Talk:Group (mathematics)/Archive 4

Formal Definition - Superfluous Information

Isn't point 1 of the formal definition superfluous? It's given as: "For all a, b in G, the result of a • b is also in G."

However, this is already given by definition as • is a binary operation on G. Mind if I change? Inflatablegarfield (talk) 18:27, 21 July 2008 (UTC)

Yes, I mind. The exposition should be as self-contained as possible, so that a layman can roughly understand what goes on without browsing the subpages. Your comment is also covered by a remark in the "Variants of the definition" section. Jakob.scholbach (talk) 11:17, 22 July 2008 (UTC)

I don't really see why a layman would be bothered by the formal definition, nor do I see the point in adding extra conditions. But as long as its mentioned in the article I suppose thats fine then. Inflatablegarfield (talk) 12:51, 22 July 2008 (UTC)

You have to remember that all people of all levels will read this article. There isn't a simple novice-expert dichotomy. There may be some student just starting university that want to know what a group is, and so would be "bothered by the formal definition". For me, the more detail the better. I'm not suggesting that people repeat themselves, but mentioning something that isn't necessarily obvious to everyone is a good idea. Declan Davis (talk) 16:36, 27 August 2008 (UTC)

More to the point, the inclusion of this point frequently improves the presentation of certain concepts by making the requirement more explicit (e.g., for showing that the integers are not a group under division). Dcoetzee 19:07, 27 August 2008 (UTC)

Two different pages

I see no interest in having two different pages: "Group (mathematics)" and "Group Theory". Many parts of them are common, and in a certain way "Group (mathematics)" is more detailed than "Group Theory" contrary to what the banner says. Thus those pages should be merged. -- fl

No. This has been discussed before, see here. Jakob.scholbach (talk) 13:30, 30 May 2008 (UTC)

Really ? I will read this the arguments must be impressive. —Preceding unsigned comment added by 194.199.4.102 (talk) 14:13, 30 May 2008 (UTC)

They are impressive indeed. You should explain them at length on the concerned pages because I'm not sure the final reader will enjoy the byzantine sophistications of this choice by himself. -- fl —Preceding unsigned comment added by 194.199.4.102 (talk) 14:18, 30 May 2008 (UTC)

Yes. As you may have noticed in the above and archived talk sections, a couple of editors including myself are determined to improve the group theory article to a similar standard than the one here. Then, the distinction between the articles will be hopefully clear(er). Jakob.scholbach (talk) 14:33, 30 May 2008 (UTC)

I think that it is important to have two articles for the object and the study, but I think that there needs to be consensus on what exactly each should cover.

I feel that there is currently too much overlap. For example, the history section of Group (mathematics) begins:

The history of group theory happened

Also, I think that it is somewhat inappropriate to have the definition appear on Group theory.

I would like to propose that:

1. Group (mathematics) focus on the definition. Thus, its history section will concern the development of the definition, alternative formulations, who first proposed the term group, etc. Mostly, the rest of the article is already in line.
2. Group theory focus on the field, including its researchers, research trends, important milestones (eg. Classification of the finite simple groups), and current goals.

« D. Trebbien (talk)

Yes, your idea is basically what I also had in mind. See also the archived threads--Geometry guy put it quite well, I think, so as to say here what groups are, in the gp th article to say where to, what for, how. (The lapse in the history section can certainly be solved easily; was just written up by me in a quick-n-dirty manner). To a certain extent, the discussion is philosophical, because it is almost impossible to neatly separate the object from its study and vice versa. One problem with this whole discussion is, that the group theory article is also evolving (I'm currently slowly progressing on this, too). I'm currently working on the Peer review comments. Feel free to join in improving either article... Jakob.scholbach (talk) 21:33, 5 June 2008 (UTC)

Okay. I'll join in when I have time. I just wanted to see if others would be amenable to this.

« D. Trebbien (talk) 03:15 2008 June 9 (UTC)

Some links: Wikipedia talk:WikiProject Mathematics/Archive4#Graph .28mathematics.29 vs Graph theory, Talk:Group (mathematics)/Archive 2#Merger proposal, Wikipedia talk:WikiProject Mathematics/Archive 35#groups.2C group theory.2C and elementary group theory (I'll have to pick this up later...)

« D. Trebbien (talk) 03:28 2008 June 9 (UTC)

Knowing the subgroups

The article currently reads:

a general principle [is] that knowing the subgroups of a group is important to understand the structure of the group in question.{{fact}}

This is probably true enough, but there are some issues that some people might be interested in.

• The set of isomorphism classes (with multiplicity) of proper subgroups doesn't determine a group up to isomorphism
• We don't know the full list of subgroups of hardly any "well understood" groups. We do for PSL(2,q), but it gets shakier very quickly after that, and for instance we don't know them for most of the sporadics.
• The set of *maximal* subgroups is extremely useful for understanding simple groups, and they are basically known
• The set of p-subgroups decorated with the homomorphisms induced from inclusion and conjugation is extremely helpful for understanding finite groups, these are called (not-exotic) fusion systems
• The lattice of subgroups of a group is very interesting, and people study groups with the same lattice of subgroups (up to isomorphism). The homomorphisms here ar called projectivities.

At any rate, hopefully this helps give an idea of the limitations on the statement. It might be easier to source and more true if it said something more like "group theorists seek to understand the subgroups of group as part of their work in understanding the group". JackSchmidt (talk) 15:22, 16 June 2008 (UTC)

Thank you for this, Jack. As you have seen, I have marked all statements (most of them were added by myself) which probably need some explanation or source by the fact template. This one was/is one of the more awkward ones, which I wasn't yet sure about what to do. Would you do the change (together with some source, and perhaps footnote)? Jakob.scholbach (talk) 21:01, 16 June 2008 (UTC)

Perhaps this is a good point to discuss some other {{fact}} tags (the rest is done by stupidly looking up an algebra book):

• "This method of dissociating the group from its concrete nature, and focussing instead on its abstract properties is a steadily recurring and deeply impacting theme in algebra, and many other mathematical domains, as well." -- seems uncontroversial to me, but may be difficult to pin down with a source (?)

replied below JackSchmidt (talk) 21:48, 16 June 2008 (UTC)

• "Group representations are particularly useful for finite groups, Lie groups, algebraic groups and (locally) compact groups." - It's easy to give tons of books about these topics, but what about this precise claim? Is there an overview over all of representation theory?
• The two claims about physics. I guess I will ask Markus Poessel for help here. Jakob.scholbach (talk) 21:08, 16 June 2008 (UTC)

P.S. Your comments give enough material for a bigger section in subgroups. Dang! What a shoreless topic. Let's also talk about a possible FAC. While we are asking the copyedit team for help, we could lift group theory to a reasonable level. As you may have seen, at the talk page there I have a rough list of topics that could/should be included. This gives a fairly straight curriculum for a while. Jakob.scholbach (talk) 21:12, 16 June 2008 (UTC)

The only controversy I can think of for:

This method of dissociating the group from its concrete nature, and focussing instead on its abstract properties...

is that this sort of ignores computational group theory and computational algebra. For instance the discrete log problem is trivial to solve up to isomorphism, and the matrix recognition project is also trivialized by up to isomorphism. In many cases, one is not given an abstract group which will answer any and all requests for information, but rather a very concrete group and one wants to know something concrete about it. This is very often the case in algebraic topology where one has a specific presentation of a group in mind and wants to solve the word problem for some very specific word. Similarly, the computation of ideal class groups in algebraic number theory is only difficult because of the concrete realization being used (where listing distinct elements is somewhat infeasible computationally). If you count homology groups as groups, then again the concrete way in which these homology groups are given forces one to tackle computationally infeasible linear algebra to answer simple questions like equality of elements. JackSchmidt (talk) 21:48, 16 June 2008 (UTC)

OK. You are right. I think we have to make clear both points of view. I.e. concrete calculations such as the above in contrast with structure-centered work. As far as I can see, the statement as such is not wrong, but has to be completed with a counterbalance consisting of some of the above. However, "group theory proceeds in a a purely abstract manner, disregarding the concrete nature of the group elements and the group operation." needs to be amended as per your comments. Perhaps the first comment could indeed stay as it is and only the second one improved? Jakob.scholbach (talk) 16:17, 17 June 2008 (UTC)

All stupid fact tags have been covered. The above still need treatment. Jakob.scholbach (talk) 17:39, 18 June 2008 (UTC)

I have tried to improve the situation by deleting most of the abstract-concrete(computational) stuff and carefully rewriting it in one place (applications section). A reference for the knowing-the-subgroups-question should also settle the other point. Everybody happy :) ? Jakob.scholbach (talk) 18:29, 20 June 2008 (UTC)

Thanks. I have had a tab open working on this for most of the week, but have not had the time to figure out exactly what I wanted to say. Your edit looks very good, so I've closed my tab.

I always worry about WP:COI problems when I do more than just list theorems or copyedit in group theory, so it takes me forever to fix something like: "please answer succinctly: what is the importance of subgroups in group theory?" Tons of what is in this article and the group theory article offend my sensibilities, but expressing the ineffable, zen-like beauty of groups to the world is not the point of this article. I think this article is doing a wonderful job of giving a broad and complete summary of what a mathematical group is. JackSchmidt (talk) 19:18, 20 June 2008 (UTC)

Lie Groups and the General Linear Group

It says in the article that GL(n,K) is a lie group over any field K. While it is obvious that it is a group over any field K, I am not sure that it necessarily is a lie group because I'm not sure there is some "obvious way" to make it into a smooth manifold. In the case where K = R or C, it's obvious because we can give it the induced euclidean topology, but what about for stranger fields like rational numbers or something like a field with two elements {0,1} with arithmetic mod 2? There may be a "right" way of making these into smooth manifolds which I am unaware of. In any case, clarification would be nice. LkNsngth (talk) 03:06, 17 June 2008 (UTC)

Indeed. I think, Silly rabbit has already fixed your point. Jakob.scholbach (talk) 16:18, 17 June 2008 (UTC)

FAC?

I'd like to hear other's opinion about a Featured Article Candidacy for our nice little article. (Prior to that, solving the above issues w.r.t. abstract approach vs. concrete calculations, and also a thorough copyediting is in order, of course). What do you think? Are there major obstacles for a positive outcome of such a FAC? Jakob.scholbach (talk) 15:46, 19 June 2008 (UTC)

Is there much point in looking at "A class" first? I think the article obviously met the A-class requirements from WP:WPM when it was accepted as a GA, so I suspect this level is not much used? I am in favor of moving to FA for this article, but still have not had time to devote to the group theory article. JackSchmidt (talk) 19:07, 20 June 2008 (UTC)

I'm not too much into the details of assessments, but the requirements of A-class "Essentially complete, well written and referenced" are certainly met. This being my first serious attempt to produce a high-quality article, I may be a bit in rush, but anyway... I have asked the WP:WPM, WP:LoCE for help with a copyedit. Hopefully, somebody will join in. I realize that I just start dreaming when I start reading the article. Jakob.scholbach (talk) 20:41, 20 June 2008 (UTC)

There are no such major obstacles AFAIK. I'm reading it and actually understanding it. It's one of the best articles on Wikipedia, and I went to this talk page just to praise it. It's considerably better than my school book, the better one of those two of mine. Said: Rursus ☻ 09:35, 4 August 2008 (UTC)

Suggestions

Here are some comments about the article as it's currently laid out. These would all be nice to resolve before you apply for FA.

1. The history section needs expansion. It says that groups turn up when you do various other kinds of math, but it does not say how or why. A reader with very little prior mathematical knowledge won't know what, say, it means for a group to act on the solutions to polynomial equations. If he clicks on the wikilinks, he can read about group actions and Galois groups. This answers the how question, but it forces the reader to stop reading the article, start reading two more advanced articles (Galois group in particular is much more difficult), stop when he gets stuck, and for some reason come back here and continue reading. But he still doesn't know why Galois introduced Galois groups. He can read that they provide "criteria for solvability", but he will get no idea at all as to what that means or why it's better to consider these groups instead of, say, staring at some equations for a long time. The same is true of the other history paragraphs: Why is the Erlangen program important? How is it related to groups? What information do they give in this context? Why would I care? The number theory paragraph says that some abelian groups turned up naturally in number theory. How and why? Even though the present article points at History of group theory for more discussion, that article isn't much better in this regard: It presents more names, more dates, and more statements of what was done, but doesn't provide any more motivation.
2. The history section should mention Abel's work on abelian integrals, especially since the article later mentions that he lent his name to abelian groups without saying why.
3. It would be good to hedge on the date of the classification of finite simple groups rather than flatly stating that it was done in 1982. There is some controversy here, and the article should reflect that.
4. Odd wikilinks: "Groups" - "Galois groups"? (in the history section) "Arsenal of basic group theory" -> "Glossary of group theory"? "Wished-for existence" -> "Fraction field"? Principle of least surprise should apply.
5. The basic concepts such as subgroups, quotient groups, and so on should be illustrated by concrete examples (such as the dihedral group defined at the beginning: List its subgroups, quotient groups, etc.).
6. Cosets need their own section. I think the numbers on a clock face are a good example here.
7. Direct and semidirect products should be defined in their own section rather than at the end of the quotient group section. They also need more explanation.
8. "Homomorphism" should mention the Greek root "homo".
9. Lay readers won't know that a "mapping" is a function. (You could even argue that "map" is inappropriate in this context as it suggests some sort of underlying geometry such as a chart.)
10. Did Poincaré really introduce higher homotopy groups? I thought that was done by Hurewicz (and he was doubtful about his definition because they were abelian).
11. The symmetric group should be placed in the "symmetry groups" section rather than the "finite groups" section.
12. Finite simple groups probably deserve a separate subsection of the "finite groups" section".
13. The mentions of monodromy and differential Galois theory in the "symmetry groups" section is going to be way over the head of an average reader. So will GIT.
14. Footnote 87, to the sentence, "Given a group action, this [representation] gives further means to study the object being acted on." reads "For example, the Leray spectral sequence relates arithmetic information to geometric information via the action of the (absolute) Galois group." I don't think that footnote helps the reader: Say I'm a bright high school student who likes math, and I click on that wikilink. I'll probably then click on spectral sequence, and looking at that page will be sort of like watching the videotape in The Ring. That's not the outcome we're hoping for.
15. The content footnotes should separated from the citation footnotes using Template:cref and Template:cnote. (See Alcibiades for an example.) Or they should be written into the text of the article.

Ozob (talk) 00:00, 21 June 2008 (UTC)

Reply

Sincere thanks, Ozob, for your thoughts. I numbered your items for easier reference. I'll work on them. Jakob.scholbach (talk) 10:22, 21 June 2008 (UTC)

1) I disagree with you about this. First of all, the section has to be short. That history of group theory is in bad shape is quite true, but a problem to be amended there and not here. I added two or three little simplifications here and there. Given the needed brevity, we cannot afford to have lengthy discussions why staring at the equations does not give the results we want. In this light "thus giving a criterion for the solvability of such equations" seems to be a very tight and concise explanation of what these groups are good for. If you convince me that we can do better in one sentence, I'll be glad to do so.

2) Forgive me my ignorance, but in what sense are abelian integrals related to abelian groups? I know that Picard groups and so on are related to this, but this is kind of a very far path, right? Again, the section has to stay focussed. Giving the three main roots of the notion, and the later unification and mentioning a few modern branches is all we should aim for at this point, I think. Jakob.scholbach (talk) 22:43, 21 June 2008 (UTC)

3 and 5) done. Jakob.scholbach (talk) 13:02, 21 June 2008 (UTC)

4) wished-for and Galois groups done. The "arsenal" is not too bad a name for a glossary, I think. Jakob.scholbach (talk) 18:06, 21 June 2008 (UTC)

6) cosets are done. I preferred to use the dihedral, which also allows normality to be exemplified.

7) I introduced a subsection instead of hanging it at the end of quotients. However, further expansion seems infeasible. This comment will also turn up in other responses: the article is already long, if not very long, possibly even too long. Thus, we have to stay very focussed. The product material strives to give a rough idea that the processus of quotients can be inverted, but, say, a definition of semidirect products, is too much at this point (and too technical to be interesting for the lay anyway). Jakob.scholbach (talk) 12:46, 21 June 2008 (UTC)

8) and 9) done

10) done. I added also an image to explain what goes on. Jakob.scholbach (talk) 12:53, 22 June 2008 (UTC)

11) Hm. It would take a moment to broaden the reader's intuition what symmetry means in this case. Also, Cayley's theorem belongs to the finite gp section, so I guess I'll leave it there. Jakob.scholbach (talk) 12:53, 22 June 2008 (UTC)

12) is done. Jakob.scholbach (talk) 12:46, 21 June 2008 (UTC)

13) and 14): I have simplified the exposition a bit, and also explained what a group action is. However, it is a fact (regrettable or not) that there are things which will be over the lay's head. Jakob.scholbach (talk) 12:53, 22 June 2008 (UTC)

15) is done. In the very end a sorting of the cnotes is necessary, but I postpone this for now, for minor reorderings are possible. Jakob.scholbach (talk) 12:46, 21 June 2008 (UTC)

1) I agree that there are severe space limitations, and that the history can only be properly treated with its own article, the still-gestating History of group theory. But I worry about the reaction of a reader with no prior knowledge of groups. When that reader reaches the history section, he'll know the definition, the two examples, Z and D₄, and nothing else about groups. So I'm betting that when he reads that Galois used groups to give criteria for solvability of polynomial equations, he won't have any idea what that means: We were just talking about numbers and squares, and this lets us solve equations somehow? If you said something like, "Galois introduced groups to represent the symmetries among the locations of the roots and proved that a polynomial equation could be solved in radicals if and only if the symmetry group of the roots had a special property, called solvability," then I think the lay reader would get some idea of what's going on.

2) Euler proved an addition formula for integrals, and Abel extended it to an addition formula for what we now call abelian integrals; there's a discussion of this in section 5 of [1]; the theorem is stated on p. 102. In modern language, I think (but am not sure that) it's a statement about periods of Riemann surfaces. I think this sort of information is in the historical sketch in the first volume of Shafarevich's Algebraic Geometry; at least, that's my best recollection as to where I read it. Unfortunately this (like the Leray spectral sequence) is also over the heads of lay readers, but I read that this was a very important example in the development of group theory. (I might be wrong, though; Bourbaki in his history doesn't mention it at all.)

13) and 14) Yes, I agree. Again, I'm worried about the lay reader. But it's probably more helpful for an actual lay reader to make these comments than for a mathematician like me to guess what's appropriate. Ozob (talk) 19:48, 22 June 2008 (UTC)

1) I have done a bit more of explanation in the history section. Should now be roughly at the point where you want to have it(?) I guess the key word was symmetry here, and this is now briefly laid out. If the reader has the willingness to scroll down, he will find more details.

2) I'll try to find this gadget in the historical literature. I still fail to see the direct link to groups (other than we are talking about a kind of (co)homology group) and in particular their crucial impact on shaping the field, but I may well be wrong.

13) and 14) Exactly. Suppressing all precise (and, I hope, beautiful) information like the one on diff Gal th would degrade the article. Letting the layman know/feel that there is more out there is certainly not wrong, but exactly right. Placed in a footnote, it does not disturb a smooth reading, on the other hand. Jakob.scholbach (talk) 20:44, 23 June 2008 (UTC)

1) That's closer to what I had in mind. I think it should be possible to improve the other paragraphs in the history section in a similar way.

2) OK, I looked in Shafarevich. He says that the earliest results of this type are due to Jacob Bernoulli. Here's what Euler proved: Let C be an elliptic curve, and let dx/y be a non-zero regular translation-invariant differential. Euler showed that if p and q are two points on C (presumably real points of C?), then $\textstyle \int _{0}^{p}dx/y+\int _{0}^{q}dx/y=\int _{0}^{p+q}dx/y$, where p + q is the sum of p and q under the group law for C. The point here is that translation invariance of the differential form allows you to change $\textstyle \int _{0}^{q}dx/y$ into $\textstyle \int _{p}^{p+q}dx/y$, and then the formula is obvious. Abel then extended this relationship to arbitrary algebraic curves using the Jacobian and linear equivalence somehow.

Of course, this requires far too much background to state in the article. I didn't see Shafarevich saying that this was important in the development of the notion of the group, but I know I read that somewhere... It may be better to leave this out, given that I can't help you find a citation. Ozob (talk) 23:05, 23 June 2008 (UTC)

Wrong

(fd	•	fv)	•	r2	=	r3	•	r2	=	r1, which equals
fd	•	(fv	•	r2)	=	fd	•	fh	=	r1.

I think there is something wrong here. fd • fv does not equal r3. It equals r1. --Abdull (talk) 10:30, 4 July 2008 (UTC)

No, this is right. You may have misinterpreted it insofar as fd . fv means "perform fd after fv" and not before. By inspection of the colored squares, fd . fv is r3 and fv . fd = r1. Jakob.scholbach (talk) 12:08, 4 July 2008 (UTC)

History of group theory

I propose to combine several sections on this topic in various places into a single article. Please, comment at Talk:Group_theory#History_of_group_theory:_proposal_to_fork. Arcfrk (talk) 04:04, 9 July 2008 (UTC)

Thoughts on taking this article to FAC

Cum grano salis magno. ;)

Jakob asked for my thoughts on whether the article was ready to go to FAC. I'll definitely do my best, but I'm no expert, so you should take my comments with a gigantic grain of salt.

First off, I think we should all applaud Jakob for such a massive improvement in the article! :) He's contributed ten times as many edits as anyone else here, and we owe a lot to his devotion.

That said, I feel compelled to say that the article does not yet seem ready for FAC. There are two major areas where I think it would be wise to lavish more care and consideration:

1. The article seems to be written by mathematicians for mathematicians, not for everyday readers. The opening sentence illustrates a general problem: "a group is a fundamental object of the field of abstract algebra." I suspect that the average English major will have no conception of what that means. Scartol, an English teacher, estimated that about 40% of the article was intelligible; he'll be sending some more comments shortly.

I sympathize that there's a tremendous amount of relevant material to cover, and that we lack the space to lead the reader by the hand through it. I suggest two things:

• Rearrange this article so that the simplest things come first, and gradually build up to the more obscure. For example, I would give a simpler explanation of the definitions, and then give many more worked examples at the beginning, examples with ever increasing complexity. The sudden leap to integer addition and thence to D₄ (even before the definition!) is too great, I believe, for most readers. Why not start with a simple finite group with three elements, and work up gradually from there? Bend and stretch your readers' minds, don't break them. ;)

• We should write another article, Introduction to mathematical groups, that dispenses with the higher math altogether and discusses groups at the level of, say, a freshman English major. I'll volunteer to help with that! :)

2. The article does not seem to be comprehensive? The discussion of the mathematics of groups is admirably thorough, but the discussion of their applications to other fields is very brief. But maybe I'm not understanding it properly? Perhaps the qualifier "(mathematics)" in the article title "group (mathematics)" means that only mathematical applications will be covered?

If that's not true, then I would give more "air-time" to non-mathematical applications, ideally picturesque ones, along the lines of Hermann Weyl's popular book, Symmetry. For example, you might have a subsection on symmetry groups, maybe with a paragraph each on frieze groups, wallpaper groups, and braid groups with illustrations drawn from art and Nature. In crystallography, you could devote perhaps a paragraph to the 243 space groups, the 64 space groups consistent with a chiral center, the possibility of non-crystallographic symmetries (e.g., a five-fold symmetry axis of the crystallized molecules), etc. The applications of group theory for molecular spectroscopy (maybe the infrared specta of CO₂ or NH₃?) seem important to discuss? The groups associated with the gauge symmetries that seem to underlie the fundamental laws of physics might be cool to mention; they're kind of sexy for many general science readers, no? You could also show how the form of the Lorentz transformations of special relativity results from the group postulates, again with bonus points for coolness. ;) You could talk more about groups in algebraic coding theory and their applications for cell-phones, DVDs, etc. I'm sure that there must be a bazillion other applications outside of math, too, plus or minus a gazillion. ;)

Perhaps the discussion of these individual applications is too detailed for this article, but I do think that the article is unbalanced at present, favoring mathematics too heavily at the expense of other fields.

3. The "parallel threads" of the History section struck me as not specific enough and not long enough? The chronology seemed unclear, as to who developed what in which order? I think you could spend a little more time on Galois and the question of polynomial roots before moving on to more general conceptions of the group. It might be worth noting that Galois' papers were rejected, not because his proofs were wrong, incomplete or ambiguous, but because they were too radical and their train of thought couldn't be easily understood even by illustrious French mathematicians of the time. Galois' death and the stories surrounding the publication of his group theory also make for good reading, no?

I'm very sorry not to be more enthusiastic about the FAC for this article as it is now, but I hope you can sense that I'm offering my suggestions with the best of intentions. :) Good luck to everyone, especially Jakob, Willow (talk) 20:15, 1 August 2008 (UTC)

Millefois merci, Willow, for your review! Good to see what a fresh mind thinks about the article.

ad 1) Whereever we can boil down the complexity without loosing too much of correctness and rigor, we should absolutely do so.

1a) lead section: the very first sentence is certainly not very entertaining, and others are not too, for a general reader. (The first one just seems to be kind of standard to tell roughly the global position of the subject matter amidst human knowledge). However, it's gonna be impossible to avoid words not everybody knows. But we should aim for a gradually increasing amount of weird notions, too.

1b) "increasing complexity": I (and others) hoped, that the intro example of the integers are as smooth a beginning as possible. The idea was an index poking out of the screen saying "You may not have called it a group, but you know a group". Seriously, this is an object everybody has seen some day, whereas I feel (possibly with "warped mental coordinates"?) a group with three elements is not fundamentally easier than the above group with 8 elements? What would we gain by adding Z/3: a relatively short description of the elements and the composition, but the axioms would have to be explained with this example too, much in the same way as with the other ones we already have. But I invite you to convince me of my being wrong ;) Also, placing the definition after the examples was done with an increasing level of abstraction in mind. Having the definition (I'm happy to see your simpler / more intuitive one!) in the beginning could (I think) stress the reader who is not acquainted to seeing symbols... What ordering of the sections would you have instead?

1c) Introduction to mathematical groups: hm... There is also examples of groups which could deserve this purpose.

2) group (mathematics) certainly should include non-math applications. (Only mathematical applications of groups (mathematics) should not :)) I guess you are right that they are currently underrepresented, which is due to my mathematical background. That's certainly a point. Overall, we have to be very restrictive with including further material, just because of space limitations. There is an article applications of group theory (currently still stuck in group theory, but that is to be moved at some point), so here we cannot give more than an overview, or perhaps not even that, but only a few glimpses.

We do have a subsection on symmetry groups (including a little image of wallpapers. Frieze groups and space groups are just the one- and three-dimensional counterparts, right? This makes my space-limit-buzzer ring...) I like braid groups, they are also easy to visualize. Perhaps another image at the right or left is nice. But perhaps it's better to stick to real-world groups such as the one you are referring to. Do you know of a specific, easy-to-visualize case of groups in chemstry. I.e. where you can interpret the spectra or something in terms of groups? Lorentz is also pretty cool, I have to say. Didn't know that. So let's include something along these lines.

Only now I realise that we actually had the Lorentz transformation there, however buried in the Poincaré group. I made it clearer (and a bit longer). Plus three cute molecules with their symmetries. Jakob.scholbach (talk) 15:40, 3 August 2008 (UTC)

I have, with the help of the chemistry guys, included a really nice application: apparently groups can also be used to show that certain molecules can not be symmetric. Does this satisfy your thirst for applications? Jakob.scholbach (talk) 19:32, 4 August 2008 (UTC)

3)OK. More clarity is a good idea, and also the degree of advance of G's work. I'll work on that. However, G's death, as picturesque as it is, is pretty much unrelated to gps, I feel. Jakob.scholbach (talk) 21:31, 1 August 2008 (UTC)

There are now some more dates and a bit more on Erlangen program. Jakob.scholbach (talk) 15:40, 3 August 2008 (UTC)

Playing with the introduction

I started playing around with the intro to try and address some of WillowW's concerns. I'm not sure how happy I am with it, but here is what I have so far ...

In mathematics, a group is a set together with an operation that acts on two elements of the set in a particular way. A familiar example is the set of integers and the operation addition (denoted +); given two integers, the operation + adds them and returns another integer. The particular way an operation must behave on the set is given as a list of conditions in the formal definition of a group below. Groups play an important part in many academic disciplines, but are often studied in their own right within the mathematical discipline abstract algebra.

• I don't really see the point in mentioning their relation to other abstract objects in the first paragraph, so I scrapped it.
• I've avoided the word field so that if we need to discuss the 'other' field we can do so without worry. I don't know about 'academic discipline' though.
• I couldn't think of a nice way to introduce the word 'set' (is that possible?), but I hope the example immediately following its use is enough to give the uninitiated a naive (I couldn't resist) feeling for the word.
• I wanted to include a very familiar example as early as possible so people could relate to the article. If you're old enough to read and use wikipedia, you're probably old enough to know what integers and addition are.
• I think it's important to say these things are important and that abstract algebra is their home, so to speak, but not before giving a feeling for what a group is.

Cheers, Ben (talk) 10:54, 2 August 2008 (UTC)

Looks fine to me. I don't know whether you refer to the status befor or after I changed the lead section a little bit, but coming up with Z in the lead looks pretty nice. You wanna perform the change? Jakob.scholbach (talk) 16:58, 2 August 2008 (UTC)

Should we wait and see what WillowW thinks too? I'm also not sure about only placing the restrictions on the binary operation. Is that strictly correct? Is the above wording suggesting that there should be a group with just one element for instance? Ben (talk) 03:18, 3 August 2008 (UTC)

I think your introduction looks nice. Focussing on the operation is usually wise. A group with 1 element is fine (trivial group), but a group with 0 elements is usually disallowed. I don't think worrying about such details in the first paragraph is needed. The mathematical fix is just replacing "set" with "nonempty set", but I think this needlessly draws attention to a minor aspect. Discipline is fine. JackSchmidt (talk) 03:56, 3 August 2008 (UTC)

Sorry, I had fields on the brain. Ben (talk) 04:22, 3 August 2008 (UTC)

By the way, a group with 0 elements is always disallowed, since one of the axioms of a group is the existence of an identity element. RobHar (talk) 16:08, 3 August 2008 (UTC)

---

In mathematics, a group is a set together with an operation that acts on two elements of the set in a particular way. A familiar example is the set of integers and the operation addition (denoted +); given two integers, the operation + adds them and returns another integer. The particular way an operation must behave on the set is given as a list of conditions in the formal definition of a group below. Groups play an important part in many academic disciplines, but are often studied in their own right, called group theory, within the mathematical discipline abstract algebra.

Most groups contain a set with an infinite number of elements like the integers, but this is not necessary. The set under consideration does not even need to be a set of numbers; symmetry groups are a set of transformations that leave a geometrical object unchanged with the operation composition, where given two transformations, composition applies one transformation after the other. An easy to visualise example is to start with a square and consider the four transformations that rotate the square by 0, 90, 180 and 270 degrees; applying any of these transformations always returns another square, composing any two transformations is the same as one of the four transformations we started with and this group has only four elements.

If G is the set under consideration, and * is the corresponding group operation, it is common to denote them as a pair and talk about the group $(G,*)$. Every group has at least one group contained inside of it, called a subgroup, and there are notions of the sum, product and quotient of groups. There are many properties a group may satisfy, for instance, a group $(G,*)$ is said to be Abelian if for any elements $g,h$ in the group, $g*h=h*g;$ so the integers together with addition form an Abelian group. In some cases it is possible to classify all the groups satisfying certain properties, and as of 2008, this is a very active area of research.

Historically, the concept of a group arose from the study of polynomial equations. These ideas were brought together by the end of the 19th century and since then, the study of groups has flourished. Modern group theory studies parts and combinations of groups, such as subgroups, quotient groups and simple groups. A particularly rich theory has been developed for finite groups, which culminated with the monumental classification of finite simple groups. Groups are also underlying many other abstract algebraic objects such as rings and modules. Symmetry groups, particularly the continuous Lie groups, play an important role in advanced theoretical physics and, to a lesser extent, in chemistry.

---

I added a little more, mostly keeping everything that was already in the original intro but rearranging it a bit more. I haven't done anything with the last paragraph yet. I want to mention topological groups and group homomorphisms at least. Still on the right track or .. ? Ben (talk) 11:01, 3 August 2008 (UTC)

I like the 1st paragraph and the last one. I think, the 2nd and 3rd are too detailed. "Every group has at least one group contained inside of it, called a subgroup" is true, but not really good, because actually every group has two subgroups (0 and the whole group). Also, this is not that important. Somehow, the lead has to be an overview of the article. So symmetry groups are important, but talking about the square is not that representative for the concept. Also, there is some redundancy between your 3rd and 4th paragraph. I tried to mingle some of your points with the existing stuff. How about

---

In mathematics, a group is a set together with an operation that acts on two elements of the set in a particular way. Since the definition of a group is so general, groups occur in many forms and have applications in numerous areas in and outside mathematics. A familiar example is the set of integers and the operation addition (denoted +); given two integers, the operation + adds them and returns another integer. Another common type is the symmetry group of a geometrical object, which consists of the set of transformations that leave the object unchanged; two transformations are combined by performing one after the other. Such symmetry groups, particularly the continuous Lie groups, play an important role in in many academic disciplines such as theoretical physics and, to a lesser extent, in chemistry.

Historically, the concept of a group arose from the study of polynomial equations. Modern group theory, a very active part of the mathematical discipline abstract algebra studies groups in their own right. Various notions, such as subgroups, quotient groups, simple groups and group representations build a framework designed for investigating deeper structures of and related to groups. A particularly rich theory has been developed for finite groups, which culminated with the monumental classification of finite simple groups. Groups underlie many other algebraic objects such as rings and modules and are a central organizing principle of algebra, and mathematics in general.

---

Jakob.scholbach (talk) 14:17, 3 August 2008 (UTC)

Given the size of this article, I thought we should be heading for a 3 - 4 paragraph lead (per Wikipedia:LEAD#Length), so I started to pad it out a bit. Maybe I did go a bit far with the examples though. I hadn't touched the fourth paragraph, it was what was left over after taking pieces from the original and writing the second and third paragraphs. Looking through the articles table of contents, I had in mind that the first two paragraphs would cover the definition and examples of the article (this is why I chose the square), the third paragraph the basic concepts part (it still needed to mention homomorphisms), and the fourth a historical/current research summary and applications (hadn't even attempted it yet). That covers all the sections of the article, and satisfies that lead guideline. What do you think? My biggest concern with a 'short but to the point' lead is that readers will be looking at a list of terms more than getting a feeling for what a group is/does/etc. Expecting the average reader to try and get that from the articles main text might be a bit harsh.

Oh, I had the trivial subgroup in mind when I wrote "one subgroup", and didn't want to have worry about including exceptions in the lead or explaining what the subgroups might be, I just wanted to introduce the term. Even if mentioning subgroups stays, I do think there must be a better way of doing it though. Ben (talk) 15:45, 3 August 2008 (UTC)

OK. The length of the lead could be longer. However, we should keep in mind that the symmetry of square example is just an example to show what a group can be. This is not really at the heart of the notion of groups or group theory (as opposed to the integers, they are not only a nice example, but crucial to groups and many other things). Here is another take. Better ? Jakob.scholbach (talk) 16:41, 3 August 2008 (UTC)

---

In mathematics, a group is a set together with an operation that acts on two elements of the set in a particular way. Since the definition of a group is so general, groups occur in many forms and have applications in numerous areas in and outside mathematics. A familiar example is the set of integers and the operation addition (denoted +); given two integers, the operation + adds them and returns another integer.

Groups share a fundamental kinship with the notion of symmetry. Any geometrical object possesses a group encoding its symmetry features, called symmetry group. It consists of the set of transformations that leave the object unchanged; two transformations are combined by performing one after the other. Such symmetry groups, particularly the continuous Lie groups, play an important role in in many academic disciplines. Matrix groups, for example, can be used to understand fundamental physical laws underlying special relativity. They also find applications in chemistry.

Starting with Évariste Galois in the 1830's, the concept of a group arose from the study of polynomial equations. Getting input from other fields such as number theory and geometry, the group notion was firmly established around 1870. Modern group theory, a very active part of the mathematical discipline abstract algebra studies groups in their own right. To explore the realm of groups, various notions have been contrived to break groups into smaller, better understandable, pieces. Basic ideas include subgroups, quotient groups and simple groups. In addition to the abstract approach, group theorists also strive to express groups in computable terms, both from a theoretical and a computational point of view. In this direction, group representations are fundamental. A particularly rich theory has been developed for finite groups, which culminated with the monumental classification of finite simple groups. Groups underlie many other algebraic objects such as rings and modules and are a central organizing principle of algebra, and contemporary mathematics in general.

---

paste Scartol's comments here

The following comments of Scartol are copied from WillowW's talk page.

1. In mathematics, a group is a fundamental object of the field of abstract algebra. A group is a set of elements with a single operation by which two elements may be combined into a third; to qualify as a group, the operation must satisfy several conditions. I'm used to getting the actual definition in the first sentence. Is there a particular reason it's second here?
2. ...and form the core of several more complex algebraic objects... Is the "more" needed here?
3. The examples are very helpful in aiding comprehension – but the section is titled "Definition and illustration", and I'm used to getting the formal definition first. Is there a reason for the inverted structure?
4. The (to me) unusual facts that the • symbol isn't multiplication (which is what I learned in high school math classes), and that the symbols are read right to left aren't given much emphasis – at least early on. I wonder if perhaps we should mention that the • is often used for multiplication (or maybe that was just my school?), and that most equations are read left to right. Or is this too simplistic?
5. I think linking square at the start of the symmetry group is excessive.
6. Historically, the group concept has evolved in several parallel threads. I'd lose the word "Historically". Doesn't add much, in my opinion.
7. We could really use more years (or at least a circa) in the first paragraph of History. Same for the third.
8. The second paragraph, meanwhile, feels cursory and rushed. Another sentence or two describing more about Klein's program?
9. All the last-name-only links are confusing. Is there a reason we don't also give first names? Maybe even nationalities? I know these are less standard in science/maths literature, but this is for a general audience, yeah?
10. Strictly speaking the closure axiom is already implied by the condition that • be a binary operation on G. Perhaps: "...a binary operation on a group G"?
11. The group operation on this set (sometimes called coset multiplication, or coset addition) behaves in the nicest way possible: "nicest" feels odd. Maybe we need a better word? (Is it possible to be POV in this situation?)
12. The study of abelian groups is quite mature, including the fundamental theorem of finitely generated abelian groups, and reflecting this state of affairs, many group-related notions, such as center and commutator, describe the extent to which a given group is not abelian. I'd vote for a semicolon instead of the comma + "and" preceding "reflecting this state...".
13. Examples and applications of groups abound. Why is the first part italicized in the article?
14. A group is called finite if it has finitely many elements. This wording feels odd. How about: "...a finite number of elements."?

1. and 2. Changed

3. The intention was to give the reader the chance to understand the concept by examples first, instead of being annoyed by symbols. Does that make sense?

Yeah, that's fine (I actually prefer the way you've done it). I just wondered if other math folks would bristle about it being non-standard. – Scartol • Tok 12:32, 4 August 2008 (UTC)

4. OK. I tried to make pretty explicit the abstract nature of the bullet symbol. You are right, usually it's multiplication. (Notice however, the slight difference between • we use for a general group composition and · for multiplication). The right-to-left thing is also explained a bit. However, it still remains an intuitively awkward thing -if that comforts you: it took me years to really digest this way of writing ....

I think it's something people can pick up easily, especially with the fixes you've made. It just threw me a bit when I first read it. – Scartol • Tok 12:32, 4 August 2008 (UTC)

5 OK

6. OK

7. Should be OK now?

Yes, looks good. – Scartol • Tok 12:32, 4 August 2008 (UTC)

8. OK, a bit more details.

9. and 10. Changed

11. Changed to "most natural"

12. OK

13. It's supposed to be kind of a section hookline: Examples of groups and Applications of groups

I was referring to the first actual sentence of that section – it's italicized and I don't think it should be. – Scartol • Tok 12:32, 4 August 2008 (UTC)

14. OK Jakob.scholbach (talk) 15:48, 3 August 2008 (UTC)

Looking good. As for the phrase "groups share a fundamental kinship with symmetry" – it's linguistically okay, but it's not the clearest way to word it, in my opinion. Although your phrase is enjoyably poetic, I suggest something simpler like "Groups are especially important with regard to symmetry" or some such. – Scartol • Tok 12:32, 4 August 2008 (UTC)

Regarding #3, I'm afraid I'm responsible for that. But I honestly think some examples first is helpful. Obviously in some contexts one may want a formal definition first, but pedagogically, the "examples first" approach has often been used in some contexts as a gentler introduction. Group theory is one of those contexts, so I thought it'd be fine. --C S (talk) 07:10, 6 August 2008 (UTC)

Regarding #4, there's no reason we have to go right-to-left. Enough people do write symmetry groups from left-to-right. If it's confusing, it should be changed. We're not trying to train readers to be able to read the literature, we're trying to explain the concepts to the lay reader. --C S (talk) 07:10, 6 August 2008 (UTC)

I would say that I prefer a phrasing along the lines of "fundamental kinship" rather than the other phrasing. The kinship describes more of a two-way relationship between groups and symmetry, whereas, to me, the other phrasing seems to say that groups are useful for studying symmetry, but not really vice versa; whereas the origins of group theory as permutation groups, and much subsequent work on the subject has been based on interpreting a specific group as an encoding of some symmetry. RobHar (talk) 16:54, 4 August 2008 (UTC)

The point is that the relation is not two ways. There are many groups that have no relation to symmetry. −Woodstone (talk) 18:21, 4 August 2008 (UTC)

Every group acts on itself (in a few ways) which already relates it to the symmetry of an object (itself), though one cannot generally get too much information out of this (though this is often the first step in studying a group). Finding other objects for it to act on is a fundamental problem in group theory, and the more interesting objects found, the better the group is understood. People are constantly trying to relate a group to a symmetry. The very important subject of Representation theory is all about studying all possible actions (i.e. symmetry) of a group on vector spaces. The theory of automorphic representations is all about looking at a group acting on itself. All I'm saying is that I believe symmetry is as useful (or pretty much as useful) to the study of groups as groups are to the study of symmetry. A group is generally much better understood when it is known to act on something, anything. Every finite group is a group of symmetries (conjecturally even a Galois group over Q), every semisimple Lie group is a group of symmetries of a geometric object, ... RobHar (talk) 18:47, 4 August 2008 (UTC)

Echoing Rob's comments, I find the comment that there is a group that "[has] no relation to symmetry" quite misleading. As far as I'm concerned, groups are symmetry. Basically every important group arises in the context of some kind of symmetry, and for ones that don't, ways are concocted to take advantage of symmetry properties. I'd be quite interested in any counterexamples to my claim because I'm sure it must be quite unusual and fascinating an example. --C S (talk) 07:10, 6 August 2008 (UTC)

right to left?

C S above, voted for changing the right to left notation currently in the article to the opposite. I have personally never seen the other version. Can you give an example? It's true that we want to access lay readers, but nonetheless we should, I think, also adress undergrad students which will profit from sticking to usual notations. Comments? Jakob.scholbach (talk) 08:28, 6 August 2008 (UTC)

It looks like the only place in this article where this arises is in the example on symmetries of the square. I would certainly say that if it is much more common when dealing with symmetries of such things to use right-to-left, then it would probably be more confusing for the average reader for wiki to use the opposite convention. I have no knowledge about which convention is more in use for this type of symmetry. However, the right-to-left vs left-to-right really only arises when a group is acting on something. It wouldn't be hard to just consider D₄ acting on the square via a right action.

Perhaps the action could just be made more notationally explicit: you could have the square (or some notation representing it) on the right and a bunch of symmetries lining up right-to-left to do their thing on it. That could justify the notation. RobHar (talk) 17:03, 6 August 2008 (UTC)

Pretty much any easily accessible mathematical example will lead to "right-to-left" (matrix multiplication, composition of functions). This is all of course because an (irrational) preference for left actions. It would be very akward to explain to people that: fg(x) = g(f(x)). Hence, a very strong vote to keep right-to-left notation. (TimothyRias (talk) 13:30, 18 August 2008 (UTC))

The lead

In some recent changes, Willow introduced the concrete axioms and proof sketch that Z is a group. I think that's contraproductive if we aim for a leisurely readable lead section (and also gives too much emphasis on Z in the lead section, compared to its portion in the text body).

Also " Groups share a fundamental kinship with the notion of symmetry." got replaced by " Groups are often used to clarify the symmetry of a geometrical or mathematical object." which I would also undo: as far as I understand (and this seems to be endorsed by RobHar and C S above), groups are more or less the same as symmetry. Perhaps one could say, they are like twins that grew up in different households, so their behaviour is a bit different, but when they look into each other's face, they have visibly the same parents. That's what I wanted to say with "kinship". The current wording does not reflect this relationship in two directions that well.

Comments? Jakob.scholbach (talk) 21:08, 6 August 2008 (UTC)

As I remarked above: the point is that the relation is not two ways. There are many groups that have no relation to symmetry. −Woodstone (talk) 21:17, 6 August 2008 (UTC)

Woodstone, your remark was rebutted by two people above (as Jakob mentions) (including me), so reiterating the same comment is unlikely to further your cause. Do you have arguments to support your statement? I am reverting the change as the discussion related to it was either unfinished or finished with the opposite conclusion. RobHar (talk) 21:59, 6 August 2008 (UTC)

Yes, I was worried that the first paragraph of the lead was maybe too technical. On the other hand, I thought it might be intelligible as it stood? We should ask Scartol or someone similar.

For me, the problem with the "fundamental kinship" wording is that, I believe, it won't mean anything to most readers. My alternative wording wasn't that great, either, but I was trying to express the idea that the symmetry of an object can be captured quantitatively by its symmetry group, as opposed to the vague notions of symmetry that all people have.

The problem with specifying "geometrical object" in sentence 2 was that some mathematical objects may not be "geometrical" but might still have a symmetry group? For example, the physics of a two-dimensional field (φ_x, φ_y) might have U(1) symmetry group in the field space; the potential energy might depend only on φ_r² = φ_x² + φ_y². Am I understanding that correctly? There might be other examples as well. However you all decide is fine with me, however. Willow (talk) 00:13, 7 August 2008 (UTC)

Comments from Randomblue

This article has obviously been much improved recently. I read a random part of it, starting at "Basic concepts", and finishing at "Abelian groups". I made a few small edits and listed a few small comments here. The main thing that seems to come up is a need for an in-depth copyedit. Hope this is helpful! Randomblue (talk) 13:08, 25 August 2008 (UTC)

1) "One may then speak simply of the set of cosets of N." Is this just notation? This seems a bit ad-hoc.

2) "a graphical device showing certain features of discrete groups" This seems misleading since Cayley graphs contain all the information about the group, not just 'certain features'. Furthermore, 'certain features' is vague, and if kept, should maybe be explained, linked, and referenced.

3) "Taking subgroups and quotients of a given group G tends to reduce the size of G." This may suggest that subgroups and quotient may be able to produce groups bigger than G. The vague word 'tends' needs to be removed. Anyway the sentence doesn't make much sense since the size of G is fixed, G is some fixed group.

4) "Several group constructions reverse this direction" needs rephrasing. 'direction' seems an unfortunate choice of word.

5) "given two groups, one constructs bigger groups" This isn't always true, the constructed groups may not be bigger than both original groups.

6) "Here, "product" has a slightly different meaning than the product of elements in a group." only slightly?

7) "it allows for the twisting of the group operation on one factor" this should be explained, linked, and referenced.

8) "The structure being determined by the group operation, this is made formal by requiring" I don't understand this sentence. Maybe check the English.

9) Why do you use the notation 'a' in the section "Group homomorphisms"? Phi and h seem to be much more standard notations. Also, a is used many times in the article to refer to an element of G (see "Abelian groups" and "Cyclic groups" just below).

10) "The study of abelian groups is quite mature" could do with some rephrasing. Maybe 'the structure of abelian groups is well understood, owing notably to the fundamental theorem of...'.

Dix fois merci :), Randomblue, for your ten comments.

1)Yes, this is just a saying. The cosets is intended to mean both the left and the right cosets. We need some way to call them later, and calling them the (equivalently left or right) cosets is more odd.

2, 3, 4, and 6) OK

5) For example?

7) We refer to semidirect product for details.

8) changed

9) I did not want to use greek letters, for they are not that widespread in general scripture. h, g are bad because they look like group elements instead. As we need two maps (here a and b), f alone is not enough ...

10) I also thought what you think now, but JackSchmidt has tought me that abelian groups are not that well understood either. See a comment of his above. (possibly archived). In light of this, I think the present wording is better than what you suggest. Jakob.scholbach (talk) 15:05, 25 August 2008 (UTC)

I've been working on copyediting the article, section by section. I'll look over your comments as I go to try to address them. — Carl (CBM · talk) 15:16, 25 August 2008 (UTC)

De rien, Jakob :). For 5) a trivial example is {e}x{e} = {e}, but there are much less trivial examples.Randomblue (talk) 17:37, 25 August 2008 (UTC)

Hm. But the semidirect product is made of some normal subgroup, which is (not necessarily strictly) "smaller", i.e. the semidirect product is bigger (your example is just when is does not get strictly bigger). Jakob.scholbach (talk) 17:56, 25 August 2008 (UTC)

Yes. Randomblue (talk) 18:00, 25 August 2008 (UTC)

11) "Because the definition of a group is so general , groups [... ] have applications in numerous areas". I disagree that generality leads to applicability. Classes that aren't sets, although extremely general, don't occur much outside mathematics (I don't know if they even occur much in mathematics!). A delicate balance between generality and structure make groups interesting.

12) "If no such n exists the order of a is said to be infinity." Maybe add a citation. Also, quite amusingly this sentence is in the "Finite groups" section.

13) "Ammonia, NH3. Its symmetry group is cyclic of order 3." It seems the molecular symmetry is bigger; isn't the symmetry group of order 6?

11) Fair enough. Better now?

12) Right.

13) No, I don't think so. Do you think of keeping one H-atom fixed and exchange the other 2? This would not keep the N-atom in the middle fixed, for it is not a plane molecule. Jakob.scholbach (talk) 18:37, 26 August 2008 (UTC)

11) Better.

13) In the Molecular symmetry article, 'planes of symmetry' are allowed. The fact that the molecule is nonplanar doesn't change much. Randomblue (talk) 20:38, 26 August 2008 (UTC)

Je suis con, quoi. Jakob.scholbach (talk) 17:40, 27 August 2008 (UTC)

Comments about the lead:

14) "Any geometrical object possesses a group encoding its symmetry features, called its symmetry group." That's a bit of a fast statement. Much of (modern) geometry deals with 'geometrical objects' defined up to homeomorphism or homotopy equivalence, where it may become difficult to give any sense of symmetry to these objects. That said, if the citation was referenced...

15) "Modern group theory—a very active part of the mathematical discipline of abstract algebra—" This might be original research, or it might not. Anyway, it needs a citation.

16) "A particularly rich theory has been developed for finite groups, which culminated with the monumental classification of finite simple groups completed in 1983." This is controversial. Maybe write 'maybe completed'. Anyway, it needs a citation. Randomblue (talk) 13:02, 27 August 2008 (UTC)

OK. In general, I tried to avoid inline citations in the lead, for aesthetic reasons, and also, because the statements made there are all repeated (more lengthily) in the main text. But the statements as such deserve a back-up, that's right. I will be out of town for a few days, but perhaps somebody else feels an itch to amend them / even yourself? Thanks. Jakob.scholbach (talk) 17:40, 27 August 2008 (UTC)

14) is amended (slightly watered down)

15) OK.

16) I see from the subpage that there have been concerns. In the 2004 overview paper of Aschbacher, so such concerns are talked about (anymore?) Unless you or somebody can give a scholar reference that there are doubts/gaps etc. we have to stick to the statement that it is proven. However, the article also does make clear (in the body) that the project as a whole is still ongoing, so I think the situation is adequately covered. Jakob.scholbach (talk) 12:08, 1 September 2008 (UTC)

History section

To comply with WP:SCG#Attribution, we need to track down citations for all the historic research publications alluded to with parenthesized years in the history section. Is there a book that collects together most of these foundational papers into a single volume? — Carl (CBM · talk) 14:49, 25 August 2008 (UTC)

Leray spectral sequence?

Footnote x currently has the mysterious sentence, "A more involved example is the Leray spectral sequence relating arithmetic information to geometric information via the action of the (absolute) Galois group." I'm sure that the Leray spectral sequence is useful for studying absolute Galois groups, but this sentence seems very obscure to me. The Leray spectral sequence with respect to which sheaf and which morphism of spaces (I presume they're varieties over Q)? (Maybe this is not the right sort of sentence for this article. Might it be better to limit the sentence to "representations of absolute Galois groups are useful"?) Ozob (talk) 17:29, 25 August 2008 (UTC)

There are certainly better examples for group actions, but as it stands it's perfectly reasonable for me. I wanted to say that the action of the Galois groups gives means to understand etale cohomology groups, for example. But I dont object scrapping it. It is not at all central, I guess. Jakob.scholbach (talk) 17:59, 25 August 2008 (UTC)

Ah, OK, that makes sense. I've edited the article so that it's clear that the group is acting on étale cohomology; the Leray spectral sequence doesn't seem to be directly relevant here, so I took that out. Ozob (talk) 19:48, 25 August 2008 (UTC)

Trimming

This article is currently very long, perhaps some of the Basic Concepts and Examples and Applications sections would be better suited to group theory? I don't think a discussion (however brief) of quotient groups, for example, is really necessary to understand what a group is and how it's used, it's just a way of studying groups and as such would be better off in group theory. --Tango (talk) 17:53, 25 August 2008 (UTC)

I'm also concerned about the ever-increasing length of the article. We definitively have to find ways of reducing the article length, either by moving / reformulating / scrapping. However, I disagree with your idea. Moving the whole "Basic concepts" section elsewhere would, first of all, make the rest of the text pretty uninviting, because explaining even the very first phenomena (such as "A nontrivial group is called simple if its only normal subgroups are the trivial group and the group itself.") would then be impossible without resorting to other articles. And also, somehow, the article would miss the flavor of groups, if we are avoiding all these notions. I would scrap the following

• Mention and prove of (a b)^-1 = b^-1 a^-1
• the paragraph on kernel and image of homomorphisms
• the subsection "Cyclic multiplicative groups", moving the notion of "free of rank 1" up to the cyclic group sections (very briefly)

Jakob.scholbach (talk) 18:28, 25 August 2008 (UTC)

I said "some of", certainly removing all of it wouldn't be good. I think it's the best section to find something to remove, though. --Tango (talk) 18:47, 25 August 2008 (UTC)

What content should be in this article, versus group theory, should not be considered in isolation from how other articles handle this division, e.g. Ring (mathematics) and ring theory, topological space and topology, vector space and linear algebra, set and set theory. In the examples given, the former is generally about the structure, while the latter is generally about the field of study. Based on these examples the content on quotient groups seems to belong here. Paul August ☎ 18:33, 25 August 2008 (UTC)

My thinking was that being a quotient group wasn't something specially about a group (you can't tell by looking at a group if it happens to be a quotient group), it's about how the group relates to other groups. As such, it would make sense under group theory. It would also make sense here, but we need to find something to cut. --Tango (talk) 18:47, 25 August 2008 (UTC)

If I may jump in here, I would think that quotient groups do belong in this article, but the classification of finite simple groups and representation theory can be merged to Group theory. A brief mention of the sporadic groups as examples of interesting finite groups should suffice. If you want to cut deeper, I don't think this article really needs a History section either -- there's nothing wrong with it, but it fits better in the other article. Melchoir (talk) 21:13, 25 August 2008 (UTC)

...Um, are there no objections to removing those sections? Melchoir (talk) 08:58, 27 August 2008 (UTC)

I think it is indeed probably better to move the topic of representation theory to the group theory article, as it is mainly concerned with the study of groups and doesn't directly say anything over groups in general. I would however keep the part about classification of groups. I would expect an article introducing a type of object to say something about what is known about all such possible objects. The small history section is probably also fine. Not having it will probably launch a whole bunch of complaints when the article goes FAC.

On a broader note I would note worry too much about the length. The article discusses a very broad topic (the avarage university library will have a bunch of shelves devoted to just groups and many more of we include more advanced group theory.) so to be comprehensive the article can d=be expected to be long. (and compared to the 150k of the recently FA promoted general relativity the 80k of this article isn't that big. (TimothyRias (talk) 10:10, 27 August 2008 (UTC))

Actually, I'm not too worried about length, but if this article were on FAC I'd have to ask, why is there a History section but not Education or Outstanding Problems? Although... this argument is somewhat weakened by the fact that Group theory doesn't have those sections either.

I guess as a reader I'm not sure in which article to look for which information. The presence of a History section here probably doesn't help this issue. Melchoir (talk) 10:37, 27 August 2008 (UTC)

I think removing either the history section, classification of finite simple groups or representation theory would be a step back. History is just interesting for people not too much into maths, the classification is one of the most important results, or probably simply the result about groups. Likewise, rep th is at the very core of what a group is. (Already the idea to express the group operation in terms of matrices is so important for groups). I'm not sure I understand what you mean by "Education" in this context. Outstanding problems would be nice, if we can present them (briefly!) so that the reader can get a rough idea of what goes on. Both this article and group theory need (IMO) a history section - however this one here should focus on the historical development of groups, whereas the other one more on the development of the endeavour of studying groups. If we remove anything, the only things I can light-heartedly agree with are "Notations" and "Products of groups". What about these ones? In general, me too, I think the length of the article is justified by the breadth and depth of its topic. Jakob.scholbach (talk) 11:13, 27 August 2008 (UTC)

Oh, in an Education section, as a reader I'd hope to learn:

• To which students is the group concept generally taught? Perhaps it was once taught in primary education in the U.S., and if so how did that work out? Do the Romanians have a better idea...?
• Where do groups fit into the curriculum? Is it considered better to have a solid grasp on matrices first? Do you need set theory? Is group theory a good introduction to proofs?
• What are the most common misconceptions or cognitive barriers about groups? What does that say about mathematical intuition? What does it say about language and notation?
• Is there any systematic research on the use of illustrations in teaching groups? The use of manipulatives? How about computers?

Melchoir (talk) 11:40, 27 August 2008 (UTC)

• Oh, and how could I forget: are there any fun controversies over which topics in group theory should be emphasized and which should be passed over? Anything like a "Down with Determinants"-type shakeup? Melchoir (talk) 11:50, 27 August 2008 (UTC)

I like some of your points (e.g. the linguistic stuff). I'm not sure, however, how the majority of them fits into an article like this. Actually I think, they tend to be a bit off-topic. Also, things like "Is it considered better to have a solid grasp on matrices first?" are difficult to answer, at least when it comes to information that can be backed up by references. On the other hand, a good article should (implicitly) answer some of your questions or at least give an idea of their answer. Let's keep focussed on the mathematical essence of groups, I propose. Jakob.scholbach (talk) 17:34, 27 August 2008 (UTC)

Right, for this article focus has to remain a priority. And the subjective questions about the best educational approach would be much harder to answer than factual questions about theory or history. It's something to consider for another time, another place... Melchoir (talk) 20:16, 27 August 2008 (UTC)