Talk:Group (mathematics)

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Featured article Group (mathematics) is a featured article; it (or a previous version of it) has been identified as one of the best articles produced by the Wikipedia community. Even so, if you can update or improve it, please do so.
Main Page trophy This article appeared on Wikipedia's Main Page as Today's featured article on November 5, 2008.
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Group TFA image possible copyvio[edit]

The image being used in the TFA for the 5th for Group is currently under deletion discussion as a derivative work. As a result it should not be used (at least until this is resolved), especially on the mainpage. There are a number of other possible images to use, if you could provide suggestions and another illustration that would be wonderful. Cheers! Mostlyharmless (talk) 04:15, 4 November 2008 (UTC)

I have done so. I also asked Raul654 if he could perhaps put the snowflake image on the main page excerpt, too, since the illustration of the group elements of the dihedral group is pretty tiny and not so beautiful as a standalone. Jakob.scholbach (talk) 10:59, 4 November 2008 (UTC)
The current rubik's revenge image does also break the image rights
The RUBIK'S CUBE® in its three dimensional form and any graphic or photographic representation of it, in any configuration, coloured or uncoloured, whether it carries the RUBIK'S CUBE® name or logo, is protected by intellectual property laws throughout the world.
--Salix (talk): 07:26, 5 November 2008 (UTC)

Mathematical set[edit]

I noticed the phrase "mathematical set" in the lede. Normally I would just change it, but I thought a longer explanation might be helpful here. A set (mathematics) is a particular type of object, and there is no useful or commonly-made distinction between "mathematical sets" and "non-mathematical sets". The phrase "mathematical set", in particular, does not mean "a set considered in the context of mathematics". The clearest way to establish context is simply to say, "In mathematics...". — Carl (CBM · talk) 13:31, 4 November 2008 (UTC)

I totally agree. Somebody changed this recently, perhaps in relation to the TFA thing(?). Jakob.scholbach (talk) 13:38, 4 November 2008 (UTC)
I think it was a well-intentioned copyedit, since the "In math..." phrase can look a little awkward at first. But I don't see any small change that makes it better while keeping the sense right. If the lede needs to be specially formatted for the front page, it's possible to edit the short blurb without editing the article itself. But personally I would prefer not to see the phrase "mathematical set" on our most prominent page. — Carl (CBM · talk) 13:48, 4 November 2008 (UTC)
Yes. Please have a look at Wikipedia:Today's_featured_article/November_5,_2008 to see the outcome of such a shrinking procedure. I'm personally not terribly happy with this shortened lead, but I'm not sure whether I should intervene again. Perhaps you could propose a better version to Raul654, who is in charge of this. We've to hurry, it's shown tomorrow. (I already asked Raul to replace the image there by the snowflake). Jakob.scholbach (talk) 14:09, 4 November 2008 (UTC)
It's 14:15 UTC, so that leaves 10 hours until the change (I believe). I guess I would suggest copying that text here, copyediting it on the talk page, and then copying it back after that. It may be easier to leave the content pretty much as is, and just work on phrasing. — Carl (CBM · talk) 14:17, 4 November 2008 (UTC)

Here is a draft. I don't know how to put "mathematics" etc. in the first sentence other than "In math, a group is ...". But perhaps it's even clear enough like this, that it belongs to math? Jakob.scholbach (talk) 14:29, 4 November 2008 (UTC)

Or simply "A mathematical group is a ..."? Is "climaxing with" grammatically correct? Jakob.scholbach (talk) 14:31, 4 November 2008 (UTC)

Raul's version[edit]

Symmetry of a dihedral group

A group is a set of elements together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and operation must satisfy a few conditions called group axioms, namely associativity, identity and invertibility. The ubiquity of groups in numerous areas—both within and outside mathematics—makes them a central organizing principle of contemporary mathematics. The concept of a group arose from the study of polynomial equations, starting with Évariste Galois in the 1830s. After contributions from other fields such as number theory and geometry, the group notion was generalized and firmly established around 1870. Modern group theory—a very active mathematical discipline—studies groups in their own right. To explore groups, mathematicians have devised various notions to break groups into smaller, better-understandable pieces, such as subgroups, quotient groups and simple groups. (more...)

Other proposal[edit]

Symmetries of a snowflake form a group.

A group, in mathematics, is a set together with a operation that combines any two of its elements to form a third element. To qualify as a group, the set and operation must satisfy a few conditions, called group axioms, that are familiar from number systems. The ubiquity of groups in numerous areas—both within and outside mathematics—makes them a central organizational tool in contemporary mathematics. The concept of a group arose from the study of polynomial equations, starting with Évariste Galois in the 1830s. After contributions from other fields such as number theory and geometry, the group notion was generalized and firmly established around 1870. Today, group theory is a very active mathematical discipline that studies groups in their own right. Symmetry groups are widely applied in molecular chemistry and various phsyical disciplines. (more...)

Comments on "other proposal"[edit]

I edited the proposal above. I think we should avoid using em dashes twice in the same paragraph, so I reworked one sentence to remove them. I removed the link to the classification of finite simple groups both because that sentence seemed a little awkward and because it's a pretty advanced topic for a 1-paragraph summary. I moved the "in mathematics" inside the first sentence instead of at the beginning. I integrated the elements link in the first sentence, as suggested by Tobias Beregmann — Carl (CBM · talk) 15:40, 4 November 2008 (UTC)

Yes, that's better. I like it. You are an admin, right? Do you have the permission to update the main page snippets? I don't know if Raul or anybody else holds a kind of priority over these things. If not, the easiest would be that you just post it there. I can't edit the protected page. Jakob.scholbach (talk) 15:57, 4 November 2008 (UTC)
Yes, I'm an admin. As far as I know any admin can edit the main page snippets; they usually follow Talk:Main page to look for suggestions. I'll wait a little while for any comments here, then copy over the text. — Carl (CBM · talk) 16:14, 4 November 2008 (UTC)
I copied it over there. — Carl (CBM · talk) 17:10, 4 November 2008 (UTC)

Image example[edit]

A small note, under Examples and applications: For example, the image at the right depicts elements of the fundamental group of a plane minus a point. It should be made clear which image to the right this is referring to. I initially looked at the wallpaper image, since it's the one immediately to the right of the text. Very nice article, good job! /skagedal... 07:53, 5 November 2008 (UTC)

[1] – that works, thanks! /skagedal... 10:43, 6 November 2008 (UTC)

Subsections on Order of a group etc.[edit]

I reverted recent changes by Delaszk, who introduced short subsections "Order of a group", "Order of an element". Here is why: the argument that other articles want to point to something specific does not count, especially not since there are subarticles (order (group theory)) which cover the content in more detail. If one would want to link to a specific paragraph of this article, one should use <cite id=labelxyz>the most important paragraph</cite> and [[Talk:Group (mathematics)#labelxyz]] etc. Most importantly, the subsection titles don't sum up the subsections' content, e.g. the paragraph titled "Order of a group" contains exactly one mention of the order, the rest is other material. Also, in the FAC process, questions like this (appropriate section length) are raised, in other words over-sectioning is decreasing the article quality. Please do not reinstate the subsection headers without further discussion and consensus. Jakob.scholbach (talk) 14:42, 8 November 2008 (UTC)

Error in D4 table?[edit]

Maybe I'm missing something, but "fh • r1 = fc" and "r1 • fh = fd" seem to be transposed; fh • r1 = fd while r1 • fh = fc. No matter what the grid says, it's obvious from looking at the diagrams above and potentially very confusing. (talk) 17:28, 17 November 2008 (UTC)

The table is right. You have to read "fh • r1" from right to left i.e. "apply fh after r1", as stated in the article. Jakob.scholbach (talk) 17:54, 17 November 2008 (UTC)
Doh! Shows how well I pay attention. Thanks. Sorry. —Preceding unsigned comment added by (talk) 23:51, 17 November 2008 (UTC)


I was surprised to see that the identity and inverse requirements are redundant. For identity, it suffices to assume only a*e = a; for inverse, it suffices to assume only the existence of b such that a*b = e. From these, one can deduce e*a = a and b*a = e. (Any book on group theory should prove this.) I hope someone who has the time will remove this redundancy. It also exists on the Glossary of Group Theory page. Milkycow~enwiki (talk) 11:27, 28 May 2015 (UTC)

There are actually a number of different ways of defining groups without explicitly mentioning the existence of an identity and inverses. But it's not really a good idea to use such a definition in this article, since it would make the definition look arbitrary and obscure its purpose. --Zundark (talk) 12:53, 28 May 2015 (UTC)

The Definition should come first for all articles about algebraic structures. I am standardizing these and moving the definition to the top. —Preceding unsigned comment added by (talk) 01:29, 25 April 2009 (UTC)

Please read my comment on your talk page. Also, I request you to comment at the bottom of the page, as this eases the difficulty in finding your comment. On your note, that which you assert is false. The editors who have significantly improved this articles, are precisely those who know best about the layout of articles in mathematics. It is this evidence, from which I conclude doubt about the correctness of your remark. --PST 01:45, 25 April 2009 (UTC)
Careful with WP:OWN, PST. I like 67's edit, but I think I recall there was some issue with starting with the definition. Can we focus on that (I forget the exact issue) rather than on who has edited the article? Cheers, Ben (talk) 01:52, 25 April 2009 (UTC)
Those who edited this article have neglected every other algebraic structure. The definitions should be standardized at the least. I feel the definition fits best at the top since Wikipedia is both a reference as well as a tool for learning. (talk) 01:55, 25 April 2009 (UTC)
I'm 67 and I just registered. I think the definition should come first. Negi(afk) (talk) 02:12, 25 April 2009 (UTC)
Welcome to Wikipedia. Like I said above I don't see anything wrong with the edit, but I guess we can wait a day or so for more comments. Cheers, Ben (talk) 02:24, 25 April 2009 (UTC)
This is a Featured Article, meaning that people worked carefully trying to make it understandable. Throwing the whole formal definition of a group into the lead will create a barrier to the less-skilled readers. Let's hear from some of the editors who worked on the featured article development, and see what they think. We are constantly being scolded by people who find WP's math articles incomprehensible. EdJohnston (talk) 02:45, 25 April 2009 (UTC)
Ed is correct. This is why the article is arranged in such a way. --C S (talk) 06:21, 25 April 2009 (UTC)
May I add that there is no question of WP:OWN as I have never edited this article before (except of course for reverting the edits made by the IP). That is not to say that I am defending my revert, but from my experience, I conclude that the IP's edit did not conform to preserving the FA quality of the article. My argument for this is that FA articles lack content, and contain a few grammatical/spelling errors in most cases. Rarely, does their layout need improving. This motivated me to revert. --PST 02:54, 25 April 2009 (UTC)
The first example is a terrible one, as it's an abelian group, which then necessitates the longish paragraph about how all groups are not abelian. I don't understand how swapping definition and illustration #1 makes any difference. The reason to move the definition up to the beginning is so we can standardize all of the related pages, namely magmas, semigroups, monoids, groups, and abelian groups. There is a natural progression in the addition of more restrictive axioms to the structure. If you don't like what I did, then you should go fix the other pages to match the way this one currently is. However, the problem you might run into is that not all of these structures have useful examples. I'm most concerned with standardizing the format so one can find the definitions easily. Negi(afk) (talk) 02:57, 25 April 2009 (UTC)
Also, Point-Set Topologist, just because an article is FA doesn't mean that its layout is perfect. The only content change I made was changing the beginning of a sentence so it fit in its new context. I maintain that the definition should go first simply for reference purposes.Negi(afk) (talk) 03:00, 25 April 2009 (UTC)

OK. I wish to discuss the issue in greater depth including an argument supporting my assertions. To begin, let me welcome the IP to Wikipedia, and stress that by no means are my reverts a way to discourage his motivation to edit. I strongly believe that he may become a valuable editor one day, but in my view, he currently lacks an understanding of the FA criterion. FA's are far more difficult to deal with, as opposed to other articles. It is certainly unfortunate that the IP was to have edited this article, and therefore, I note that no problems will arise should you to edit any other article in group theory. The other related point that I wish to comment upon, is that changes of the nature of your edits, made to FA's, should always be discussed. I am sure that you were unaware of this prior to editing this article, and therefore in no way criticize you for your edits. But it is important to understand that a large part of the Wikipedia community see no significant problems with this article, let alone its layout, so it is with this evidence from which I conclude that your edits should have been discussed.

Now, the reason for the motivating example of the integers, lies in the fact that this is the one set that is familiar to a large proportion of laymen. Most simple examples of groups tend to be abelian, and this is why the integers have been chosen. On the same note, it is important to understand that many laymen cannot imagine the mere existence of a non-abelian group let alone understand what it is. And this comment draws back to the same point - this article is intended for everyone, not just experienced mathematicians like you, as it is felt that the concept of a group is one that should be understandable to all.

On yet another similar note, I am proud to say that it is enjoyable for me to read GA's and FA's irrespective of their standard. I find it quite interesting to see how a simple topic, such as the number pi or 0.999... is presented to laymen, though keeping in mind the broader mathematical community. I am sure that many people feel this way when it comes to reading Wikipedia articles, and I, for one, am in no rush to find out the definition of a group, before reading the finest discussion prior to it. It is often clear to me that professional mathematicians have played a role in presenting such an article, and in particular, I respect User:Jakob.scholbach for this. This lengthy comment draws, yet again, to the same point - we intend to motivate the concept of a group, and draw interest to the minds of laymen and mathematicians alike. I feel strongly, that to attract interest to this field, is far more important that hurrying to include the definition. It is with certainty that I assert that other mathematicians feel the same way.

To conclude, I greatly appreciate your desire to improve an article like this one, but contributing to Wikipedia takes more experience obtained than that just from mathematics. One needs to be able to present the topic in a manner that appeals to all - both mathematicians and laymen, and it is this skill that takes time to achieve. I feel, that even in my few months or so experience of editing Wikipedia, I have not yet achieved it. --PST 03:18, 25 April 2009 (UTC)

I second the remarks by PST above. I think readability should be a greater concern than some grand scheme of standardization. Someone that actually is just looking for the definition can easily find it by section heading in the table of contents. I don't see what the problem is with finding the definition. --C S (talk) 06:21, 25 April 2009 (UTC)
Did you even see what the revision would have looked like? I feel it was just as readable but organized more logically. I mean, literally, the only thing I did was split the "Illustrations and Definition" section into two sections, and then transposed the definition and the first example. It's just nitpicking. The reason why one would want to put the definition on top is so that anyone reading through different algebraic structures can see precisely how each one is constructed from the previous one. Negi(afk) (talk) 08:27, 25 April 2009 (UTC)
Did you even see what the revision would have looked like? Yes of course. I looked through your edits on other articles to see what you were doing regarding standardization. Why do you assume I would voice an opinion on your edits without looking at them? I would say as far as exposition goes, your edits are not merely a matter of sectioning. You removed an explanatory paragraph and added a sentence on monoid. On abelian group, I find your listing of "abelian group axioms" rather than first explaining in prose that an abelian group is a commutative group a definite regression in readability. --C S (talk) 15:05, 25 April 2009 (UTC)
Anyone who is confused by that completely understandable defition can click the word COMMUTATIVE. It's not that difficult. I maintain that that definition is better than what was there before.Negi(afk) (talk) 17:24, 25 April 2009 (UTC)
If that's your attitude, it's not surprising you think what you wrote is better (even given the natural bias to think one's edit is always better). To make a math article readable as possible, it is necessary to first understand one's audience(s) and secondly, to try to, as much as possible, explain what you can without making the reader click on a wikilink to go to a different article that treats the topic in greater generality. --C S (talk) 17:39, 25 April 2009 (UTC)

Let me add that I greatly appreciate the fact that you have not reverted my revert, but rather discussed it on the talk page. Few very Wikipedians tend to do this, let alone IP's. --PST 03:21, 25 April 2009 (UTC)

I have to agree with PST in this case. We're in the unfortunate position of having to serve multiple audiences simultaneously without any ability to adapt to the reader - as a reference work for mathematicians, who are already familiar with the underlying concepts, it makes perfect sense to have a very short article leading up with the definition. However, as an article intended to teach people about groups who have never seen them before, it's critical in this and any topic about an abstraction to describe motivating concrete examples. After all, historically it's not like someone sat down and defined groups one day - the definition itself is derived from motivating examples. It's often a good idea for individual learning to parallel history. In fact, if it were up to me, I would lead with two concrete examples, so that the definition could "pick out the common bits" - at the moment, it's not sufficiently motivated why some properties of the integers are selected and not others. Dcoetzee 08:35, 25 April 2009 (UTC)

I was surprised when I first read this article to see that it began with an illustration, not a definition. However, I think it can be justified; the illustration is not the very first thing in the article -- the lead is. By the time we reach the end of the lead I think it is OK to digress to a brief illustration; there is no doubt this is a teaching approach instead of a purely reference approach, but the article is going to educate as well as inform. As for the issue of whether the illustration is a good one, given that it's abelian, I think the familiarity of the example makes it a very useful example, and finding a non-abelian example would not avoid the need to explain the distinction later. Negi's edits aren't unreasonable, but there is some justification for the previous state, and I think Negi is wrong to assert so definitely that the definition has to come first. Standardization should serve the reader, and in this case there's a reasonable argument it would not. Mike Christie (talk) 10:00, 25 April 2009 (UTC)

It's good to see the article being under scrutiny of several people. As one of the editors involved in the article for a while I would like to explain my stance of the article organization regarding the issues brought up above.
As pointed out already by several people, the article is for a wide audience, say starting with an interested 12 or 14 year old kid, moving along to highschool and college audience, ending up with grad students (the latter for few parts of the article). Therefore, compromises between the expectations of the groups of readers have to be made.
I'm open to discussion but I'm fairly sure that an example first will serve most readers better than the abstract definition. This layout was done with the idea in mind that probably any reader will know the integers, and can kind of look up the meaning of every axiom presented in the actual definition in this familiar case (it might be interesting to have a look at the discussion for the FA promotion and take into account comments by (apparently) lay readers there). It is a matter of fact that many lay readers will have a hard time to digest even the meaning of an abstract symbolism like "a • b". You would have to explain what "•" means; if you don't have an example at hand, this has to be done abstractly, which is, IMO, leading astray.
Finally, for choosing Z as first example: firstly, again, you have to take something people know, you can't just start out with GL(F_q) or whatever, secondly, Z does play a fundamental role, at least in abelian groups. Negi, what other group would you prefer in place of Z? Jakob.scholbach (talk) 10:22, 25 April 2009 (UTC)
GL2(R) should be understandable to any 14 year old who knows how to multiply matrices. It's a much better example because it's not commutative. If you phrase it in terms of determinants, anyone can understand it. It would be something like: GL2(R) the set of real-valued 2x2 invertible matrices. They're invertible iff they have nonzero determinant and determinant is easy to characterize like det(AB)=det(A)det(B) = 0 iff det A or detB is 0 and therefore not invertible. The group structure is super easy to verify because linear maps are associative, the identity is obviously there, etc. In my experience, students frequently forget that not all groups are abelian. I think it's important to emphasize that they're not in the absolute first example.
Then again, That's not my main point of contention. I feel that the abstract definition should come first. If you insist on doing it the way you're doing it, could we at least have a consensus to fix the other articles for algebraic structures and standardize their definitions and formats? I guess if everyone is dead-set against having the definition first, I'll try to start working on fixing and standardizing the pages on magmas, semigroups, and monoids to fit with the conventions you guys choose. Negi(afk) (talk) 12:22, 25 April 2009 (UTC)
14 year old & knowing about matrices? Determinants? If you want, write a section in your sandbox about GL2(R) that mentions every single bit and motivate, state and prove(!) the group properties in a way understandable to somebody knowing nothing about algebra. (That's what we do with Z and D_4). You will see how long the section will grow and how far it carries you away. With all due respect, what you suggest looks undoable. The dihedral group following the definition right away does give a non-commutative group, so the article does convey the picture that there are nonc. groups.
About "standardizing": WP is a kind of a jungle. You may be able to pave some crossroads, but standardizing the articles should not be the principal aim. Instead, it should be, IMO, oriented towards improving every single article individually. Probably, magma will be oriented towards a more specialised audience, so it may be better to follow a different scheme of organization. Imposing overly rigid schemes may not be helpful in all cases. Jakob.scholbach (talk) 12:40, 25 April 2009 (UTC)
Jakob, I understand that the principal aim is not standardizing, but for those of use who use Wikipedia as a reference site, it would really be nice if some of the articles had a standard so one could find information quickly without having to sift through the hundreds of different possible permutations for content arrangement. If you maintain that a motivating example is necessary to be placed before the definition,why shouldn't we have that hold for magma, semigroup, monoid too? Then at least when you're done reading about groups, you can learn about monoids in the same framework as groups since their format is standardized. Negi(afk) (talk) 17:25, 25 April 2009 (UTC)
Yet again you claim that the organization of this article is so confusing it undermines Wikipedia as a reference site. I see no evidence for this. You yourself see no problem with making a reader confused by an article like abelian group parse through a list of "abelian group axioms" to make him/her click on "commutative" at the end of the passage on to understand what an abelian group is. So I don't see how you can see a problem in skipping down one subsection or looking at the table of contents first to see the definition of a group. Your arguments would be more persuasive if you didn't resort to hyperbole like claiming hundreds of permutations simply because there are some articles that have an illustrative example before the definition. --C S (talk) 17:46, 25 April 2009 (UTC)
Those are the axioms for an abelian group. If you want to explain why they shouldn't be stated as such, go ahead. I have no problem with someone adding a motivating example beforehand, but the explanation that was there before repeated all of the ideas from group and commutativity. I believe my choices there made excellent sense —Preceding unsigned comment added by Negi(afk) (talkcontribs) 17:56, 25 April 2009 (UTC)
There once was a time when matrices were taught to 14 year-olds, in the UK that time is long gone. One of the primary functions of this page is for many people from different disciplines to get a basic understanding of groups. Introducing any concept which we cannot be sure of their knowing is going to make that aim harder. Indeed the only structure many people are going be familiar with is the integers. --Salix (talk): 17:02, 25 April 2009 (UTC)
I should say I'm not at all against putting examples to the magma (algebra) article, which would improve it, I guess. I'm just saying that it is difficult to find schemes along which many articles can be taylored. Jakob.scholbach (talk) 17:51, 25 April 2009 (UTC)

I appreciate it, User:Negi(afk), that you are commenting with a desire to improve the article. Your main argument seems to be, that you desire other articles in the contexf of algebra, to follow the same structure as this one. There are two possibilities to achieve this goal. The first, is to modify this article, to obtain a structure similar to that of the other articles. The second, is to modify the other articles on algebra to obtain the structure of this article. Let me add, that you have chosen the first, which by no means enables us to be against you. It is certain that you have modified this article with unawareness that it was an FA (or unawareness of the meaning of "FA"). Although the second seems feasible if the first is given not to be, the second is clearly the most difficult option to take. To achieve the second option, one must make every other article on algebra a FA; the task of which is spoken, being far too difficult. Furthermore, the other articles are intended for audiences distinct to that of this article - in the other articles, we may need to assume the existence of a computer scientist reading the article or even physicists, with not as much weight given towards the assumption that laymen may read the article, in which case, implies that it would not be correct to include a "motivation".

On the other point, you assert that a 14 year-old should be able to understand the concept of a matrix. I do not doubt this, and in fact support you in this assertion. The concept of a matrix is not at all too difficult enough for a 14 year-old to grasp. However, you must understand the essential wording in that which I have asserted. School systems are no longer as they used to be, and I can assert with certainty that the concept of a matrix is sometimes introduced as late as to students of 17 years of age (or elders!). Furthermore, it is not the concept of a matrix that might lead to difficulties but rather the theory. Although it seems trivial to understand that a given set of matrices (under certain restrictions) actually forms a group, it is suprisingly non-trivial to see for many students, even though they may fully understand the concept. If a matrix group were to be given as the first example, this would not only suggest that the concept of a group was derived from the concept of a matrix, but would also significantly reduce the possible audience who may understand this article.

Lastly, I would encourage you to improve the other articles in algebra. For example, there exist many stub articles, the format and structure of which, will be decided by he/she who first expands it. The page at which you may find articles of this nature is at this page. I am certain that you will not encounter any problems, should you edit these articles (unless of course, if you make a mathematical error). --PST 02:13, 26 April 2009 (UTC)

M. Artin introduces groups with GLnNegi(afk) (talk) 15:27, 26 April 2009 (UTC)
I have not heard of the author referenced, but I can assert with certainty that he/she assumes the reader to have a prior knowledge of linear algebra. In basic ad important articles of Wikipedia (group (mathematics) is certainly basic and important), it is tried to present the topic in a manner that is accessible to the maximum audience possible. Readers of mathematics textbooks are most likely those who intend to become professional mathematicians, or those professionals who need the knowledge of a particular concept to apply in their field. Whatever the case, textbooks will therefore assume some prior knowledge on the reader's part. Contrary to this, Wikipedia is intended for everyone (as it is an encyclopedia) and it is therefore necessary to assume minimum knowledge of the reader. --PST 00:42, 27 April 2009 (UTC)
Negi(afk) is referring to Michael Artin's "Algebra". It is indeed true that one of the first examples Artin gives of a group is GL(n) (in chapter 2). However, he spends the entire first chapter of the book developing the basic theory of matrices.
I would also like to add that I disagree with Negi(afk)'s point of view. To assert that the definition should come first is to make two incorrect assumptions. The first is that the goal of wikipedia is to serve as a reference for mathematicians. The second is that the axiomatic definition of a mathematical object is somehow the most important thing about it. I am certainly all about having definitions of objects in the article (and shudder when I read articles that don't have them), however they are not generally the essence of the object, nor are they usually terribly useful in understanding the object, its uses, its importance, etc. Furthermore, when a definition is present in an article i have never had trouble finding it as it is always in a section called "Definition" or "Formal definition". RobHar (talk) 02:13, 27 April 2009 (UTC)


If one put together a bunch of group theory lemmas, is there room on Wikipedia for a page of "Small group theory theorems with proofs"? Negi(afk) (talk) 20:33, 28 April 2009 (UTC)

I'm not sure exactly what you mean. But we are not trying to replace textbooks, so we don't want pages that consist of nothing but minor results and proofs. — Carl (CBM · talk) 20:45, 28 April 2009 (UTC)
We have glossary of group theory and elementary group theory. As Carl says the latter should actually not exist per WP guidelines, but it does... Jakob.scholbach (talk) 20:57, 28 April 2009 (UTC)
Negi, you may be interested in [2]. Ozob (talk) 22:48, 28 April 2009 (UTC)
I agree. Furthermore, many of the important theorems in mathematics already correspond to an article in Wikipedia. Basic results are generally not considered as high in importance as that which is included in Wikipedia. On the other hand, the sole purpose of Wikibooks is to include theorems of all kind, and therefore, your assertion suggestions that it would be great if you were to have a look at Wikibooks, as User:Ozob comments. --PST 00:12, 29 April 2009 (UTC)

Reading to help bring it down from +- 90k[edit] (talk) 00:12, 14 May 2009 (UTC)


The generalisation section says: For example, if the requirement that every element has an inverse is eliminated, the resulting algebraic structure is called a monoid. That seems clear enough, but when I look at the accompanying table I see that groups and monoids differ by "division". Do we mean "inverse" instead? Whatever we mean the table needs to be clearer and match the text. --Michael C. Price talk 08:39, 16 May 2009 (UTC)

I've changed the table to clarify this (see Template talk:Group-like structures#'Divisibility' changed to 'invertibility') and have renamed and rewritten the 'Division' section as well: Group (mathematics)#Invertibility of the group operation. It only took seven years! splintax (talk) 10:46, 28 September 2016 (UTC)

Looking for a special name[edit]

Hi, I would like to know, if there is a certain special name for the set G of e.g. a group (G, •) or the set of algebraic structures in general?! I hope you can help me. Thanks :) --WissensDürster (talk) 19:10, 14 July 2009 (UTC)

PS: If there is such an name please add an interwiki-link to the german arictle de:Trägermenge, I watch this site more often. Greetz --WissensDürster (talk) 19:40, 14 July 2009 (UTC)
The standard name in my experience is underlying set, but here it redirects to forgetful functor. There's always been a confusion between a group (etc.) and its underlying set, with most mathematicians now referring to "a group G" without carrying the group operation along in the notation. Category theory has encouraged this by reducing the primacy of the category of sets, and emphasising morphisms over objects. Thus one only needs to speak of the underlying set of a group G (perhaps denoted |G|) if one is interested in studying functions between groups which are not group homomorphisms. That rarely happens. Even the concept of an "element of G" is perhaps better captured by the group theoretic notion of a homomorphism from Z to G, rather than the naive set theoretic meaning. Geometry guy 19:58, 14 July 2009 (UTC)

Wow, first thanks for your answer, but that explanation sounds as difficult as it does in my lectures.... i thought it would be a more trivial way for distinguishing between the name of the algebraic structure and the set where it is based on, I therefore added some examples on the german page:

  • underlining:
  • Calligraphy:
  • Fraktur:

In german we could call the structure (e.g. group) and its "Trägermenge" which means the "set that bears/makes" the structure. All in all it looks like there isnt a name for such banal things in English =) --WissensDürster (talk) 20:34, 14 July 2009 (UTC)

In model theory one would view a group as a structure, and write |G| for the set that makes up its domain. But regular mathematical English doesn't have any term other than "underlying set" that I am aware of. Part of this is likely the category-theoretic view that Geometry guy mentions: one is not really interested in groups, but in isomorphism classes of groups. — Carl (CBM · talk) 20:43, 14 July 2009 (UTC)
I've come across the name "carrier", or "carrier set" for the underlying set of an algebraic structure. I'm not sure how common it is. --Classicalecon (talk) 21:07, 14 July 2009 (UTC)
"carrier set" is one possibility i found in a leo-forum too (, google hits), apart from the intereset in groups, "carrier set" would be a direct translation for "Träger(carrier )-Menge(set)". Common or not, if there isnt an article on this lemma in English, there's no way for interwiki-linkin it ^^ --WissensDürster (talk) 13:31, 15 July 2009 (UTC)

Recent addition[edit]

Recently, Charvest added two mentions of generalizations to the article, hyperstructures and association schemes. I wonder how strongly these are connected to groups and, thus, whether they should really be here. From the subarticles (which are fairly stubby), I can see that a finite group gives rise to an association scheme. Is the same true for arbitrary groups? Secondly, what about hyperstructures? I don't know the concept, but from the article it seems to be just one of many algebraic structures, so it might be better to mention it there? What do others think? Jakob.scholbach (talk) 20:50, 21 July 2009 (UTC)

Association schemes look interesting, but they are fundamentally combinatorial and the finite group examples don't seem to be the most interesting ones (distance regular graphs are much more so); arbitrary groups would not give rise to examples without a radical change in the definition. So, interesting, but only indirectly and partially connected with groups. I see no reliable secondary source for hyperstructures: at present it appears as a generalization for generalization's sake; better not to mention it at all and possibly consider the article for deletion as unnotable/unencyclopedic. Geometry guy 22:20, 21 July 2009 (UTC)
I added link to association schemes after reading this bulletin review. A couple of quotes: "One would like to know to which extent basic group theoretic definitions and results can be generalized to scheme theory ... There is a Lagrange Theorem for schemes, there is a Homomorphism Theorem for schemes, there are two Isomorphism Theorems, there is a Jordan-H¨older Theorem for schemes; and even Sylow’s Theorem" and "The fact that, indeed, large classes of schemes arise from groups seems to be a hint that there is more behind the relationship between groups and schemes than we presently know." See also An Algebraic Approach to Association Schemes
As regards hypergroups the article is indeed very stubby. There is a sizable number of references for hypergroups: google. See also Applications of hyperstructure theory. Charvest (talk) 23:12, 21 July 2009 (UTC)
The association scheme mention may be more appropriate on the Finite group page. But that page itself is fairly short and doesn't say a great deal more than the main group article. Should Finite group be merged into the main article. ? Charvest (talk) 23:31, 21 July 2009 (UTC)
If such topics are to be considered in this article, representation theory/character theory should also be included, for they form an important part of group theory (and finite group theory). More generally, I cannot see many of the important topics in group theory - the two quoted topics (hypergroups and association schemes) are some of the last topics I think should be included. The more general subject of algebraic combinatorics should be included first as a topic, if anything. --PST 00:52, 22 July 2009 (UTC)
Count me against merging finite group into this article. Finite groups have all sorts of wonderful properties not shared by general groups. E.g., think of all the counting you can do with orbits and stabilizers of groups acting on themselves by conjugation. That sort of technique is meaningless in the general case, and it's inappropriate for this article; but it's vital for finite groups. (Where else would we get graduate algebra exam problems? :-) Ozob (talk) 18:35, 22 July 2009 (UTC)
I would also not merge the finite group and group articles. Finite group should be much expanded, and merging the two would just hinder people in doing so. If it is true that "large classes of schemes arise from groups seems to be a hint that there is more behind the relationship between groups and schemes than we presently know" I prefer removing the mention in this article. There are tons of things whose connection to groups is firmly established, and we should keep focused here. The same applies to hyperstructures. (I also wondered whether that article should be nominated for deletion.) Jakob.scholbach (talk) 20:20, 22 July 2009 (UTC)
I've reverted the mention. The hyperstructure article should be improved rather than deleted though. Charvest (talk) 21:25, 22 July 2009 (UTC)
I agree with the removal and with not merging finite groups here. I've added the association scheme article as a "see also" at finite group and added the BAMS ref to association scheme. Hyperstructures may not need to be deleted, but it certainly needs to be improved. Subject to some tolerance of temporary imperfections, no reliable secondary sources=no article. Geometry guy 22:26, 22 July 2009 (UTC)

Note on the definition[edit]

Recently, a lengthy discussion of a variant of the axioms using only left identity and left inverses is given. While I think the material is correct etc., I feel, there is no space for this in the article. Also, few references actually make that point, and if so, usually at most in the form of some beginner's exercise etc. I think we should trim down the addition to at most one sentence, possibly moving the added material to Elementary group theory. Any other opinions? Jakob.scholbach (talk) 08:21, 15 October 2009 (UTC)

Completely agree. There is no need to explicitly do what is basically a freshmen exercise in axiom manipulation. It is good to note that there exists other equivalent sets of axioms, but actually showing the equivalence is not something that should be within the scope of this article. After all, wikipedia is not an undergrad mathematics textbook. I see Gandalf has already implemented the change.(TimothyRias (talk) 09:44, 15 October 2009 (UTC))
Usually, such large edits, followed by trimming, tends to lead to lengthy disputes (I remember having reverted an IP earlier in the year which lead to the evolution of a huge thread). Therefore, I think that we should move the material to Elementary group theory as Jakob suggests above, in order to utilize the recent additions (which I think were done with good faith). On the other hand, I believe that the material should be trimmed slightly (although not to a great extent) before it is moved. --PST 11:46, 15 October 2009 (UTC)

Operator ambiguity[edit]

In the Group homomorphisms section, I think it should be emphasized that in this expresison a(gk) = a(g) • a(k). the • at the left of = is an operator of G and the • at the right of the = is an operator of H. [xissburg 00:41, 17 March 2010 (UTC)]

Done. Jakob.scholbach (talk) 13:35, 17 March 2010 (UTC)

Since this is emphasized at the top of the Group homomorphisms section, do you think it would be nice to use the same notation at a(g) * a(g) = 1H:

"For example, proving that g • g = 1G for some element g of G is equivalent to proving that a(g) • a(g) = 1H, because applying a to the first equality yields the second, and applying b to the second gives back the first." — Preceding unsigned comment added by Dagcilibili (talkcontribs) 09:08, 29 August 2011 (UTC)
Given the notation in that section, you're right, the section was wrong. I've fixed it. Ozob (talk) 11:22, 29 August 2011 (UTC)

Recent edit[edit]

I mildly disagree with the recent edit of Gandalf introducing "When acting on the set" in the lead section. I think the mathematical meaning of "action" is not known to the lay (or even non-lay) reader. Jakob.scholbach (talk) 18:39, 25 June 2010 (UTC)

While the last three edits to the article were done in good faith, I think that they have decreased the overall clarity of expression in the lead. [3] In general, much time is invested in writing the lead of a featured article, and changes to this part of the article should never be substantial without discussion. I agree that "When acting on the set" is imprecise, for a binary operation acts on ordered pairs of elements rather than on a single element, and perhaps more importantly as Jakob point out, the statement, even in its precise form, would not be readily understood by the lay reader. PST 13:15, 26 June 2010 (UTC)

Also recently, Jrtayloriv removed

"While these are familiar from many mathematical structures, such as number systems—for example, the integers endowed with the addition operation form a group—the formulation of the axioms is detached from the concrete nature of the group and its operation. This allows one to handle entities of very different mathematical origins in a flexible way, while retaining essential structural aspects of many objects in abstract algebra and beyond."

Jrtayloriv got reverted and re-reverted. I am inclined to revert his removal, since the sentence is an important piece of information. Thoughts? Jakob.scholbach (talk) 21:55, 27 June 2010 (UTC)

I agree. PST 02:39, 28 June 2010 (UTC)


The notation in this article is inconsistent, I think. For example, the identity element of a group is variously referenced as e, id, 1G, and 1 throughout the article. Sometimes we see gh; sometimes it's g · h. I understand that these variants have different functions in different places, but they seem to be used indiscriminately at times in this article. The different notations need to be explained more fully, and their usage requires review. Most of my knowledge of group theory comes from Wikipedia itself, so I don't feel comfortable making such changes myself. Thanks. —Anonymous DissidentTalk 11:16, 24 August 2010 (UTC)

I've fixed a few abnormalities. But in most cases the different notations mean different things. For example, in this article id denotes the identity operation on the square, i.e. the operation that leaves the square invariant. In the example, it is shown that this element satisfies the axiom for the identity element of the group D4, calling this 1 would be confusing. The only other case of using a different symbol for the identity other than 1 is in the proof of the uniqueness of the identity and inverse. In this case, this notation is needed because we need to show that e = e'. (That statement doesn't really work with a notation 1 and 1').TimothyRias (talk) 11:52, 24 August 2010 (UTC)
Thanks for the clarification. I fixed a few abnormalities yesterday, but thought it best to raise the issue here. The reason each variant is used in each context still needs to be explained, in cases where it has not been already. —Anonymous DissidentTalk 12:28, 24 August 2010 (UTC)

Should articles be readable by non-mathematicians?[edit]

Recently, on the talk page of the article ring (mathematics), there was discussion to the effect that Wikipedia policy requires that at least the first paragraph of articles be readable by non-specialists. I did some work here, and on ring and field (mathematics) with that goal in mind. I also think there should be some uniformity in these three articles. Do you disagree? Rick Norwood (talk) 12:29, 27 August 2010 (UTC)

While getting this article ready for FA, a lot of attention has been spent on getting the article as accessible as possible for a general audience. In the course of the process the article has been reviewed by a number of "lay" readers, who found the article hard but accessible. If there are specific concerns about the accessibility of the lead I would be happy to here them. If there are any problems, then a complete rewrite of the first paragraph might not be the best solution as it is likely to be a step ahead one front and a step backwards on five other fronts.
As for uniformity between the group, ring and field articles: I don't think that is necessary at all. In particular, I think it is contra productive in terms of accessibility to mention rings and fields at all in the lead of this article.TimothyRias (talk) 12:49, 27 August 2010 (UTC)
Read WP:LEAD where these issues are explained well and in detail. An opening paragraph is generally supposed to start with the definition of the subject of the article. That definition is typically phrased in fairly general language, but not so general as to sacrifice its substance and correctness. There are many types of WP articles addressed at different classes of readers. It is certainly not desirable to have the situation where an 8-th grader is able to read and understand every single article on Wikipedia or even every lead section of every article on Wikipedia. We have lots of highly technical articles that are addressed at more narrow and specialized groups of readers, and such articles are still extremely useful. The more general the subject of an article is the more accessible the lead needs to be, but again that certainly does not mean sacrificing substance. This particular article, Group (mathematics) is a good example of what a WP article needs to be (and it is deservedly rated as FA class), even though I am pretty sure that if you take an average high school graduate in the U.S., they'll probably have trouble getting through the opening paragraph - although maybe that is more of a commentary on the general sad state of affairs in the U.S. educational system. I have looked at your edit to Field (mathematics), sotored here[4]. I must say, and I mean it as gently as I could, the lead paragraph that you wrote is pretty horrible, both stylistically and mathematically and you were rightly reverted by another user. For example, "Three important classes of numbers considered in abstract algebra are groups, rings, and fields" is mathematically incorrect (groups, rings and fields are not classes of numbers and need not have anything to do with numbers). Basically you tried to distill the notion of a field down to a fifth-grader level - it is neither possible nor desirable to do that. Nsk92 (talk) 13:11, 27 August 2010 (UTC)
I think you underestimate the level of knowledge necessary to read even the first sentence of the article. I don't think anyone can understand that sentence who has not had at least an undergraduate course in abstract algebra. I suggest you run it past, say, a Ph.D. in English. However, I yield to what is clearly a consensus. Rick Norwood (talk) 13:20, 27 August 2010 (UTC)
Is a grad student in 18th century British literature also OK?TimothyRias (talk) 13:58, 27 August 2010 (UTC)
I understood the entire article, and I haven't finished even high school mathematics yet. I don't attribute that to unnatural precociousness – I just found the article very clear and digestible. Concepts are introduced carefully and in stages, so anyone with good concentration should be able to comprehend this material. —Anonymous DissidentTalk 14:20, 4 September 2010 (UTC)

I agree with everyone else that the lede (at least) should be written with non-experts in mind. But the lede needs to focus directly on the topic at hand, and it has severe length restrictions. So I think that the sort of discussion given by the first paragraph of this version [5] should generally be avoided in lede sections. Introductory material like that fits better in lower sections of the article. I do think that there should be links to the articles on rings and fields somewhere in this article, just not at the beginning of the lede. The general point about abstract algebra is already made in the first paragraph of the current version. — Carl (CBM · talk) 13:31, 27 August 2010 (UTC)

So, let me ask TimothyRias. What brought you to this article? Have you read much mathematics prior to coming here? Do you understand the first sentence? Does the article give you a clear idea of what a group is? Rick Norwood (talk) 11:46, 28 August 2010 (UTC)
I'm the wrong person to ask since I have a graduate degree in mathematics. But please review the FA candidacy comments linked at the top of the page. In particular, the comments by user:Awadewit.TimothyRias (talk) 16:48, 28 August 2010 (UTC)

I read Awadewit's comments, which were obviously a big help in making the article a FA. If only every editor had a non-expert willing to spend that much time and effort. Rick Norwood (talk) 12:21, 29 August 2010 (UTC)

Order of operation inconsistent?[edit]

In the symmetry group example (sec. 1.3) the article says that to perform "first a then b" one writes "b•a". But when discussing the associativity axiom, the article represents "a then b then c" with "a•b•c". Shouldn't this be "c•b•a"? Similarly, in describing "(a•b)•c" the article says: "composing first a after b, and c to the result a • b thereof". Seems to me these words are actually describing c•(a•b). (talk) 15:46, 3 September 2010 (UTC)

There's a distinction between multiplying group elements (which we can think of as happening left to right, but which is arbitrary due to the associativity axiom) and applying the group elements to the square (which in this example we defined to happen right to left, and which is certainly not arbitrary). I've edited the section so that it is hopefully clearer. Please take a look. Ozob (talk) 12:47, 4 September 2010 (UTC)

This is much clearer. Thanks! (talk) 02:55, 7 September 2010 (UTC)

Consistency in notation[edit]

The one real problem with this article is that the notation is inconsistent. I know that for trained mathematicians, this is rarely a problem, because they've all seen the various equivalent notations (e.g the various representations of the group identity or of the group operation), but for a layperson or student, this can be very confusing. Someone with a little more understanding of Algebra than I ought to go through the article and standardize all the notation to make it more readable. —Preceding unsigned comment added by (talk) 00:15, 10 September 2010 (UTC)

The notation in the article is standardized. When talking about abstract groups the multiplication is always "•" (except when discussing group homomorphisms in which case there is a second abstract group with a different operation "*", and the identity is always "1G" or "1" (dependent on whether the group has been given a name).
Explicit examples use the customary notation for their operation/identity. This needed to be able to discuss whether these operation/identity satisfy the group axioms. (Also it would be very confusing to denote the sum operation on the integers as "•" and the corresponding identity (zero) as "1".)TimothyRias (talk) 06:29, 10 September 2010 (UTC)

Combine vs. Compose[edit]

I propose to change the word 'combine' in the first sentence to 'compose'. The latter sounds to me less implicitly abelian. I am however not a native English speaker, so I decided to ask here first. What do you think? tzanko (talk) 19:10, 16 September 2010 (UTC)

I don't think "combine" sounds implicitly abelian at all, and I think it is better than "compose". Although you can compose functions or relations, that is quite a technical use of the word, and not one that the average reader would immediately understand. And "composing" elements does not sound natural. Gandalf61 (talk) 19:27, 16 September 2010 (UTC)

Incorrect explanation of symmetry inverse?[edit]


"The rotations r3 and r1 are each other's inverse, because rotating one way and then by the same angle the other way leaves the square unchanged"

The explanation here implies that the direction of rotation is changed (implying a transformation not in the set), rather than pointing out that the two inverse rotations add up to a complete rotation which is the same as the id...

Rpicolet (talk) 07:46, 24 September 2010 (UTC)

Good catch. This phrase is a left over from an earlier version of the example, when r3 was rotation of 90 degrees in the opposite direction. I'll fix it. TimothyRias (talk) 09:02, 24 September 2010 (UTC)

Group-Like Structures[edit]

The table of Group-Like Structures seemed flawed, since Quasigroup and Magma are identical. Probably the inverse no on Quasigroup should be changed to a yes.

Jfdavis (talk) 18:18, 6 February 2011 (UTC)

A quasigroup without an identity element can't have inverses. The problem you point out is really because the table doesn't ask the right questions to distinguish between quasigroups and magmas. Adding a "Cancellation" column would provide a distinction: quasigroups have cancellation, magmas don't. --Zundark (talk) 18:45, 6 February 2011 (UTC)
Zundark, you beat me a by few seconds...
According to our article on quasi-groups, in quasi-groups multiplication is right and left divisible, but the two don't need to be equal. I don't have a reference at hand, but if this is indeed right, it does not qualify as an inverse in the usual sense, since they don't need to have a neutral element and since the two divisions don't in general agree. Jakob.scholbach (talk) 18:47, 6 February 2011 (UTC)
Or, one could replace the column title "Inverses" with "Division". Since "Identity" + "Division" implies "(unique) inverses" anyway, perhaps division is the right concept to include in this table. RobHar (talk) 18:51, 6 February 2011 (UTC)

Operations as elements[edit]

I find the second example difficult to read. As a complete novice, having read the previous passages, I look for elements and an operation. The circumstance that the elements in this group are operations (on geometric objects ?) and that the defining operation of the group is mentioned very late, makes it difficult to identify the elements and the operation of the group. It gets especially confusing when stating that the element identity operation is mentioned. This is the identity element of the group, but not the operation of the group. Maybe it would be easier if those identities were specified at the outset, something like: The possible symmetry operations on a square can be seen as the elements of a group, where the operation of the group is the combination of pairs of symmetry operations. --Ettrig (talk) 07:34, 18 May 2011 (UTC)

I've tried to clarify this a bit. Does this help you?TR 08:08, 18 May 2011 (UTC)
Yes, in my eyes this is much better. Maybe it's just that I'm getting used to the stuff  ;-) --Ettrig (talk) 11:10, 18 May 2011 (UTC)
I recently read this article through for the first time (as an intro to Groups and Category theory, thank you!) and discovered the same issue. I agree that Ettrig's recommendation: TimothyRias's change helped; however, I would suggest being even more explicit. I added the following bold text: The possible symmetry operations on a square can be seen as the elements of the underlying set of a group, where the operation of the group is the combination of pairs of symmetry operations. This change was then rolled back by Pmanderson with the comment (The rigid mappings from a square to itself are a group.) However, as stated in the paragraph preceding the introduction to the D4 group, what is actually meant is "a subset of the underlying set G of the group (G, •)." I think that given the context of having just described how to be more precise, the article should go ahead and use that precision. To me, at least, it wasn't clear that the D4 operations were the elements of the underlying set until much further in (I thought I was being shown a group with 8 operations, which was clearly contradictory to the definition). Duane Johnson (talk) 01:25, 29 May 2011 (UTC)
Well, I agree with you, but I think I see Pmanderson's point, too. I've rewritten the start of this section. I think it's clearer than before, but I welcome comments, corrections, improvements, etc. Ozob (talk) 18:00, 29 May 2011 (UTC)

Classification of finite simple groups completed in 1983 or 2004?[edit]

There seems to be an error in this piece of text:

...which culminated with the monumental classification of finite simple groups completed in 1983.

The article "classification of finite simple groups" states:

Daniel Gorenstein announced in 1983 that the finite simple groups had all been classified, but this was premature as he had been misinformed about the proof of the classification of quasithin groups. The completed proof of the classification was announced by Aschbacher (2004) after Aschbacher and Smith published a 1221 page proof for the missing quasithin case. — Preceding unsigned comment added by (talk) 05:23, 8 September 2011 (UTC)

Well spotted. I've changed "completed" to "announced". Jowa fan (talk) 06:42, 8 September 2011 (UTC)

Note format[edit]

Currently the article uses a somewhat archaic {{cref}} and {{cnote}} templates to format the notes. This has a couple of downsides including

  • Notes are separated from the sentence they are a note to, which makes editing harder.
  • The notes are not automatically numbered, leading to the need to use labels like aa for some notes that were added at a later date.

I suggest that the notes be switched to using the

<ref group=note>Content of the note.</ref>


Since this is a quite involved change, I wanted to discuss before starting. Any objections? TR 08:19, 8 September 2011 (UTC)

I'm not familiar with "group=note". Will this keep content footnotes separate from citation footnotes? If so it sounds like a good idea. Ozob (talk) 10:36, 8 September 2011 (UTC)
Yes, the "group=note"refs will be displayed separately (by calling {{reflist|group=note}}) from the normal refs. See the speed of light article for an example where this is used.TR 11:24, 8 September 2011 (UTC)
I took a look at speed of light. That's an excellent system! We should switch. Ozob (talk) 22:48, 8 September 2011 (UTC)
I'd be in favour of this. Apologies for my previous mildly WP:POINTy edit to the page: the difficulty of inserting a new footnote didn't put me in a good mood. Jowa fan (talk) 11:07, 8 September 2011 (UTC)


Shouldn't this page be called "introduction to group theory", since, as it says at the top of the article, at heart it's a layman-friendly version of the group theory article? This would bring it in line with introduction to special relativity etc. — Preceding unsigned comment added by (talk) 15:09, 1 June 2012 (UTC)

In short, no. The name is fine. (The article covers the basic properties of the mathematical objects known as groups. The group theory article more broadly covers the area of mathematical research concerning these objects.)TR 09:15, 4 June 2012 (UTC)
+1 to TimothyRias' conclusion and reasons. Rschwieb (talk) 12:08, 4 June 2012 (UTC)

Definition - identity element symbol - e vs. e_G[edit]

In the definition section, the identity element is first introduced as '' (in subsection 'Identity element'), then later used as '' (in subsection 'Inverse element'). Should these either be the same symbol, or the difference explained? All Clues Key (talk) 04:31, 5 September 2012 (UTC)

They should be the same symbol. I've fixed this. — Quondum 05:18, 5 September 2012 (UTC)

"Second example: a symmetry group" - clarifying the role of the square[edit]

This is an excellent example, and was extremely helpful for me to understand groups (and especially symmetry groups); but, there should be some additional comment on the role of the square in this group. i.e. this sentence is a great conclusory statement about what is being exemplified:

"The underlying set of the group is the above set of symmetry functions, and the group operation is function composition."

But I feel as though it should include mention of how 'the square' fits into the group---specifically, how you relate the group to the square in the technical language. The group, as described, is a group of its own right---but is it only a symmetry group for the set of squares? All Clues Key (talk) 16:23, 5 September 2012 (UTC)

Recent addition of modular addition[edit]

I think the recent addition of a lengthy discussion of modular addition is somewhat out of place: we already discuss cyclic groups and modular arithmetic (multiplication, mainly) later in the article. In the interest of a concise exposition, I am actually leaning towards removing this example (or at least it should be trimmed and moved to modular multiplication). Any thoughts? Jakob.scholbach (talk) 23:02, 4 March 2013 (UTC)

P.S. I think the Z/n example does not add anything that a novice reader would need to know at this point: the text underlines that there are also finite groups, which is exemplified with a more important example right afterwards. Mentioning the notion "cyclic group" is at this point unnecessary and potentially distracting. The group tables are already (and more beautifully, IMO) shown later... Jakob.scholbach (talk) 23:06, 4 March 2013 (UTC)

Continuation of the discussion on examples in Wikipedia talk:WikiProject Mathematics#Broadest possible audience[edit]

First example: It starts with One of the most familiar groups is. IMO the Rubik's cube is familiar to a much broader example. IMO, the first example should be the Rubik's cube group and should explain informally why it is a group (one may compose and reverse the movements).

I disagree. Virtually every reader of this article (i.e., unless a 6-year old child is reading this without his parents...) will know the integers. Explaining the group axioms with the integers is easy, short, and concise. Rubik's cube is nice to look at, but even precisely describing the group elements takes a bit of time. Moreover, Rubik's cube is quite similar to the symmetry group given below. Thus, replacing Z by Rubik's cube would be a huge step back, IMO. Jakob.scholbach (talk) 11:45, 5 March 2013 (UTC)
Sorry I was not clear. I did not suggest to replace the example of the integers, but to place the example of Rubik's cube before. I think that describing the elements of the group and the operations is easy: an element is a succession of elementary movements and the operation consists in making a succession of movements after the other. This is sufficiently precise to show that it is a group, and it is enough here. It may be worth to call this example "motivating example" and to say that group theory allows to show that there are exactly 12 ways to mount the cube in configurations that are not linked by a succession of movements. Why so many WP editors try to make algebra more abstract than necessary? D.Lazard (talk) 12:44, 5 March 2013 (UTC)
Rubik's cube to most people is a synonym for impenetrably complex. Showing that group theory can make it tractable would be useful, but hardly as the first example to introduce the subject.--agr (talk) 16:09, 5 March 2013 (UTC)
@D.Lazard: I think we don't make anything more abstract than necessary here. We have the group axioms, they are, well, unavoidable to state in a precise way. Explaining these axioms with the integers is about as non-abstract as this can be. Jakob.scholbach (talk) 17:52, 5 March 2013 (UTC)

Second example: modular addition: IMO it should be removed, as too technical here. Moreover the introduction of the table of operations is misplaced: Unless maybe in teaching, it is never used when dealing with groups (it may be written only for very small groups and it is difficult to recognize a particular group from its table).

I agree (see above). However, at least one group table in this article seems OK, because it can be used to graphically display subgroups and cosets etc. Jakob.scholbach (talk) 11:45, 5 March 2013 (UTC)
Modular arithmetic is not too technical here. It is commonly taught at the high school level and the clock face example is concrete and familiar. There are other ways to explain these groups as well, e.g. rotations of a regular polygon. Note that the cyclic groups are covered in this article, so we are only discussing where to place them. Right now they are first mentioned deep in the article, after subgroups, cosets and quotient groups, and there they are introduced as the Nth roots of unity. Talk about being unnecessarily technical. The cyclic groups are the simplest examples of finite groups and they are often covered early in elementary courses. The leap from the integers to a symmetry group is too big for beginners. Remember that this article is supposed to be the introductory one. As for the tables, I included them primarily to introduce the concept before the much more complicated example in the next section. As you say they are mostly used in teaching. Why is that bad in the introduction to an encyclopedia article?--agr (talk) 16:09, 5 March 2013 (UTC)
It is not bad, but in the interest of a concise article, we must be careful not to repeat ourselves (as your addition of two(!) group tables) shows.
The "leap" between Z and D_4 does not become smaller after inserting Z/n. The essence of the leap consists in realizing that the axioms of a group are so versatile: they can be used to add numbers and to compose rotations etc. Again adding numbers, but this time modulo n is not fundamentally new at this stage. Moreover, it requires a relatively lengthy digression about modular arithmetic which is not central to the notion of a group. Thus, I stand by my earlier statement that introducing this example is not improving the article. As I said, what might be done is to add the "clock" explanation and maybe picture to the section on cyclic groups. Maybe then sections 5.1.3 and 5.2 should be merged into one named "Modular arithmetic" or the like. Also, when editing FA articles like this, you might want to familiarize yourself with WP:MOS which, for example, says that one should not address the reader directly (no "Note that..."). Also watch out for typos ("later"). Jakob.scholbach (talk) 17:49, 5 March 2013 (UTC)
Calling the integers a group is just giving a fancy name to properties everyone is familiar with. The first leap is realizing that there are many other systems that obey the group axioms. In my original edits, I included the set of even numbers under addition as the next example. Sticking with numbers but operating on them in a different way, e.g Z/nZ, is a useful intermediate step before getting to symmetry operations as group elements, which is a much bigger leap.--agr (talk) 18:19, 6 March 2013 (UTC)
Well, I disagree with you. We have little space, so we need to pick examples that are as helpful and prototypical as possible. Z/n is clearly not prototypical. Z is also not prototypical, but helpful in that it lets us explain the axioms with something that every reader already knows. A non-abelian symmetry group is already more prototypical. Other editors have argued that a finite matrix group such as SL_2(F_p) is the most prototypical example, but this is too far away from the audience. Jakob.scholbach (talk) 09:17, 7 March 2013 (UTC)
On what basis do you say we have little space? This is an electronic document. We can certainly afford a paragraph or two if it helps our readers. I have to focus on another project so I will have to leave this article for now. I encourage you to take to heart D.Lazard's comments on the symmetry group presentation. Cheers.--agr (talk) 12:50, 8 March 2013 (UTC)
I agree with Jacob on the addition of Z/nZ. The clock explanation is a definitely a good one that should be added to the article at the appropriate point, but I don't think that the example of a finite cyclic group is especially clarifying to someone who is willing to spend time to learn from our article what a group is. The example of a nontrivial symmetry group is imminently more suitable for this, as I have already said at WT:WPM. Jacob's comment that even a simple cyclic group involves a lengthy digression on modular addition clinches this for me. Perhaps this is what Professor Lazard means as well when he says that it's too technical. Sławomir Biały (talk) 22:55, 5 March 2013 (UTC)
Modular addition is too technical, but presenting the same groups as Nth roots of unity, as is done later in the article is not???--agr (talk) 18:19, 6 March 2013 (UTC)
If you read my reply from the beginning to the end, then you will see that I agree that our presentation of cyclic groups could be improved. I just don't think that it needs to be one of the first things the article presents. Sławomir Biały (talk) 22:07, 6 March 2013 (UTC)
I'm sorry I misunderstood you. If we can agree on improving the example of the cyclic groups, that would be a good place to start making this article more accessible. We can discuss placement later. Again, there is no need to introduce modular arithmetic to present these groups. The clock example can lead into the symmetries of a regular polygon, for instance. I'd note though that this article already uses modular arithmetic, with nothing more than a link to its article, in the example on multiplication mod p, which precedes the cyclic groups section. --agr (talk) 00:58, 7 March 2013 (UTC)
(unindent) I have moved and integrated the example Z/n in the text. I have removed the group tables since they don't give any particular information other than what is already obvious from the definition. (The second group table of units in Z/8 was particularly little connected to the text.) Jakob.scholbach (talk) 09:12, 7 March 2013 (UTC)

Third example: a symmetry group: This is a good example but awfully presented. It should be named Third example: the symmetries of the square. The tables of operation should be removed as not useful for showing that it is a group. It is a strange idea to present the square as a labelled graph instead of a piece of surface (tile) with an asymmetric drawing on it, as it is usually done. This allows to present the elements of the group as the possible movements of the tile and makes evident that they may be composed and reverted. Presented in this way this example allows also to introduce the notion of subgroup (movements that do not return the tile). Note that the article refers to symmetries as functions on the corners of the tile, while "movement" refers to isometries of the whole tile, which seems more natural for a broad audience.

There may be many ways of depicting the group elements. I don't see anything bad in the way it is done here. Depicting the corners of the square with numbers is a quick way of telling "what goes where". Note that we don't discuss movements of any tiling here. The symmetry group of a (periodic) tiling you refer to is much bigger. In the interest of a finite group that is useful to explain further notions with, the D_4 group is just fine. Jakob.scholbach (talk) 11:45, 5 March 2013 (UTC)

Group action: It is strange that this article does not say anything about the fundamental notion of group action. Galois introduced groups as the permutations acting on the roots of a polynomial. At the origin, the additive groups of vector spaces were viewed as groups of translation, and every symmetry group is defined by its action.D.Lazard (talk) 10:16, 5 March 2013 (UTC)

This is mentioned in the section on symmetry groups. In general (and this is not meant to be an offense!) I invite everyone trying to improve the article to read it first entirely. When writing this article it was my aim to choose examples/explanations etc. that can be picked up later in the article (where appropriate). Thus, the choice of this and that may (even if I partly disagree with you above) seem strange, but hopefully become better motivated when reading on. Jakob.scholbach (talk) 11:45, 5 March 2013 (UTC)

Have any of you tried showing this article to someone who does not have a strong background in mathematics? I have. It's hopeless. I'm trying to make it better. I haven't even gotten to the article's intro. It is far more dense and jargon-filled than the intro to group theory, which is supposed to be the advanced article. As I said in the WikiProject discussion, this is a perfect example of a math article that could and should be readable by a layperson. --agr (talk) 16:21, 5 March 2013 (UTC)

Yes, of course I did. Specifically, for the FA candidacy, a non-math reviewer read the article and said she learned something. I repeatedly and specifically got the feedback that this article is unusually well-readable. (Please have a look in the FAC archive and the talk pages back then, if you consider this statement as self-advertisement.) I think you may be missing the point that to read the entire article may take some time (in terms of sitting down and thinking about the content). All of us, who now understand what a group is, needed some time to get to this point. (Remember?!) I know no magic solution getting around this necessary investment of time and effort. Jakob.scholbach (talk) 17:49, 5 March 2013 (UTC)
See e.g. the comment of Anonymous Dissident above. Jakob.scholbach (talk) 17:56, 5 March 2013 (UTC)
I have great respect for Anonymous Dissident, but his statement above that he learned most of the group theory he knows from Wikipedia does not help your argument. If you look at User:Anonymous Dissident/Pages created, you will see that he is the original author of Smooth functor, Logarithmic differentiation, Hyperfinite set, and Fundamental increment lemma, among other articles. This editor is hardly a typical reader.--agr (talk) 18:34, 6 March 2013 (UTC)

Where to put the definition[edit]

In recent edits, the definition of a group was moved before the motivating and most elementary example. I personally think this is a step back in the comprehensibility of the article to a wide audience, which is probably not used to the (necessary) abstract notation. I prefer having Z first. Comments? --Jakob.scholbach (talk) 20:35, 12 November 2013 (UTC)

I agree. See also Gowers on putting examples first. —David Eppstein (talk) 20:59, 12 November 2013 (UTC)

Repetition in the first paragraph[edit]

The page states that a group combines "any two of its elements to form a third element also in the set", aka closure. Yet the article in the same sentence says that closure is a requirement for a set to be a group under an operation. Shouldn't this repetition be fixed? — Preceding unsigned comment added by 2601:1:9180:94A:F84A:B92B:DA42:57F9 (talk) 02:34, 20 March 2014 (UTC)

I've removed the repetition. Ozob (talk) 02:56, 20 March 2014 (UTC)

What is a "flip"?[edit]

The article uses the term "flip" several times, but does not explicitly define it. If "flip" is a synonym for "reflect" or "reflection", the article should say so. The example of a symmetry group is particularly inconsistent. -- (talk) 15:35, 15 September 2014 (UTC)

The article on reflections does not mention the term "flip", however, mathematicians do indeed sometimes use "flip" to mean "reflect" or "reflection":
-- (talk) 17:10, 15 September 2014 (UTC)
(Edit conflict) I agree. I have never seen "flip" used in math. with this meaning (Flip (mathematics) refers to another meaning). I agree to replace everywhere in this article "flip" (noun) by "reflection" and "flip" (verb) and "flipping" by "reflect" and "reflecting". After reading your second post, I suggest to edit also Reflection (mathematics) to add "flip" as a synonym. D.Lazard (talk) 17:22, 15 September 2014 (UTC)
I think "flip" is being used informally here, as an ordinary English word. But "reflect" seems to be both more precise and to preserve the same ordinary English connotations, so I've changed the article to use "reflect" throughout. Ozob (talk) 00:50, 16 September 2014 (UTC)
The notation in the example should be adjusted accordingly. (i.e. it is weird to use the letter f to represent a reflection). This poses a problem because r is already used for a rotation. It think this may be the reason why the term flip was used in first place. (Don't remember precisely)TR 09:33, 16 September 2014 (UTC)
I suggest to use s instead of f (for "symmetry with respect to a line"): this has also the advantage to use two consecutive letters (r and s), which is always less confusing for denoting related objects. D.Lazard (talk) 13:22, 16 September 2014 (UTC)
The letter "f" is the third letter of "reflection". Anyway, the notation should follow that of a reliable source. -- (talk) 15:59, 16 September 2014 (UTC)
Gallian uses H, V, D, and D' for the four reflection symmetries of a square. He also uses the term "flip". See Figure 1.1 on page 32.
-- (talk) 16:39, 16 September 2014 (UTC)
And, for completeness, Gallian denotes rotations by R0, R90, R180, and R270.-- (talk) 17:05, 16 September 2014 (UTC)

Nathan Carter also uses "flip" in his award-winning book Visual Group Theory. The problem with "reflection" is that while technically accurate, it starts with the same letter as "rotation". (talk) 07:20, 18 April 2015 (UTC)

Mismatch title: ISBN 978-0-89871-510-1[edit]

In this article there is a book in the reference section: Artin, Michael (1991), Algebra, Prentice Hall, ISBN 978-0-89871-510-1 But OpenLibrary shows that the ISBN 9780898715101 belongs to a different book: "Ordinary Differential Equations (Classics in Applied Mathematics)". Tuanminh01 (talk) 03:57, 3 March 2015 (UTC)

Amazon lists the ISBN as 978-0130047632. Ozob (talk) 14:57, 3 March 2015 (UTC)

This article is a little too basic[edit]

It never mentions conjugation (or the class equation), nor it ever gives the formula (xy)-1 = y-1x-1 which is heavily involved in that. This material is found in intro books to groups so its total omission here is pretty dubious. (talk) 07:08, 18 April 2015 (UTC)

Also the article never mentions orbit or stabilizer, which while technically related to group actions is how the symmetry-image porn that fills this article is actually explained mathematically, i.e. the symmetry group of a figure is its stabilizer group. Frankly even very slow-paced textbooks like that of Jordan and Jordan explain this much pretty early on. (talk) 07:08, 18 April 2015 (UTC)

You did see the line at the top, "This article is about basic notions of groups in mathematics. For a more advanced treatment, see Group theory.", right? You might also want to see WP:TECHNICAL, about trying to make our material accessible to the widest part of the audience. This doesn't mean avoiding the more technical details, but it does mean starting out more gently when possible and saving the more technical parts for later (and per the line at the top, this entire article should be considered as the starting out phase, with the more technical parts in other articles). —David Eppstein (talk) 07:32, 18 April 2015 (UTC)
That "advanced" page doesn't mention conjugation either. Epic win for Wikipedia. (talk) 07:42, 18 April 2015 (UTC)
In my brain the equality of left and right cosets is a less intuitive way to explain the nature of a normal subgroup than it being its own self-conjugate. YMMV. Conjugation also has very visual properties like changing a flip into another flip (or a rotation into a another rotation) etc. (talk) 07:45, 18 April 2015 (UTC)

What this page has too much of[edit]

Blah blah like: Not only are groups useful to assess the implications of symmetries in molecules, but surprisingly they also predict that molecules sometimes can change symmetry. The Jahn-Teller effect is a distortion of a molecule of high symmetry when it adopts a particular ground state of lower symmetry from a set of possible ground states that are related to each other by the symmetry operations of the molecule. If I know nothing about the physics of molecules beyond high-school, that paragraph is completely incomprehensible to me. It doesn't actually explain anything. You might as well have put the Jahn-Teller effect in a bullet list of "applications of group theory". By all means create a page of applications of group theory in natural sciences. (talk) 07:58, 18 April 2015 (UTC)

Basically this article has a scent of New Math. After reading it you'll have a lot of "I've heard of X" in your brain, but not having learned/understood much, including of the basics. (talk) 08:09, 18 April 2015 (UTC)

Wikipedia is an encyclopedia, not a textbook (see WP:NOTTEXTBOOK). Perhaps you are looking for something more like our sister project Wikibooks, which has a book on abstract algebra; see in particular this section on group theory, which contains the identity you were looking for in the section above. Mark MacD (talk) 11:05, 18 April 2015 (UTC)
The identity in question is not just some identity that happens to hold in a group. It's a rather important one (just like division) to the extent that (like division which taken separately leads to quasigroups) the identity in question leads to semigroup with involution of which are also plentiful examples. I've seen various lame excuses above for the sorry state (in certain respects) of this "awesome" article. They usually involve talking down any concerns and general hand-waving toward some WP:THISORTHAT, just like you did. Since I actually enjoy improving articles much more than the sterile discussion that characterizes (this) FA work... all I have left to say to you guys is sayonara. (talk) 11:29, 18 April 2015 (UTC)
I agree that all articles can be improved (but I don't think this one is in a sorry state). The Jahn-Teller effect is an example of abstract mathematics being used for something in the real world, and therefore I think it's being given due weight in the article; one should keep in mind that this article is not written exclusively for pure mathematicians. But you're right, maybe it could be explained better. Under the generalization section something more could be added about semigroups, along the lines you suggest; that seems reasonable to me. Mark MacD (talk) 16:45, 18 April 2015 (UTC)

Some diagrams missing from PDF version of article[edit]

Neither the table of diagrams with the caption "The elements of the symmetry group of the square (D4). The vertices are colored and numbered to distinguish between them.", nor the following table with the title "Group table of D4", appear in the article when rendered as a PDF. Is it possible to fix this? — Preceding unsigned comment added by Domiel42 (talkcontribs)

There is a general problem with the software to export Wikipedia articles to PDF and other E-book formats. See Help:Books/Feedback and Help talk:Books/Archive 1. — Tobias Bergemann (talk) 07:30, 23 January 2016 (UTC)

Blue element in figure 'Group table of D4'[edit]

In the figure 'Group table of D4', there are various sections coloured with different concepts related to the group. There is one colour - blue - which covers the element fh•rd, but this is not explained in the caption. What concept does the blue colour represent here? --Fearedinlasvegas (talk) 13:33, 8 December 2016 (UTC)

As explained in the text, it is just highlighted to emphasize the order of the product.

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Section Request[edit]

This page should discuss some more tools for constructing groups and give specific examples. This should include

  • Finding subgroups of using a finite set of conjugations
  • Constructing groups as limits and colimits
    • Such as the additive group of -adic integers
    • Construction of from
    • Semi-direct products using matrices

Group homomorphisms - Cayley's theorem[edit]

The last paragraph of the Group homomorphisms section describes Cayley's theorem applied to finite groups:

One useful theorem about finite groups is that every finite group admits an injective group homomorphism into the symmetric group of a finite set. This can be constructed by taking the finite set as the underlying set of the group {\displaystyle G} G and sending a {\displaystyle g\in G} g\in G to the set-automorphism given by {\displaystyle g\cdot } {\displaystyle g\cdot }.

This paragraph seems out of place given the level of the rest of the article. If we want to keep it, we should find a way to re-write it to be as readable to the lay-person as the surround text. I also think that (if we keep this), we should add an explicit mention and link to Cayley's theorem and remove the restriction to finite groups. Having said that, ignoring the readability issues, Cayley's theorem seems like it is still a bit to advanced to warrent inclusion in an otherwise 2 paragraph long introduction to homomorphisms. Homura1650 (talk) 07:23, 5 December 2017 (UTC)

I agree with you. The edit was certainly done in a constructive mood, but we do mention Cayleys theorem in the § about finite groups (where it belongs). I don't think a more detailed explanation why Cayleys theorem holds is in order there.
I suggest reverting the edit you quote. Jakob.scholbach (talk) 10:02, 5 December 2017 (UTC)

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