Wikipedia:Featured article candidates/Group (mathematics)


 * 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.

Group (mathematics)

 * 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, and. Jakob.scholbach (talk) 12:24, 1 September 2008 (UTC)

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, 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 &zeta; 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) Comments
 * Superscript OK. I have also changed zeta to z, and also omega to z, too. So this is now coherent and easier to read (the explanation of the group is just given in the text). Jakob.scholbach (talk) 18:12, 1 September 2008 (UTC)
 * But zeta is the standard mathematical convention for these values, and now the text doesn't match the notation in the figure. —David Eppstein (talk) 20:27, 1 September 2008 (UTC)
 * OK. I think, it should be OK for both people knowing about roots of unity and those who don't to use z instead of zeta. For the other issue, this is proba. your cache. I had changed the image accordingly. Jakob.scholbach (talk) 21:13, 1 September 2008 (UTC)
 * Yes, it was my cache — it went away after a forced reload. As for z vs zeta, I tend to agree with you that readability for non-mathematicians trumps pedantic adherence to minor stylistic conventions. —David Eppstein (talk) 02:11, 2 September 2008 (UTC)
 * References with links to JSTOR/etc articles need to note that a subscription may be required.
 * I've seen many subscription-only links to references on articles here, and never seen such a note. Where does this supposed requirement come from? In regards to WP:V please note that most such references may be checked by anyone by walking into a public university library, or by asking someone in Category:Wikipedians who have access to JSTOR (including myself) for a copy. On the other hand, your point below about language is well taken. —David Eppstein (talk) 17:25, 1 September 2008 (UTC)
 * It's not a "requirement" (unless it's in the MOS somewhere, which it could be for all I know) but it is common courtesy, same as the language part below. Most folks when they see a link, assume that they can freely access it. It's along with the requirement to note that registration is required for some other sites, or that you show when a link is to a .doc or .pdf file. If you're adamantly opposed to saying "JSTOR subscription required" or similar notation, you can remove the links or you can leave it alone for someone else to note that they think you should do it. I certainly won't oppose based on it LACKING that information. I'm kinda puzzled by the fact that you're opposed to putting it in though, it's just not that big a deal to me. Ealdgyth - Talk 17:36, 1 September 2008 (UTC)
 * I have no strong opinion on this, but mentioning it every time seems a bit over-explicit to me. AFAIK the JSTOR articles cited are research papers, which will be mainly interesting to scholars, who will usually know that JSTOR requires a subscription (which many universities have, btw). Jakob.scholbach (talk) 18:12, 1 September 2008 (UTC)
 * Heh. Maybe I wasn't quite clear. I was only referring to it being mentioned in the "References" section with the full journal article listing once. Not in the notes section each time the article is referred to. But that's an editorial choice, so I'll mark it resolved on my end. It is, however, the usual practice for FAs, which is why I brought it up. Ealdgyth - Talk 19:48, 1 September 2008 (UTC)
 * The reason "fee required" or "subscription required" is entered as a courtesy on subscription sources is this. Although you're citing the actual journal article, you're providing a websource that those who have access to a subscription can view.  Sandy Georgia  (Talk) 03:44, 2 September 2008 (UTC)
 * I have a hard time taking WP:EL seriously in this context: a strict reading would seem to indicate that we should just omit all links to commercially published journal articles, leaving only the unlinked textual reference to them, but this would make verifying the content much more difficult and WP:V as a policy takes precedence over any guideline. Regardless of that, what is the proper way of formatting these within a reference? The obvious method would be to use JSTOR stable URL in the id= field, in place of the link to the jstor site in the url= field, but that doesn't work for new-style JSTOR urls such as http://www.jstor.org/stable/2690312 (the proper replacement for the older url listed for Kleiner's “The Evolution of Group Theory: A Brief Survey.”) And anyway that still wouldn't do anything about including the warning notice “DANGER: LINK GOES TO A SITE THAT IS NOT FREELY AVAILABLE TO EVERYONE” written in large bold letters at the start of each citation. —David Eppstein (talk) 04:48, 2 September 2008 (UTC)
 * What I usually do is shown at Hubert Walter where two of the Turner refs are from JSTOR. Same for one of the Gillingham refs. Ealdgyth - Talk 15:35, 2 September 2008 (UTC)
 * While "fee required" has the benefit of being short, it is also inaccurate; those of us coming from site-licensed IP addresses do not need to pay individual fees to access each paper. —David Eppstein (talk) 16:06, 2 September 2008 (UTC)
 * Then use "subscription required". Or don't use it, I was merely pointing out that it is usual to note that not all people will have access to the online version of the article. Ealdgyth - Talk 18:26, 2 September 2008 (UTC)
 * References in a non-English Language should probably note that.(I'm assuming that journal articles with French or German titles are written in those languages...)
 * Indeed. I have noted it along with a translation in these cases. Jakob.scholbach (talk) 17:51, 1 September 2008 (UTC)
 * 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)
 * 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)

Comments by : Dabomb87 (talk) 14:43, 1 September 2008 (UTC)
 * I see first-person pronouns, a no no in encyclopedia articles.
 * Butting in: Per WP:MSM, quote, Article authors should avoid referring to "we". This was last discussed at WT:MOS in February, I believe.  AFAIK, the mathematicians on Wikipedia are generally happy with this guideline.  On the general principle, whenever experts tend to use a kind of language that isn't generally recommended on Wikipedia, the compromise tends to be to use more generally accessible language in the more generally accessible articles. - Dan Dank55 (send/receive) 15:31, 1 September 2008 (UTC)
 * Fixed, I think. I found two instances of "we", in First example: the integers and Uniqueness of identity element and inverses, and fixed both cases. Are there any more first-person pronouns ? Gandalf61 (talk) 15:50, 1 September 2008 (UTC)
 * Nope, you got them. - Dan Dank55 (send/receive) 19:01, 1 September 2008 (UTC)
 * "Thus they have applications in numerous areas, both within and outside mathematics." "Numerous" is vague. "Both" is unnecessary, unless you're putting stress on that idea.
 * I'm not a native speaker, so I may get things wrong. However, the word "numerous" and its vagueness reflects the fact that applications of group theory can not be counted (first, because they are many, second because it is difficult to sharply tell whether something is an application or not). Also "both" is intended, in order to emphasize the generality of the concept. This is further explained in the following two paragraphs of the lead. Jakob.scholbach (talk) 17:51, 1 September 2008 (UTC)
 * "The abstract symbol '•' is to be understood as a general placeholder for a concretely given operation." "to be" is not necessary here.
 * Done. Jakob.scholbach (talk) 17:51, 1 September 2008 (UTC)
 * "The original motivation for group theory was the quest of solutions of polynomial equations of degree higher than 4." of-->for.
 * Covered by somebody else already. Jakob.scholbach (talk) 17:51, 1 September 2008 (UTC)
 * "Being an open subset of the space of all n-by-n matrices, it is a Lie group." "Being"-->Because it is.
 * Covered by somebody else already. Jakob.scholbach (talk) 17:51, 1 September 2008 (UTC)

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)
 * Are you referring to introduction/overview paragraphs in every section? In that, I did not sometimes did not put a ref at these paragraphs for this is just a summary of the section content, which is ref'd at the places where the stuff is explained in greater detail. If I understand you right, putting a ref there would just be repeating the same refs that show up a little bit later (which is, IMO, not necessary). Jakob.scholbach (talk) 17:55, 1 September 2008 (UTC)

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)
 * The audience is supposed to be mixed ;) "Definitions and illustration" should be, I hope, accessible to a mature high-school kid (this subject is in the first place pretty much unrelated to calculus, though). Laymans are also supposed to get the right feeling of everything w.r.t. "history" and "examples and applications" (there are some technical words, which have to be either swallowed, or looked up in sub-pages, or also in the rest of the text, though, but the essence should come through). "Simple consequences..." should be fine, too, but may not be too interesting for the layman, "basic concepts" probably requires stronger interest/prior knowledge or an exposition in greater detail (which is impossible due to total length restrictions, but the subpages would be a good place for this). What does your test tell? Jakob.scholbach (talk) 18:23, 1 September 2008 (UTC)
 * 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)

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)

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:
 * It would make sense to introduce homomorphisms earlier, right before or after subgroups. Needed to talk about isomorphic groups, make more sense of quotient groups,...
 * Should include kernel of a homomorphism (painful to leave cokernel out, but that could be done if space is an issue)
 * Quotient groups would be more understandable (at least more motivated!) if introduced by the way of equivalence relations compatible with group law. One could explain that often there is a reason to "identify" or "consider the same" a number of elements in a group. Then being able to unambiguously define ("descend") the group operation to equivalence classes dictates that the equivalence class of e is a (normal) subgroup and the rest follows nicely. Currently the definition of cosets is unmotivated, as is the desire to define group operations for them.
 * With homomorphisms defined, the universal property of quotients would make them even more natural
 * Semidirect products jump out from the more elementary topics, and are indeed not described in similar detail -> consider cutting out

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)

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)
 * All right. Then we are on the same track. Jakob.scholbach (talk) 17:53, 2 September 2008 (UTC)

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

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)


 * 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)
 * 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)
 * 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)

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)
 * 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)
 * History section good now. Stca74 (talk) 13:47, 7 September 2008 (UTC)

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

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)
 * 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)
 * 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)
 * 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)

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

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)


 * 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)
 * 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)
 * 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)

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)
 * 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)
 * Better. However, still quite a lot on rather simple examples this late in the article Stca74 (talk) 13:47, 7 September 2008 (UTC)

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

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)
 * 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)
 * 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)
 * 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)
 * See below for further commentsStca74 (talk) 13:47, 7 September 2008 (UTC)
 * 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)

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)
 * 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)


 * Please review WP:ACCESSIBILITY regarding the image layout relative to section headings and main templates. Sandy Georgia  (Talk) 01:58, 2 September 2008 (UTC)
 * 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))

 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)
 * OK, I tried to address your concern in my recent edit. I did not start with history, basically because I think the historical first steps are, while important, pretty far from what groups are (now). Also, it is nicer to have the historical thread in the last paragraph. However, the first paragraph should be more accessible now. Also it emphasizes the importance of groups at an earlier stage (criticized by Awadewit below). OK now? Jakob.scholbach (talk) 18:05, 5 September 2008 (UTC)

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)


 * Groups underlie many other algebraic objects such as rings and modules; they are a central organizing principle of algebra and contemporary mathematics - This statement seems like it should be near the beginning of the lead - when I got to the end of the lead, I was like "oh, that's how important groups are!"
 * OK, I will work on the lead, along with GrahamColn's remarks about that. Jakob.scholbach (talk) 15:43, 3 September 2008 (UTC)
 * Please see my comment to Graham above. Jakob.scholbach (talk) 18:05, 5 September 2008 (UTC)


 * For every integer a, there is another integer b = −a such that a + b = b + a = 0. −a is called the inverse element of the integer a. - I found this notation confusing. My roommate (who is much more mathy than I) came up with two alternatives that made more sense to me:
 * 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))


 * The associativity constraint is the natural axiom to impose in order to make composing more than two symmetries well-behaved - "well-behaved" sounds strange to a lay person
 * OK, reworded. Jakob.scholbach (talk) 15:43, 3 September 2008 (UTC)


 * given three elements a, b and c of G, there are two possible ways of computing "a after b after c". - G needs to be defined - this is the first time it is mentioned in the article.
 * Fixed. (Should have been D4 rather than G. (TimothyRias (talk) 08:57, 3 September 2008 (UTC))


 * The x and y in axiom 2 of the "Worked example" hinder understanding - they don't seem to be necessary, either.
 * Reworded. Is it better understandable now? Jakob.scholbach (talk) 21:11, 3 September 2008 (UTC)
 * Yes. Awadewit (talk) 18:17, 5 September 2008 (UTC)


 * The "Worked example" was much harder for me to grasp than the integer example. I had to get out a book and rotate it to really understand everything. :) I was wondering if putting the definition between the integer example and the worked example would help. It would have helped me.
 * Aha. How would it have helped? (We had the definition at the very beginning at some point, now it is at the very end. We haven't tried the middle so far...)Jakob.scholbach (talk) 15:43, 3 September 2008 (UTC)
 * I kept looking down at the definition while I was trying to understand the "Worked example". For me, the integer example was more intuitive. I would like to be able to suggest putting the definition first, but several years of teaching has suggested to me that concrete examples are often the easiest way to explain a new concept (sadly, not the most accurate). However, the second example started to confuse me a little and I got bogged down in the details of the example. But that is just one reader's reaction. I am not "everybody". :) Take what you will from that. Awadewit (talk) 18:17, 5 September 2008 (UTC)


 * The definition of a group is an abstract formulation incorporating the essential features common to the integers and the above symmetry group - This reads as if the integers and the D4 group are somehow essential to the definition of "group"
 * Reworded. Jakob.scholbach (talk) 21:11, 3 September 2008 (UTC)
 * The integers and the symmetries of the square are just two out of many entities having essential features in common - Why do we even need to mention the integers and the symmetries of the square in this section? Why not just begin with the definition? I found this beginning confusing. I kept thinking that I was missing something - that perhaps integers and symmetries were essential to groups in a way that I had not grasped. Awadewit (talk) 18:17, 5 September 2008 (UTC)
 * OK. One the one hand, these examples are in a sense crucial, but the aim is not at all to tell this, rather to tell that these are two examples of something which one encounters pretty often in maths, so there is a general definition comprising many things (including these two). My next take: "The integers and symmetries of the square are just two of numerous mathematical objects having essential structural aspects in common. The ubiquity of similar entities calls for a general definition which opens up the way to understanding these items disregarding the concrete provenience of the object in question. A group (G, •) is a set ..." What about that? Jakob.scholbach (talk) 22:36, 5 September 2008 (UTC)
 * What about something a little less wordy: "Integers and symmetries of the square are just two of many mathematical objects that share the same structural aspects, demanding a general definition for the class. A group..." Awadewit (talk) 16:05, 6 September 2008 (UTC)


 * Birth and death dates of mathematicians are not consistent in the "History" section - either include them or don't.
 * Made this more consistent. (i.e. removed birth and death dates for Galois) (TimothyRias (talk) 10:12, 3 September 2008 (UTC))


 * To prove the uniqueness of an inverse element of a, suppose that a has two inverses, denoted l (left inverse) and r (right inverse). Then l = l • e = l • (a • r) = (l • a) • r = e • r = r, so l and r must be equal. - If this proves there is only one inverse, I can't see it. Also, it was quite difficult for me to figure out how this proved uniqueness. Perhaps some more explanation? I feel like the reader is doing a lot of work here.
 * OK, now explained. Better? Jakob.scholbach (talk) 15:43, 3 September 2008 (UTC)
 * The thing is, I really have trouble looking at the equation and figuring out what it is saying. I rarely look at equations. :) Without some accompanying words to explain it, I just kind of sit here, scrolling back and forth, trying to remember what every part of the equation means and how they fit together. I don't how much this section matters for non-mathy people, but I can't just jump into it, apparently. I'm really sorry. I feel really silly right now. Awadewit (talk) 18:23, 5 September 2008 (UTC)
 * Please don't feel sorry, the incapacity is at my side. Have another look, I added an explanation of every single step in the chain of equalities, so no need to scroll hither and thither. Better now? Jakob.scholbach (talk) 18:52, 5 September 2008 (UTC)
 * If the equation were presented like this:


 * 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))
 * I tried this. Let's see what Awadewit says. Jakob.scholbach (talk) 22:36, 5 September 2008 (UTC)
 * I think that is easier, yes. Awadewit (talk) 16:08, 6 September 2008 (UTC)
 * }
 * 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))
 * I tried this. Let's see what Awadewit says. Jakob.scholbach (talk) 22:36, 5 September 2008 (UTC)
 * I think that is easier, yes. Awadewit (talk) 16:08, 6 September 2008 (UTC)


 * Similarly, if the equation n • m = e holds (or m • n = e), that suffices to conclude that n is the inverse element of m - At this point, I felt like the article slipped into textbook writing rather than encyclopedic writing. What are n and m? How does this prove the uniqueness of the identity element?
 * I have dropped this one. Jakob.scholbach (talk) 15:43, 3 September 2008 (UTC)


 * Left inverse and right inverse are never clearly defined in the article.
 * Has been dropped now. Jakob.scholbach (talk) 15:43, 3 September 2008 (UTC)


 * Many authors (such as Lang (2002)) therefore omit this axiom. - Odd to suddenly mention textbooks.
 * OK. Jakob.scholbach (talk) 15:43, 3 September 2008 (UTC)


 * These basic concepts form the standard introduction to groups (see, for example, the books of Lang (2002, 2005) and Herstein (1996, 1975)). - Perhaps make the parenthetical a footnote?
 * OK. Jakob.scholbach (talk) 15:43, 3 September 2008 (UTC)


 * The group operation on this set (sometimes called coset multiplication, or coset addition) behaves in the most natural way possible - Behaves in a natural way? This language is confusing to non-mathy people.
 * OK Jakob.scholbach (talk) 15:43, 3 September 2008 (UTC)


 * A major theme in contemporary mathematics is to study given objects by associating groups to them.  - awkward phrasing
 * OK Jakob.scholbach (talk) 15:43, 3 September 2008 (UTC)

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)

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)
 * I did not understand the "Nonzero integers modulo a prime" section but this is my fault because I didn't take the time to understand modular arithmetic.


 * I understood paragraphs 1,2 but not really paragraphs 3 and 4 of "Finite groups" but I think this is ok. The other material seemed a bit too advanced.


 * So, when people say "nontrivial" on Wikipedia, are they referring to Trivial group? Wow. Cool.
 * Yes, somehow. In maths, the "trivial" blahblah is often something which is zero, either as a function or in general an object which contains just zero (such as the trivial group). Jakob.scholbach (talk) 19:52, 6 September 2008 (UTC)


 * The monstrous moonshine conjectures, proven by 1998 Fields Medal winner Richard Borcherds, provide a surprising and deep connection between the largest finite simple sporadic group, called the monster group, and modular functions and string theory. - Not that I would understand the connection if it were explained, but it is worth pointing out that said connection is not mentioned. :)
 * OK, I tried to do better. Jakob.scholbach (talk) 19:52, 6 September 2008 (UTC)


 * This remark is formalized and exploited using the notion of group actions, which means that every group element performs some operation on another mathematical object, in a way compatible to the group structure. - wordy
 * OK. Jakob.scholbach (talk) 19:52, 6 September 2008 (UTC)


 * This way, the group leaves its footprints on the mathematical object. - Poetry or something I am missing? It is odd to suddenly have this evocative image in the middle of an article largely written without such language.
 * Was poetry. More sober now. Jakob.scholbach (talk) 19:52, 6 September 2008 (UTC)


 * In the example below, a group element of order 7 acts on the tiling by permuting the highlighted warped triangles (and the other ones, too). - It is not really clear what "example" is being referred to.
 * OK. Jakob.scholbach (talk) 19:52, 6 September 2008 (UTC)


 * Not only are groups useful to get hold of symmetries of molecules - "get hold of" is vague
 * OK. Jakob.scholbach (talk) 19:52, 6 September 2008 (UTC)


 * Not only are groups useful to get hold of symmetries of molecules, but surprisingly they sometimes also predict that molecules will not be perfectly symmetric, as in the Jahn-Teller effect, the name given to the distortion of molecule of high symmetry when it adopts a particular ground state of lower symmetry from a set of possible ground states that are related to each other by the symmetry operations of the molecule - sentence is too long Awadewit (talk) 16:02, 6 September 2008 (UTC)
 * OK. Thanks again muchly for your thorough scanning. Jakob.scholbach (talk) 19:52, 6 September 2008 (UTC)


 * Comment A bit more history seems in order; we are left with the impression that nothing much happened between the formalization of group theory and Gorenstein. Perhaps as little as "Sophus Lie expanded the subject to continuous [or "differentiable"] groups"; no need to be technical about Lie groups there. Septentrionalis PMAnderson 16:41, 4 September 2008 (UTC)
 * Please see my recent comment to Stca74 above. (Point 4) Jakob.scholbach (talk) 18:00, 4 September 2008 (UTC)
 * I also replied above. I think your comment was very accurate, and I think Jakob has worked a miracle addressing it.  The new text gives a much smoother path from von Dyck to Gorenstein.  Of course, adding Lie's name to the list including Weyl and Cartan might not hurt, but any summary confined to just a few paragraphs will have to omit virtually all of the giants.  He is at least mentioned earlier in the section, so perhaps in a section under such space-constraints, mentioning him twice would be too much! JackSchmidt (talk) 21:15, 4 September 2008 (UTC)

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)
 * "a few conditions" in the lead: Why not enumerate them? There are only two, after all.
 * "mathematical origin" in the lead - I'm unsure if this should be "mathematical origins".
 * Why is "Variants of the definition" a separate level-one header? Shouldn't it be under "Definition"?
 * "Galois' ideas were first rejected by his contemporaries, and published only posthumously." - why "first"?
 * "group of n-th complex roots of unity" - I'm fairly sure that a certain article is needed there. I'm also unsure why "complex" is necessary, but that part's your call.
 * Plenty of passive voice in the "history" section which makes it a bit difficult to read.
 * 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.
 * Reworded the first paragraph anyway. Jakob.scholbach (talk) 22:06, 5 September 2008 (UTC)
 * Not sure on this. As a non-native speaker I'm not quite sure whether it should be plural or not.
 * originS now. Jakob.scholbach (talk) 22:06, 5 September 2008 (UTC)
 * 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)
 * 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.
 * I removed some passive voice. Should be fine now. Jakob.scholbach (talk) 22:06, 5 September 2008 (UTC)

Thanks again. (TimothyRias (talk) 08:06, 5 September 2008 (UTC))
 * 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)
 * OK, now it reads "At first, Galois' ideas ...". OK? Jakob.scholbach (talk) 19:52, 6 September 2008 (UTC)
 * All right.
 * It's better now.
 * Nousernamesleft (talk) 22:11, 5 September 2008 (UTC)

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.
 * Arb break (temporary break added by SG on request)

Structural concerns:
 * I still do think that the space taken by very detailed examples early in the article is not fitting to an encyclopaedia being a reference work. On the other hand, examples are important in introducing an abstract topic, so much should be retained. But in any case, I think the actual definition should appear before the detailed examples - now the reader must work through a significant amount of text and pictures before getting to the definition, which in turn is (at least as far as maths go) very simple. If some preparatory material should go before the definition, it should be there to motivate the definition, explaining that "operations" each having and "inverse operation" occur in many situations and that it has turned out to be useful to abstract the [following] definition to work with the common features of such situations. Readers should also be better able to focus when studying the examples if they knew what features to look for.
 * 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)


 * Simple consequences of the group axioms is a bit of an orphan after some recent restructuring. The few claims made could well be moved (in a condensed form - no need for proofs here) to the end of the Definition subsection as remarks.
 * 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)
 * Fine with that. Structure around the section now cleaner. Stca74 (talk) 17:20, 12 September 2008 (UTC)


 * Abelian groups and cyclic groups are in an odd place in Basic concepts: the definition of abelian groups fits arguably best in the Definition section. On the other hand, any results on Abelian groups + main article template, as well as the subsection on cyclic groups, belong more naturally in the same section with Finite groups etc.
 * 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)


 * Examples and applications lacks clear structure and flow of presentation. Galois groups and fundamental groups (now only in passim) as well as symmetry groups do indeed qualify as examples in addition to the (currently overemphasised, in my view) groups of various types of numbers. However, finite groups are more a subclass of groups, while Lie groups are rather groups with additional structure. I would propose the following:
 * Split Finite groups into an own higher level section - the topic is clearly important enough.
 * OK. Jakob.scholbach (talk) 20:02, 7 September 2008 (UTC)


 * Condense the first examples, and consider adding one or two examples from other parts of mathematics than algebra (e.g., fundamental groups expanded into a subsection)
 * I trimmed down (Z, +) a bit. I think, spacewise, we can't afford more content. We have Examples of groups, which could come up with things like that. Jakob.scholbach (talk) 20:02, 7 September 2008 (UTC)


 * Create a new section for "Groups with additional structure". After introducing the point that it is often natural to endow a group with additional structure (the additional structure sometimes being the "more immediate" one), I would introduce, following WP:SUMMARY, the three subsections below:
 * 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.
 * Lie groups: could be moved from the present article
 * 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)
 * 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))
 * 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)

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

Details:
 * It would be good to emphasise the existence of inverse elements as the crucial part of a group's definition already in the lead.
 * 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)
 * 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)


 * "Arsenal of group theory" does not sound very encyclopaedic in style.
 * OK. Jakob.scholbach (talk) 20:02, 7 September 2008 (UTC)


 * Reiterate the proposal to discuss quotients before cosets and explaining why the definitions are set out the way they are (what can an equivalence relation compatible with group operation be like?"
 * 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 &rarr; 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)
 * 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)
 * 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)


 * 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)
 * 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)
 * 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)
 * Happy to go with the consensus here. Stca74 (talk) 17:20, 12 September 2008 (UTC)


 * Lie groups should mention the crucial point that the group operations must be smooth functions - i.e., spell out the meaning of the word compatible. The fact that one can consider continuous paths is hardly the reason Lie groups have been called continuous groups.Stca74 (talk) 14:49, 7 September 2008 (UTC)
 * 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))
 * 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)
 * 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))
 * 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)
 * 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)
 * I have removed this phrase. Jakob.scholbach (talk) 19:50, 10 September 2008 (UTC)

Small comments support (from Randomblue (talk) 10:32, 10 September 2008 (UTC))
 * 1) "From an abstract point of view, isomorphic groups carry practically the same information." What does 'practically' suggest? If it does suggest anything, it should maybe be made explicit.
 * I deleted that word. Jakob.scholbach (talk) 15:10, 10 September 2008 (UTC)


 * 2) The disambig links tool gives 4 links to be disambiguated.
 * OK, done. (There are 3 left Janko group, left inverse and hyperbolic plane, where the DAB page is actually more appropriate than the offered subpages). Jakob.scholbach (talk) 19:50, 10 September 2008 (UTC)
 * I created Janko group (disambiguation), left inverse (disambiguation), and hyperbolic plane (disambiguation) and linked the article to these as per WP:DAB. Ozob (talk) 22:10, 11 September 2008 (UTC)


 * 3) "a subset with an unrelated group law does not qualify as a subgroup" please clarify what is meant by 'unrelated' (I see this is explained just after, but it is still a bit disturbing).
 * 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)
 * 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)
 * 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)


 * 4) "This choice is dictated by the desire" is a bit of a poetic formulation.
 * OK. Jakob.scholbach (talk) 19:50, 10 September 2008 (UTC)


 * 5) "The counterpart to injective maps are surjective maps" I don't think 'counterpart' is appropriate here.
 * Why not? Just the usual dual category thing. Jakob.scholbach (talk) 15:10, 10 September 2008 (UTC)
 * Yes, this is what I thought: replace 'counterpart' with 'dual'. Randomblue (talk) 15:20, 10 September 2008 (UTC)
 * OK. I added a footnote (dual category may frighten people). Jakob.scholbach (talk) 19:50, 10 September 2008 (UTC)


 * 6) The dashes and hyphens are inconsistent throughout.
 * 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)
 * " 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)
 * Done. Jakob.scholbach (talk) 14:42, 12 September 2008 (UTC)

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)


 * Images (temporary break added by SG at request)


 * Comments regarding images:
 * Image sandwiching in "Examples and applications" section (MOS:IMAGES); perhaps consider multiple image in a vertical alignment?
 * Is this a hard criterion? I have to say, it looks better, IMO, if one is right, one is left, instead of two at the same side. I would personally leave it like this. OK? Jakob.scholbach (talk) 08:25, 13 September 2008 (UTC)
 * Well, criterion 2 does say "It follows the style guidelines"; I'll continue this on the talk page... Эlcobbola  talk 17:30, 13 September 2008 (UTC)
 * See here. Эlcobbola  talk 17:37, 13 September 2008 (UTC)
 * Sigh. It's done. Jakob.scholbach (talk) 23:54, 13 September 2008 (UTC)
 * Criterion 3 requires "succinct captions". The "fundamental group of a plane minus a point" and "unit circle in the complex plane" captions appear overly verbose and, for me, are in TLDR territory.  Please consider integrating the detailed explanations into the surrounding prose or image summaries.
 * In fact, a solution here might be to remove the picture altogether: in the example the hole gets detected already by the set of homotopy classes of loops (with base point), the group structure (infinite cyclic group) is irrelevant for the question at hand. Cases where the different group structures on the same-cardinality set distinguish spaces from each other (simplest? one detecting 2 holes vs. one), could require long explanations, I'm afraid. But still, if this example is wanted in the article, it would be worth at least mentioning that often the group structure is needed to tell spaces apart. Stca74 (talk) 06:56, 13 September 2008 (UTC)
 * OK, I have trimmed the caption and explained the image in the text. (Wouldn't want to remove the image, since it helps understanding what loops are talked about). Jakob.scholbach (talk) 08:25, 13 September 2008 (UTC)
 * Image:Cubane-3D-balls.png and Image:Hexaaquacopper(II)-3D-balls.png do not have source information or image summaries (WP:IUP)
 * Image:Uniform tiling 73-t2 colored.png does not have source information
 * Image:Matrix multiplication.svg is derived from an image with no source information
 * In the case of self-made images, which these presumably are, an explicit assertion of authorship is needed (this image, also used in the article, is a good example); we can't rely solely on the implications of a self-variant copyright tag. Эlcobbola  talk 15:17, 12 September 2008 (UTC)
 * The images have now been tagged by their authors. Jakob.scholbach (talk) 08:25, 13 September 2008 (UTC)

}}


 * Support Good article, adapted to today's requirements.
 * In the external links, maybe adding the Springer article, it's a different point of view than Mathworld and Planetmath.
 * In the historical references, Historically important publications in group theory may use another formatting, similar to to differentiate it from actual publications.
 * There is a blank space in the Generalizations section, maybe reducing the size of the template or adding more text.
 * In the see also section also, it may be worth to make two colums or add other relevant articles. Group ring, group algebra, Grothendieck group doesn't seem less relevant thant CPT symmetry (these are examples of constructions of groups or based on groups). The template affords already several examples of groups, maybe Euclidean group and Poincaré group should be moved to the template. Cena rium  Talk  14:20, 13 September 2008 (UTC)
 * All done. Jakob.scholbach (talk) 16:24, 13 September 2008 (UTC)
 * 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.  Cena rium  Talk  22:44, 15 September 2008 (UTC)
 * 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)


 * Note pls review the dab finder at the top of this page; there are three dab links that need repair. Sandy Georgia  (Talk) 16:43, 13 September 2008 (UTC)
 * Dab discussion moved to the talk page here. Sandy Georgia  (Talk) 01:19, 14 September 2008 (UTC)
 * All right, I have replaced the link by the four concrete pages. Jakob.scholbach (talk) 09:58, 14 September 2008 (UTC)


 * 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 ( and, 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.)  Sandy Georgia  (Talk) 01:09, 14 September 2008 (UTC)
 * I changed the cite books to citations. —David Eppstein (talk) 06:20, 14 September 2008 (UTC)
 * 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)


 * Looks better now (last time I looked I saw lists in italics, when lists are bolded). Sandy Georgia  (Talk) 23:35, 14 September 2008 (UTC)


 * 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.  Sandy Georgia  (Talk) 23:35, 14 September 2008 (UTC)
 * 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. Cena rium  Talk  00:30, 15 September 2008 (UTC)
 * 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. Cena rium  Talk  00:40, 15 September 2008 (UTC)
 * All right, if you all are satisfied with that as the definition for people who only read the lead. Sandy Georgia  (Talk) 00:42, 15 September 2008 (UTC)

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)
 * Group (mathematics) contains many citations that are hard to follow. Merely citing a book isn't enough; we need book and page number, or at least book and chapter name.
 * Group (mathematics) seems to contain many references that are not cited. I suggest these be removed, or moved to a subsidiary article ("List of references for group theory", or something like that). For example, Devlin 2000 does not seem to be cited (this is the 2nd reference listed; I did not check them all). There's no need for this section to list material that is not needed to support the claims in this article.
 * Group (mathematics) should be removed. The external links in this section all seem to satisfy the first criterion given in WP:LINKS, "Any site that does not provide a unique resource beyond what the article would contain if it became a Featured article." They all seem to be merely short introductions to groups, which are a subset of what's in this page. There's no need to have external links to pages like that.
 * 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.) Cena rium  Talk  10:57, 15 September 2008 (UTC)


 * 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)


 * 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)


 * 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 or WP:LINKS . 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)  should cite O'Connor & Robertson. If there's no reason for Group (mathematics)  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)
 * 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)
 * 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)

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.  Sandy Georgia  (Talk) 15:46, 16 September 2008 (UTC)
 * 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)
 * 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)
 * 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.  Sandy Georgia  (Talk) 18:25, 16 September 2008 (UTC)
 * 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)
 * The above discussion is preserved as an archive. Please do not modify it. No further edits should be made to this page.