# Talk:Group theory

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Field: Algebra

## Scope and structure

Now that Group (mathematics) has reached GA status, I feel that it is time to start working on this article and reduce overlap between the two articles. To do this I think we need to firmly establish what the scope of the article should be.

I feel that this article should mainly deal with the field of research in mathematics dealing with groups. So it should deal with the history of the subject (starting with Galois and abel probably), the major players (past and present), major results, open questions, and current branches of research. It should probably also say something about connections with others fields of research and thus applications. (note that these should be applications of the theory of groups-i.e. situations in which the results obtained by considering abstract groups are useful- not merely situations where groups are applied) (TimothyRias (talk) 08:05, 15 May 2008 (UTC))

Sounds good. I have a little trouble envisioning the result -- each major research area has its own history, players, results, and open questions, and similarly in each era, the division of group theory into subfields was different. It is probably a good idea to sketch out the table of contents here. JackSchmidt (talk) 12:33, 15 May 2008 (UTC)

Mock TOC:

• History
• Major results
• Classification of finite simple groups, etc.
• Groups vs. geometry. Groups vs. symmetry
• Branches of group theory
• Open problems/current branches of research.
• Applications
• Applications in other fields of mathematics:
• Applications in other fields:
• cryptography (e.g. using ab. varieties over finite fields, in particular elliptic curves, Diffie-Hellman, fast exponentiation)
• coding theory
• Physics
• Chemistry (there are whole books about, e.g. Group Theory and Chemistry by Bishop)

Feel free to mess around with this.(TimothyRias (talk) 12:46, 15 May 2008 (UTC))

I left a comment at Talk:Group (mathematics). In short that article comments on what is a group, whereas here the focus is on why, how, when, where to. The suggested table of contents is a good step in this direction. Geometry guy 20:33, 15 May 2008 (UTC)

Fleshed out the TOC a little bit. (TimothyRias (talk) 15:15, 16 May 2008 (UTC))

Added some items to the TOC. It's getting longer and longer ... Jakob.scholbach (talk) 15:39, 16 May 2008 (UTC)

The List of group theory topics has a fairly good (and long) list of what could show up here. Jakob.scholbach (talk) 12:15, 18 May 2008 (UTC)

## list of gp th topics

I copy below the content of the list of gp th topics. Items already mentioned or very close to ones covered are striked out. Indented items seem (to me) to be less relevant or irrelevant to the article. Jakob.scholbach (talk) 20:51, 2 June 2008 (UTC)

## restructuring

I did some restructuring of the article involving mainly the lead and the history section. I am considering also removing the part giving the definition of group, this adequately dealt with in the group article and having this here adds to the critisims that the two articles have too much overlap. This article (at least in my view) should not be about groups but about the field that studies them. (TimothyRias (talk) 09:25, 22 May 2008 (UTC))

You are right, the lead was still totally out of shape. Probably this is the last we need to do, when everything else is pretty much done. History in front seems good to me. (I deleted the 2nd copy at the end. It also needs lots of improvement still, btw).
As for the overlap: as a matter of fact, groups and group theory (as corpora of certain knowledge) do overlap. My aim with the basic stuff section was to give a condensed intro to some definitions which are central to group theory and will be used later in the article (such as group (obviously ;)), subs, quotients, homomorphisms, normal and composition series etc.) Some of them could theoretically be eliminated because they are already in the group article. But I think it is much nicer for a reader to be reminded of what a normal subgp is when reading about normal series.
I would like to propose the following procedere: let's write up the article (as complete and comprehensive as possible), and then decide which content is moved where. Ideally, the gp article should be readable without prerequisites, the gp th article does have to refer to the knowledge of certain notions, but should nonetheless be somewhat self-contained. Jakob.scholbach (talk) 12:25, 22 May 2008 (UTC)

Agreed. The lead should probably be re-edited at the end. My current restructuring was mainly to make clear to new editors coming into the article what the difference between the article should be. (pre-empting any new merger/deletion proposals from editors wandering by noticing that the lead of group theory deals mainly with groups instead of the theory.

About the basic stuff. This should probably be dealt with on a case by case basis. I.e. introducing concepts when needed and referring to the relevant articles for a more indepth discussion. If we don't do this I feel the size of this article could quickly get out of hand. (TimothyRias (talk) 13:58, 22 May 2008 (UTC))

## History of group theory: proposal to fork

I was going to fork out the section on early history of group theory (that I wrote a while ago) from Abstract algebra when I realized that presently there are at least two other places on Wikipedia with subtantial coverage of the topic, namely, here and at Group (mathematics). I think that from the point of view of maintenance and verifiability, it is vastly preferable to have a dedicated article titled "History of group theory" (currently, the title redirects here), with the others using the summary style and referring to it. (For example, there are some inconsistencies between the lead and the history section here, relatively easy to fix in this instance, but having a separate article which can be referenced and consulted for verification purposes seems to me a good long-term solution.) Therefore, I propose to create such an article by importing the already existing material. The three sections that I've mentioned are quite similar in their approach (perhaps, because of the reliance on the same sources?), but there are subtle differences in the scope and emphasis that will require nontrivial amount of effort to reconcile. For example, the version here has the widest scope and it is decently referenced, but it omits any mention of Gauss and seems to be rather sketchy beyond the initial period. If there are no objections to forking, I can help with the merging process. Arcfrk (talk) 04:00, 9 July 2008 (UTC)

Yes, I agree. I think both this article and the group article need to have a history section. It was my intention to have a separate History of group theory article, which incorporates the material here and also of the group article. The only reason that I did not yet fork it out was lack of time (and motivation, a bit). In the several articles, as you say, the emphasis should be slighlty different. How did the group notion evolve (--> group)? How did the building now called group theory was edified (--here). If you are up to it, just go ahead. Jakob.scholbach (talk) 12:12, 9 July 2008 (UTC)
P.S. From a quick glance at abstract algebra, it seems that the history material in the "early group theory" section belongs almost completely to the history of group theory, so trimming it down there and moving it to h.o.g.t. is in order, I guess. Jakob.scholbach (talk) 12:26, 9 July 2008 (UTC)

## Assessment

The suggestions above concerning the scope etc have all been very good, but the implementation, of course, will be difficult and protracted: this is a formiddable project, perhaps, the most complex that I've worked on so far. Here are a couple of comments.

• What struck me most upon reading the present version of the article is the incredible level of detail the section "Groups" goes into: this would, perhaps, be appropriate in a textbook, but within wikipedia, a better solution would be to simply state that the basic notions are explained in the article "Group (mathematics)". I think that the only subsection worth keeping in its present place is "Groups with additional structure". "Finiteness conditions" can, perhaps, be moved further down, but frankly, I think that it's too technical a topic for a top-level article. The rest just unnecessarily duplicates the material at the main group article (or should be moved there).
• The section "Applications" is basically a glorified list. I think one has to be much more discriminating about what to include there. How about confining ourselves to strictly mathematical applications for a start, and moving the rest to separate articles, like "Applications of group theory in physics" (chemistry, etc)? As has already been pointed out, whole books have been filled with this type of material. The next step would be to agree on which mathematical applications to include, and the size and level of the capsules. Arcfrk (talk) 06:24, 9 July 2008 (UTC)
This sounds like you want to join in! Great. Actually I have half-heartedly started writing the article, but got frightened by the size of the topic. Perhaps the two of us (and others, for example JackSchmidt also wanted to join in, but apparently is busy otherwise) can do a good job together...
• I think the level of detaillation (does this word exist?) is OK. There are only the basic definitions plus a number of standard theorems. If we end up with a terribly long article, we can (and have to) decide what to scrap / move elsewhere, but I believe, normal series, say, does form a core part of group theory, and so has to be here.
• I strongly dislike the idea of moving any material that is more advanced and/or technical than the current content of group (mathematics) to this introductory article (for example, commutators, exact sequences). What I have (and still have) in mind for groups is an article which is interesting, and hopefully understandable to the layman.
• We can discuss trimming down the material on quotient groups etc., whiich is already covered in the group article. In case of space limitations (which are sure to turn up) we can perhaps do that, but before actually having to do so, I would not do it.
• Apps: I think an article Applications of group theory in physics or Applications of group theory in science would be premature unless we have a very good and long Applications of group theory article (which we don't have yet, you are right). Probably it is best to do the fork right now with an honest separate article and replace the section here by a overview of the newly expanded article. This is basically the same as with the history of group theory above.
• In general, I feel it tempting to write a nice article on a subtopic of group theory, say computational group theory. This helps the subarticles (many of which are in crappy state), but less the one here. Jakob.scholbach (talk) 12:25, 9 July 2008 (UTC)
I want to mention again how important I think it is to think of the topic of this article as the human endeavor called group theory, rather than the contents of a book called group theory. I think two topics of importance are: that "group theory" is used as an introduction to abstract algebra (unlike loops or Lie algebras, but like commutative rings), and that "group theory" is currently a large research area (but was not during various periods of history). For the education topic: I believe one of the reviews of the recent reprinting of van der Waerden has some description of how this curriculum came about. It compares Galois theory to Euclidean geometry: a core topic perhaps more out of tradition than anything else, but one that establishes a common background amongst all mathematicians. For instance for the research area, "finiteness conditions" is very important, but not to explain what they are, but rather to explain that group theorists by and large divide into two camps, those who deal predominately with finite, and those with finiteness conditions. I have heard on a number of occasions that Robinson's text on Finiteness Conditions ushered in a new era in group theory in the 1970s. The section can be rounded out by mentioning that in modern work, both areas have a large interplay (for instance, a famous example from the 70s is the odd order theorem being used to prove theorems about infinite groups, and the current influence of profinite groups and zeta functions on finite groups, or even algebraic groups, or the amalgam method). In other words, I think a significant space should be used to describe the field of group theory, rather than the subject matter it studies. JackSchmidt (talk) 13:37, 9 July 2008 (UTC)
I have trouble understanding the difference between the human endeavour and the totality of textbook content about g.th. Do you say that some facets of the endeavour are not covered in the textbooks? What exactly would that be? and how can we write about them, in a referenced-based manner? I should probably refrain from any philosophical aberrations, but aren't the books are written thoughts, and the thoughts are the endeavour?
I think, if we are able to give an overview of all textbooks (and research articles) on group theory, we will have a fantastic and unique article. If we are even able to convey some of the spirit of the domain, even better. Jakob.scholbach (talk) 16:46, 9 July 2008 (UTC)
Yes, some facets of the endeavour are not covered in the textbooks (about group theory, or algebra). Many collected works, birthday proceedings, and obituaries describe not only a single mathematician, but the world in which lived and worked. There are history books and articles, and even the introductions to many texts contain statements about the atmosphere.
I'll try to give an example: "The Feit-Thompson odd order paper proved that ...." versus "The Feit-Thompson odd order paper (was part of a larger movement that) changed the way mathematicians thought about mathematical proof." Actually, Feit–Thompson theorem handles this pretty well, "Perhaps the most revolutionary and important new idea was that of the very long paper." Before this, good papers were short. After this an entirely new research method was led by Gorenstein: hundreds of workers, meticulously checking special cases and chipping away at a problem. Compare this to the Bourbaki or Grothendieck research method of "slowly dissolving the problem in water" (or so) by generalizing and abstracting so far beyond the original unsolvable problem, that somehow you have solved it, perhaps like the FLT proof.
Another type of example is the idea "schools of thought". I think in Germany this is very clear, a single faculty member at a university establishes a (changing) set of visiting scholars and students all of whom approach the same set of problems in at least roughly the same way. For instance at Aachen, Joachim Neubueser had a school of computational group theory, but it has been said that "Burnside never created a school of group theory in England." I think Hilbert (section link) is another example.
Hopefully this analogy is useful (even if a little absurd): The article physician does not try to summarize the PDR (or the sum total of medical school textbooks), but rather describes the history, culture, and social impact of physicians. JackSchmidt (talk) 18:07, 9 July 2008 (UTC)

OK, now I get it. It seems we are slightly disagreeing on the tune of this topic. Perhaps this is because I'm not really that much into group theory, in particular not in the presence/absence of any schools whatsoever etc. In general, I think mathematics articles such as group theory (in possible contrast to history of group theory) are primarily concerned with mathematics, and to a far lesser extent, with its historical, psychological or social concomitants. The thing I'm afraid of is getting a stack of mumbling statements concerning group theorists -- this is exaggerating your vision, obviously. So, I stop and formally and solemnly propose: let's wash our hands and start increasing our edit counter in the main space :) Both points of view will necessarily turn up anyway. Jakob.scholbach (talk) 21:11, 9 July 2008 (UTC)

Just editing is a good idea. Definitely, the more "human" we make the article, the worse the problem of drive-by editors adding unsourced statements will be. I'm still trying to stick to wikignome edits on this article to avoid pushing my vision. Oddly, I'm finding it a lot easier to write on topics I no longer really care about (maybe this is why you have three times as many edits as me on these articles!). I started a little list of applications of group theory, but I realized that I think traffic lights and breakfast cereal are applications of group theory, so I stopped. JackSchmidt (talk) 21:52, 9 July 2008 (UTC)
Aha. Enjoy your cerals, but please do not indulge in your (good) excuse w.r.t. to this article any longer :) I think pushing one's vision is not too bad, as long as the article improves. I have been pushing my ideas into the group article (without mentally caring too much, but also without knowing too much). In the next days, I will try to have a look at symmetry and/or computational group theory and write a word about this. Jakob.scholbach (talk) 17:43, 10 July 2008 (UTC)

I still think that it's better to keep to higher level of exposition ("bird's eye view") in this article. In addition to the main article Group (mathematics), Glossary of group theory may the appropriate place to refer to for the basic definitions. Perhaps, the material on exact sequences and normal series can be incorporated there?

While "human dimension" is interesting, it is also notoriously unencyclopaedic (except in biographical articles, of course). There are serious OR and personal opinion issues involved, "drive-by edits" already mentioned are just one of the aspects. I propose to work "top-down" and stick to the branches/subject areas and main lines of development at first. As a first approximation, take a look at the Springer EOM "Group" entry. Unfortunately, Britannica has only one page (as part of abstract algebra article) devoted to group theory, never going beyond finite groups, so it's of little use. Perhaps, it would be helpful to consult a few standard textbooks before mapping out the article? (I wish I knew what they were!) Scanning through MathSciNet (books with "Group theory" in the title), it seems that there is quite a bit being published on group/representation theory for physicists (mostly, dealing with Lie groups); apart from that, active areas are finite groups (with classification of finite simple groups being the most important topic), combinatorial group theory, geometric group theory. There are some proceedings, but no new textbooks (as far as I can tell) on computational group theory and abelian group theory. Some general books perhaps worth looking at for inspiration are Kurosh's "The theory of groups", Marshall Hall's "The theory of groups", William Scott's "Group theory", Joseph Rotman's "An introduction to the theory of groups" (GTM 148). We need to decide whether Lie groups and lattices in them should be given major treatment here or not. Ditto for various branches of representation theory (finite groups, unitary representations, etc). Arcfrk (talk) 04:17, 11 July 2008 (UTC)

OK, I quickly removed the basic material, it is now in the glossary, which was indeed a poor version of the content here. So, you have enough space for tons of bird's eyes. Jakob.scholbach (talk) 07:14, 11 July 2008 (UTC)
The plan sounds fine to me. Covering the basic areas of group theory research suffices for my "human element".
• CGT is definitely still active. Holt and Seress both have 21st century computational group theory textbooks. Sims and Butler are other good introductory texts. The recent book by Michler is pure CGT, but less expository. Most of CGT is in the literature though. Students are usually trained as group theorists or computer scientists, but work with an adviser in CGT. A huge trend in the 1980s and 1990s was the coclass conjectures; the classification of all finite p-groups. A huge trend in the 1990s and 2000s is the matrix group recognition project.
• For finite groups, other important areas are called "classes of groups" (Doerk and Hawke is the best current textbook, but there are two other recent texts too), "subgroup lattices of groups" (Schmidt is the current best textbook). There are also tons of areas related to the classification, mod rep theory, cohomology of finite groups, fusion systems, subgroup geometries, etc. but the people involved there have a heavy overlap with the old CFSG crowd. There is a recent "proceedings", but fully half, if not 3/4 of the book is expository and has been recommended to me and served as a textbook in these areas.
• Finiteness conditions: Infinite soluble groups/locally finite groups/subnormal subgroups together form a very active area of group theory (basically, the area in which you use amazing new methods in order to prove analogues of finite group results for infinite ones). There are recent texts by Robinson, Lennox, and Stonehewer. There are other important contributors, but this isn't my area.
• Abelian groups is still active, though it sometimes goes by different names now. A recent abelian groups conference was called "Abelian groups, rings, and modules", since abelian groups are usually studied through their (full) endomorphism rings. Fuch's and Salce's Non-noetherian domains is actually basically the study of abelian groups with valuations. Mader and Faticoni have recent expository books on torsion-free abelian groups. Rudiger Goebel, ALS Corner, Shelah are other big names in the area. A recent expository textbook "Exercises in abelian group theory" is a good introduction, but Fuchs's and Kaplansky's books are still the gold standard. Check out 06A, 06F, as well as 20K if you are searching directly on MSC (and beware lots is mingled in 16D, especially 16D70). Basically, there are ring theorists who use abelian groups, and there are abelian group theorists who use rings, and their work often gets mixed up as module theory. Really it just depends on which you start with, the "module" or the ring.
Let me know if you want me to add a bunch of these books to a references section (either on the talk, or on the article). JackSchmidt (talk) 05:30, 11 July 2008 (UTC)
Thanks for the overview of recent research directions! (Although now I feel like "What am I doing at this page with my hopelessly obsolete knowledge of the state of the subject?") It will probably be a while before we get to expounding these new developments, but annotated references of the main textbooks will be nice to have in any case (either here, or at the corresponding more specialized pages, like Computational group theory). Arcfrk (talk) 09:15, 13 July 2008 (UTC)

## Scope of the subject

I am starting a new section, since it's getting harder to read/edit tbe preceding one. I've attempted to give snapshots of different types of groups, in preparation to introducing different subject areas within group theory. I am not satisfied with the present result, but what about the idea itself? Comments are welcome! Arcfrk (talk) 09:24, 13 July 2008 (UTC)

Looks fine. The material on presentations of groups has to be merged with the content of combinatorial gp th at some point. Jakob.scholbach (talk) 10:37, 13 July 2008 (UTC)

I've rewritten the section on groups with extra structure. In the process of rewriting, I removed the following material on algebraic groups — it's more pertinent to algebraic geometry than to group theory. Arcfrk (talk) 04:50, 15 July 2008 (UTC)

The parallel situation in algebraic geometry deals with algebraic groups, i.e. algebraic varieties possessing a compatible (i.e. given by regular maps) group structure. They fall into two regimes: projective and affine ones. The group underlying the former can be shown to be necessarily abelian, therefore they are called abelian varieties[1], which is in marked contrast to the affine case. Affine algebraic groups are necessarily groups of matrices, for example the SLn, which are commutative only in trivial cases.[2]

## Periodic Table in light of the group theory

I would like to add following link: Periodic Table in light of the Group Theory I think it is very illustrative.

Any objections? Drova (talk) 21:10, 25 September 2008 (UTC)

I don't formally object, but I think this does not have too much to do with the content of the article. Perhaps consider putting it to Sphere packings etc. Jakob.scholbach (talk) 06:33, 26 September 2008 (UTC)

Thanks for the advice. I'll do just that.Drova (talk) 12:14, 26 September 2008 (UTC)

## top image

isn't there a more representative image for group theory than the rubik's cube? 212.200.205.163 (talk) 01:29, 1 November 2009 (UTC)

i see there was discussion on this before [1] 212.200.205.163 (talk) 01:34, 1 November 2009 (UTC)
i find this interesting: [2]. didn't see a single cover that "sends a message" 212.200.205.163 (talk) 18:42, 1 November 2009 (UTC)
Rubik's Cube is a perfect ikon/logo for group theory. I can't believe someone has the nerve to complain about it. Don't ever think of getting into politics.
Pazouzou (talk) 02:52, 30 March 2010 (UTC)

## Consistency of dates

This article states ".. mostly published between 1960 and 1980, that culminated in a complete classification of finite simple groups." where the linked Classification of finite simple groups article states "made up of tens of thousands of pages in 500-odd journal articles written by about 100 authors, published mostly between 1955 and 1983.". Which is correct? Does anyone have a source for this? —Preceding unsigned comment added by S-1-5-7 (talkcontribs) 16:16, 20 January 2010 (UTC)

## Emmy Noether

Emmy Noether does not seem to be even mentioned in this article, whereas from her wikipedia article it seems that she has made many fundamental contributions to Group Theory.Pratik.Mallya Talk! 19:57, 2 March 2011 (UTC)

## Failed

"Failed to parse" appears. — Preceding unsigned comment added by 86.181.9.123 (talk) 12:30, 7 February 2014 (UTC)