Talk:Group (mathematics)

2724:, with inline citations. Wish-lists/prayers like {{Cotton&Wilkinson}} are of no use; instead the text book "Advanced Inorganic Chemistry. A Comprehensive Text by Cotton F.A., Wilkinson G. (3rd edition)" can be found and read. If this is to be comprehensible as an article on mathematics, there should be some attempt to reconcile the terminology of physical chemistry with the standard language of theoretical physics and mathematics. In the case of the section "Symmetry"—a brief overview of a general topic—there has so far been no consensus to create separate brand new sections. Here they were sometimes done by copy-pasting content from the section on "Symmetry"; deleting the content, cited to 2858:

sentences of the penultimate paragraph, which currently reads: "As of the 20th century, groups gained wide recognition by the pioneering work of Ferdinand Georg Frobenius and William Burnside, who worked on representation theory of finite groups, Richard Brauer's modular representation theory and Issai Schur's papers. The theory of Lie groups, and more generally locally compact groups was studied by Hermann Weyl, Élie Cartan and many others. Its algebraic counterpart, the theory of algebraic groups, was first shaped by Claude Chevalley (from the late 1930s) and later by the work of Armand Borel and Jacques Tits.". —

1726:, from a time when groups were thought of in some specific formulation of what their elements should be and how they would combine (permutations and composition of permutations) rather than as anything obeying an abstract system of axioms. It's saying that the axiomatic point of view was an improvement because it allowed us to apply group theory more widely in a less cumbersome way rather than having to repeatedly translate one kind of group to another kind of group or re-prove the same theorems for every different kind of group. But if that's the intention, I don't think it expresses it very clearly. — 2372:(as you wrote), but rather a service provided by the American Math Society (one of the, if not the most prestigious national mathematical societies). It lists all mathematical papers that have been peer-reviewed, contains secondary reviews of these papers, and contains their classifications into the several areas of maths. This information is in no way a synthesis of other knowledge that has been partly assembled here and there, it is simply a number that is out there. Questioning that 1700+ papers indicates a high level of activity strikes me as being a bit off. 279: 269: 248: 215: 2932:

accept that getting agreement to changes to the article will be harder than you are used to. If you still think that you want to invest the time in achieving structural changes to the article, I recommend you put in some time and familiarise yourself with the changes that were made to the article over the last year, which has seen quite a big change in the degree of conformance with the style guide due to the push to get the article to FA level. —

21: 360: 206: 2917:, which is why the original diagrams were removed. The example of vibrations in methane illustrated the importance of group theory in relation to spectroscopy. For these reasons, I split the original section, without changing anything in the general part, and amplifying the chemical applications, albeit very briefly. The reversion should be undone so that the new material can be properly discussed, if needed. 54: 3322:"The fundamental group of a plane minus a point (bold) consists of loops around the missing point. This group is isomorphic to the integers." I know very little about groups that I didn't learn from this page... but... the integers are a set, not a group, right? So "isomorphic to the integers" is a vague way of saying "isomorphic to some group that has the integers as the underyling set"? 2882:. The finite generation of invariants of finite groups goes back to Felix Klein ("Lectures on the Icosahedron"), David Hilbert and Emmy Noether (1916). Her short, elementary and constructive proof is presented in Weyl's "The Classical Groups, Their Invariants and Representations" (Pages 275–276, 2nd edition). I don't believe it can be found on wikipedia. OTOH 2800:

is a problem with the article that I rasied during the FA process that it does not make enough of the applications outside mathematics. While the concept might fundamentally be a mathematical one, its most exciting applications lie in chemistry and physics and the article should not assume that the interest of the reader primarily comes from mathematics. —

1703:

to be an improperly constructed sentence: according to my reading it seems to implicitly suggest that the "concrete nature of the group and its operation" (I don't know what this means) has some manner of existence prior to the "formulation" (?) of the axioms. I assume we all agree that the opposite is the case.

3435:

every left inverse is also a right inverse of the same element as follows.", which you reverted with the comment "It must be clear that a proof is behind the assetrion". I do not understand the need for including "one can show that". Of course it has been shown. That is the reason we know it is true.

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There's also the problem that the blue and orange curves in the image don't show two elements of the fundamental group: they show two different (free homotopy classes of) maps from the circle into the space. A loop representing the fundamental group has (although usually only implicitly) a fixed base

2799:

I think this is overly harsh. You were right to revert the changes, but that's because we should be conservative with FAs. But there were not CIR-level problems with the changes proposed. If this was not FA quality, I'd say this is what we should expect from the BRD cycle. Additionally, I think there

1702:

I just came across this page, and came here to say that this sentence is very confusing to me (I am mathematician, familiar with groups, group actions, group representations, etc). Whatever exactly it is supposed to mean (Jakob.scholbach's explanation above did not clarify it for me), it seems it has

1622:

I still find it too difficult, which is a disservice to the rest of the article. I'm unfortunately not great with prose, but I see two problems with "While these are familiar from many mathematical structures, such as number systems—for example, the integers endowed with the addition operation form a

492:

I would propose the following, which is slightly longer, but more explicit about the role of closure, which really should be separate from the group axioms. This also breaks the definition into more manageable chunks: first understand what a binary operation is, and then understand the definition of

3182:

Your example is incoherent: the object you have presented is not a set with an operation on it (because what is -1 + 1?). Assuming you had not made this error, you would be wrong that an identity exists: the operation is defined (only) on the set, things outside the set cannot be combined using the

3158:

Note to 100.36.106.199 who removed (2 days ago) my words "the set contains an identity element" and returned to the previous wording "an identity element exists": The point is that it is not sufficient for an identity element to exist; it must be part of the set or else the set does not constitute a

2435:

I have included a link to the MSN page (again, there are no issues / page numbers, this is an electronic database). I continue to see absolutely no problem with taking a number of 1700+ papers as an indication that this branch is highly active. If that number wouldn't make it so evidently clear that

2043:

Yes, the list of groups of order <= 2000 is complete (and the result is in a sense independent of who does it). GAP does not offer a list of groups of order up to 3000, as far as I have seen. Nor does any other site (according to a search I did the other day). In this sense, the citation is still

821:

About "closure": the term is normally used for the restriction of a binary operation to a subset. Using it as it was done is thus an error. I guess that editors were confused by the usual definition of a subgroup as a nonempty subset on which the group operation and the inverse operation are closed.

3197:

I mean, it is true that students first learning abstract algebra suffer from the confusion that you are expressing here. But I think it is instructive that the first time you made the change, you did not even notice that the same argument applies to inverses as to the identity. That's because the

2967:

OK, so be it. This means that, for people like me, the article is sub-standard and should never have been promoted to FA. I've checked with a number of chemistry texts (University level) and they all have something about symmetry; most include or discuss applications that depend on the use of point

2931:

While it's natural to be disappointed when changes you've put substantial work into are rejected, my impression is that you lack experience of FA-quality editing. That's OK - little of post-high-school science is FA quality on WP - and I think the impulse behind your changes are OK, but you need to

1779:

It is true that the monster simple group has not introduced there (and is hardly introduced further down), but Borcherd's description "a huge and extraordinary mathematical object" strikes me as highly appropriate for a layman to grasp a bit of the depth out there... Other than that the quote talks

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acting on {a,b,c} or {1,2,3} are the same group). Or that everything that satisfies the axioms is a group (which it really seems to be saying), but that is kinda too implicit in the idea of what axioms are for to be using such abstruse language. If the latter, wouldn't "Any set and operation that

957:

In a similar vein, I modified the leading sentence to mention that the binary operation is closed (defined on the set). Seeing as the original sentence didn't call it a "binary operation" and instead called it an "operation that combines any two elements to form a third element", I would argue that

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Being fully precise would obviously not be desirable in the context of the page, but perhaps a talented writer could find a way to rephrase the paragraph and image/image caption in a way that remains concise and readable but is also fully accurate. (I'm not talented enough.) Maybe it would help to

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As for requiring parallelism in wording for identity element and inverse elements, I actually agree that the wording should be parallel. So I will now make it parallel by adding that the inverse elements also must be part of the group (although you said you hoped not). Again for the integers under

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General references for broad topics that do not need page numbers: Curtis 2003 (footnote 21), Weyl 1952 (footnote 50), Bishop 1993 (footnote 52), Mumford et al (footnote 63), Fulton & Harris (footnote 66), Serre 1977 (footnote 67), Rudin 1990 (footnote 68), Artin 1998 (footnote 70), Ronan 2007

1036:

So, to be clear, this is a purely mechanical and syntactic imposition, completely divorced from any understanding of the content? It would be satisfied if we found a basic textbook on group theory and tacked it on as a footnote at the end of every paragraph? You do notice that the FA requirements

921:

Including both left and right identity and inverse is very common mistake. The existence of the left identity and inverse can be proven using the right identity and inverse and vice versa. So it is sufficient to present only one of each in the list of the axioms. Here there are some proves, for

2857:

I'll note that, independently of Petergans's motives for changing the history section, we have failed to have any women in that section in the the maths FA where there would be least tokenism in avoiding that failing. Noether's contributions are as worthy of mention as anyone in the last three

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The sections of level 4 do not appear in the table of content because of the limit parameter in the template {{TOClimit|limit=3}} that appears at the end of the lead. This is a choice for having a table of content that is not too large. This choice may be discussed, but the table of content is

840:

The above wording of the last two axioms combines an axiom (one sentence) with consequent properties (e.g. uniqueness of the identity element) that is not part of the axiom. It would be good if this separation was made clearer to the reader, since the current presentation does not adequately

2822:

can be traced back to Weyl, Heisenberg, Schrödinger, Wigner, von Neumann, M.H. Stone, Dirac, Bargmann and Harish-Chandra (cf Wiener's 1933 Cambridge book or Mackey's Chicago and Oxford lecture notes). Specific examples of character tables are undue here, compared to the character formulas of

939:

You are right that some of the axioms could be deduced from the others, but this is not a "mistake". The standard textbooks intentionally require the identity be a two-sided identity and so on, presumably because it is more natural not to favor one side. Therefore we should leave it as is.

2403:

And I am not questioning that it is a high level of activity; I am looking at it from the perspective of a non-mathematician reading this: How does one get a sense of what the figure means when one has no reference, other than the claim made in WP's voice? In non-technical contexts and for

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of loops. The main text avoids this by saying that "elements of the fundamental group are represented by loops" which is perfectly correct but maybe overly evasive or obscure for most readers. Also, the loops don't have to go around the missing point - they just have to avoid

2007:

I think I may have misinterpreted what it says. So I was thinking, does a more modern source have "The groups of order at most 3000 are known"? From your comments it seems like there is a specific collection of groups, called the GAP small groups? Can that be clarified?

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The formulation of the axioms is, however, 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 mathematical

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the category of groups: I think this entire section is quite difficult, and uses terms that an applied mathematician won't be familiar with. I don't think it falls under domain-specific common knowledge. Can it be slightly simplified (what is a category?) and

2908:

The reversion is very disappointing. The uses of group theory in chemistry are extensive and were properly documented with references to relevant books. Symmetry in molecules is an essential part of the undergraduate curriculum in chemistry. For example

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Consider the first example: the integers under addition. If we consider the set without the identity element zero: ..., -3, -2, -1, +1, +2, +3, +4, ... then we have a set which is NOT a group. Zero still exists but it has to be included in the group.

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Re 2: this is meant to say that the group axioms don't make reference to the nature of the group elements, nor to "what" the group operation actually is. This is a critical piece of information. If you have a better way of saying this, let me know!

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1. such as and for example is quick succession makes it more difficult to read. I think leaving out "such as number systems" works, considering that "number systems" may not be familiar to everybody reading this. I can guess what it means, but not

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I have reworded it slightly, but yes, in some sense it is true that groups of order 2001 (not the 2001st group though, this makes no sense) are "unknown" in the sense that there is (to the best of my knowledge) no list available listing them all.

2648:

Why are all the main section titles double indented ==title==? They should be single indented as the menu only shows 3 levels of indentation. Currently ====items==== are present in the article, but are not shown on the menu. This will require

1861:

This is one of those things that's probably stated in some form in just about any intermediate quantum-physics book. I've added an older reference that I had at hand, and I'll poke around for a more recent one that seems particularly good.

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Please don't: "closure" is a property of subsets, and there is no subset here. The fact that the result of the operation belongs to the group is a part of the definition of an operation. By the way, I have removed the use of "closure" in

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point, and these two loops obviously have no common base point. So the picture is not quite illustrative of the fundamental group, even though any reader already familiar with the concepts can easily see what it's trying to communicate.

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Hm, this is a case where some problem is super-hard, and is known to be super-hard to anyone studying this. Therefore, researchers seem not to restate this too frequently again. At least I didn't find a more recent source for that.

1967:

I decided to trim this § down to a sentence which is now placed in the paragraph on homomorphisms. Elaborating on the notion of a category is IMO better left to articles where this has a stronger effect (e.g., for abelian groups).

2358:

of the worst kind, even if you will not get many people disagreeing with the conclusion. It invites the query from a reader: "Are you sure?" Two: the number itself needs citation. It should not be claimed out of the blue.

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Re 1. I have broken the sentence into two. I think leaving out number systems makes the lead less informative. The directly following example of the integers should convey enough implicit meaning about number systems to be OK

2353:

I am uncomfortable with this on two levels. One: a high level of interpretation and judgement is needed to (a) classify papers and (b) to translate the number into a conclusion of how "active" the field is. In short, it is

1722:", right? Material in the lead is supposed to be a summary of something. I suspect this is (or should be) thought of as a summary of the 19th-century notion of a group touched on in the History section and in more detail in 841:

distinguish for the reader who is not already familiar with the exact axioms. The parts that do not form part of the axiom could be moved to under the listed axioms, for example, or preceded by "This implies that ...". —

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MOS allows for italics for emphasizing things. Since associativity is such an important piece of this concept I believe it is worth highlighting it. Generally my feeling is that italicization is not used too excessively.

1116:, so we will need a list of sentences that need citations. From a quick read, it seems the article has very good bones, and it shouldn't take much time to bring it up to modern FA standards. A few points of improvement 2823:

Frobenius, Schur and Weyl (which have been widely applied in theoretical physics and mathematics). Charles Stewart is completely correct that the section can be improved, but that should be done in an incremental way.

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Subject-specific common knowledge: Material that someone familiar with a topic, including laypersons, recognizes as true. Example (from Processor): "In a computer, the processor is the component that executes

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Finally about 2): of course we can include a link to the Mathscinet page, but I frankly don't quite see the need for that. It is (to anyone with a subscription to MSN) a trivially verifiable information.

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I essentially agree, and I have edited the article accordingly. By the way, I have copy-edited the whole section for clarification and for using a simpler wording that is also more common in mathematics.

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in order to make this expansion clear and precise, it's required to mention that the domains/codomain are all in the set. So therefore I modified it to "an operation that combines any two elements

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The wording still implies that group 2001 is unknown. Could it be reworded as: "Up to isomorphism, there are about 49 billion groups of order below or equal to 2000", or something in that sense?

3119:, and so that composition goes well. The reason why I didn't do the changes myself is that I don't know where to put it, or if it could only be a redirection to the (quite scarce) examples from 335: 1154:

In my understanding this is used to support that group theory is an active mathematical discpline, not how it impacts other fields. For that purpose this note is perfectly appropriate, IMO.

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I don't think this needs a citation: the kernel is the identity element if and only if a homomorphism is injective (see the subarticle). This is domain-specific standard knowledge, IMO.

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Yes, that would make sense and would be good to communicate. It seems a little tricky to formulate clearly in an lead-appropriate way, unfortunately I don't have any good suggestion.

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I think having 1700 scientific papers published per year in some domain is quite aptly proving that this domain is active, no? I see absolutely no problem with this claim.

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The statements about injective homomorphisms use several notational elements that have not been introduced previously and that will not be intuitive to a general reader:

2968:

group character tables. The applications don't belong in the same place as the theory (as is the case at present). For me, that means that this discussion is now closed.

2404:

schoolgoers, that might seem like a tiny number or an enormous number. Also, to cite the issue and page number that provides the mentioned list would not be strange. —

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For example, group theory is used to show that optical transitions between certain quantum levels cannot occur simply because of the symmetry of the states involved. cn

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Although it is important to mention closure, there are a few things that disturb me about the way the definition of group is currently written. What is an operation,

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I've tweaked the article so that footnote is only used once, and now we point to the "Examples and applications" section to show how group theory has applications.

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check throughout for statements like this that don't make sense on phones (where that image is above the text). The majority of our readers now use mobile devices.

3052: 3005: 1670:, could I have a third opinion here? I know a lot of your work is comprehensible, so wonder whether you can simplify or assure me it does not need simplifying. 497:. Finally, there are many modern textbooks at all levels that present the definition along these lines (e.g., Artin, Lang, ...); I would add such references. 219: 2417:

was a journal (of reviews), it stopped being published as a journal well before the date given in the note, and became a database, under the different name

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Done for those where (IMO) the nuisance of the link outweighs the benefits. If you see some more that you specifically think should go, please let me know.

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from a mathematical standpoint; and is explained in standard text books on chemistry & group theory (e.g. "Chemical Applications of Group Theory",

2770:; why delete images from the section on "Symmetry"; why favour chemistry above physics? Here are diffs of recent problematic edits, including today's. 1871: 1472: 1434: 1339: 1192: 3500: 862:. Technically the only difference is the commutativity of the operation, so it doesn't make sense to list closure as an axiom of one but not another. 923: 3510: 2038: 1405: 3525: 2239:

I have moved this § down to topological groups and added a ref for the second sentence you are asking about. I will check for the first later.

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I concerned that this article no longer meets the FA criteria. The are large sections of uncited text. Can this be resolved without a formal

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Re 2, it might help to figure out what the concept to be conveyed here is. It could be that you can have groups in various contexts (e.g. S

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the closure axiom is imposed? A function from G × G to some unspecified set? Not only is this a little vague, but it also contradicts the

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are often first encountered in undergraduate courses on finite groups and angular momentum in quantum mechanics (see e.g. the treatment by

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Barging in here, from the FAR, if that's okay. I've never been taught group theory, so please bear with me when I say stupid things here.

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page it links to. Also, technically speaking, the sentence defining group is wrong, because it ends before any of the axioms are imposed.

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I agree with all of Charles Stewart's comments. Since article is FA, before making such edits it is good to discuss on talk page first.

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For example, an element of the (2,3,7) triangle group acts on a triangular tiling of the hyperbolic plane by permuting the triangles, cn

1601:

The last sentence of the first paragraph of the lede is difficult to understand. I'm not sure whether splitting it in two is sufficient.

3431:: I removed the bold text from "...assuming associativity and the existence of a left identity and a left inverse for each element , 2629:. These tables used extensively in chemistry: see, for example, "Chemical Applications of Group Theory", F.A. Cotton, 3rd. edn., 1990. 3333: 2495:

Has a page number, but in the full reference not the footnote and should probably be made more consistent: Bersuker 2006 (footnote 54)

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I would welcome advice about which defined terms should be bold and which should be italicized; I'm not sure what the convention is.

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In the rightmost example below: with people reading on phones, this should be phrased differently (last example? May be too unclear)

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addition: the set 0, +1, +2, +3, +4, ... is NOT a group without the negative integers. The fact that they exist is not sufficient.

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that the best solution to the issue you raise is that closure should not be listed an axiom either for group or for abelian group.

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In all respect, I think the edit you made indicates that you are not very familiar with the situation here: Math Reviews is not a

2232:

It is also useful for talking of properties of the inverse operation, as needed for defining topological groups and group objects.

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Good idea! I tried to implement your suggestion, by adding it to the discussion of groupoids in the Generalizations section.

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footnote a is a bit outdated, and is used to support that group theory impacts other fields, which isn't immediately clear

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It was in an unacceptable state, given the last diff. In physics, the group-theoretic approach to quantum mechanics and

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is encyclopedic and explained clearly on the tables in mathematics, physics and chemistry (230 cases); mathematically,

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Done for Kurzweil. Welsh does not mention the Mathieu group, so this might better be replaced by some other reference.

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cannot be taught without reference to symmetry operations. The designations of many point groups are illustrated at

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https://math.stackexchange.com/questions/65239/right-identity-and-right-inverse-in-a-semigroup-imply-it-is-a-group

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Using this definition, it is a theorem that a subgroup is a group, and that the groups axioms are thus satisfied.

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it is a highly active branch, it would require us to give further references, but this is not the case here.

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To me this comes across as "I don't want to answer that so I'm going to do the most hostile thing I can". —

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There's some sort of error in the citation "{{Harvard citations|nb=yes|year=2003|last1=Simons|loc=§4.2.1"

2029:. I'm not clear on why this footnote would need updating; once the exhaustive search was done, it's done. 1820:

Found one more: if the hour hand is on 9 and is advanced 4 hours, it ends up on 1, as shown at the right.

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just about the simplicity of these axioms, which is what this § is all about. I suggest leaving it there.

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I removed this second half of the sentence, none of the sources I looked at mentioned that. Added a ref.

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group—the formulation of the axioms is detached from the concrete nature of the group and its operation."

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Kernel and image of group homomorphisms and the first isomorphism theorem address this phenomenon .. cn?

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equipped with a binary operation ⋅ satisfying the following three additional requirements, known as the

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Some of these books refs it would be nice to have page numbers, if possible to help with verifiability

2276:"Via Euler angles, rotation matrices are used in computer graphics" - appears to be a sentence fragment 278: 1688:

acting on {a,b,c} or {1,2,3} are both groups) or that all isomorphic groups are the same group (e.g. S

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But Galois theory isn't standard knowledge for laypeople with a keen interest in mathematics (I quote

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Looking at this picture, I agree it's weird. We probably instead want something like the pictures in

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operation with things in the set and so in particular they cannot be an identity or an inverse. --

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the formulation of the axioms is detached from the concrete nature of the group and its operation.

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on Knowledge. If you would like to participate, please visit the project page, where you can join

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Should the quote from Borcherds be moved down? Those technical terms have not been introduced yet

1745: 1708: 1088: 1060: 1023: 994: 941: 904: 886: 827: 804: 494: 451: 284: 268: 247: 3288: 1041:, right? See in particular the "Subject-specific common knowledge" bullet point at that link. — 3441: 3172: 2893: 2848: 2789: 2748: 2475: 2405: 2360: 2323: 2303: 2265: 1694: 867: 842: 508: 44: 3314:

The latter also appears in the Presentations section, along with reference to the free group

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I'm very tempted to add Closure as one of the four group axioms, as it's already one of the

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Needs page numbers: Welsh 1989 (footnote 62), Kurzweil & Stellmacher 2004 (footnote 74)

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group. Also, this would bring this page more in line with other Knowledge pages, such as

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Well, frankly, I understood little of this, so I may just be plain wrong on my comments.

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Can probably be replaced by a better reference: Lay (footnote 64), Kuipers (reference 65)

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move the paragraph to its own subsection "algebraic topology" or "fundamental group".

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Added references. I pinged WP Chemistry about the JT-effect, will add one there, too.

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for your comments. I have addressed some of them and will work on the remainder asap.

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I added the Standard Model to an appropriate spot in the body text and rephrased the

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You are mistaken. I chose not to answer your rude assertion about my understanding.--

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emphasize that citations are needed "where appropriate", with a link that points to

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It is generally preferred for computing with groups and for computer-aided proofs.

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rest of these images probs also need citations (mentioned above, but XOR'Easter's

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the wording a little; however, a citation for this claim would be appropriate. —

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In this case, the relevant group is the integers under the addition operation. –

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gave Hilbert's non-constructive proof using the averaging or Reynolds operator.

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The edits today to the article are a combination of vandalism, incompetence and

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A presentation of a group can also be used to construct the Cayley graph .. cn?

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This is an article that will have many paragraphs that fall squarely under the

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This article appeared on Knowledge's Main Page as Today's featured article on

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have been reverted. That was due partially to the misuse of indentation, see

2421:. I have corrected the note to reflect its name as of the referred-to date. — 2910: 2759:. It unbalances the article. The new image without citations is unhelpful. 1355:

I am not convinced this needs a specific citation. Basically you could name

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This makes sense to me now; I apparently lost my ability to read briefly.

1359:(contemporary) book on Galois theory (such as the ones we do cite above). 2099:

a problem too hard to be solved in general (footnote r), needs updating?

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Research is ongoing to simplify the proof of this classification -: -->

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What sections, specifically, do you think require additional citation?

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Is there a way it could be true without having been shown to be true?

1508:. With a technical topic as this, many are defensible, but please use 2394:. I took that it is described as a journal from the linked article 2025:

is software for doing group theory. It includes implementations of

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consists of loops and the fundamental group crucially consists of

2736:, was unhelpful; similarly for the citation to Graham Ellis. 2155:

Simons, Jack (2003) listed in specific references, but not used.

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though the double bonds reduce this to pyritohedral symmetry, cn

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meaning is not actually ambiguous or otherwise problematic. --

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I tried rewriting the explanation here. Is that any clearer? –

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I feel that the "category" point of view is missing : a group

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This is true, but it is not an evidence. See your talk page.

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Featured articles that have appeared on the main page twice

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Having said all that: the revised wording seems fine. --

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It is referenced when talking about symmetry of Ammonia.

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What is the fundamental group of a plane minus a point?

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Point groups in three dimensions#Finite isometry groups

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Why? The GAP small groups lists still the same number?

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Identity and also inverse elements must be part of set

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Not sure whether needs pages: Shatz 1972 (footnote 81)

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This seems to have been done by someone else already.

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Featured articles that have appeared on the main page

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I think some of the footnotes need citations: g, j, p

574:; it must be checked if it is not known initially.) 296:, a collaborative effort to improve the coverage of 3356:The given quote could be considered incorrect: the 2720:; but also changes to content must be supported by 1934:

response directly below may have caused confusion )

687: 3303: 3277: 3257: 3111: 3072: 3046: 3026: 2999: 2751:). Separate new sections at the moment seem to be 899:, for pointing out the discrepancy. I agree with 1443:The various molecules and their properties .. cn 3516:Knowledge level-4 vital articles in Mathematics 2670:Help:Section#Creation and numbering of sections 1693:satisfies the axioms is a group" be clearer? — 1479:Please add alt text to images for accessibility 3123:, in which case I would try and extend these. 2313:Can we get a more exact citation for note a? 1634:Not entirely sure what this is meant to say. 460:This page has archives. Sections older than 8: 43:. Even so, if you can update or improve it, 39:as one of the best articles produced by the 33:; it (or a previous version of it) has been 1382:). Can it be found in simpler sources too? 757: 3323: 1544:I've removed a few more. Hope that works. 1396:I added a citation to a textbook chapter. 242: 72: 15: 3290: 3270: 3258:{\displaystyle {\stackrel {\sim }{\to }}} 3247: 3242: 3240: 3239: 3237: 3091: 3085: 3059: 3039: 3018: 3012: 2992: 2766:: why delete references to physicists or 1221:this one also has been taken care of :). 2492:(footnote 77), Husain 1966 (footnote 79) 1169:Ah, now I see it is used a second time. 1120:Standard Model not mentioned in the body 203: 3506:Knowledge vital articles in Mathematics 3228:Undefined terms and notational elements 2413:Relatedly, although it is correct that 1582:Brilliant, thanks for your swift work! 470:when more than 10 sections are present. 244: 3521:FA-Class vital articles in Mathematics 3389:Winding number § Intuitive description 3330:2404:4408:6A6E:7000:E48B:A59:8E82:2FCF 2915:Molecular symmetry#Common point groups 1718: 1460: 1378: 1252:Included a more up-to-date reference. 2625:A major omission is any reference to 2202:Knowledge:Manual of Style/Mathematics 1717:This is regarding the two sentences " 1348:The problem can be dealt with ... cn? 7: 2837:crystallographic restriction theorem 1210:Sorry, what is your objection here? 290:This article is within the scope of 2653:indents to be changed in the text. 748:is the identity element. For each 233:It is of interest to the following 14: 3531:Top-priority mathematics articles 2605:No worries, that's just alright. 1114:Subject-specific common knowledge 686:hold. Such an element is unique ( 514:is a rule for combining any pair 464:may be automatically archived by 310:Knowledge:WikiProject Mathematics 3501:Knowledge level-4 vital articles 3278:{\displaystyle \hookrightarrow } 2739:As far as groups are concerned, 2390:You are presumably referring to 2302:here; it was a little clumsy. — 778:Formally, a binary operation on 358: 313:Template:WikiProject Mathematics 277: 267: 246: 213: 204: 52: 19: 3511:FA-Class level-4 vital articles 1798:group table at the right -: --> 1597:Comments from my second read: 1059:and let the community decide.-- 330:This article has been rated as 3272: 3243: 3103: 1: 3526:FA-Class mathematics articles 2833:crystallographic point groups 1750:06:27, 15 February 2022 (UTC) 1736:06:20, 15 February 2022 (UTC) 1713:05:57, 15 February 2022 (UTC) 1512:to remove the improper ones. 913:23:01, 29 December 2021 (UTC) 891:15:00, 29 December 2021 (UTC) 872:14:20, 29 December 2021 (UTC) 832:10:44, 16 December 2020 (UTC) 813:02:49, 16 December 2020 (UTC) 760:) and it is commonly denoted 304:and see a list of open tasks. 3491:Old requests for peer review 3148:18:42, 1 November 2022 (UTC) 3133:14:47, 1 November 2022 (UTC) 3112:{\displaystyle f_{g}:a\to a} 3080:corresponds to isomorphisms 950:00:03, 23 January 2023 (UTC) 935:00:06, 20 January 2023 (UTC) 3476:Knowledge featured articles 3415:22:13, 3 October 2023 (UTC) 3399:19:37, 3 October 2023 (UTC) 3383:19:31, 3 October 2023 (UTC) 3352:17:24, 3 October 2023 (UTC) 3338:08:21, 3 October 2023 (UTC) 2488:I analyzed a few of these: 1608:OK, I have rephrased this. 1152:I have updated it to 2020. 1022:At least every paragraph.-- 540:. (The property "for all 526:to form another element of 3547: 3304:{\displaystyle \ker \phi } 3007:can be seen as a category 2978:20:11, 26 March 2022 (UTC) 2960:15:25, 26 March 2022 (UTC) 2944:18:23, 26 March 2022 (UTC) 2927:15:13, 26 March 2022 (UTC) 2898:20:42, 26 March 2022 (UTC) 2870:18:47, 26 March 2022 (UTC) 2853:16:43, 26 March 2022 (UTC) 2812:14:19, 26 March 2022 (UTC) 2794:14:12, 26 March 2022 (UTC) 2698:12:03, 26 March 2022 (UTC) 2682:11:46, 26 March 2022 (UTC) 2663:10:48, 26 March 2022 (UTC) 2639:08:47, 15 March 2022 (UTC) 2200:in italics compliant with 1577:09:42, 30 April 2021 (UTC) 1554:19:35, 30 April 2021 (UTC) 1540:09:42, 30 April 2021 (UTC) 1522:16:50, 27 April 2021 (UTC) 1473:21:47, 26 April 2021 (UTC) 1453:18:48, 26 April 2021 (UTC) 1369:09:49, 30 April 2021 (UTC) 1326:17:13, 30 April 2021 (UTC) 1312:09:42, 30 April 2021 (UTC) 1287:09:49, 30 April 2021 (UTC) 1231:19:28, 30 April 2021 (UTC) 1137:09:42, 30 April 2021 (UTC) 1097:16:36, 26 April 2021 (UTC) 1083:16:32, 26 April 2021 (UTC) 1069:07:52, 26 April 2021 (UTC) 1051:06:50, 26 April 2021 (UTC) 1032:06:46, 26 April 2021 (UTC) 1018:04:05, 26 April 2021 (UTC) 1003:11:09, 20 April 2021 (UTC) 980:06:58, 14 March 2022 (UTC) 708:, there exists an element 690:), and thus one speaks of 421:4 (21 July - 27 August 08) 158:Featured article candidate 3222:13:54, 10 July 2023 (UTC) 3208:13:52, 10 July 2023 (UTC) 3193:13:49, 10 July 2023 (UTC) 3177:02:01, 10 July 2023 (UTC) 2027:all groups of small order 1986:footnote q needs updating 426:5 (25 August - 21 Oct 08) 329: 262: 241: 186: 75: 71: 3461:13:26, 26 May 2024 (UTC) 3446:13:20, 26 May 2024 (UTC) 3034:with 1 objeect (call it 2583:21:50, 11 May 2021 (UTC) 2565:09:56, 12 May 2021 (UTC) 2537:09:56, 12 May 2021 (UTC) 2512:09:56, 12 May 2021 (UTC) 2446:20:04, 14 May 2021 (UTC) 2196:Is having terms such as 1510:User:Evad37/duplinks-alt 879:abelian group#Definition 642:There exists an element 411:2 (Oct 06 - 23 April 08) 336:project's priority scale 3496:FA-Class vital articles 2615:13:03, 8 May 2021 (UTC) 2600:22:05, 7 May 2021 (UTC) 2479:23:11, 7 May 2021 (UTC) 2431:19:42, 9 May 2021 (UTC) 2409:16:13, 9 May 2021 (UTC) 2386:12:28, 9 May 2021 (UTC) 2364:13:52, 8 May 2021 (UTC) 2349:13:06, 8 May 2021 (UTC) 2327:23:11, 7 May 2021 (UTC) 2307:23:11, 7 May 2021 (UTC) 2290:23:00, 7 May 2021 (UTC) 2269:23:11, 7 May 2021 (UTC) 2249:13:03, 8 May 2021 (UTC) 2220:13:03, 8 May 2021 (UTC) 2182:13:03, 8 May 2021 (UTC) 2165:20:26, 7 May 2021 (UTC) 2149:16:13, 3 May 2021 (UTC) 2131:19:13, 3 May 2021 (UTC) 2117:18:50, 3 May 2021 (UTC) 2083:13:03, 8 May 2021 (UTC) 2068:20:25, 7 May 2021 (UTC) 2054:06:54, 4 May 2021 (UTC) 2039:23:49, 3 May 2021 (UTC) 2018:19:12, 3 May 2021 (UTC) 2003:18:50, 3 May 2021 (UTC) 1978:18:50, 3 May 2021 (UTC) 1951:18:07, 7 May 2021 (UTC) 1922:18:07, 7 May 2021 (UTC) 1897:03:53, 4 May 2021 (UTC) 1872:23:56, 3 May 2021 (UTC) 1844:13:03, 8 May 2021 (UTC) 1830:20:21, 7 May 2021 (UTC) 1816:18:50, 3 May 2021 (UTC) 1790:18:50, 3 May 2021 (UTC) 1724:History of group theory 1698:20:54, 7 May 2021 (UTC) 1680:20:20, 7 May 2021 (UTC) 1663:20:51, 4 May 2021 (UTC) 1644:19:06, 3 May 2021 (UTC) 1618:18:50, 3 May 2021 (UTC) 1592:16:13, 3 May 2021 (UTC) 1496:10:53, 2 May 2021 (UTC) 1435:20:35, 2 May 2021 (UTC) 1406:03:35, 4 May 2021 (UTC) 1392:16:13, 3 May 2021 (UTC) 1340:20:47, 2 May 2021 (UTC) 1262:19:11, 2 May 2021 (UTC) 1193:21:02, 2 May 2021 (UTC) 1179:11:04, 2 May 2021 (UTC) 1165:10:53, 2 May 2021 (UTC) 846:11:33, 4 May 2021 (UTC) 293:WikiProject Mathematics 177:Featured article review 3305: 3279: 3259: 3113: 3074: 3073:{\displaystyle g\in G} 3048: 3028: 3001: 2704:Restructuring article? 1375:Knowledge:When to cite 1039:Knowledge:When to cite 467:Lowercase sigmabot III 3306: 3280: 3260: 3114: 3075: 3049: 3029: 3027:{\displaystyle A_{G}} 3002: 2820:representation theory 2741:representation theory 1245:cited to a 2004 study 650:such that, for every 416:3 (2 May - 16 May 08) 220:level-4 vital article 3289: 3269: 3236: 3084: 3058: 3038: 3011: 2991: 2983:Groups as categories 2688:already very large. 2415:Mathematical Reviews 2397:Mathematical Reviews 406:1 (July 05 - Oct 06) 316:mathematics articles 120:Good article nominee 101:Good article nominee 3362:equivalence classes 2835:. It describes the 2121:Thanks for trying. 431:6 (Nov 08 - Nov 17) 41:Knowledge community 27:Group (mathematics) 3301: 3275: 3255: 3109: 3070: 3044: 3024: 2997: 2708:The edits to this 1155: 1055:I'll nominate for 285:Mathematics portal 229:content assessment 151:September 17, 2008 76:Article milestones 3433:one can show that 3423:One can show that 3340: 3328:comment added by 3252: 3054:) where elements 3047:{\displaystyle a} 3000:{\displaystyle G} 2942: 2868: 2810: 2749:Jean-Pierre Serre 1933: 1461:rightmost example 1153: 694:identity element. 474: 473: 350: 349: 346: 345: 342: 341: 198: 197: 194: 193: 3538: 3310: 3308: 3307: 3302: 3284: 3282: 3281: 3276: 3264: 3262: 3261: 3256: 3254: 3253: 3251: 3246: 3241: 3118: 3116: 3115: 3110: 3096: 3095: 3079: 3077: 3076: 3071: 3053: 3051: 3050: 3045: 3033: 3031: 3030: 3025: 3023: 3022: 3006: 3004: 3003: 2998: 2938: 2887: 2880:invariant theory 2864: 2841:F. Albert Cotton 2806: 2745:character theory 2722:reliable sources 2710:featured article 2627:character tables 2555:Is OK, I think. 2473: 2321: 2301: 2263: 2234:need references? 2233: 2229: 1932:(brilliant name) 1931: 1425:Citation added. 985:Featured Article 966:a third element 795: 774: 743: 729: 685: 671: 658:, the equations 639:Identity element 561: 539: 505:binary operation 487:binary operation 469: 453: 362: 354: 318: 317: 314: 311: 308: 287: 282: 281: 271: 264: 263: 258: 250: 243: 226: 217: 216: 209: 208: 200: 189:Featured article 187:Current status: 172: 153: 134: 115: 96: 73: 61:November 5, 2008 56: 31:featured article 23: 16: 3546: 3545: 3541: 3540: 3539: 3537: 3536: 3535: 3466: 3465: 3425: 3320: 3287: 3286: 3267: 3266: 3234: 3233: 3230: 3156: 3087: 3082: 3081: 3056: 3055: 3036: 3035: 3014: 3009: 3008: 2989: 2988: 2985: 2935:Charles Stewart 2883: 2861:Charles Stewart 2803:Charles Stewart 2706: 2646: 2623: 2621:Character table 2607:Jakob.scholbach 2598: 2557:Jakob.scholbach 2529:Jakob.scholbach 2504:Jakob.scholbach 2469: 2438:Jakob.scholbach 2378:Jakob.scholbach 2341:Jakob.scholbach 2317: 2297: 2288: 2259: 2241:Jakob.scholbach 2212:Jakob.scholbach 2190: 2174:Jakob.scholbach 2109:Jakob.scholbach 2075:Jakob.scholbach 2046:Jakob.scholbach 1995:Jakob.scholbach 1970:Jakob.scholbach 1943:Jakob.scholbach 1914:Jakob.scholbach 1836:Jakob.scholbach 1808:Jakob.scholbach 1782:Jakob.scholbach 1691: 1687: 1655:Jakob.scholbach 1610:Jakob.scholbach 1569:Jakob.scholbach 1532:Jakob.scholbach 1506:duplicate links 1504:There are many 1488:Jakob.scholbach 1361:Jakob.scholbach 1318:Jakob.scholbach 1304:Jakob.scholbach 1279:Jakob.scholbach 1254:Jakob.scholbach 1212:Jakob.scholbach 1171:Jakob.scholbach 1157:Jakob.scholbach 1129:Jakob.scholbach 987: 783: 771: 731: 717: 697:Inverse element 673: 659: 562:belongs to the 553: 531: 522:of elements of 479: 465: 454: 448: 367: 315: 312: 309: 306: 305: 283: 276: 256: 227:on Knowledge's 224: 214: 168: 149: 130: 111: 92: 12: 11: 5: 3544: 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3509: 3507: 3504: 3502: 3499: 3497: 3494: 3492: 3489: 3487: 3484: 3482: 3479: 3477: 3474: 3473: 3471: 3462: 3458: 3454: 3450: 3449: 3448: 3447: 3443: 3439: 3434: 3430: 3422: 3416: 3413: 3410: 3406: 3402: 3400: 3397: 3394: 3390: 3386: 3385: 3384: 3380: 3376: 3371: 3367: 3363: 3359: 3355: 3353: 3350: 3347: 3343: 3342: 3341: 3339: 3335: 3331: 3327: 3317: 3315: 3312: 3298: 3295: 3292: 3248: 3227: 3223: 3219: 3215: 3211: 3209: 3205: 3201: 3196: 3194: 3190: 3186: 3181: 3180: 3179: 3178: 3174: 3170: 3164: 3160: 3153: 3149: 3145: 3141: 3140:Ebony Jackson 3137: 3136: 3135: 3134: 3130: 3126: 3122: 3106: 3100: 3097: 3092: 3088: 3067: 3064: 3061: 3041: 3019: 3015: 2994: 2982: 2980: 2979: 2975: 2971: 2961: 2957: 2953: 2949: 2945: 2941: 2937: 2936: 2930: 2929: 2928: 2924: 2920: 2916: 2912: 2907: 2899: 2895: 2891: 2886: 2881: 2877: 2876: 2875: 2874: 2871: 2867: 2863: 2862: 2856: 2855: 2854: 2850: 2846: 2842: 2838: 2834: 2830: 2826: 2821: 2817: 2816: 2815: 2814: 2813: 2809: 2805: 2804: 2798: 2797: 2796: 2795: 2791: 2787: 2784: 2782: 2780: 2778: 2776: 2774: 2772: 2769: 2765: 2760: 2758: 2754: 2750: 2746: 2742: 2737: 2735: 2732:et al, or to 2731: 2727: 2723: 2719: 2715: 2711: 2703: 2699: 2695: 2691: 2686: 2683: 2679: 2675: 2671: 2667: 2666: 2665: 2664: 2660: 2656: 2652: 2643: 2641: 2640: 2636: 2632: 2628: 2620: 2616: 2612: 2608: 2604: 2603: 2602: 2601: 2597: 2593: 2584: 2580: 2576: 2572: 2566: 2562: 2558: 2554: 2553: 2552: 2551: 2547: 2544: 2543: 2538: 2534: 2530: 2526: 2525: 2524: 2523: 2519: 2518: 2513: 2509: 2505: 2501: 2500: 2499: 2498: 2494: 2490: 2489: 2487: 2486: 2484: 2480: 2477: 2472: 2468: 2467: 2465: 2464: 2447: 2443: 2439: 2434: 2433: 2432: 2428: 2424: 2420: 2416: 2412: 2411: 2410: 2407: 2402: 2399: 2398: 2393: 2389: 2388: 2387: 2383: 2379: 2374: 2371: 2367: 2366: 2365: 2362: 2357: 2352: 2351: 2350: 2346: 2342: 2338: 2337: 2336: 2335: 2334: 2333: 2328: 2325: 2320: 2315: 2314: 2312: 2308: 2305: 2300: 2299:made a change 2295: 2294: 2291: 2287: 2283: 2279: 2278: 2277: 2274: 2270: 2267: 2262: 2258:I think I've 2257: 2256: 2254: 2253: 2250: 2246: 2242: 2238: 2237: 2225: 2224: 2221: 2217: 2213: 2208: 2207: 2203: 2199: 2198:associativity 2195: 2194: 2193: 2187: 2183: 2179: 2175: 2171: 2170: 2167: 2166: 2162: 2158: 2153: 2152: 2151: 2150: 2146: 2142: 2132: 2128: 2124: 2120: 2119: 2118: 2114: 2110: 2105: 2104: 2103: 2102: 2098: 2097: 2084: 2080: 2076: 2071: 2070: 2069: 2065: 2061: 2057: 2056: 2055: 2051: 2047: 2042: 2041: 2040: 2036: 2032: 2028: 2024: 2021: 2020: 2019: 2015: 2011: 2006: 2004: 2000: 1996: 1992: 1991: 1990: 1989: 1985: 1984: 1979: 1975: 1971: 1966: 1965: 1964: 1963: 1958: 1957: 1952: 1948: 1944: 1940: 1939: 1938: 1937: 1929: 1928: 1923: 1919: 1915: 1911: 1910: 1909: 1908: 1904: 1903: 1898: 1894: 1890: 1886: 1885: 1884: 1883: 1879: 1878: 1873: 1869: 1865: 1860: 1859: 1858: 1857: 1853: 1852: 1845: 1841: 1837: 1833: 1832: 1831: 1827: 1823: 1819: 1817: 1813: 1809: 1805: 1804: 1803: 1802: 1797: 1796: 1791: 1787: 1783: 1778: 1777: 1776: 1775: 1771: 1770: 1751: 1747: 1743: 1739: 1738: 1737: 1733: 1729: 1725: 1721: 1716: 1715: 1714: 1710: 1706: 1701: 1700: 1699: 1696: 1683: 1682: 1681: 1677: 1673: 1669: 1666: 1665: 1664: 1660: 1656: 1651: 1647: 1646: 1645: 1641: 1637: 1633: 1629: 1625: 1621: 1620: 1619: 1615: 1611: 1607: 1606: 1605: 1604: 1600: 1599: 1598: 1593: 1589: 1585: 1581: 1580: 1579: 1578: 1574: 1570: 1566: 1555: 1551: 1547: 1543: 1542: 1541: 1537: 1533: 1529: 1528: 1527: 1526: 1523: 1519: 1515: 1511: 1507: 1503: 1502: 1497: 1493: 1489: 1485: 1484: 1483: 1482: 1478: 1477: 1474: 1470: 1466: 1462: 1458: 1457: 1454: 1450: 1446: 1442: 1441: 1436: 1432: 1428: 1424: 1423: 1422: 1421: 1417: 1416: 1407: 1403: 1399: 1395: 1394: 1393: 1389: 1385: 1381: 1376: 1372: 1371: 1370: 1366: 1362: 1358: 1354: 1353: 1352: 1351: 1347: 1346: 1341: 1337: 1333: 1329: 1327: 1323: 1319: 1315: 1313: 1309: 1305: 1301: 1300: 1299: 1298: 1294: 1293: 1288: 1284: 1280: 1276: 1275: 1274: 1273: 1269: 1268: 1263: 1259: 1255: 1251: 1250: 1249: 1248: 1243: 1242: 1241: 1232: 1228: 1224: 1220: 1219: 1217: 1213: 1209: 1208: 1207: 1206: 1202: 1201: 1194: 1190: 1186: 1182: 1181: 1180: 1176: 1172: 1168: 1166: 1162: 1158: 1151: 1150: 1149: 1148: 1144: 1143: 1138: 1134: 1130: 1126: 1125: 1124: 1123: 1119: 1118: 1117: 1115: 1098: 1094: 1090: 1089:Graham Beards 1086: 1085: 1084: 1080: 1076: 1072: 1071: 1070: 1066: 1062: 1061:Graham Beards 1058: 1054: 1053: 1052: 1048: 1044: 1040: 1035: 1034: 1033: 1029: 1025: 1024:Graham Beards 1021: 1020: 1019: 1015: 1011: 1007: 1006: 1005: 1004: 1000: 996: 995:Graham Beards 992: 984: 982: 981: 977: 973: 969: 965: 961: 951: 947: 943: 942:Ebony Jackson 938: 937: 936: 932: 928: 925: 920: 914: 910: 906: 905:Ebony Jackson 902: 898: 894: 893: 892: 888: 884: 880: 875: 874: 873: 869: 865: 861: 860:group axioms" 859: 853: 852: 847: 844: 839: 838: 837: 836: 833: 829: 825: 820: 817: 816: 815: 814: 810: 806: 805:Ebony Jackson 802: 798: 794: 790: 786: 781: 777: 773: 769: 763: 759: 755: 751: 747: 742: 738: 734: 728: 724: 720: 715: 711: 707: 703: 699: 696: 693: 689: 684: 680: 676: 670: 666: 662: 657: 653: 649: 645: 641: 638: 635: 631: 627: 623: 619: 615: 611: 607: 603: 599: 595: 593:Associativity 592: 591: 590: 588: 584: 580: 575: 573: 569: 565: 560: 556: 551: 547: 543: 538: 534: 529: 525: 521: 517: 513: 510: 506: 501: 498: 496: 490: 488: 484: 476: 468: 463: 458: 457: 443: 442: 437: 436:7 (Dec 17 - ) 434: 432: 429: 427: 424: 422: 419: 417: 414: 412: 409: 407: 404: 403: 402: 401: 398: 394: 390: 386: 382: 378: 374: 371: 370: 366: 361: 356: 355: 352: 337: 333: 327: 324: 323: 320: 303: 299: 295: 294: 286: 280: 275: 273: 270: 266: 265: 261: 255: 252: 249: 245: 240: 236: 230: 222: 221: 211: 207: 202: 201: 190: 185: 181: 179: 178: 174: 171: 167: 166: 162: 160: 159: 155: 152: 148: 147: 143: 141: 140: 136: 133: 132:June 15, 2008 129: 128: 124: 122: 121: 117: 114: 110: 109: 105: 103: 102: 98: 95: 94:April 7, 2008 91: 90: 86: 83: 80: 79: 74: 70: 66: 62: 58: 55: 51: 50: 46: 42: 38: 37: 32: 28: 25: 22: 18: 17: 3432: 3426: 3361: 3324:— Preceding 3321: 3313: 3231: 3165: 3161: 3157: 2986: 2966: 2934: 2860: 2802: 2768:Hermann Weyl 2761: 2757:WP:consensus 2738: 2707: 2650: 2647: 2624: 2589: 2414: 2395: 2369: 2275: 2191: 2154: 2139: 2044:up to date. 1631: 1596: 1562: 1356: 1330:Done for g. 1316:Done for p. 1302:Done for j. 1239: 1113: 1111: 988: 967: 963: 959: 956: 857: 803: 799: 792: 788: 784: 779: 775: 772: 770: 767: 761: 753: 749: 745: 740: 736: 732: 726: 722: 718: 713: 709: 705: 701: 691: 682: 678: 674: 668: 664: 660: 655: 651: 647: 643: 633: 629: 625: 621: 617: 613: 609: 605: 601: 597: 587:group axioms 586: 582: 578: 576: 570:" is called 567: 563: 558: 554: 552:, the value 549: 545: 541: 536: 532: 527: 523: 519: 515: 511: 502: 499: 491: 482: 480: 461: 364: 351: 332:Top-priority 331: 291: 257:Top‑priority 235:WikiProjects 218: 188: 175: 170:May 22, 2021 156: 137: 118: 113:May 14, 2008 99: 45:please do so 34: 26: 2825:Space group 2764:POV pushing 2714:mathematics 2644:Indentation 2157:FemkeMilene 2141:FemkeMilene 2123:FemkeMilene 2060:FemkeMilene 2010:FemkeMilene 1822:FemkeMilene 1672:FemkeMilene 1636:FemkeMilene 1584:FemkeMilene 1565:Femkemilene 1546:FemkeMilene 1514:FemkeMilene 1445:FemkeMilene 1384:FemkeMilene 1240:Citations: 1223:FemkeMilene 895:Thank you, 756:is unique ( 612:, one has ( 307:Mathematics 298:mathematics 254:Mathematics 139:Peer review 3470:Categories 3358:loop space 2755:, with no 2419:MathSciNet 2031:XOR'easter 1864:XOR'easter 1465:XOR'easter 1427:XOR'easter 1398:XOR'easter 1332:XOR'easter 1185:XOR'easter 968:of the set 960:of the set 716:such that 530:, denoted 106:Not listed 36:identified 3409:jacobolus 3393:jacobolus 3346:jacobolus 3125:GLenPLonk 2970:Petergans 2919:Petergans 2911:chirality 2655:Petergans 2631:Petergans 2392:this edit 2319:clarified 2261:addressed 922:example: 758:see below 700:For each 688:see below 581:is a set 223:is rated 63:, and on 3453:D.Lazard 3429:D.Lazard 3405:Gumshoe2 3375:Gumshoe2 3326:unsigned 3121:Category 2952:Gumshoe2 2753:WP:UNDUE 2730:Thurston 2690:D.Lazard 2674:D.Lazard 2592:Hog Farm 2356:WP:SYNTH 2282:Hog Farm 2264:this. — 1742:Gumshoe2 1720:objects. 1705:Gumshoe2 1563:Thanks, 927:Andrewsk 901:D.Lazard 883:D.Lazard 824:D.Lazard 744:, where 596:For all 462:365 days 365:Archives 225:FA-class 163:Promoted 144:Reviewed 3438:Nuretok 3169:Dirac66 3159:group. 2890:Mathsci 2845:Mathsci 2786:Mathsci 2476:Quondum 2406:Quondum 2370:journal 2361:Quondum 2324:Quondum 2304:Quondum 2266:Quondum 1887:Done. — 1834:Fixed. 1695:Quondum 964:produce 897:IBugOne 864:IBugOne 858:abelian 843:Quondum 572:closure 507:⋅ on a 477:Closure 334:on the 84:Process 3285:, and 2940:(talk) 2885:R.e.b. 2866:(talk) 2808:(talk) 2726:Conway 2718:WP:CIR 2502:Done. 1960:cited? 1806:Done. 1486:Done. 1463:line. 1057:WP:FAR 991:review 752:, the 483:before 231:scale. 125:Listed 87:Result 2471:Fixed 2296:I've 1649:here. 1627:sure. 972:Quohx 579:group 212:This 29:is a 3457:talk 3442:talk 3379:talk 3334:talk 3218:talk 3204:talk 3189:talk 3173:talk 3144:talk 3129:talk 2974:talk 2956:talk 2923:talk 2894:talk 2878:Ah, 2849:talk 2790:talk 2743:and 2734:Weyl 2694:talk 2678:talk 2668:See 2659:talk 2635:talk 2611:talk 2596:Talk 2579:talk 2561:talk 2533:talk 2508:talk 2474:. — 2442:talk 2427:talk 2382:talk 2345:talk 2286:Talk 2245:talk 2230:and 2216:talk 2178:talk 2161:talk 2145:talk 2127:talk 2113:talk 2079:talk 2064:talk 2050:talk 2035:talk 2014:talk 1999:talk 1974:talk 1947:talk 1918:talk 1893:talk 1868:talk 1840:talk 1826:talk 1812:talk 1786:talk 1746:talk 1732:talk 1709:talk 1676:talk 1659:talk 1640:talk 1614:talk 1588:talk 1573:talk 1550:talk 1536:talk 1518:talk 1492:talk 1469:talk 1449:talk 1431:talk 1402:talk 1388:talk 1365:talk 1336:talk 1322:talk 1308:talk 1283:talk 1258:talk 1227:talk 1216:talk 1189:talk 1175:talk 1161:talk 1133:talk 1093:talk 1079:talk 1065:talk 1047:talk 1028:talk 1014:talk 1010:Ozob 999:talk 993:? -- 976:talk 970:". 946:talk 931:talk 909:talk 887:talk 868:talk 828:talk 809:talk 730:and 672:and 620:) ⋅ 566:set 564:same 495:ring 182:Kept 81:Date 3412:(t) 3396:(t) 3391:. – 3365:it. 3349:(t) 3293:ker 2843:). 2712:on 2651:all 2226:Do 2023:GAP 1630:2. 1357:any 962:to 712:in 704:in 692:the 654:in 646:in 628:⋅ ( 608:in 548:in 509:set 326:Top 3472:: 3459:) 3444:) 3381:) 3336:) 3311:. 3299:ϕ 3296:⁡ 3273:↪ 3265:, 3249:∼ 3244:→ 3220:) 3206:) 3191:) 3175:) 3146:) 3131:) 3104:→ 3065:∈ 2976:) 2958:) 2925:) 2896:) 2851:) 2792:) 2728:, 2696:) 2680:) 2672:. 2661:) 2637:) 2613:) 2581:) 2563:) 2535:) 2510:) 2444:) 2429:) 2384:) 2347:) 2316:I 2247:) 2218:) 2188:HF 2180:) 2163:) 2147:) 2129:) 2115:) 2081:) 2066:) 2052:) 2037:) 2016:) 2001:) 1976:) 1949:) 1920:) 1895:) 1870:) 1842:) 1828:) 1814:) 1788:) 1748:) 1734:) 1711:) 1678:) 1661:) 1642:) 1616:) 1590:) 1575:) 1552:) 1538:) 1520:) 1494:) 1471:) 1451:) 1433:) 1404:) 1390:) 1377:: 1367:) 1338:) 1324:) 1310:) 1285:) 1260:) 1229:) 1218:) 1191:) 1177:) 1163:) 1135:) 1095:) 1081:) 1067:) 1049:) 1030:) 1016:) 1001:) 978:) 948:) 933:) 911:) 889:) 881:. 870:) 830:) 811:) 791:→ 787:× 776:b: 739:= 735:⋅ 725:= 721:⋅ 681:= 677:⋅ 667:= 663:⋅ 636:). 632:⋅ 624:= 616:⋅ 604:, 600:, 589:: 577:A 557:⋅ 544:, 535:⋅ 518:, 503:A 395:, 391:, 387:, 383:, 379:, 375:, 3455:( 3440:( 3427:@ 3403:@ 3377:( 3332:( 3216:( 3202:( 3187:( 3171:( 3142:( 3127:( 3107:a 3101:a 3098:: 3093:g 3089:f 3068:G 3062:g 3042:a 3020:G 3016:A 2995:G 2972:( 2954:( 2921:( 2892:( 2847:( 2788:( 2692:( 2684:÷ 2676:( 2657:( 2633:( 2609:( 2577:( 2573:— 2559:( 2531:( 2506:( 2440:( 2425:( 2380:( 2359:— 2343:( 2243:( 2214:( 2204:? 2176:( 2159:( 2143:( 2125:( 2111:( 2077:( 2062:( 2048:( 2033:( 2012:( 1997:( 1972:( 1945:( 1916:( 1891:( 1866:( 1838:( 1824:( 1810:( 1784:( 1744:( 1730:( 1707:( 1690:3 1686:3 1674:( 1657:( 1638:( 1612:( 1586:( 1571:( 1548:( 1534:( 1516:( 1490:( 1467:( 1447:( 1429:( 1400:( 1386:( 1363:( 1334:( 1320:( 1306:( 1281:( 1256:( 1225:( 1214:( 1187:( 1173:( 1159:( 1131:( 1091:( 1077:( 1063:( 1045:( 1026:( 1012:( 997:( 974:( 944:( 929:( 907:( 885:( 866:( 856:" 826:( 807:( 796:. 793:G 789:G 785:G 780:G 764:. 762:a 754:b 750:a 746:e 741:e 737:a 733:b 727:e 723:b 719:a 714:G 710:b 706:G 702:a 683:a 679:e 675:a 669:a 665:a 661:e 656:G 652:a 648:G 644:e 634:c 630:b 626:a 622:c 618:b 614:a 610:G 606:c 602:b 598:a 583:G 568:G 559:b 555:a 550:G 546:b 542:a 537:b 533:a 528:G 524:G 520:b 516:a 512:G 397:7 393:6 389:5 385:4 381:3 377:2 373:1 338:. 237:: 67:. 47:.

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