Talk:Group (mathematics)

Template:Current-Math-COTM

The difference between group and group theory

Grubber raises a good point about the difference between Group (mathematics) and Group theory which is worth discussing. We could actually take a radical step of merging the two, indeed this would turn two OK articles into one article which could be put forward to GA. Group theory does the history well and briefly mentions more advanced topics. Alternativly we could have group theory as a more advanced article sumarising the more advanced concepts like group representations and clasification of finite simple groups. --Salix alba (talk) 17:58, 31 October 2006 (UTC)

You might want to look at some discussion we had on this subject before: see Wikipedia talk:WikiProject Mathematics/Archive4#Graph (mathematics) vs Graph theory. -- Jitse Niesen (talk) 00:42, 1 November 2006 (UTC)

Great discussion. Thanks. I tried a few different topics to get some perspective (ring, field, probability, matrix) and those articles were just as "messed up". I do like the idea that the broader perspective belongs on the Theory page. - grubber 01:17, 1 November 2006 (UTC)

Rational numbers

I don't think the rational number example has a place in this article.

It is true that the rational numbers (without 0) form a group under multiplication. But the existence of this group is not trivial to prove.

Even if we assume all the properties of the rational numbers to be given, the definition remains unclear. What is clear, in any case, is what I wrote: that the structure (Z, x) can be extended in such a way that it includes inverses - so long as it does not contain 0.

I will add back the very pertinent remarks about 0.

Please comment; I'm an amateur mathematician and not a very good encyclopedist. --VKokielov 15:27, 7 June 2007 (UTC)

In what way is it non-trivial to prove that non-zero rationals form a group under multiplication?

• Closure:

$\ (a/b)*(c/d)=([ac]/[bd])$

• Associativity:

$\ {\Big (}(a/b)*(c/d){\Big )}*(e/f)=([ac]/[bd])*(e/f)=([ace]/[bdf])$

$\ (a/b)*{\Big (}(c/d)*(e/f){\Big )}=(a/b)*([ce]/[df])=([ace]/[bdf])$

• Identity element:

$\ (1/1)*(a/b)=([1.a]/[1.b])=(a/b)$

• Inverse element:

$\ (a/b)^{-1}=(b/a)$

As the article said, all of these follow from the properties of integers, and the definition of a rational number. Oli Filth 20:38, 7 June 2007 (UTC)

But, you see, when it is presented in that form it runs around on itself, and misses the most important point, which is that the rational numbers (provided they even exist) are an extension of the integers. I think that even the most basic presentation must stand in view of that. --VKokielov 00:51, 8 June 2007 (UTC)°

By "runs around on itself", do you mean that it's based on circular logic? If so, at what point does this occur?

I'm afraid I don't know enough about group theory and monoids to know whether the extension property is the most important point. Oli Filth 08:38, 8 June 2007 (UTC)

No - there's no circular logic, you're right. But the definition which assumes the properties of the rational numbers. viz. "the smallest field which contains the integers with all their quotients" is unconvincing -- all the more here, where it is obvious that Q is an extension of Z. The procedure which extends Z to Q can be generalized and used to extend other algebraic structures.

We can restore the separate section, if you want, and mention the extension there. --VKokielov 13:26, 8 June 2007 (UTC)

• The idea that the rational numbers without 0 form a group is well-defined and well-established. I'm not sure what the issue is? I can give you a very formal and exhaustive explanation if you like. And, I can also show how the rationals must be the smallest field containing the integers (proof by contradiction; also, read field of fractions). - grubber 20:58, 9 June 2007 (UTC)

My issue is that there was no proof of the existence of Q -- only a claim about the group, assuming all the properties. By a stroke of luck I know your formal construction (I wouldn't have ventured any of this if I didn't). I think that it is for us to lay it forward in the article. --VKokielov 03:59, 10 June 2007 (UTC)

The existence of Q is better discussed in the context of rational number. This article as I see it is only discussing some common examples of groups among the well-known sets. Dcoetzee 08:06, 10 June 2007 (UTC)

It is not the purpose of this article to prove the existence of one set or another. What this article deals with is, given a particular set, can it be shown to be a group? Oli Filth 11:07, 10 June 2007 (UTC)

If it seems convincing to you to do it this way, then I will restore the old text. --VKokielov 19:39, 10 June 2007 (UTC)

Lay definition

I added a section for non-mathematicians that I found on the German Wikipedia page, found useful, and translated. I'm new to Wikipedia so I'd be grateful for any improvements. 151.197.38.55 17:56, 24 September 2007 (UTC)Lucas

The prose is actually pretty good. It duplicates the formal definition part quite a bit, but I'm sure we can fix that. Thanks for the edits! - grubber 06:18, 25 September 2007 (UTC)

Top image

The image at the top of the article doesn't seem very helpful. It shows a clock, and arrow, and then a clock showing a time four hours later than the first one. This is supposed to illustrate that the integers for a group under addition mod 12. However, the introduction has not defined "group" or "modular arithmetic", so I don't see how useful the picture is here. It seems like the picture should be lower down, perhaps near the discussion of cyclic groups. Then, we would need a new picture for the top. Maybe a dodecahedron or something, and the caption could point out that its symmetries form an interesting group? LeSnail (talk) 20:16, 25 January 2008 (UTC)

I think your critique would apply to the new image as well. I think the current picture would be nice near the cyclic group section, but it might be overwhelming to the article as a whole to have pictures for each section. However, the main article Cyclic group could definitely use this image!

I'm going to add some {{main}} to some of the shorter sections, and see how it looks from there. Certainly more interesting groups exist than clock arithmetic. I think there are even some animated images somewhere on wikipedia showing the dihedral group of order 8 acting on something like an envelope. JackSchmidt (talk) 20:27, 25 January 2008 (UTC)

I added it to Modular arithmetic, which looked like it could use it. Cyclic group is a dense mess, and could use a lot of work. LeSnail (talk) 20:38, 25 January 2008 (UTC)

Thanks. It is hard to believe it was not already there! I agree on Cyclic group. The definition section needs to have its nothing-to-do-with-definition material moved to a nice section, and the properties section is just crazy dense. Some of my previous edits on the properties section of that page were part of a delicate consensus process, so I want to avoid making too many drastic changes myself now that I think everyone is happy. I'm happy to help tidy, especially if you want to split up the definition and properties section into smaller sections like the representation and endomorphism ring sections. JackSchmidt (talk) 20:51, 25 January 2008 (UTC)

Table for group generalizations

Group-like structures

•	closure	associativity	identity element	division possible
Group	Y	Y	Y	Y
Monoid	Yes	Yes	Yes	No
Semigroup	Y	Y	N	N
Loop	Y	N	Y	Y
Quasigroup	Y	N	N	Y
Magma	Y	N	N	N
Groupoid	N	Y	Y	Y

What do you think about this table I put together? I personally have a lot of trouble keeping track of which structures satisfy each axiom, and the explanation at the bottom of the page is a little hard to follow because they are all defined in terms of each other. The table, however, doesn't make everything as clear as I had hoped it would. Please comment on what would make it more effective. Actually, maybe the whole discussion of group generalizations should be removed here, since it is probably more fitting for the Group theory page. LeSnail (talk) 23:23, 26 January 2008 (UTC)

Looks fine to me (except that I would change green to cyan). It's suitable for inclusion here, and I think it belongs here more than at group theory. Silly rabbit (talk) 23:27, 26 January 2008 (UTC)

Why cyan? Just for readability? I thought the green suggested the idea "yes", like a green light, so maybe just a lighter green? Also, how about the dot in the upper left? It looks odd, but I don't know what to replace it with. LeSnail (talk) 23:32, 26 January 2008 (UTC)

I like it too. I added appropriate links for most of the headings, which I think is helpful in a table (quick ref + quick click). "Division possible" is tricky, since in a quasigroup division and inverse elements are different. Is there a better article to link it to? I would prefer a softer color scheme, but green and red seem good. Maybe pea fuzz and winter's blush with black text? I can go either way on the dot. JackSchmidt (talk) 23:39, 26 January 2008 (UTC)

Oohh... pretty. Go for it. Silly rabbit (talk) 23:55, 26 January 2008 (UTC)

Nice job! It looks lovely! LeSnail (talk) 04:36, 27 January 2008 (UTC)

May I suggest using the Wikipedia standard {{yes}} and {{no}} templates in the table instead of specifying custom colours, so any changes made to those templates (e.g. for colour-blindness issues, which are constantly debated in the template talk pages) will apply Wikipedia-wide? I've changed the Monoid row of the table to use the templates for easier comparison to Lesnail's colours. -- simxp (talk) 22:21, 31 January 2008 (UTC)

Well, I suppose my shades were a little hazy. The full "Yes" versus "Y" is a definite improvement, and the table is much easier to edit when using the templates, so I made the change on the main article too. JackSchmidt (talk) 00:55, 1 February 2008 (UTC)

How about if "division possible" links to Division (mathematics) or Division (mathematics)#Division in abstract algebra, since as you point out division and inverse elements are different? LeSnail (talk) 04:41, 27 January 2008 (UTC)

Much better link; I made the change. I had only scanned the top of the division article, and it looked focused on examples that didn't seem to fit with quasigroups. JackSchmidt (talk) 05:20, 27 January 2008 (UTC)

Some remarks

Much of the article is either technical or is only interesting for a mathematical reader, such as "A group (G, *) is often denoted simply G where there is no ambiguity as to what the operation is."

I think most of the concepts can and should be illustrated by an example. The group of symmetries of the square is eligible to show: order of an element, subgroup (of rotations, which is cyclic), abelian-nonabelian. I will also try to do this, but feel free to make this as elementary and readable as possible.

The section "constructing new groups from given ones" is good, but duplicates material from above. Jakob.scholbach (talk) 14:56, 13 March 2008 (UTC)

I'd like to emphasize my opinion about the accessibility of the article. In this recent edit, User:Woodstone reinstated the definition in a pure form, starting with "A group (G, *) is a set G with a binary operation * that satisfies the following four axioms:"

IMHO this is not the best way to address a general audience. A lay reader will not read further than "binary operation". I like the way, for example, manifold is defined (see Manifold#Mathematical_definition). First, an informal introduction is given, secondly the formal definition is given. What do others think? Jakob.scholbach (talk) 22:32, 13 March 2008 (UTC)

I completely agree. Far too many of the maths articles are unreadable unless you're already very familiar with the article's topic; which very much detracts from Wikipedia. We are, after all, writing an encyclopedia, not a modern all-encompassing Euclid's Elements! That said, the Group article is nowhere near as bad as some -- at least it has some sort of introduction before the descent into formality; many don't. Something else to bear in mind the difference between Group (mathematics) and Group theory, since some people are of the opinion the most of the mostivation and history should go into the theory article, leaving this article as quite basic and definition-oriented. Personally, I don't agree (I think the articles should be merged), but It's something to bear it in mind. -- simxp (talk) 14:56, 14 March 2008 (UTC)

I agree that an article should be accessible to non-experts. However do you really think that the given example makes the concept of a group clearer than the definition? Would the casual reader have any idea what the meaning would be of (a*b)*c applied to this example, let alone the difference with a*(b*c). The editor leaves that completely in the dark. He also keeps talking about negative rotations, which are not strictly part of the exhibited operands in the group. In my preference a short formal definition should come first, followed by elucidations on simple examples. The current example is not clear enough for this purpose. Some simple clock-arithmetic would do better. −Woodstone (talk) 15:38, 14 March 2008 (UTC)

(unindent) OK. I personally also think that group and group theory should be merged. In fact the overlap of the articles is about 60% of each article. But perhaps we should postpone thinking about merging the two until they (or at least this page) is somewhat more stable. As for the clarity of the example I introduced:

• In the inverse element property, I wrote counter-clockwise vs. clockwise because I thought it clearer that these are the mutual inverses. But one can also replace 90° ccw by 270 cw, obviously. In the intro, I just mentioned ccw, because a reader might wonder about them (not realizing the 90°cw=270°ccw etc).

Replying to "However do you really think that the given example makes the concept of a group clearer than the definition? ": Yes, I absolutely do. A chemist, for example, who will be repelled by the definition, but will be invited to read further if the content is underlaid with an example. For us math-guys the definition is probably nice and neat, but that's exactly the difference.

As this is a CotM, perhaps we should try to find a good answer for the following 2 questions:

1. Is the formal (non-layman accessible) definition to come first or not?
2. What kind of introductory example do we want to take?

Woodstone opined for "first", "simple clock-arithmetic" or other simple examples. I'd say the definition is only the mathematical abstraction of the intuition we want to convey, therefore the abstract naked definition should not be first. As for the group, I propose an elementary, yet sufficiently ample example. The symmetry group I chose has the advantage of being simple to understand (as is modular arithmetics), but nonetheless shows many features that occur in the article (semidirect product, subgroup, (non-)commutativity, order). Jakob.scholbach (talk) 16:59, 14 March 2008 (UTC)

I'm still curiously waiting to see how you show in an easy and convincing layman's way that (a * b) * c = a * (b * c) for the example symmetry group. Or actually even in clear terms, what (a*b)*c really means. Of course one could work out the complete group, but that is not elegant. −Woodstone (talk) 22:33, 15 March 2008 (UTC)

You are right, this is challenging. I'm spending sleepless nights :). But, the question I raised does not seem to be covered by your pointing out this difficulty. If I am (and nobody else is) not able to clarify this axiom, it does not quite prevent explaining the other ones, especially the term "binary operation", right? Jakob.scholbach (talk) 21:06, 16 March 2008 (UTC)

(By "explain" I meant: explain before throwing the abstract definition into the readers mouth, as outlined above) Jakob.scholbach (talk) 21:08, 16 March 2008 (UTC)

You got rid of "binary operation", but essentially got back a "function composition". The elements in the group are not just something like numbers, but functions (mappings if you like). Associativity of the composition is so trivial, that it becomes difficult to explain in a layman's fashion. That is one of the reasons I would prefer clock arithmetic (although that is abelian). −Woodstone (talk) 21:58, 16 March 2008 (UTC)

Sorry to barge in. I hope the following two cents are somewhat related and not overly redudant with your discussion. All groups are transformation groups in some sense (if anything, they're transformation groups of their own underlying set), and the fact that associativity is "trivial" for function composition is precisely the reason why it is an important property giving rise to such a deep concept. If the layman reading this article somehow gets the idea that groups are made up of (invertible) mappings, and that the group law operator is function composition, then they definitely got things right. Of course there's a more abstract side to group theory, but most of the groups we encounter in everyday life (and this even applies to almost all mathematicians) do arise naturally as transformation groups of some mathematical object. Bikasuishin (talk) 23:03, 16 March 2008 (UTC)

Interesting thoughts. But still, in order to show associativity, using this example, one would have to define what a function composition is. That sort of defeats the original purpose of the example: explaining a group while avoiding a formal definition. The definition of a*b by (a*b)(x) = a(b(x)) would look awfully circular to the layman in the context of associativity. −Woodstone (talk) 08:57, 17 March 2008 (UTC)

Well, what prevents us from saying "a * b" means performing the symmetry b and then the symmetry a? I think that's perfectly understandable and correct. Jakob.scholbach (talk) 09:27, 17 March 2008 (UTC)

Might work. It's starting to sound less trivial:

• (a*b)*c means: perform c, perform a*b and so: perform c, perform b, perform a
• a*(b*c) means: perform b*c, perform a and so: perform c, perform b, perform a

Perhaps it would be possible to indeed show the whole group table. As for naming of the elements, we might use shorter ones. How about I, L (left 90), R (right 90), M (180), H (horizontal flip), V (vertical flip), D (diagonal flip), C (counter diagonal flip). Then L*R=I, H*H=I, L*M=R etc. −Woodstone (talk) 11:22, 17 March 2008 (UTC)

(<--) Sure, go ahead. When I get the chance I will also try to improve the images so that they also show the operation which has been done (starting from the identity configuration). Actually, we could also stick to the symmetries of a isosceles triangle, which comes down to the dihedral group of order 6 already in the article (below). I don't have a clear preference, but naming the flips in the triangle would be a little bit more cumbersome, I guess. Perhaps we can also use your remark "

• (a*b)*c means: perform c, perform a*b and so: perform c, perform b, perform a
• a*(b*c) means: perform b*c, perform a and so: perform c, perform b, perform a

" to indicate that the associativity requirement is natural to impose. Showing the group table would also show another facet of the group description. I think you can copy it from the dihedral group article. Jakob.scholbach (talk) 14:30, 18 March 2008 (UTC)

I have performed the ideas sketched above. Hope you like it. Jakob.scholbach (talk) 11:31, 21 March 2008 (UTC)

Thanks, it's starting to take shape. I rearranged the pictures because they did not fit on the screen. I added the identity transformation as well. I cannot control the sizing. Jakob, perhaps you can take care. The colored corners of the square are difficult to follow. I would prefer (like in one of the referenced articles) the big letter F painted on top of the square. That makes it unnecessary to add the small original square each time. Do you agree that as constants, the 8 elements should not be italicised? The table would look better with the ordering: I, H, V, D, C, M, L, R, because then the first 6 rows have "I" on the diagonal. −Woodstone (talk) 14:37, 21 March 2008 (UTC)

Hm, I also thought about the "F"'s. The problem is a little bit, that rotations by non 90-multiples have to be excluded in order to get this group. We don't necessarily have to stick to a finite group, too, but I think it is easier to digest. How would you define the symmetries in case of the "F"-shape? As for the italics: I prefer italics. You have a point that the H, V, D, etc. are kind of constant, but in the normal text such as "The symmetry I leaving everything" the letters are better distinguished if they are italic. If you want to change the order of the letters, just do it. (Be careful when doing it, though, I messed up the stuff several times). Jakob.scholbach (talk) 17:26, 21 March 2008 (UTC)

I was thinking of keeping the squares, but just paint a (faint) F on top, to show the orientation (taking the place of the colored corners). How about renaming I to E to make it stand out better in text? Is in your browser also the small square missing in operation M? And the "vertical flip" is actually the flip around the horizontal axis (is ok, but should be made consistent). −Woodstone (talk) 17:49, 21 March 2008 (UTC)

Well, if you can do it, why not. But actually I think, now, with the original configuration in the image, it is not too bad(?). Yes, the "M" image is weird, I definitely uploaded the new version (which you see when clicking on the image in the article). Is there a Wikipedia cache etc? Jakob.scholbach (talk) 14:01, 22 March 2008 (UTC)

dihedral group of order 6 vs. 8

Currently, the introductory example (dihedral group of order 8) I added and the dihedral group of order 6 in the examples section have considerable overlap. I prefer the order 8 one, because it shows a little bit more of what can happen (different orders of elements, for example). Does anybody oppose against trimming down the order 6 example in favor of the other one (which is already in the intro section, but could get some more material in the examples section, but much of this stuff is also well-covered in the subarticles)?

The space won by trimming down at this point could be used to write something about less-everyday groups, such as an example of a Galois group. Jakob.scholbach (talk) 21:03, 16 March 2008 (UTC)

I think it shouldn't be lost completely, because it is the place where the symmetric groups are mentioned. But trimming down sounds like a good idea. One could also mention that the introductory example is a subgroup of S₄. --Hans Adler (talk) 11:42, 17 March 2008 (UTC)

Why did you, in fact, use D8 as the introductory example? What does it show that D6 doesn't? D6 is non-commutative, has elements of order 2 and 3, has a commutative subgroup, etc. And on top of that it has less elements for the reader to keep track of. I might be biased a little, cause the my first introduction to groups used D6 as the prime example. But the current example is fine, so I don't think it is worth the hassle of changing it. (TimothyRias (talk) 08:43, 31 March 2008 (UTC))

question

Can somebody make sense of

"{aⁿ, n ∈ Z/mZ}"?

(unless a is a root of unity)? It shows up in the article Jakob.scholbach (talk) 18:01, 20 March 2008 (UTC)

References

To make this a good article, probably we need some more references. Especially I crave for some introductory texts accessible for the layman. Can somebody add something in this direction? Jakob.scholbach (talk) 23:50, 23 March 2008 (UTC)

Example

If find de labels for the transformations in the illustration of the definition to be a bit unintuitive. If I look at the multiplication table I keep to have to look back at the definitions to figure out what they refer to. Might it be a better idea to relabel them to something that makes it a bit more intuitive which operations do what. I was thinking more along the line L-->r_L M-->r_M R-->r_R for the rotations and H-->f_H V-->f_V C-->f₁ D-->f₂ for the various rotations. How do other people feel about this?

Sure, why not. Actually we had a similar notation earlier, but Woodstone preferred the current one because of its shortness(?). If you change it, be sure to change it also in the lower sections, whereever the symbols show up. Jakob.scholbach (talk) 18:27, 28 March 2008 (UTC)

In what way do you think f₁ is any more intuitive than D? I'm not against this per se, but wouldn't f_d be better? Also you might consider to change "id" into r₀, bringing the rotation subgroup in view. −Woodstone (talk) 09:12, 29 March 2008 (UTC)

I see that you changed id to r₀. I personally don't like it. I feel that statements like f_v² = r₀ are not really intuitive, especially since we are trying to explain the group axioms. In which the identity is singled out as a special element and thus deserves a special notation. Even in the case of the rotational group I wouldn't use r₀ as the identity. (I would probably use r, r², r³ and 1=r⁰.) But if others like this, I guess it is just a matter of taste. (TimothyRias (talk) 08:53, 4 April 2008 (UTC))

I partly agree f_d is slightly better than f₁. But f_c on the other hand is not. (at least to me.) But feel free to change it. I would keep id for the identity though. (Or at least some notation that stresses that the identity is special.) I also have some doubts about the letter f for flips, since typographically it looks a lot like the r for rotations. (In Dutch it would have been an s for spiegeling which is much nicer but makes no sense in English) Any suggestions. (TimothyRias (talk) 09:39, 29 March 2008 (UTC))

I changed the unity symbol mostly because it is unusual to have mathematical constants of more than one letter (sub and superscripts aside). So I chose r₀ to bring out the rotational subgroup. Using r would be fine as well. I would oppose using r² for r₂, because that makes it impossible to express what r squared is. The idea is to express the elements as constants, so the group structure can be explained. If you don't like the f for flip, you might change to m for mirror. −Woodstone (talk) 06:43, 6 April 2008 (UTC)

Typical textbook notations for the identity of a group include, e, id, 1, 0, etc. So if you don't like the two letter id (which is fairly standard, I would suggest either 1 or e. (and by the way r squared IS r₂)

Anyway I had some time on my hand this weekend so I tried out how the example would work if we did D6 instead of D8. I've tested it here. Any comments?

(TimothyRias (talk) 10:11, 6 April 2008 (UTC))

I looked at the D6 example. Looks pretty too. however, I have some objections against "mirror in bisector of red corner". This definition will cause trouble in function composition. The red corner has moved and the definition moves with it. It should be more like "mirror in bisector of left (right, top) corner". That way the definition remains stable after preceding operations. I have no strong preference for either D6 or D8. Perhaps smaller is better for an introductory example. I have less trouble mentally following 3 than 4 colored dots. −Woodstone (talk) 04:44, 7 April 2008 (UTC)

I see your point. Do you have a better idea for labeling the 3 reflections. The only other thing I have come up with is the rather abstract labeling them m1, m2, m3. But that feels very unintuitive. I found the color labeling kinda cute since you can also color the symbols for each reflection, which is kinda pretty, but I see how it also can be a bit confusing. (TimothyRias (talk) 07:58, 7 April 2008 (UTC))

This kind of problem suggested to me that handling the square is slightly easier nomenclature-wise. Vertical flip or reflection seems to be easier to understand than reflection along the line which crosses the ... edge (in the triangle), doest't it? Jakob.scholbach (talk) 09:50, 7 April 2008 (UTC)

Well the same problem comes back in the case of the square in the somewhat ambiguous terms diagonal and counter-diagonal. (which was one of the reasons I started to explore the triangle case. In that case the description "reflection in the bisector of the bottom right corner" is completely unambiguous. But labeling that reflection with something intuitive seems to be hard. (But the same goes for labeling reflections in the diagonals of the square.) I currently thinking of just labeling the bisectors just in a general way. (TimothyRias (talk) 11:17, 7 April 2008 (UTC))

I also prefer a more particular notation for the identity element. Currently a reader will wonder why there is a distinction between three rotations (r1, r2, r3) and the identity (r0). r0 as a symbol is in no ways preferrable over f_x or a similar notation (based on the idea that the identity is a certain flip, too), because the identity rotation is also a identity flip, if you want. I second the feeling of Timothy that this element is better represented with something like "id" or "i". As for "it is unusual to have mathematical constants of more than one letter": I think, even if it is unusual, "id" fits the purpose we want, namely a good mnemonic for "identity". I also oppose r², for the same reason as Woodstone. We need to make clear that (r₁)² is on the one hand a symbolic expression, namely performing the rotation twice, but on the other hand, it is a new element of the group, called r2 (or whatever). Jakob.scholbach (talk) 11:48, 6 April 2008 (UTC)

OK, I've modified my D6 example a bit to reduce the confusion about mirroring across the bisector of a certain corner. I now like it better than the current D8 example. The current one has some naming issues regarding: 1) vertical flip vs. flipping across the vertical 2) distinguishing between diagonal and counter diagonal. At least these issues have been resolved in the new example. If others agree i'd like to replace the current example with the D6 one since the later is a little more concise. (The pictures might need some tweaking, I think the dots might be a little small.) (TimothyRias (talk) 12:59, 23 April 2008 (UTC))

Forget the D6 example. Anyway, we never attained any consensus with regard to the notion for the identity element. I strongly object to r0 as it highly unusual. I propose we go with either e or id both are fairly common in textbook treatments of the subject. I personally prefer the latter, since it is more descriptive. (TimothyRias (talk) 08:39, 14 May 2008 (UTC))

I also dislike r0 for the same reasons. I may want to change it back to id. The only reasons against it are "multi-letter symbols are uncustomary" (which is a minor reason) and "the subgroup of rotations is more clearly visible". None of the two outweighs, I think, the benefit of having a functional notation for the identity element. Jakob.scholbach (talk) 09:24, 14 May 2008 (UTC)

I have no objection against changing the identity to e (but maintain my objections against id). Next to that I still prefer f_d (diagonal) and f_c (counterdiagonal) over the vacuous names f₁ and f₂. −Woodstone (talk) 10:22, 14 May 2008 (UTC)

OK, I changed r0 to id (this may disputable over e, but I thought it is good to have a distinction between the abstract identity e from the axioms and the concrete one in the example group), and f1, f2 to fd, fc respectively. I hope everybody is happy! Jakob.scholbach (talk) 14:32, 14 May 2008 (UTC)

GA Review

• Wikipedia:Manual of Style (dates) - months and days of the week generally shouldn't be linked. Years, decades, and centuries can be linked if they satisfy WP:CONTEXT, though.

I now delinked the only occurence of a year.

• You may wish to consider adding an appropriate infobox for this article, if one exists relating to the topic of the article. ^[?] (Note that there might not be an applicable infobox; remember that these suggestions are not generated manually)

The article already contains one box at the very top, another one containing some basic notions has now been included. Jakob.scholbach (talk) 16:39, 7 April 2008 (UTC)

• Is there anything on density in the article? I guess that that it only concerns specific types of groups, though.

I don't understand what you mean by this. Is density a group-theoretic notion? If so, I presume it is a pretty specific thing. This article really covers only the very first basics of groups and group theory. More involved features shall be covered in group theory.Jakob.scholbach (talk) 17:08, 7 April 2008 (UTC)

Okay, it wasn't very important anyways.

• Wikipedia:Manual of Style (headings) - headings generally don't start with the, a(n), etc.

Done.

• Wikipedia:Guide to layout - reorder the last few sections, please.

Done.Jakob.scholbach (talk) 16:47, 7 April 2008 (UTC)

• Redundancy - there's a bit of that in the article. Try to clean it up.

I removed a few redundant statements. Some pieces, however, which occur more often (such as the cyclic group generated by an element) are basic principles which underlie many situations in group theory, so cannot be avoided.

• Footnotes would be nice.

Do you want to have certain statements be backed up by footnotes? As this articles contains no statement which is not repeated at the corresponding subpage (e.g. Lagrange's theorem) (and should be referenced there), this article currently contains only some general references (which contain all of the statements made here, anyway). Jakob.scholbach (talk) 16:53, 8 April 2008 (UTC)

The lead, the first section, and the "notation" section. Nousernamesleft^{copper, not wood} 17:02, 8 April 2008 (UTC)

I have now added some footnotes for statements which contain some meta-information and the like. Jakob.scholbach (talk) 19:31, 14 April 2008 (UTC)

• Maybe you could shorten the image captions?

The image caption " The possible rearrangements of Rubik's Cube form a group, called the Rubik's Cube group" is the one you refer to? I think it is not very long, actually, and every piece of information (reaarangemenets, Rubiks Cube, group, Rubiks Cube group) is worth staying at this place, I believe. Jakob.scholbach (talk) 16:59, 7 April 2008 (UTC)

It's mostly a good job, but there's some issues. Nousernamesleft^{copper, not wood} 15:35, 7 April 2008 (UTC)

• • Sorry about the infobox one; I didn't mean for that to be there, actually. What I do is auto-generate some suggestions, weed out the ones I don't think are right, then add some of my own. I forgot to remove that one. Nousernamesleft^{copper, not wood} 16:54, 7 April 2008 (UTC)

As a conclusion, I feel that all issues (most of which were pretty formal anyway or do not exactly apply) raised above have been addressed. Jakob.scholbach (talk) 16:53, 8 April 2008 (UTC)

In response to the GA fail, the ones which didn't actually apply to the article for the automatic ones I weeded out. The ones that don't apply here I actually came up with on my own, not through an automatic script. I've raised a few points there; after those are addressed, this should pass GAN. Best wishes. Nousernamesleft^{copper, not wood} 17:02, 8 April 2008 (UTC)

function group?

Currently function group links to something in chemistry, not at all related to a group consisting of functions. Is there any other term widely used for a group consisting of functions, other than "function group"? Jakob.scholbach (talk) 11:31, 11 April 2008 (UTC)

I've personally never met this notion. Is it really notable? If so, can somebody remove the redirect from function group to functional group and write a sentence and an example of a function group, please? Otherwise, I'think it's better to remove this from the group article. Jakob.scholbach (talk) 11:35, 11 April 2008 (UTC)

Function group is really a very bad name. For one usually a function is a map into field. Altough these can indeed form a group, and usually do, the notation used suggests that this is not what is intended. Since they are using the notation for composition as a group operation, my guess is that the orignal author was referring to automorphism groups i.e. groups of automorphisms of some object in some category. This type of group is indeed very common. (the example we are using is in fact of this type, eventhough we are using multiplicative notation in that instance) On the other hand the subject is not notable enough to have its own article. (Mentioning it in the group and maybe category articles should suffice.) I'll change the reference to automorphism, since it is probably useful to include this notation convention. (TimothyRias (talk) 12:46, 11 April 2008 (UTC))

I should think a more direct translation would be transformation group. Typically an automorphism group involves some kind of categorical ideas. Transformation groups do not. It's splitting hairs, I know, but this seems to be closer to the intended meaning, and is certainly a more accessible notion than automorphism group (in fact, there is at least a bona fide article on transformation groups). 14:58, 11 April 2008 (UTC)

Transformation groups would be a very particular example of such a group, while very common (and important examples that fall in this class are absolutely not covered. Automorphism groups of vector spaces, manifolds, bundles, groups, etc. all play important roles in both mathematics and physics. The concept that was meant was groups consisting of maps with as an operation composition of maps. The group axioms then immediately imply that these maps are automorphism. (The name indeed stems from categroy theory, but it is very easy to naively understand what these are.) (TimothyRias (talk) 15:10, 11 April 2008 (UTC))

Yes, a symmetry group (is this a synonym for transformation group?) consisting of isometries of a given geometric object is an automorphism group. Jakob.scholbach (talk) 15:21, 11 April 2008 (UTC)

Not exactly. An automorphism group consists of morphisms of an object in a category: thus one generally talks about automorphism groups of some particular algebraic structure: e.g., the group of automorphims of a ring (or topological space, etc.). A transformation group consists of a given collection of functions on a particular manifold, topological space, or some other kind of set. These transformations are typically automorphisms of some sort, but a transformation group may be a quite small subgroup of the automorphism group in the relevant category. Anyway, like I said, it is splitting hairs. silly rabbit (talk) 15:29, 11 April 2008 (UTC)

:-) To split the hair even finer: you can look at the category consinsting of the one object X and its morphisms X --> X Jakob.scholbach (talk) 15:36, 11 April 2008 (UTC)

Anyway, I'm just going to drop it. I don't actually have strong feelings about it. One thing I have just noticed though is that the symmetry group article has an absurdly limited scope (Euclidean geometry in dimensions 2 and 3). I'm not sure what should be done about that. I will add a hatnote to the article directing readers to the automorphism article. I'd appreciate it if others would have a look at it. silly rabbit (talk) 15:42, 11 April 2008 (UTC)

merge remarks into basic stuff section?

Actually, I dislike the notation and remarks section as it is. I think notation should be below Examples, because it uses notation from this section. The "remarks" (esp. concerning identity and inverses) are in a sense belonging to the "Elementary group theory" stuff, right? Anybody against merging this up there? Jakob.scholbach (talk) 15:31, 11 April 2008 (UTC)

Free groups

Does anyone think there should be a section briefly talking about free groups and free products and presentations as generators and relations? LkNsngth (talk) 03:12, 13 April 2008 (UTC)

Well, there is one sentence talking about this (in the quotient group section). I think we should be careful when expanding, because the article is already pretty long. But if you want to write a little bit more, go ahead. Jakob.scholbach (talk) 10:24, 13 April 2008 (UTC)

Merger proposal

The group and group theory articles overlap quite a bit. This has already been suggested at WP:MATHCOTM Indeed123 (talk) 15:08, 19 April 2008 (UTC)

Thanks for the formal proposal, the idea had been in the air for a while. At first I didn't know what to think about it, but after carefully comparing the two articles I think they should be merged for the moment.

Both articles are fairly elementary. Where they treat things in a parallel way, each article has some ideas that could be used to improve the other. If we do this, several passages will be almost identical. In which of the two articles a particular topic is treated seems to be basically random. E.g. group (mathematics) discusses a specific smmetry group in detail, but only uses the word "symmetry group" once (in passing), without definition or wiki link. Group theory has a section on the topic, but no example.

Group theory has 20 KB and group (mathematics) has 45 KB. I think group theory should be merged into group (mathematics), to form a single article that gives the subject a bit more justice. If that article grows further, we can split it into two articles with more clearly distinguishable profiles. --Hans Adler (talk) 11:51, 20 April 2008 (UTC)

Please also see the wikiproject discussion.

Personally I think the group theory article should discuss the field of study called group theory, developing its history, distinguishing the current research fields within it, linking to the major publications, conferences, institutions, mathematicians, and results of which it consists and which it has produced. The current article is more like a chapter called "group theory" in an algebra text. JackSchmidt (talk) 15:18, 20 April 2008 (UTC)

While it is true that the two pages have a considerable overlap right now, I nonetheless disagree with merging the two articles. Here is why: As it is, this article is already very long, containing basic material which explain some of the first steps.

In writing the article I tried to make clear the distinction I thought of between the two articles. I see three levels of knowledge:

• Fooling around with the axioms to get such things as "there is exacly one identity" etc.
• Introducing basic structures as in every category (without necessarily emphasizing that we are in a cat.): sub/quotient groups, homomorphisms etc.
• Thirdly, and this is the stuff I think just does not fit here spacewise: involved statements about groups, like the ones arising from representation theory etc.

The third item, together with applications and history of group theory, will give enough material for a long (and possibly beautiful) extra article.

So, my proposal: reduce the overlap by trimming down group theory, perhaps introduce a "basic stuff" section which points to this article (we also haveglossary of group theory, btw), instead of merging the two pages, which will come at the cost of reducing some of the (I believe interesting and necessary for an intro text) material. Jakob.scholbach (talk) 07:39, 21 April 2008 (UTC)

I agree with Jakob. Merging with group theory is a bad idea. The group article discusses (and should discus) the concept of a group (and some basic properties having to do with the definition). Group theory on the other hand should talk about the mathematical field of research group theory. It should discus the history of this field (abel, galois, etc.) and about techniques used (reprentation theory) and should state and then link some basic results. (like the classification of finite groups). (TimothyRias (talk) 08:42, 21 April 2008 (UTC))

Yes, incorporating the good ideas from group theory that belong here into this article and replacing the parallel parts there by a summary of this article sounds like a good solution. But then I would also like to see the sections on Lie groups, Galois groups and Generalisations moved to group theory, to make the respective roles of the articles clearer. --Hans Adler (talk) 08:53, 21 April 2008 (UTC)

OK. Including Galois and Lie groups etc. was done with the intention that this elementary article should somehow convey a sense of what is beyond integers and rationals. Perhaps here a little section "Advanced examples" etc. would be in order, containing a little bit about the ones mentioned right now. But on the other hand, what we have now is already a fairly tight description, so shortening further may be problematic. Jakob.scholbach (talk) 11:11, 21 April 2008 (UTC)

I would suggest that while merger proposals are being discussed, the article should be withdrawn from good article nominations. Thanks, Geometry guy 06:41, 23 April 2008 (UTC)

Actually I think, instead of withdrawing the GAN we should rather withdraw the merger proposal. This issue had already been discussed (see the thread above in this talk page), not only for group theory vs. groups but also graph vs. graph theory and the like. Consensus was reached at the time (and also seems to be reached this time(?)). Does anybody object removing the merger proposal? Jakob.scholbach (talk) 07:56, 23 April 2008 (UTC)

As the only one so far who has shown any enthusiasm for the merger proposal, I agree with closing it. The nominator doesn't seem to log in very often, so we shouldn't wait more than a day or so to get their opinion. Or perhaps just close it per WP:IAR? (WP:SNOW almost applies.)

I have incorporated a few minor ideas from group theory into this article. That article should later be partially rewritten so it has a section that is an accurate summary of those aspects of this article which it really uses, and nothing else. We also have the problem that the lede of group theory would be better suited for group (mathematics), and vice versa. But I am not sure it's a good idea to just swap them. --Hans Adler (talk) 08:12, 23 April 2008 (UTC)

I agree the official merger proposal can be closed. I suspect major changes to the group theory article will be made that remove or shorten its duplicated material, but I think this article, Group (mathematics), is in a good state and does not need to absorb much if anything from the GT article. JackSchmidt (talk) 12:35, 23 April 2008 (UTC)

Done.Jakob.scholbach (talk) 13:01, 23 April 2008 (UTC)

Thanks for resolving this. Good luck with the GA nomination! Geometry guy 19:10, 23 April 2008 (UTC)

Survey

WP:Good article usage is a survey of the language and style of Wikipedia editors in articles being reviewed for Good article nomination. It will help make the experience of writing Good Articles as non-threatening and satisfying as possible if all the participating editors would take a moment to answer a few questions for us, in this section please. The survey will end on April 30.

• Would you like any additional feedback on the writing style in this article?

Sure, this is appreciated. I wrote a considerable amount of the current version of the article, I'm personally not a native speaker, so style issues are good to know.Jakob.scholbach (talk) 07:42, 21 April 2008 (UTC)

• If you write a lot outside of Wikipedia, what kind of writing do you do?

I'm sometimes writing mathematical papers.Jakob.scholbach (talk) 07:42, 21 April 2008 (UTC)

• Is your writing style influenced by any particular WikiProject or other group on Wikipedia?

Well, I'm hanging around in the Wikiproject Mathematics, but I think it does not influence my style very much. Jakob.scholbach (talk) 07:42, 21 April 2008 (UTC)

At any point during this review, let us know if we recommend any edits, including markup, punctuation and language, that you feel don't fit with your writing style. Thanks for your time. - Dan Dank55 (talk)(mistakes) 04:11, 21 April 2008 (UTC)

The lead

Yes, Hans, you are right. The lead section is not really well corresponding to the articles content. Are you up to improving it? I hope so... Jakob.scholbach (talk) 11:44, 23 April 2008 (UTC)

Hmmm, I hoped that you would have a plan. But now I will have a look. --Hans Adler (talk) 11:53, 23 April 2008 (UTC)

Having looked a bit more closely I would say the lede of this article is fine; deals only with groups themselves and not with the wider connections. The problem is that group theory is totally underdeveloped, and that includes the lede. Perhaps we should tackle that article next, although not necessarily to good article standard. Sorry for the mistake.

By the way, when I print the article the SVG images have a black background. I have asked for help at WP:SVG Help#Requests for assistance. --Hans Adler (talk) 12:40, 23 April 2008 (UTC)

(I think the lead on this article is good. I know nothing about the SVG problem.) I plan on making a push in June for a more human group theory article. I have Festschriften and obituaries and some history books, that should give all the sources needed. However, I am not familiar with infinite group theorists, and only vaguely familiar with loopers and finite geometers, so I think I can only survey 35-40% of group theory without making (sourced) caricatures of some areas. I can probably get help on infinite solvable groups and combinatorial group theory, but I don't have any good sources on profinite groups and Lie groups. I think I can get at least that viewpoint to GA status in June, but there probably needs to be a section on group theory in mathematics education ("modern algebra" from van der Waerden to Gallian), and a section on applications of group theory (statistics as highly symmetric block designs, chemistry as crystallography, physics as symmetry principles, computer science as network design, crypto, and some arguments in complexity, can we do biology? medicine?). Does this sound like an interesting collaboration? JackSchmidt (talk) 13:00, 23 April 2008 (UTC)

Sounds good. I can probably help out for the applications in physics part. Any further discussion should probably continue on the group theory talk page. (TimothyRias (talk) 13:12, 23 April 2008 (UTC))

Good article review

1. It is well written.
   • Not yet
   • For example, the powers of any element a and their inverses (that is, a0 = e, a, a2, a3, a4, …, a−1, a−2, a−3, a−4, …) always form a subgroup of the larger group, the so-called cyclic subgroup generated by a, see below under Cyclic groups. Link "cyclic subgroup" and get rid of the "see below" part. Also, it might be useful to write a¹ in the list.
   • The operation between the cosets behaves in the nicest[7] way possible: (Ng) • (Nh) = N(gh) for all g and h in G. Move the citation to the end of the sentence.
   • "Lie groups" has a one sentence paragraph that should be merged into the one before it
2. It is factually accurate and verifiable.
   • Not fully verified
   • Try to aim for at least one reference per section. Usually a footnote for the first sentence of each section does the trick.
3. It is broad in its coverage.
   • Pass
4. It is neutral; that is, it represents viewpoints fairly and without bias.
   • Pass
5. It is stable; that is, it is not the subject of an ongoing edit war or content dispute.
   • Pass
6. It is illustrated, where possible, by images.
   • Pass

Let me know when you're done.-Wafulz (talk) 14:32, 2 May 2008 (UTC)

Thanks for your review. The issues mentioned above have now been fixed. I believe that adding additional further footnotes would decrease the readability of the article. So, not every single subsection contains a footnote. But I tried to back up every statement which may not be directly clear to the reader. Jakob.scholbach (talk) 18:04, 6 May 2008 (UTC)

It looks good to me. However, I do combinatorics and number theory by trade, so I'm requesting a second opinion to (hopefully) get someone who has more expertise with this topic. Thanks for your patience.-Wafulz (talk) 12:32, 7 May 2008 (UTC)

I've been watchlisting this page during the review process. I've made minor contributions during the review, but am not a significant contributor. I have expertise in geometry and am very familiar with undergraduate and some advanced group theory, but also have some experience of the GA process, so I hope I can provide a useful second opinion. The "list/do not list" decision remains with the original reviewier however.

My main comment is that I think this is an excellent introduction to the concept of a group. It meets the challenge to be broad, accessible and encyclopedic very well. It is also well written and neutral point of view. I list below the problems that I noticed.

1. Prose style: generally this is very good in my view, but there are some lapses, and some issues with encyclopedic tone.
   • First example: the identity element discussion is repetetive. I suggest dropping the first sentence, and clarifying the third, placing "additive" in brackets, as in the inverse element part.
   • Variants of the definition: try to find a better subject for these sentences than "one".
   • Basic concepts in group theory: the first paragraph contains the only really bad prose I have found in the article; please don't start paragraphs with long noun phrases, or use gerunds as subjects of sentences. This paragraph needs a complete rewrite.
   • "By counting cosets, one can show Lagrange's theorem...": rephrase with an identifiable subject, preferably in the active tense; one approach would be "Lagrange's theorem, which states..., follows by counting cosets."
   • "Quotient groups, also known as factor groups, form the counterpart to subgroups." This conveys no meaning to the reader who does not already know what it means.
   • What it is the relation between subgroups and presentations?
   • Simple groups: the second definition appears to include the trivial group, whereas the first does not. (Good solution!)
   • Group homomorphisms: "Groups together with their morphisms are assembled in the category of groups." This is not properly explained (what is a morphism?) and the prose is weak.
   • Cyclic groups: why the need to assemble them into a set?
   • Order of a group: o(a) is a common notation for the order of an element.
   • Notations: "Groups can use different notation...". Please fix. I could probably fix this one myself, but I'm not completely sure what you are trying to say. A lot of the prose in this section needs improvement.
   • In general try to avoid the use of "we" and "one": sometimes using these words is the best option, but often one/we can find a better subject for the sentence :-)
2. Citations: the scientific citation guidelines (a GA requirement) encourage providing one general citation per section so that the reader knows that the material is general knowledge and where to find it. I fail to see how this would diminish the readability. In the first section, a citation for the particular form of the axioms chosen would be reassuring (in particular, the "closure" issue). The second section would benefit from a general reference and a reference to a book on group representation theory. As mentioned above, the first paragraph of "Basic concepts in group theory" needs to be rewritten: I suggest taking the inspiration from a source and citing it. The "Notations" section is a bit weak and so really needs a source. In the generalizations section, I would think references are needed to cover each and all of the generalizations presented.
3. Images and tables: these need captions. In particular, the parenthetic aside on the coloring of the vertices of the square would be much more use as part of a caption for the image. The group table of d4/8 could use a caption as well, to explain the shading. (Captions could be improved, but are okay.)
4. The lead. The lead fails to summarize the article because it does not refer at all to any of the material described in the "Basic concepts in group theory" section.
5. There are one or two other broadness issues, but I think the above is already too much food for thought!
6. Taking this forward to FA. There are lots of MoS issues that need to be fixed, e.g., em-dashes are now required to be unspaced (much to my regret). There is also a question of comprehensiveness. I suggest developing the Group theory article to GA first. This article reads a bit like an Introduction to group theory article. A comprehensive article on groups and/or group theory could/should include a lot of advanced material. Lie groups and representation theory, for example, get fairly short shrift here. Geometry guy 20:41, 8 May 2008 (UTC)

Issues raised by Geometry guy

Prose style

I believe I have addressed the majority of the bulleted points for 1.

• Basic concepts intro should be looked at: no reason the new must be better than the old.
• Relationship between presentations and subgroups (is amazingly interesting, but) has not been made clearer. The editor just mean the normal subgroup generated by the relations... is a subgroup.
• For simple groups, I think generally the trivial group is not called simple, but I think it is important to remember "who cares", and so have phrased the definition to technically exclude the trivial group from the definition, by refusing to address it. I believe this is a common method.
• Category theory prose is still weak, but at least the only new word is blue linked now. This is basically a "see also" stuck into the text. fixed by the other leading JS.
• Notations has not been changed. It is unclear how useful it is in an encyclopedia, rather than in a textbook. I am not sure how to phrase its waffling statements more strongly. ditto.
• We should be careful how often one is distracted by change in voice, but I have not done anything about it. ditto

Thanks again for the two detailed reviews. JackSchmidt (talk) 21:39, 8 May 2008 (UTC) and updated 20:59, 12 May 2008 (UTC)

• First example: the identity element discussion: done Jakob.scholbach (talk) 13:58, 9 May 2008 (UTC)
• "We" and "one": I eliminated every occurence (in one place (Quotient groups) I wasn't sure what to take else) and tried to get directly to the point of the phrase in question. Hopefully an improvement.... Jakob.scholbach (talk) 19:29, 9 May 2008 (UTC)

  This is not a matter of principle: we/one can be better than an awkward passive construction and is certainly not forbidden, but often looking for a good subject for each sentence leads to better prose. Geometry guy 20:05, 9 May 2008 (UTC)

• Lagranges theorem...: this is now reformulated (and moved to finite groups).
• Notations: this is now much shorter (I eliminated some less important stuff + redundancy). Hopefully also better structured. One citation still missing. Jakob.scholbach (talk) 19:56, 12 May 2008 (UTC)

I believe this section (prose, #1) is now done. JackSchmidt (talk) 20:59, 13 May 2008 (UTC)

Citations

• I have added a general reference for all of the "examples" subsections except the integers (under + and *). Jakob.scholbach (talk) 11:08, 9 May 2008 (UTC)

  I haven't yet checked this carefully, but a cite for the axioms presented is still missing. Do you agree that this is worth providing? Geometry guy 20:05, 9 May 2008 (UTC)

Yes, I agree. I had a somewhat different idea earlier, namely that almost all of the content is covered in almost every book in the ref list. But you are perfectly right. At the moment, I don't have access to books, but as soon as I do, I will try to cover the rest of the sections still missing a reference. Or, obviously, if somebody is faster ... Jakob.scholbach (talk) 20:25, 9 May 2008 (UTC)

• Now, every section except "Division" and "First example - the integers" contains at least one general reference. I hope this is enough. Jakob.scholbach (talk) 15:01, 13 May 2008 (UTC)
• The newly added section about symmetry groups (this article is a mess, btw) now needs a general reference. Jakob.scholbach (talk) 21:03, 13 May 2008 (UTC)

Done. Jakob.scholbach (talk) 15:04, 14 May 2008 (UTC)

Images and tables

• I added two captions. I don't know how to get nicer formatting, though. Any formatting experts in the house? Jakob.scholbach (talk) 14:12, 9 May 2008 (UTC)

I have been happy with the formatting. I don't like lots BR tags captions. I have to use larger font sizes than many editors, so the margins for me are different than for the anonymous uebermenschen browsers with perfect eyesight. Captions with lots of BRs and such just look ragged to me. The position on the page also looks fine to me. Were there any in particular you were worried about? JackSchmidt (talk) 20:51, 12 May 2008 (UTC)

I wasn't exactly happy with the BR tags, too. I introduced them because without them the table gets very large (as to display the whole caption in one line). I wasn't able to circumvent this behaviour. The caption not wrap automatically into several lines. Jakob.scholbach (talk) 13:09, 13 May 2008 (UTC)

I fixed the column widths to the width of the (current) pictures. Now, the captions wrap natural without the need of BR tags. Yeah. (TimothyRias (talk) 13:41, 13 May 2008 (UTC))

Lead section

• I tried to improve the lead. Contentwise it's now closer. But my limited English abilities prevent me from more brilliant prose... Jakob.scholbach (talk) 20:03, 9 May 2008 (UTC)

It looks good to me as well. I think anything too brilliant detracts from the article. An encyclopedia article should catch the interest of the reader, but not try to dazzle them or anything. Otherwise, we should have some sort of flash image of a tiger leaping out of one of the early illustrations, hopefully roaring something that sounds like "grrrrrroups", and then the screen fades away to "behold its fearful symmetry!" JackSchmidt (talk) 20:55, 12 May 2008 (UTC)

You seem to have a lot of unused creative energy :-). Perhaps you are up to writing a comic like the ones on WP:POST? OK, then let's consider this issue fixed for now. Jakob.scholbach (talk) 13:10, 13 May 2008 (UTC)

Broadness issues

• GGuy, could you please give a hint what you'd like to see here? Jakob.scholbach (talk) 20:04, 9 May 2008 (UTC)

  Sure. By and large, I think this article is probably broad enough for GA, which is one reason I didn't detail. Another reason is the overlap with the still-to-be-well-developed Group theory article. At the FA level, "broadness" is replaced by "comprehensiveness", and it is probably easier to say what is needed in a comprehensive article than a broad one. Here are a few issues.

  • Apart from the discussion of the classification of finite simple groups, the article barely touches upon advances in group theory made in the last 50 years. Maybe that is the job of the Group theory article, but some advances probably deserve a mention. One example is computational group theory, and computer algebra (there is a hint of this at List of small groups). Another example, already mentioned, is group representation theory (especially Lie groups). It is also worthwhile to tell the reader that this is work in progress (one of the common questions asked of mathematicians by lay folk is "I thought it was all known"/"what is more to prove?")
  • A related issue is the absense of any history: again this is covered by the group theory article, and so is probably not required at GA level for this article, but an introductory article to groups would certainly be enhanced by an introduction to the colourful history of the subject.
  • There is no section on applications. My feeling is that this really is a GA-broadness issue. Groups are fundamental in physics. The article mentions this in the (old and new) lead, but fails to elaborate it in the article. Such an applications section is vital, in my view.

  I hope that helps. Again, I emphasise that you should trust your own judgement: I have the impression that at least one regular editor here is a real expert on group theory. Make this article sing and leave it to the judgement of the main reviewer to assess whether the article is GA. Geometry guy 21:02, 9 May 2008 (UTC)

Applications: I can think of only one type of application that fits in the current article, but it should be sufficient. It is summed up as "symmetry is good for science".

• Highly symmetric block designs save money: You don't want to test your expensive new drug on everyone, but you know your population is multidimensional, how can you eliminate correlation in multiple dimensions with small sample sizes?
• Highly symmetric subsets of vector spaces like Golay codes.

and then the only one which is "worked":

• Symmetry saves time: if you need to count something and a group acts on it, count orbit sizes instead. How many distinct rearrangements of MISSISSIPPI are there? How many colorings of a necklace? I intended for this to be at the level covered in very basic classes on counting, perhaps for ages 15 to 20.

If someone wants to add these feel free. I don't have the references handy, but most discrete math books should have something like this. Most of the good applications I know require a little more than is covered in this article. JackSchmidt (talk) 20:46, 12 May 2008 (UTC)

OK. We have some (very short) glimpses of applications (cryptography for finite gps, molecular symmetries and physical theories for Lie groups, and gauge theory and error corrections for finite symmetry groups). I think this should be enough for a "good" article. This belief is also based on the idea, that in fact one is not really applying groups (which are abstract in a sense), but group theory (the knowledge about groups, so to say). This, and the idea that group theory shall contain (and already contains) more application material, make me think this way. Jakob.scholbach (talk) 15:01, 14 May 2008 (UTC)

Conclusion

I think we have covered all points in the reviews above. If nobody is against, I will probably ask Wafulz, the first reviewer (of the 2nd review) to give his opinion, or if he doesnot want to decide, we probably need a 3rd opinion... Jakob.scholbach (talk) 15:03, 14 May 2008 (UTC)

In my opinion, you've met the Good Article criteria, so I'm promoting the article. For "next steps", I suggest you go for a peer review and maybe aim for FA. Congrats and thanks for your hard work.-Wafulz (talk) 18:59, 14 May 2008 (UTC)

Rational number example in first section

I couldn't find when this example was added to the first section, but its addition is unnecessary and, in my opinion, undesirable. Someone went to a lot of trouble to explain the idea of a group in terms of symmetries using those colored diagrams, and this is way better than the rational numbers example, which is too technical for this article (at least that high up in it anyway). I propose completely deleting this reference. Xantharius (talk) 19:02, 6 May 2008 (UTC)

I added it after I saw it in group theory, in preparation to making this article a subarticle of the other one. In my opinion the example (or a similar one) is essential to this article:

Groups are algebra, and algebra is just generalised numbers. If we don't give an example with numbers very early in the article, many non-mathematicians who try to read it and have some half-knowledge will be very confused. (It might even make sense to extend this into another example section. Proving the associativity of matrix addition is probably just the right level of abstraction and mathematical sophistication for most of our readers. But then this could detract from the nice square symmetries example, and I agree that that would be a pity. And such arguments would be even more appropriate in field (mathematics) anyway.) --Hans Adler (talk) 19:28, 6 May 2008 (UTC)

I think the current rational number example is short, to the point, and gives someone (horribly atypical) examples of groups to give them some comfort. So I think the example should stay (I am fine changing "rational" to "real" or whatever people think most readers assume "number" means).

I did want to mention my strong disagreement with the idea that groups are generalized numbers. Numbers, ordered tuples of numbers, simplicies, finite geometries, manifolds, abstract elements of sets -- these are the passive carriers for group actions, the setting. The groups are the actors. The symmetry example is reasonable as it distinguishes the setting (the square) from the actor (the group). Aschbacher's and Alperin and Bell's textbooks present groups in this light. The next example after symmetry of a shape (a geometric object), should be symmetry of a vector space (an algebraic object), also known as GL. Perhaps another typical example would be the symmetry group of an infinite binary tree. JackSchmidt (talk) 19:55, 6 May 2008 (UTC)

I still disagree. What groups are fundamentally about is symmetry: their very formulation deals with the nature of symmetry, and groups are the perfect vehicle for describing how symmetries operate on other objects. To have the example of rational (or real) numbers appearing before the beautifully done and accessible example of the symmetries of the square might be appealing to some mathematicians (not this one), but I think for most readers the rational numbers without zero under multiplication is a horrible introduction of the idea of group. By all means have some of these examples later, but not before this very well thought-out presentation of the idea behind groups: that they are about symmetry. I still propose removing this example from its place before the symmetries of the square (particularly in light of it being borrowed from group theory, which is a much more technical article on groups). Xantharius (talk) 20:06, 6 May 2008 (UTC)

The numbers example is needed for a certain type of non-technical readers (for their "comfort", as JackSchmidt put it, and I agree with that formulation), because otherwise they will never read the beautiful example. Yes, of course groups are mainly about symmetries. We have to pick our readers up where they are (that's a German idiom; don't know if it works in English), and some of them are precisely there, at the numbers. Once we are holding their hands we can take them elsewhere. Especially after seeing multiplication signs and 1s, and reading about "associativity", people expect to see numbers. They are not going to scroll down several pages to understand where the notation comes from. So it will be completely abstract and meaningless for them, and there is nothing more scary for the general reader than abstract notations. --Hans Adler (talk) 20:17, 6 May 2008 (UTC)

It's important to give a familiar example of a group up front, but I actually think a better example is the integers with addition. I suggest something like this:

A familiar example of a group is set of all integers Z, where the operation • is addition (+). It satisfies these four axioms because:

1. The sum of any two integers is an integer.
2. Addition of integers is associative.
3. Zero is the identity element, since adding zero to an integer always yields the same integer (e.g. 2 + 0 = 2).
4. The inverse of an integer is its negative, since adding a number to its negative always gives the identity element, zero (e.g. 2 + (−2) = 0).

I agree that groups are more fundamentally about symmetry, but before exploring how abstract groups can get it's good to ground it in something familiar and concrete. Dcoetzee 20:21, 6 May 2008 (UTC)

(e/c) If the idea is to give a completely easy "throwaway" example, then why not use the integers under addition? At least this is of fundamental importance as an abelian group, and doesn't require excluding any elements (like zero for the rationals). The rationals under multiplication seem to me to be both atypical and fussy as a first example of a group. silly rabbit (talk) 20:26, 6 May 2008 (UTC)

Integers under addition sounds good to me. Every torsion-free group has it as a subgroup, so I would be hard pressed to call it absolutely atypical. It certainly should be familiar to readers: I think integers are covered by the age of 12 years old even in slower math programs. It is certainly fundamental to abelian groups, which perhaps my viewpoint is biased against. It will also be *even shorter* than the rational numbers example. Even with the numbered list, the ideas are so clear, that in some sense the example is not there: it is just restating the definition in a comfortable language.

(to Xantharius) I hope it is clear we agree that groups are fundamentally about symmetry, and our only disagreement is whether a "comfortable" example is worthwhile even if it does not display the idea of symmetry. I think such a simple example is worthwhile (especially the integers), even though it does not express this important idea. JackSchmidt (talk) 20:42, 6 May 2008 (UTC)

Integers are fine for me, too. --Hans Adler (talk) 21:02, 6 May 2008 (UTC)

OK, why don't we make something like: Definition, First Example, Second Example? In the first example subsection, I'd like to have the fact stressed that Z is a very special group. There would be no notion of group if everything was as simple as Z. Who's up to it? I'm done for today...Jakob.scholbach (talk) 21:34, 6 May 2008 (UTC)

I've taken a shot at implementing your suggestion. Please edit my rough addition mercilessly ;-) silly rabbit (talk) 22:01, 6 May 2008 (UTC)

I like it. Would it be too silly to include the "e.g."s? Something like:

• 2 + 3 = 5 is an integer.
• (2 + 3) + 4 = 5 + 4 = 9 = 2 + 7 = 2 + (3 + 4)
• 2 + 0 = 2
• 2 + (-2) = 0

If we only use letters, then suddenly it is *algebra* and we lose the "even a 10 year old will understand". JackSchmidt (talk) 22:05, 6 May 2008 (UTC)

Integers under addition sounds like a good plan! If e.g.'s are done, care must be taken not to lose the encyclopedic tone: we aren't writing a textbook here :-) Geometry guy 22:41, 6 May 2008 (UTC)

section headers

Two ideas about sections headers: - "First Example" and "Second example" (suggested by myself) are not terribly informative. How about "Integers -- an everyday group: integers" and "Worked example -- a symmetry group"? - secondly, is it possible to hide some sections from the table of contents? I'd like to hide the third-level subsubsections. They just clutter up the toc, I think.

Comments? Jakob.scholbach (talk) 17:17, 8 May 2008 (UTC)

{{TOClimit}} added.

• First example: the integers
• Worked example: a symmetry group

perhaps sounds better? Headings should be short. I think "first example" is a known phrase, meaning the easiest, most everyday example that is worth mentioning. "Second example" might sound silly, I agree. "Worked example" should catch the student's eye. I think indicating which groups are in the example is a good idea. JackSchmidt (talk) 18:33, 8 May 2008 (UTC)

What do you mean by "worked" example? I think it might be a germanism. To me, "worked" suggests "forces" of "far fetched" which is probably not your intention. (TimothyRias (talk) 08:22, 9 May 2008 (UTC))

It means "exercise and solution", as in this quote from Schaum's Outlines: "each title typically has introductory explanations of topics, plus many fully-worked examples, and further exercises for the student". Wikipedia actually has an article on this method of teaching, Worked-example effect. "'A worked example is a step-by-step demonstration of how to perform a task or how to solve a problem'", "learners that studied worked examples, performed significantly better than learners who actively solved problems". etc. JackSchmidt (talk) 12:40, 9 May 2008 (UTC)

Done. Jakob.scholbach (talk) 13:55, 9 May 2008 (UTC)

SVG images for the example

I've prepared SVG images for the example, with some minor changes:

Any comments and/or changes that need to be implemented? (TimothyRias (talk) 13:49, 9 May 2008 (UTC))

For easy comparison:

They should probably be split for easy captioning. I can't tell if the arrows look really good or really confusing. On the one hand, the colored dots show the result of the action, but the arrows show (to my mind) what the action will do. For instance the bottom left picture says that you should take the red in the top left corner to the green in the bottom left corner, but really what it means is take the red in the top right corner to the green in the top left corner.

If making these images is easy, would you make some similar to the current "double" images in the article so that it is easier to compare? I like your soft gray and large arrows better, but superimposing the action and the result confuses me.

Would it be possible to catch the group elements "in the act" like the rubik's cube picture? Maybe each group element gets two pictures, the first is the result of applying 1/3 of the action (in R^3 or SO(3)), with your soft gray arrows to indicate what is happening, and the second is the plain old result. JackSchmidt (talk) 14:08, 9 May 2008 (UTC)

I also find it a bit distracting to join the operation and the result. I do like the shapes of the arrows. I prefer the colors I chose (they are from User:KSmrq's page, somehow supposed to be perfectly opposite). Jakob.scholbach (talk) 14:15, 9 May 2008 (UTC)

Oh, I had not noticed the color change. I like TR's heavier lines for the square's edges, and I like his larger, softer arrows. I think I like KSmrq's colors better, and are definitely for them if they are better for accessibility. However, the primary colors used by TR are very clear to me, and I think both will punish the color blind reader.

Probably the images are too small, but something I've done in the past when teaching is to put numbers inside the dots, as well as color. I was only trying to assist our color blind printer so that both the web version and the handouts were useful, but this might help readers as well. Of course if they don't fit, no worries. It's not like Hall-Janko graph has labelled its vertices :) JackSchmidt (talk) 14:28, 9 May 2008 (UTC)

I updated the colors. Als splitting the picture is little trouble, I just put them together for this test so that I wouldn't have to upload 8 new pictures for every change I made. But for the final version we should definately have seperate pictures.

Currently, I am looking into over laying (and slightly offseting) the original and the result with the original slightly faded into the background.(TimothyRias (talk) 15:04, 9 May 2008 (UTC))

New colors definitely look good. I'm looking forward to offset "shadow". It sounds neat. I looked at adding numbers, and noticed you did this in the sanest, nicest way possible, by actually using SVG to apply the group elements. It makes the text look funny, but I can't decide if it is a bad thing or not. Try adding:

       <text x="-54" y="-45">1</text>                                                                                       
       <text x="46" y="-45">2</text>                                                                                        
       <text x="46" y="55">3</text>                                                                                         
       <text x="-54" y="55">4</text>

to the id="square" guy. The numbers themselves get rotated and flipped, which looking at it again, looks awesome to me. I don't know how to tell SVG that I want text centered, but at least on my computer the numbers are basically perfectly centered. I suspect it is browser dependent though. Sorry, I don't know much about uploading images to wikipedia. JackSchmidt (talk) 15:19, 9 May 2008 (UTC)

Oh, and if we have the numbers in there, then it makes it easy to talk about the permutation representation, expressing D8 as a Sylow 2-subgroup of S4. Maybe one would just use the pictures of two generators in some other article. I think D8 might even have its own article. JackSchmidt (talk) 15:22, 9 May 2008 (UTC)

Shadow rotations:

(Image moved)

The 90 right looks very nice, but 180 and 90 left look akward. I haven't figured out what to do with the flips yet. I also looked at adding the text, but it looks funny for 180 turn. (TimothyRias (talk) 15:44, 9 May 2008 (UTC))

I tried some stuff. I like the result, but I've no idea if others will. Thanks for producing such sane SVG, btw. I'm sorry if mine is not so sane. Feel free to adjust it as needed.

The artist formerly known as JackSchmidt (talk) 21:04, 9 May 2008 (UTC)

How about this:

I fixed the horizontal text positioning. The vertical one is still a bit sketchy, but I am confident that it should work on all platforms. (And at the moment wikipedia is passing SVG through rsvg anyway so there is no problem.) (TimothyRias (talk) 14:20, 10 May 2008 (UTC))

I know it is a lot of fun generating nice SVGs like this, but please bear in mind accessibility issues and the kiss principle (less is more). Geometry guy 14:59, 10 May 2008 (UTC)

I like the bolder font. I think the changes do help accessibility (the current in-article pictures are useless to color blind). Each of the flips looks nice. I like the numbers to make it less dependent on color. I am quite visually impaired, so I do tend to consider WP:ACCESS. Unfortunately, I think people are so different, and math is so far from mainstream, that deciding exactly what makes something more accessible is hard.

I tend to think in full page handouts, not tiny little tables. I think the ideal handout would have the stationary, original square with the gray arrows indicating the motion to be performed, then the "in action" version next to it, and then stationary, transformed square. I am not sure what is best in the article. I think some sort of picture is definitely wise.

• Would it be reasonable to mostly use this pictures on a symmetry of the square article (I think one already exists), and just have an excerpt in this article with a link to the full article for full glorious images?

This might address the KISS concern. JackSchmidt (talk) 15:53, 10 May 2008 (UTC)

Good points about the colour blind and the numbers. Another option is to use an actual animation rather than these faux animations. But I leave it to other editors to decide what the best compromise is: I only wanted to draw attention to the issue, and am glad to hear that accessibility is being taken into consideration. Geometry guy 17:45, 10 May 2008 (UTC)

I'm impressed by you guys graphic abilities! The animes are interesting, but (except perhaps for the 90 and 270 deg. rotation) the current version is way easier to digest. One thing I could imagine: the "forefront" square showing the result of the operation stays as it is and the starting-point config is shown in grey and arrows show the movement of the vertices toward their final positions. Jakob.scholbach (talk) 17:56, 12 May 2008 (UTC)

I tried something else. (see above) Maybe, this is a little clearer. (some of the arrows may need tweaking.) (TimothyRias (talk) 21:47, 12 May 2008 (UTC))

Those are simple, and to my mind, clear. I like them, but we should probably hear from others, as I've more or less liked all of them. JackSchmidt (talk) 01:44, 13 May 2008 (UTC)

I think just about the simplest thing you could do that would be immediately clear would be to show two squares side by side with an arrow between them, and the second one showing the result of the transformation. The operation itself is implicit but clear for all the rigid operations considered here (for rotations I would also rotate the labels). Dcoetzee 07:10, 13 May 2008 (UTC)

Excellent work on the images. To my opinion especially the superimposed animation steps (above left, "group test2") reflect the mappings clearly. The rotated (mirrored) corner labels add to the clarity (but not all are correct yet). −Woodstone (talk) 12:49, 13 May 2008 (UTC)

I prefer the right version (group test3) for the above reasons. I'd opt for including these (properly sized) in the article. Is it possible to get the white numbers with a tiny little black border, so that they are easier distinguishable from the faded green bullets etc.? Jakob.scholbach (talk) 13:04, 13 May 2008 (UTC)

Black borders have been added. This is as thin as they go.(TimothyRias (talk) 13:15, 13 May 2008 (UTC))

Great. Many thanks! I think that's quite an improvement over the current version. Would you add it to the article? Jakob.scholbach (talk) 21:07, 13 May 2008 (UTC)

Done.(TimothyRias (talk) 06:21, 14 May 2008 (UTC))

Examples of groups: Lie groups => group objects

I think we might want consider replacing the Lie groups section with one talking about group objects in general and providing Lie groups as a prime example. We could include a small table of the most well-known examples. (Lie groups, topological groups, Algebraic groups, etc.) Currently, the article might overemphasize Lie groups a little, which could be due to the background bias of the editors. (including myself) (TimothyRias (talk) 08:05, 13 May 2008 (UTC))

I think Lie groups are is not overemphasized right now. The section does mention group objects. If anything, then algebraic groups etc. are mentioned pretty shortly, but we need to be careful and even restrictive when adding more and more content. None of the examples sections, except the very easy ones, can be more than a glimpse of what is out there. So, I would not change the subsection substantially (except perhaps adding details about applications as requested above in the review). Jakob.scholbach (talk) 13:02, 13 May 2008 (UTC)

I don't think the total length of the article would increase much. Currently the section on Lie groups starts by talking about group objects and then moves to Lie groups. I think it would be more natural to switch the emphasis of the first paragraph completely to group objects and move on to Lie groups. Group objects are important in mathematics and need a more prominent mention here. (TimothyRias (talk) 13:54, 13 May 2008 (UTC))

Hm, my only concern is that a section having one sentence about group objects and 5 about Lie groups should be titled "Lie groups". But, this is a wiki, so do what you see fit. Jakob.scholbach (talk) 21:01, 13 May 2008 (UTC)

Symmetry groups

Partly in response to the concerns about broadness (in particular w.r.t. applications of gps) I added a little section about symmetry groups. Does anybody know of a nice example where a geometrical object was used to prove something about an abstractly defined group? (I know something like growth of the fundamental groups of manifolds with certain curvature, but I'm not sure whether this is used rather the other way round).

Building (mathematics) is sort of a famous version. Coxeter groups more or less fall in this category. Hyperbolic groups have combintorial group theoretic properties because they act on special manifolds. Hurwitz groups satisfy a presentation because they act on special manifolds. The existence of many of the sporadic simple groups was proven by constructing them as symmetries of finite geometries. JackSchmidt (talk) 21:23, 13 May 2008 (UTC)

Let me know if this is not clear. Basically using geometric objects to prove things about groups is one of the fundamental methods in group theory. It is hard to think of any major result that does not use this technique. This is sort of the point of the first chapter of Aschbacher's text on group theory, and a point belabored in most geometric group theory texts. The classification of the finite simple groups hinges on the idea that simple groups tend to have a natural geometry associated with them, that geometry is almost always over a field, and that field is almost always the base field of the lie group that your simple group must be equal to. Current research in modular representation theory is bizarrely geometrical (see Benson's texts), and homotopy theory plays a huge rule. A literature search for "quillen complex" or "bouc complex" or "p-local geometry" should give a hint at how big just that tiny subsection of this is. JackSchmidt (talk) 21:34, 13 May 2008 (UTC)

OK. I added a tiny little piece of the above to the article. Any more complete elaboration of group theory vs. geometry should be deferred to gp. th. Jakob.scholbach (talk) 14:57, 14 May 2008 (UTC)