Talk:Double group

General revision

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)

I don't understand why that link goes to direct product of groups. Double groups are not, in general, nontrivial direct products of groups. IpseCustos (talk) 11:00, 21 June 2022 (UTC)

A link to relevant mathematical theory is needed. Can you suggest a better one? Petergans (talk) 13:27, 21 June 2022 (UTC)

Spin group#Discrete subgroups? I don't understand quite what the difference is between a (finite) double group and a binary polyhedral group. Frankly the only case I can kind of understand is that of a cyclic group generated by a rotation by ${\frac {2\pi }{n}}$, with its double group cyclic of order 2n (so, in general, not a direct product).

And then I get lost as soon as the character is given a physical interpretation, I'm afraid, so if I'm part of the intended audience for this article (and I suspect I'm not), I'd appreciate it if more detail about that could be provided.

(I do not understand either version of the article well enough to have an opinion on your rewrite of it) IpseCustos (talk) 14:04, 21 June 2022 (UTC)

There is no connection at all with Spin group. It is unfortunate that the term spin can have many different meanings. Some time ago I changed the name of the current article to Double Group (Magnetochemistry) but the name change was later reverted. I will make some enquiries to see if the article can be given a less ambiguous name. Petergans (talk) 19:16, 21 June 2022 (UTC)

There is no connection at all with Spin group

I am very surprised to hear that. The article before you edited it certainly was about certain finite subgroups of SU(2) = Spin(3) and their images in SO(3). If what you're writing about is so different from that, why not create the double group (magnetochemistry) article anyway? IpseCustos (talk) 19:35, 21 June 2022 (UTC)

revised for greater clarity

I don't think that worked, sorry.

Details on mathematical aspects have been removed

Those "details" appear to include any attempt to actually define what is being talked about.

In mathematics, the term "double group" can be applied to any group which is the direct product of two groups

Can you give a reference for that? As every group is a direct product of two groups, the statement seems nonsensical.

In general, I preferred the old version, and think we should edit that instead of rewriting it completely. It doesn't even appear clear that we're still talking about the same thing, since you don't define double groups and don't think they're related to Spin(3). IpseCustos (talk) 10:56, 22 June 2022 (UTC)

Purely mathematical aspects will be incomprehensible to non-mathematicians. Note that there is no mention of any SU.. in Cotton's book, which devotes a chapter to double groups. The purely mathematical aspects of double groups belongs elsewhere in Wikipedia, with pointers in this article to relevant WP articles.. Petergans (talk) 13:27, 22 June 2022 (UTC)

If you consider providing a definition of "double group" a mathematical aspect of double groups which does not belong in double group, I must disagree.

But I'll wait for others to express an opinion. IpseCustos (talk) 19:51, 22 June 2022 (UTC)

This article is focused on one kind of application for the concept of a "double group" in chemistry. Nothing wrong with that but personally I think we need two separate pages on double groups. One entitled "Double groups in magnetochemistry" or some such (as was more or less originally the case), and a second, more general one entitled "Double groups" designed for a mathematical physics audience - specifically, the one which was replaced by Petergans would be a good starting point. Qflib, aka KeeYou Flib (talk) 14:53, 23 June 2022 (UTC)

Also, this current page needs to start with a definition of what a double group **is** in this context, not what it's **used for.** Qflib, aka KeeYou Flib (talk) 14:55, 23 June 2022 (UTC)

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)

Future developments

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 C_2h, 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 σ_h. 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 → C_4h), 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)

What you're referring to is, in mathematics, called a group extension. "Combining two groups together" is not an established mathematical term for anything. Which is why I changed it, which you just reverted.

While direct products of groups are examples of group extensions, the interesting thing about double groups is that they are not, in general, direct products of the cyclic group of two elements and the original group. The link is thus misleading.

Double groups are precisely what is being described in Spin group#Discrete subgroups. That is why I exchanged the link to "mathematical details" to point to a page which will, at least to mathematicians, provide the mathematical details, rather than one which will not be useful to anyone.

"Double groups" is not an alternative term for "direct product". That statement also needs to be removed. Again, I did that and was reverted, and I now ask you to please provide references for this statement if you think it should remain.

You were bold and made a general revision. The new article is hardly an improvement. Unless you can fix at least the worst of the mathematical confusion, I'm afraid that restoring the old article and discussing any changes to it that you wish to make is the way to go. IpseCustos (talk) 18:03, 24 June 2022 (UTC)

As I implied above, it does not make sense to include pure mathematical details here, as scientists not specifically trained in the mathematics of group theory won't understand a word of it.

It is a regrettable fact that there is such a large gap between pure and applied mathematics in this case, which is illustrated by the statement above: "combining two groups together is not an established mathematical term". That may be factually true, but nevertheless a double group is clearly related to a point group and the group {E,R} in that all its symmetry operations, following the group closure property, are "combinations" of the operations of them. In geometrical terms, "combination" signifies one operation followed by another, in 3-D space. Scientists are familiar with this idea as, for example, an improper rotation, S_n, is often described as a proper rotation, C_n followed by a reflection in a "horizontal" plane. Petergans (talk) 22:47, 24 June 2022 (UTC)

it does not make sense to include pure mathematical details here

I removed mathematical details (that also happen to be inaccurate), which you reverted.

a double group is clearly related to a point group and the group {E,R}

Yes, it's a group extension, of one by the other. Which is why I replaced "combining" by "extending" and linked to the appropriate article, which you reverted.

That the point group is a quotient rather than a subgroup of the double group is absolutely essential, and that's why it's a bad idea to talk about E and R being in different classes when they represent the same class under the equivalence relation induced by the projection. Again, reverted.

If we can't fix the confusing and incorrect mathematics in the current article, I'm going to revert the general revision edit and we can discuss which parts of it to reinstate. IpseCustos (talk) 08:13, 25 June 2022 (UTC)

There is no mathematics in this article, only statements of universally accepted formulae. A general reversion will be treated as vandalism and reported as such. The case for the inclusion of more mathematical detail in this article has not been made. It is sufficient that there are links to articles which cover relevant mathematical details. Petergans (talk) 08:45, 25 June 2022 (UTC)

There is no mathematics in this article

But of course there is!

A general reversion will be treated as vandalism

Reverting a change that there is no consensus for isn't vandalism.

I agree it would be better to keep some of the new content, FWIW. If you want to do that, please identify which parts you think are uncontroversial.

The case for the inclusion of more mathematical detail in this article has not been made

The case for removing most mathematical details from this article has not been made, either.

there are links to articles which cover relevant mathematical details

There are no such links. The mathematical articles linked to aren't relevant, in most cases. IpseCustos (talk) 10:09, 25 June 2022 (UTC)

Okay, I tried, but I'm unable to merge the two articles (pre and post "General Revision") without performing too much general editing. I am therefore going to simply restore the pre-"General Revision" version, in the hope that manageable chunks of changes can then be discussed individually.

There are certainly good parts in the General Revision, but the burden of identifying them and restricting ourselves to those edits should be on the person wishing to make the change, not the one objecting to it. IpseCustos (talk) 11:05, 25 June 2022 (UTC)

Dubious

I'm attaching the {{dubious}} tag to the following mathematical statements:

• For mathematical details see Direct product of groups

Double groups aren't, in general, direct products. The doubling operation is described in detail in spin group#Discrete subgroups.

• In mathematics, the term "double group" can be applied to any group which is the direct product of two groups.

If such terminology exists, it is not about double groups as the article describes them. Double groups aren't, in general, direct products.

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)

RfC: Should this article exclude mathematics?

Should this article exclude mathematical content, such as a definition of "double group"? IpseCustos (talk) 18:48, 26 June 2022 (UTC)

Hmm. The Background section has math content and wouldn't be much good without it. However the concept of a double group isn't mentioned in the page, "Direct product of groups," and there no mention of group products here. That said, neither topic is in my area of expertise. (randomly invited by a bot) Jojalozzo (talk) 22:36, 26 June 2022 (UTC)

• As formulated, the question is nonsensical: "double group" is a mathematical concept used in magnetochemistry. So, excluding mathematical content implies excluding from the article its own subject.
  The true questions are thus how much mathematics should be in this article?, and which mathematical style should be used?.
  As "double group" seems to be a mathematical terminology that is specific to magneto chemistry, the minimum seems to define the concept (per WP:Verifiability) and to relate it to the standard mathematical terminology (per WP:AUDIENCE). So, the article must, at least, contain a sentence such as "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.)
  Clearly the quoted sentence is not sufficient: the linked article is too technical for chemists, and the reasons for which double groups are useful in chemistry are totally obscure for non-specialists of magnetochemistry. So, (1) the article must explain to non-mathematicians the part of the mathematical theory that is used here, and (2) the article must explain to everybody why this mathematical theory is useful in chemistry (the first version of the article used "needed" instead of "useful", without any explanation). D.Lazard (talk) 09:07, 27 June 2022 (UTC)

  I agree the mathematics-free approach is nonsensical, but it's what I think Petergans meant when he said 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 rotations I'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)

  This is exactly what I meant by "I am not even sure that this formulation is correct". Nevertheless, it is a problem that is still open to find a formulation that is both mathematically correct, and not too technical for people who do not know well spin(3). D.Lazard (talk) 10:45, 27 June 2022 (UTC)

I agree. I created the article based on section 4.7, "Double Groups", in Cotton's book, "Chemical Applications of Group Theory". Clearly the target audience of the book was "students" of chemistry. Some introductory material will be added from earlier sections of that book, as context for the general reader.
P.S. A point group is a sub-group of the corresponding double group; for example D₄ is a sub-group of D'₄. "Spin3" is a red herring. Petergans (talk) 10:52, 27 June 2022 (UTC)

A point group is a sub-group of the corresponding double group See binary tetrahedral group for an example that is not. IpseCustos (talk) 11:17, 27 June 2022 (UTC)

(As this article used to state, the binary tetrahedral group is T′ in the notation of this article. So T is not a subgroup of T′). IpseCustos (talk) 18:07, 27 June 2022 (UTC)

To what do you agree? It was very obscure, even before I restored the chronological order of the posts.

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)

• Before I answer the RfC question, an aside: ideally I think we could rank goals for the presentation of the article:
  1. Most importantly, it should accessible to mathematically inclined chemists. If most undergraduate chemists with an interest in the topic find the first paragraph contains sentences they cannot grasp, we have failed here;
  2. Then it should help algebraists find concrete applications in finite group theory; and
  3. Finally it should be a resource that helps us navigate the differences between group theory as it is described in theoretical chemistry and as it is described by mathematicians. This should not be done in a way that injures overall readability.

The current lead seems to be good with respect to the first point; in the past, attempts to have the kind of language we use in maths articles has meant the we had an article that I guess chemists would have found off-putting. The lead and article is also OK with respect to the second point. The article provides no help with regards to the third point; I think we can handle this by adding a section "Relationship to symmetry groups" with no new language in the lead.

So, w.r.t. the question, I think we should not completely exclude the kind of mathematics done by mathematicians, but we should ensure the meat of the article is accessible to undergraduate chemists. — Charles Stewart (talk) 16:19, 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)

Reference 3 "Spin-Orbit Coupling and Double Groups" (pdf available) contains more detail and will be understandable to non-mathematicians. Is this what is wanted? Petergans (talk) 17:00, 30 June 2022 (UTC)

That article defines the double group as something obtained by "adding" a 360-degree non-identity rotation to the point group. I think, at a minimum, we should avoid chemists leaving with the impression that the point group is a subgroup of the double group (which would mean that every representation of the double group restricts to one of the point group and the character tables would not be required).

I also think that it's vital to know that, but not necessarily to understand why, a rotation by 720 degrees is still the identity.

IMHO it would still be best to start over with the pre-"General Revision" version, but failing that, we can write something like "a double group is obtained from a point group through a mathematical process which distinguishes rotations by 360° from the identity element" and explain why it's physically necessary to do that (this isn't pure theory, fermions really do change when you rotate them through 360°). IpseCustos (talk) 12:26, 1 July 2022 (UTC)

Character table: double group D'₄

D'4	E		C4	C43	C2	2C'2	2C''2
		R	C4R	C43R	C2R	2C'2R	2C''2R

The header for a double group is written in two rows, for convenience. In this example the first of these rows has the operations of the point group D₄, which is a sub-group of D'₄. There is always a unique point group that is a subgroup of a given double group in this way. Closure is a general property of groups: any two successive symmetry operations must be equivalent to a single operation of the group, see Group (mathematics)#Elementary consequences of the group axioms. Therefore, rotation by 720° is never mentioned explicitly. Petergans (talk) 09:36, 3 July 2022 (UTC)

"Rotation by 720° is never mentioned explicitly": The closure property means that rotations can be composed, and the composition of a rotation of 360° with itself should be a rotation of 720°. Apparently, you consider that readers must not asking themselves what is this rotation. I do not know much of magnetism, but I believe that, if a rotation of 720° would not be the identity, there would be more poles than the North and the South ones. So, the fact that you do not mention rotations of 720°, does not mean that the knowledge of their nature is not important. D.Lazard (talk) 10:34, 3 July 2022 (UTC)

Interested readers must know, by the group closure property, that the operation of either E or R followed by another symmetry operation must be equivalent to one of the symmetry operations of the group. The 4 possibilities for rotation by 720°, (E,E), (E,R), (R,E) and (R,R), must therefore all be equivalent to a single operation of the group. Petergans (talk) 17:23, 3 July 2022 (UTC)

the point group D₄, which is a sub-group of D'₄ But it is not. There's a single self-inverse element in D'₄, but there are several in D₄.

I cannot agree. "Product" in this context means "the result of combining". D'₄ can be viewed as the result of combining the groups D₄ and {E,R}, which is what Bethe did. Petergans (talk) 17:52, 3 July 2022 (UTC)

You cannot agree to what? This is mathematics, there's little room for personal opinion. Rotations by 180° are self-inverse in D₄, but not in D'₄. Therefore, since self-inverse elements of a subgroup are self-inverse in the containing group, too, D₄ isn't a subgroup of D'₄. The "combining" must be of a different nature, and Spin(3) explains how it's actually done. IpseCustos (talk) 15:54, 6 July 2022 (UTC)

I agree with IpseCustos's comments here. Gumshoe2 (talk) 16:18, 6 July 2022 (UTC)

To clarify: This is not a personal opinion: D₄ is a sub-group of D'₄ as the operations in the first row (partial table above) are those of D₄. Put the other way around, the group D'₄ may be obtained by combining the operations of the point group with the operation R, resulting in operations such as C₄R being operations of the double group. Petergans (talk) 15:58, 7 July 2022 (UTC)

As you do not say how you got this table, and how you define the notations in it, the table shows nothing. Maybe your D₄ is a sub-group of your D'₄, but this says nothing without definining what is your D₄. D.Lazard (talk) 08:36, 8 July 2022 (UTC)

rotation by 720° is never mentioned explicitly Of 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)

Please supply an example which can be cited. Petergans (talk) 17:52, 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)

Is this something to do with space dimensions? A square will be classed as C₄ in 2-D, and D_4h in 3-D. I had renamed the article "Double group (magnetochemistry), precisely to remove ambiguity with usage in pure mathematics. Unfortunately the name was reverted back to the original. Note that Bethe used D₄ rather than D_4h because spin wave functions can be classified with the smaller group. Petergans (talk) 10:39, 7 July 2022 (UTC)

A square will be classed as C₄ in 2-D, and D_4h in 3-D ... Again, this is non sensical if you do not define what do you mean by C₄ and D_4h Also, from what you have written before, I guess that C₄ and D_4h 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)

It appears that mathematicians and scientists are speaking different languages. The point that I was trying to make was that group theory can be applied to any-dimensional space. A square can be considered as an object on a plane, a 2-dimensional structure, point group C₄. For some amusement see Flatland. A molecule exists in 3-D space. space-time is 4-dimensional. Clearly I did not understand the original comment "dihedral groups with 4 or 8 elements". I assumed that D stands for Dihedral. In modern character tables there are groups D₂ with 4 symmetry operations and D₄ which has 8 symmetry operations. Both D_4d and D_4h have 16 operations. Petergans (talk) 09:49, 8 July 2022 (UTC)

I don't know if this helps or not, but as a chemist, when some one says that a molecule has $D_{4}$ symmetry, I immediately think they are referring to this: https://en.wikipedia.org/wiki/List_of_character_tables_for_chemically_important_3D_point_groups#Dihedral_groups_(Dn) Qflib, aka KeeYou Flib (talk) 17:08, 6 July 2022 (UTC)

Perhaps this link is more to the point, actually. https://en.wikipedia.org/wiki/Point_groups_in_three_dimensions#The_seven_infinite_series_of_axial_groups Qflib, aka KeeYou Flib (talk) 19:33, 6 July 2022 (UTC)

Finally, we have the Otterbein symmetry library, which shows tetrathiacyclododecane as an example of a molecule with D4 symmetry. https://symotter.org/gallery Qflib, aka KeeYou Flib (talk) 19:36, 6 July 2022 (UTC)

None is a group of rotations I 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)

The comments by Qflib, aka KeeYou Flib make perfect sense in regard to the application of group theory in chemistry. Petergans (talk) 14:17, 8 July 2022 (UTC)