Talk:Group (mathematics)

Closure

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

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

A binary operation ⋅ on a set G is a rule for combining any pair a, b of elements of G to form another element of G, denoted a ⋅ b. (The property "for all a, b in G, the value a ⋅ b belongs to the same set G" is called closure; it must be checked if it is not known initially.)

A group is a set G equipped with a binary operation ⋅ satisfying the following three additional requirements, known as the group axioms:

Associativity

For all a, b, c in G, one has (a ⋅ b) ⋅ c = a ⋅ (b ⋅ c).

Identity element

There exists an element e in G such that, for every a in G, the equations e ⋅ a = a and a ⋅ e = a hold. Such an element is unique (see below), and thus one speaks of the identity element.

Inverse element

For each a in G, there exists an element b in G such that a ⋅ b = e and b ⋅ a = e, where e is the identity element. For each a, the b is unique (see below) and it is commonly denoted a⁻¹.

^ b: Formally, a binary operation on G is a function G × G → G.

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

Ebony Jackson (talk) 02:49, 16 December 2020 (UTC)

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.

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. Using this definition, it is a theorem that a subgroup is a group, and that the groups axioms are thus satisfied. D.Lazard (talk) 10:44, 16 December 2020 (UTC)

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 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 ...". —Quondum 11:33, 4 May 2021 (UTC)

I'm very tempted to add Closure as one of the four group axioms, as it's already one of the "abelian group axioms". 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. IBugOne (talk) 14:20, 29 December 2021 (UTC)

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 abelian group#Definition. D.Lazard (talk) 15:00, 29 December 2021 (UTC)

Thank you, IBugOne, for pointing out the discrepancy. I agree with D.Lazard 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. Ebony Jackson (talk) 23:01, 29 December 2021 (UTC)

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 example: https://math.stackexchange.com/questions/65239/right-identity-and-right-inverse-in-a-semigroup-imply-it-is-a-group Andrewsk (talk) 00:06, 20 January 2023 (UTC)

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. Ebony Jackson (talk) 00:03, 23 January 2023 (UTC)

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 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 of the set to produce a third element of the set". Quohx (talk) 06:58, 14 March 2022 (UTC)

Character table

A major omission is any reference to character tables. These tables used extensively in chemistry: see, for example, "Chemical Applications of Group Theory", F.A. Cotton, 3rd. edn., 1990. Petergans (talk) 08:47, 15 March 2022 (UTC)

Indentation

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 all indents to be changed in the text. Petergans (talk) 10:48, 26 March 2022 (UTC)

See Help:Section#Creation and numbering of sections. D.Lazard (talk) 11:46, 26 March 2022 (UTC)÷

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 already very large. D.Lazard (talk) 12:03, 26 March 2022 (UTC)

Restructuring article?

The edits to this featured article on mathematics have been reverted. That was due partially to the misuse of indentation, see WP:CIR; but also changes to content must be supported by reliable sources, 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 Conway, Thurston et al, or to Weyl, was unhelpful; similarly for the citation to Graham Ellis.

As far as groups are concerned, representation theory and character theory are often first encountered in undergraduate courses on finite groups and angular momentum in quantum mechanics (see e.g. the treatment by Jean-Pierre Serre). Separate new sections at the moment seem to be WP:UNDUE, with no WP:consensus. It unbalances the article. The new image without citations is unhelpful.

The edits today to the article are a combination of vandalism, incompetence and POV pushing: why delete references to physicists or Hermann Weyl; why delete images from the section on "Symmetry"; why favour chemistry above physics? Here are diffs of recent problematic edits, including today's. [1][2][3][4][5][6][7] Mathsci (talk) 14:12, 26 March 2022 (UTC)

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 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. — Charles Stewart (talk) 14:19, 26 March 2022 (UTC)

It was in an unacceptable state, given the last diff. In physics, the group-theoretic approach to quantum mechanics and representation theory 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 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. Space group is encyclopedic and explained clearly on the tables in mathematics, physics and chemistry (230 cases); mathematically, Point groups in three dimensions#Finite isometry groups covers the 32 crystallographic point groups. It describes the crystallographic restriction theorem from a mathematical standpoint; and is explained in standard text books on chemistry & group theory (e.g. "Chemical Applications of Group Theory", F. Albert Cotton). Mathsci (talk) 16:43, 26 March 2022 (UTC)

• 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 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.[21] The theory of Lie groups, and more generally locally compact groups was studied by Hermann Weyl, Élie Cartan and many others.[22] 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.[23]". — Charles Stewart (talk) 18:47, 26 March 2022 (UTC)

• Ah, invariant theory. 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 R.e.b. gave Hilbert's non-constructive proof using the averaging or Reynolds operator. Mathsci (talk) 20:42, 26 March 2022 (UTC)

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 chirality cannot be taught without reference to symmetry operations. The designations of many point groups are illustrated at Molecular symmetry#Common point groups, 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. Petergans (talk) 15:13, 26 March 2022 (UTC)

• 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 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. — Charles Stewart (talk) 18:23, 26 March 2022 (UTC)

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. Gumshoe2 (talk) 15:25, 26 March 2022 (UTC)

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 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. Petergans (talk) 20:11, 26 March 2022 (UTC)

Groups as categories

I feel that the "category" point of view is missing : a group $G$ can be seen as a category $A_{G}$ with 1 objeect (call it $a$) where elements $g\in G$ corresponds to isomorphisms $f_{g}:a\to a$, 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 Category, in which case I would try and extend these. GLenPLonk (talk) 14:47, 1 November 2022 (UTC)

Good idea! I tried to implement your suggestion, by adding it to the discussion of groupoids in the Generalizations section. Ebony Jackson (talk) 18:42, 1 November 2022 (UTC)

Identity and also inverse elements must be part of set

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 group.

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.

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 addition: the set 0, +1, +2, +3, +4, ... is NOT a group without the negative integers. The fact that they exist is not sufficient. Dirac66 (talk) 02:01, 10 July 2023 (UTC)

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 operation with things in the set and so in particular they cannot be an identity or an inverse. --100.36.106.199 (talk) 13:49, 10 July 2023 (UTC)

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 meaning is not actually ambiguous or otherwise problematic. --100.36.106.199 (talk) 13:52, 10 July 2023 (UTC)

Having said all that: the revised wording seems fine. --100.36.106.199 (talk) 13:54, 10 July 2023 (UTC)

Undefined terms and notational elements

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: ${\stackrel {\sim }{\to }}$, $\hookrightarrow$, and $\ker \phi$.

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

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

"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"? — Preceding unsigned comment added by 2404:4408:6A6E:7000:E48B:A59:8E82:2FCF (talk) 08:21, 3 October 2023 (UTC)

In this case, the relevant group is the integers under the addition operation. –jacobolus (t) 17:24, 3 October 2023 (UTC)

The given quote could be considered incorrect: the loop space consists of loops and the fundamental group crucially consists of equivalence classes 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 it.

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 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.

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 move the paragraph to its own subsection "algebraic topology" or "fundamental group". Gumshoe2 (talk) 19:31, 3 October 2023 (UTC)

Looking at this picture, I agree it's weird. We probably instead want something like the pictures in Winding number § Intuitive description. –jacobolus (t) 19:37, 3 October 2023 (UTC)

@Gumshoe2 I tried rewriting the explanation here. Is that any clearer? –jacobolus (t) 22:13, 3 October 2023 (UTC)

One can show that

@D.Lazard: I removed the bold text from "...assuming associativity and the existence of a left identity [...] and a left inverse [...] for each element [...], one can show that 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. Is there a way it could be true without having been shown to be true? Nuretok (talk) 13:20, 26 May 2024 (UTC)

This is true, but it is not an evidence. See your talk page. D.Lazard (talk) 13:26, 26 May 2024 (UTC)

Error in examples: division over reals has quasi-group structure despite no closure.

In the final section of the article (Generalizations) there is a table on operations over different sets. Division under the reals is listed as closed but the group structure is given as "quasi-group" despite the table above clearly stating that closure is necessary for a quasi-group structure. Perhaps it could be made more clear what is meant by the table.

Edit: After looking over the table a bit more there are a few confusing things about it. It is not explained why something is "N/A". I found myself double checking many things as I am not a group theorist. Nathalene (talk) 21:11, 24 October 2024 (UTC)