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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, 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. [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:
^ 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)
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)
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 review? -- Graham Beards ( talk) 11:09, 20 April 2021 (UTC)
This is an article that will have many paragraphs that fall squarely under the Subject-specific common knowledge, 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
Citations:
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 instructions."). Can it be found in simpler sources too? FemkeMilene ( talk) 16:13, 3 May 2021 (UTC)
rightmost exampleline. XOR'easter ( talk) 21:47, 26 April 2021 (UTC)
Thanks, Femkemilene for your comments. I have addressed some of them and will work on the remainder asap. Jakob.scholbach ( talk) 09:42, 30 April 2021 (UTC)
Comments from my second read:
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 objects.", 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 History of group theory, 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. — David Eppstein ( talk) 06:20, 15 February 2022 (UTC)
FemkeMilene ( talk) 16:13, 3 May 2021 (UTC)
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.
Well, frankly, I understood little of this, so I may just be plain wrong on my comments. Hog Farm Talk 22:05, 7 May 2021 (UTC)
@ Imaginatorium: I saw you reverted my edit of the short description. My edit removed content, but it was in-line with the purpose of the short description, see WP:SHORTDES. I made a few such edits recently and there is currently a discussion over in the Project Math talk page where I elaborate on my reasoning. To summarize here: the purpose of the short description is to briefly indicate the field covered by the article, and ( explicitly) not to define the subject of the article. Notable examples exhibiting a similar degree of brevity include "American baseball player" for Babe Ruth or "U.S. State" for Florida. Feel free to add to the discussion if you wish to. Whether or not you agree with me, your opinion is welcome. Donko XI ( talk) 11:24, 21 January 2022 (UTC)
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)
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)
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)
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)
I feel that the "category" point of view is missing : a group can be seen as a category with 1 objeect (call it ) where elements corresponds to isomorphisms , 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)
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)
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: , , and .
The latter also appears in the Presentations section, along with reference to the free group
"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)
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. | |||||||||||||||||||||||||
This article appeared on Wikipedia's Main Page as Today's featured article on November 5, 2008, and on March 14, 2022. | |||||||||||||||||||||||||
|
This
level-4 vital article is rated FA-class on Wikipedia's
content assessment scale. It is of interest to the following WikiProjects: | |||||||||||
|
1, 2, 3, 4, 5, 6, 7 |
This page has archives. Sections older than 365 days may be automatically archived by Lowercase sigmabot III when more than 10 sections are present. |
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. [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:
^ 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)
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)
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 review? -- Graham Beards ( talk) 11:09, 20 April 2021 (UTC)
This is an article that will have many paragraphs that fall squarely under the Subject-specific common knowledge, 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
Citations:
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 instructions."). Can it be found in simpler sources too? FemkeMilene ( talk) 16:13, 3 May 2021 (UTC)
rightmost exampleline. XOR'easter ( talk) 21:47, 26 April 2021 (UTC)
Thanks, Femkemilene for your comments. I have addressed some of them and will work on the remainder asap. Jakob.scholbach ( talk) 09:42, 30 April 2021 (UTC)
Comments from my second read:
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 objects.", 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 History of group theory, 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. — David Eppstein ( talk) 06:20, 15 February 2022 (UTC)
FemkeMilene ( talk) 16:13, 3 May 2021 (UTC)
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.
Well, frankly, I understood little of this, so I may just be plain wrong on my comments. Hog Farm Talk 22:05, 7 May 2021 (UTC)
@ Imaginatorium: I saw you reverted my edit of the short description. My edit removed content, but it was in-line with the purpose of the short description, see WP:SHORTDES. I made a few such edits recently and there is currently a discussion over in the Project Math talk page where I elaborate on my reasoning. To summarize here: the purpose of the short description is to briefly indicate the field covered by the article, and ( explicitly) not to define the subject of the article. Notable examples exhibiting a similar degree of brevity include "American baseball player" for Babe Ruth or "U.S. State" for Florida. Feel free to add to the discussion if you wish to. Whether or not you agree with me, your opinion is welcome. Donko XI ( talk) 11:24, 21 January 2022 (UTC)
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)
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)
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)
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)
I feel that the "category" point of view is missing : a group can be seen as a category with 1 objeect (call it ) where elements corresponds to isomorphisms , 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)
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)
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: , , and .
The latter also appears in the Presentations section, along with reference to the free group
"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)