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The article has been extensively revised for greater clarity. The emphasis is on the interpretation of experimental data. Details on mathematical aspects have been removed as they were all but unintelligible to non-mathematicians. Interested readers are pointed to mathematical articles by the re-direct at the head of the article. Petergans ( talk) 10:17, 21 June 2022 (UTC)
There is no connection at all with Spin group
revised for greater clarity
Details on mathematical aspects have been removed
In mathematics, the term "double group" can be applied to any group which is the direct product of two groups
IMHO, Double group should be designed for a double audience (!); firstly chemists and physicists who want to better understand a mathematical concept that they have encountered elsewhere. Secondly mathematicians who want to understand how very abstract concepts of group theory can be useful in chemistry. For both audiences, the mathematics must be accurate. This is as a tentative in this direction that I have written user:D.Lazard/Double group as a project for the first paragraph of the lead. This is what I have understood from the different versions of the article. If I have not misunderstood them, this is the kind of lead that I would expect for knowing whether I am interested in the article.
Clearly, such a lead must be completed with a section explaining the mathematical background, and another section explaining the relations with the mathematical concepts and the physical properties (I still do not understand the physical role of the characters). I believe that the solution of the concerns of Qfkib and IpseCustos pass through such an approach, even in the case were my text is wrong. D.Lazard ( talk) 16:30, 23 June 2022 (UTC)
Petergans, I am sorry that you had to deal (and perhaps still have to deal) with mathsci's nonsense for so long; I know from experience how unrewarding it is. I fully agree with you that, as per usual, much of what he added was not right for this page or not comprehensibly explained and that he was not able to justify its presence here on the talk page. But I think that SU(2) (also called Spin(3) since it falls into the general context of spin groups) and SO(3) are highly relevant to this page and that it is a major omission to not include them explicitly - especially since (despite what you might conclude from mathsci's contributions) they are not terribly complicated and only require the barest rudiments of group theory to say something sensible about. There is a natural and remarkable group homomorphism from SU(2) to SO(3), the molecular point group (if I understand correctly) is a finite subgroup of SO(3), and the corresponding double group is its preimage as a finite subgroup of SU(2). Since the mapping from SU(2) to SO(3) maps two inputs to every output, this "double group" has two points for every point of the original finite subgroup. This seems like a bare minimum of mathematics which should be communicated on the page. (Also, mathsci/ipsecustos are correct to say that "direct product" is not the right keyword for this, as it is an example of the broader concept of "group extension".)
Moreover, you say that it is not good to present mathematics which is unintelligible to non-mathematicians. But what about chemistry which is unintelligible to non-chemists? (At least, I am not a chemist and the page at present is all but unintelligible to me.) Gumshoe2 ( talk) 08:11, 26 June 2022 (UTC)
Many thanks to Qflib and D.Lazard for the constructive comments, above.
The distinguishing feature of a "double group" is that the symmetry operation of rotation by 360° is classed as an operation which is distinct from an "identity" or any other point group operation. This is specific to magnetochemistry. It is needed to take account of the half integer value of spin quantum number of an electron in a metal ion that is at the center of a "complex". Character tables for many double groups are given in the booklet by Salthouse & Ware.
Any two groups can be combined together to create a third group. For example the point group C2, when combined the group containing the symmetry operations identity and a mirror plane (E,σ) results in the formation of the point group C2h, assuming that the the mirror plane is perpendicular to the rotation axis. In this example the group C2 has the two symmetry operations E and C2; the product has 4 operation, E, C2, i and σh. The product group is not considered to be a double group. As this example shows, this is a relatively trivial situation. In all similar cases (e.g. C4 → C4h), the number of operations in the resulting group is double the number of operations in the larger original group.
What is needed is a new section in the article Group theory to describe the process and consequences of combining two groups together. Petergans ( talk) 10:12, 24 June 2022 (UTC)
it does not make sense to include pure mathematical details here
a double group is clearly related to a point group and the group {E,R}
There is no mathematics in this article
A general reversion will be treated as vandalism
The case for the inclusion of more mathematical detail in this article has not been made
there are links to articles which cover relevant mathematical details
I'm attaching the {{dubious}} tag to the following mathematical statements:
Such obvious nonsense should, of course, simply be deleted. Unfortunately, that will have to wait for dispute resolution.
IpseCustos ( talk) 15:59, 25 June 2022 (UTC)
Should this article exclude mathematical content, such as a definition of "double group"? IpseCustos ( talk) 18:48, 26 June 2022 (UTC)
how much mathematics should be in this article?, and
which mathematical style should be used?.
"Double group" is the term used in magnetochemistry to refer to a finite subgroup of spin(3) that is not a subgroup of the group of the rotations.(After three months of hard discussions in this talk page, and edit war on the article, I am not even sure that this formulation is correct.)
There is no mathematics in this article, and when he kept reverting any changes to the article's mathematical content.
not a subgroup of the group of the rotationsI'm not sure what that's supposed to mean. Spin(3) doesn't contain SO(3). The double group of a point group is its pre-image in Spin(3), so a finite subgroup of Spin(3) is a double group iff it contains an even number of elements iff it contains the involution element of Spin(3) iff it is closed under multiplication with the involution element of Spin(3). IpseCustos ( talk) 10:09, 27 June 2022 (UTC)
A point group is a sub-group of the corresponding double groupSee binary tetrahedral group for an example that is not. IpseCustos ( talk) 11:17, 27 June 2022 (UTC)
I created the article based on ..... Per WP:OWN, this is out of scope of this discussion. IMHO, the main problem that motivated this RfC is that you behave as if you were the owner of this article. Wikipedia does not work this way. So, please read WP:OWN, and use it for helping to reach a consensus. D.Lazard ( talk) 12:25, 27 June 2022 (UTC)
I have now received a copy of Bethe's article. He uses the adjective "Zweideutige" (two distinct) for some representations in what we now call double groups and gives character tables for some of them. Unfortunately there are no physical examples in his article.
I'm sorry if I give the impression of wanting to be the owner of this page. My intention is, rather, to make it intelligible to all interested readers. To this end I had renamed the article "Double group (magnetochemistry)" and now suggest that it be reverted back to this title, in line with the two book chapters used as principal sources. Hopefully, that would put and end to all this controversy.
A
binary tetrahedral group has been used in relation to nuclear physics. That article will, unfortunately, be completely incomprehensible except to mathematicians. It would not sit well with the subject matter of this article.
Petergans (
talk)
15:42, 27 June 2022 (UTC)
I agree with the goals that Charles just enumerated, and the suggestion of adding a third section makes sense to me. Qflib, aka KeeYou Flib ( talk) 15:57, 28 June 2022 (UTC)
D'4 | E | C4 | C43 | C2 | 2C'2 | 2C''2 | |
---|---|---|---|---|---|---|---|
R | C4R | C43R | C2R | 2C'2R | 2C''2R |
the point group D4, which is a sub-group of D'4But it is not. There's a single self-inverse element in D'4, but there are several in D4.
rotation by 720° is never mentioned explicitlyOf course it is, and it should be. There are good mathematical reasons for that, which I accept may be too specialized for this article, but the fact itself must be mentioned. IpseCustos ( talk) 16:16, 3 July 2022 (UTC)
What is the group D4 that is considered in the article and this discussion? In Wikipedia, there are two groups called D4, the dihedral groups with 4 or 8 elements. None is a group of rotations. So I believe that the group called D4 by Petergans is the cyclic group of order 4, commonly called C4. However, I may be wrong, but, please, define clearly your notations in the article, as well in this discussion. D.Lazard ( talk) 16:56, 6 July 2022 (UTC)
A square will be classed as C4 in 2-D, and D4h in 3-D ...Again, this is non sensical if you do not define what do you mean by C4 and D4h Also, from what you have written before, I guess that C4 and D4h should be groups; as a square is not a group, this makes the sentence even more non-sensical. D.Lazard ( talk) 08:15, 8 July 2022 (UTC)
None is a group of rotationsI don't understand that comment. All dihedral groups are groups of 3D rotations, since a reflection is merely a 180° rotation. In this case, D4 is the dihedral group with 8 elements, 4 of which are self-inverse (the reflections). D'4 has 16 elements, only one of which is self-inverse: both of the preimages of a reflection have order four (and there's no natural choice of which one is the correct one, so it's very lazy to use the same notation for one of the preimages and the reflection in D4). IpseCustos ( talk) 11:20, 8 July 2022 (UTC)
This article was nominated for deletion on 17 March 2022. The result of the discussion was keep. |
This article is rated Start-class on Wikipedia's
content assessment scale. It is of interest to the following WikiProjects: | |||||||||||
|
|
|
The article has been extensively revised for greater clarity. The emphasis is on the interpretation of experimental data. Details on mathematical aspects have been removed as they were all but unintelligible to non-mathematicians. Interested readers are pointed to mathematical articles by the re-direct at the head of the article. Petergans ( talk) 10:17, 21 June 2022 (UTC)
There is no connection at all with Spin group
revised for greater clarity
Details on mathematical aspects have been removed
In mathematics, the term "double group" can be applied to any group which is the direct product of two groups
IMHO, Double group should be designed for a double audience (!); firstly chemists and physicists who want to better understand a mathematical concept that they have encountered elsewhere. Secondly mathematicians who want to understand how very abstract concepts of group theory can be useful in chemistry. For both audiences, the mathematics must be accurate. This is as a tentative in this direction that I have written user:D.Lazard/Double group as a project for the first paragraph of the lead. This is what I have understood from the different versions of the article. If I have not misunderstood them, this is the kind of lead that I would expect for knowing whether I am interested in the article.
Clearly, such a lead must be completed with a section explaining the mathematical background, and another section explaining the relations with the mathematical concepts and the physical properties (I still do not understand the physical role of the characters). I believe that the solution of the concerns of Qfkib and IpseCustos pass through such an approach, even in the case were my text is wrong. D.Lazard ( talk) 16:30, 23 June 2022 (UTC)
Petergans, I am sorry that you had to deal (and perhaps still have to deal) with mathsci's nonsense for so long; I know from experience how unrewarding it is. I fully agree with you that, as per usual, much of what he added was not right for this page or not comprehensibly explained and that he was not able to justify its presence here on the talk page. But I think that SU(2) (also called Spin(3) since it falls into the general context of spin groups) and SO(3) are highly relevant to this page and that it is a major omission to not include them explicitly - especially since (despite what you might conclude from mathsci's contributions) they are not terribly complicated and only require the barest rudiments of group theory to say something sensible about. There is a natural and remarkable group homomorphism from SU(2) to SO(3), the molecular point group (if I understand correctly) is a finite subgroup of SO(3), and the corresponding double group is its preimage as a finite subgroup of SU(2). Since the mapping from SU(2) to SO(3) maps two inputs to every output, this "double group" has two points for every point of the original finite subgroup. This seems like a bare minimum of mathematics which should be communicated on the page. (Also, mathsci/ipsecustos are correct to say that "direct product" is not the right keyword for this, as it is an example of the broader concept of "group extension".)
Moreover, you say that it is not good to present mathematics which is unintelligible to non-mathematicians. But what about chemistry which is unintelligible to non-chemists? (At least, I am not a chemist and the page at present is all but unintelligible to me.) Gumshoe2 ( talk) 08:11, 26 June 2022 (UTC)
Many thanks to Qflib and D.Lazard for the constructive comments, above.
The distinguishing feature of a "double group" is that the symmetry operation of rotation by 360° is classed as an operation which is distinct from an "identity" or any other point group operation. This is specific to magnetochemistry. It is needed to take account of the half integer value of spin quantum number of an electron in a metal ion that is at the center of a "complex". Character tables for many double groups are given in the booklet by Salthouse & Ware.
Any two groups can be combined together to create a third group. For example the point group C2, when combined the group containing the symmetry operations identity and a mirror plane (E,σ) results in the formation of the point group C2h, assuming that the the mirror plane is perpendicular to the rotation axis. In this example the group C2 has the two symmetry operations E and C2; the product has 4 operation, E, C2, i and σh. The product group is not considered to be a double group. As this example shows, this is a relatively trivial situation. In all similar cases (e.g. C4 → C4h), the number of operations in the resulting group is double the number of operations in the larger original group.
What is needed is a new section in the article Group theory to describe the process and consequences of combining two groups together. Petergans ( talk) 10:12, 24 June 2022 (UTC)
it does not make sense to include pure mathematical details here
a double group is clearly related to a point group and the group {E,R}
There is no mathematics in this article
A general reversion will be treated as vandalism
The case for the inclusion of more mathematical detail in this article has not been made
there are links to articles which cover relevant mathematical details
I'm attaching the {{dubious}} tag to the following mathematical statements:
Such obvious nonsense should, of course, simply be deleted. Unfortunately, that will have to wait for dispute resolution.
IpseCustos ( talk) 15:59, 25 June 2022 (UTC)
Should this article exclude mathematical content, such as a definition of "double group"? IpseCustos ( talk) 18:48, 26 June 2022 (UTC)
how much mathematics should be in this article?, and
which mathematical style should be used?.
"Double group" is the term used in magnetochemistry to refer to a finite subgroup of spin(3) that is not a subgroup of the group of the rotations.(After three months of hard discussions in this talk page, and edit war on the article, I am not even sure that this formulation is correct.)
There is no mathematics in this article, and when he kept reverting any changes to the article's mathematical content.
not a subgroup of the group of the rotationsI'm not sure what that's supposed to mean. Spin(3) doesn't contain SO(3). The double group of a point group is its pre-image in Spin(3), so a finite subgroup of Spin(3) is a double group iff it contains an even number of elements iff it contains the involution element of Spin(3) iff it is closed under multiplication with the involution element of Spin(3). IpseCustos ( talk) 10:09, 27 June 2022 (UTC)
A point group is a sub-group of the corresponding double groupSee binary tetrahedral group for an example that is not. IpseCustos ( talk) 11:17, 27 June 2022 (UTC)
I created the article based on ..... Per WP:OWN, this is out of scope of this discussion. IMHO, the main problem that motivated this RfC is that you behave as if you were the owner of this article. Wikipedia does not work this way. So, please read WP:OWN, and use it for helping to reach a consensus. D.Lazard ( talk) 12:25, 27 June 2022 (UTC)
I have now received a copy of Bethe's article. He uses the adjective "Zweideutige" (two distinct) for some representations in what we now call double groups and gives character tables for some of them. Unfortunately there are no physical examples in his article.
I'm sorry if I give the impression of wanting to be the owner of this page. My intention is, rather, to make it intelligible to all interested readers. To this end I had renamed the article "Double group (magnetochemistry)" and now suggest that it be reverted back to this title, in line with the two book chapters used as principal sources. Hopefully, that would put and end to all this controversy.
A
binary tetrahedral group has been used in relation to nuclear physics. That article will, unfortunately, be completely incomprehensible except to mathematicians. It would not sit well with the subject matter of this article.
Petergans (
talk)
15:42, 27 June 2022 (UTC)
I agree with the goals that Charles just enumerated, and the suggestion of adding a third section makes sense to me. Qflib, aka KeeYou Flib ( talk) 15:57, 28 June 2022 (UTC)
D'4 | E | C4 | C43 | C2 | 2C'2 | 2C''2 | |
---|---|---|---|---|---|---|---|
R | C4R | C43R | C2R | 2C'2R | 2C''2R |
the point group D4, which is a sub-group of D'4But it is not. There's a single self-inverse element in D'4, but there are several in D4.
rotation by 720° is never mentioned explicitlyOf course it is, and it should be. There are good mathematical reasons for that, which I accept may be too specialized for this article, but the fact itself must be mentioned. IpseCustos ( talk) 16:16, 3 July 2022 (UTC)
What is the group D4 that is considered in the article and this discussion? In Wikipedia, there are two groups called D4, the dihedral groups with 4 or 8 elements. None is a group of rotations. So I believe that the group called D4 by Petergans is the cyclic group of order 4, commonly called C4. However, I may be wrong, but, please, define clearly your notations in the article, as well in this discussion. D.Lazard ( talk) 16:56, 6 July 2022 (UTC)
A square will be classed as C4 in 2-D, and D4h in 3-D ...Again, this is non sensical if you do not define what do you mean by C4 and D4h Also, from what you have written before, I guess that C4 and D4h should be groups; as a square is not a group, this makes the sentence even more non-sensical. D.Lazard ( talk) 08:15, 8 July 2022 (UTC)
None is a group of rotationsI don't understand that comment. All dihedral groups are groups of 3D rotations, since a reflection is merely a 180° rotation. In this case, D4 is the dihedral group with 8 elements, 4 of which are self-inverse (the reflections). D'4 has 16 elements, only one of which is self-inverse: both of the preimages of a reflection have order four (and there's no natural choice of which one is the correct one, so it's very lazy to use the same notation for one of the preimages and the reflection in D4). IpseCustos ( talk) 11:20, 8 July 2022 (UTC)