation is also an element of this set, then it's not needed to say "this is called closure". Finiteness being not a group axiom but a property of groups shouldn't been mentioned straight away, but since the class of finite groups is so important, it should be noted in the intro, like the Lie groups. Cenarium Talk 00:40, 15 September 2008 (UTC)[reply]
All right, if you all are satisfied with that as the definition for people who only read the lead. SandyGeorgia (Talk) 00:42, 15 September 2008 (UTC)[reply]

Support (was Comment). A wonderful article. I made some minor editorial changes. I'd like to change this to "Support" but there are some (I hope easily fixable) problems:

Eubulides (talk) 07:56, 15 September 2008 (UTC)[reply]

I agree with the remark on external links. There's also wikibooks, which shouldn't be left alone in an external links section. So we should remove it altogether, or put it in the See also section. (I'd prefer the former option.) Cenarium Talk 10:57, 15 September 2008 (UTC)[reply]
As regards reference sources that are not explicitly cited, WP:CITE allows general references which are "not linked to any particular part of the article". Alternatively, these sources could be moved to a "Further reading" section, as per WP:LAYOUT. Moving them to a separate "list" article is a bad idea, because it would surely be nominated for deletion by the anti-list brigade. Gandalf61 (talk) 11:17, 15 September 2008 (UTC)[reply]
External links: I removed 2 of them. The remaining two do provide additional information.
"Superfluous" refs: I removed two of them and corrected a small number of formatting errors. The Devlin reference I kept, because this is actually the only layman exposition among the references given (though not explicitly cited). I think making up a Further reading section for this one ref is a bit exaggerated.
Unprecise references: I'm a fan of valuable reference information, as you may have seen from the length, depth and quality of the ref sections. However, I disagree with you on this point. As you can see, we did care for exact references where possible and/or necessary. I.e. if a particular fact (e.g. "An infinite cyclic group is isomorphic to (Z, +),") is cited, then the precise ref with page and theorem number is given. However, making reference to the entire (more or less) oeuvre of Sophus Lie, for example, or general statements such as "... as do adele rings and adelic algebraic groups, which are basic to number theory." does not make sense by providing a particular chapter or even page. Many of the advanced references are just pointers to big topics, of which whole books are written. Any reader willing to consult the reference will face the situation that there is more to know than just a single page. Often it is even more than a single chapter. The last resort: if it should be a single chapter in a particular book, a quick glance at a book's content or glossary. OK? Jakob.scholbach (talk) 15:47, 15 September 2008 (UTC)[reply]
  • Thank you for explaining the point about citations to entire books. I was incorrect, and I struck that comment.
  • I mentioned Devlin only because I did a spot check and found it wasn't cited. I just now did another spot check and found that the next entry in that section, Dummit & Foote 2004, is also not cited. Since I've checked only 3 entries in that section, and 2 were not cited, I suspect that the problem occurs more often than twice in that section. I suggest going systematically through the section, finding every entry that is not cited, and moving it to a Further reading section (or to some subarticle).
  • The cases for the two remaining external links are weak. Both of these links look like citations, not like external links. Neither link meets the criteria in WP:LINKS #What should be linked or WP:LINKS #Links to be considered. If it's important, for example, to link to O'Connor & Robertson 1996 in order to give the reader info about the history of group theory, then Group (mathematics) #History should cite O'Connor & Robertson. If there's no reason for Group (mathematics) #History to cite O'Connor & Robertson, that suggests that O'Connor & Robertson do not need to be linked to here (though perhaps the subarticle History of group theory should link to it). Similarly for the other external link.
Eubulides (talk) 23:08, 15 September 2008 (UTC)[reply]
Sorry for the superfluous ref one: I did check the refs one by one yesterday, but somehow missed the "General refs" section. Dummit & Foote now removed. (I hope you don't prove my complete idiocy by spotting another one :)) Also removed the two links. I think they don't hurt, but perhaps you are right, they don't add that much content, and all of the tiny little pieces of additional material is certainly covered 15x in the other references. Jakob.scholbach (talk) 15:30, 16 September 2008 (UTC)[reply]
Thanks for following up; this addresses all the issues I raised, so I changed "Comment" to "Support". Again, this is a wonderful article; I'm jealous! Eubulides (talk) 17:09, 16 September 2008 (UTC)[reply]

I apologize for the endless nitpicky questions, but the more technical articles present special challenges :-) Have you all reviewed the Group table of D4 and the chart in "Generalizations" for WP:ACCESSIBILITY and color-blindness? A question on the talk page there usually gets a quick answer. SandyGeorgia (Talk) 15:46, 16 September 2008 (UTC)[reply]

I just "checked" this by displaying the images in monochrome. The Group table of D4 indeed has problems; maybe this can be fixed by using different stipple patterns as well as colors? The chart in "Generalizations" is OK, though, because it uses ((yes)) and ((no)) and the text suffices to disambiguate. Eubulides (talk) 17:09, 16 September 2008 (UTC)[reply]
I added a more precise caption, so colorblind etc. can still figure it out. (Sandy, does this mean that the less people understand of the topic, the more they are into nitpicking? I wonder, because the FAC here has for the most part brought to day lots of formatting aspects, but, except for Awadewit and Stca74, few fundamental improvements). Jakob.scholbach (talk) 18:13, 16 September 2008 (UTC)[reply]
If you're asking if I don't understand Groups, since I'm doing the nitpicking, I wonder how I ever got through grad school :-) I try to avoid opining on content because of my position of FAC delegate: I've expressed my views on our Math FAs in past FARs. If you want my opinion, yes, I believe that some of our past Math FACs and FARs did not get enough scrutiny, particularly to the quality of the prose, possibly because some of the reviewers did not understand the content or believed it was the Math they didn't understand, when in fact, it may have been a prose issue. If I misunderstood the implication in your reply, please disregard :-) What I meant about technical articles presenting special challenges, in this case, is that the images used here are not typical of other kinds of articles, so there are more issues to consider. SandyGeorgia (Talk) 18:25, 16 September 2008 (UTC)[reply]
Oh, no, I didn't mean to judge your comment. Was just a general thought. As for understanding or not, the truth is mostly somewhere in the middle between writers (not only in WP) shielding themselves with fancy vocabulary, and readers shielding themselves with a (not necessarily extant) capability of understanding. OK, I'll stop musing. Jakob.scholbach (talk) 19:09, 16 September 2008 (UTC)[reply]
The above discussion is preserved as an archive. Please do not modify it. No further edits should be made to this page.
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