Talk:Dihedral group

Untitled

I always thought that D_infinity is the symmetry group of a circle, i.e. the semidirect product of R and C₂. But maybe not. Is there a name for this symmetry group then?

The infinite dihedral group is usually defined as having presentation {{a,b}; {a^2, b^2}}, from which it can be seen to be countable (unfortunately, the only references I have handy are various sci.math and web hits).

Consider, as an extension of the usual definition of dihedral, a group with presentation {{c,b}; {b^2, (bc)^2}}; then b c b = c^-1, b c^n b = c^-n, and so (b c^n)(b c^m) = c^(m-n); similarly, (c^n b)(c^m b) = c^(n-m), and thus all elements are of the form c^n, b c^n or c^n b. By substituting ab for c, we then get strings of the form c^n = ababab...ab, b c^n = babab...ab and c^n b = ababab...aba; so the two presentations are equivalent to Z (semidirect product) C_2.

A geometric definition is to start with two axes of symmetry which are separated by an angle which is not a rational multiple of pi; the resulting set of symmetry axes forms the (countable) infinite dihedral group again.

I think the symmetry group you're thinking of is called O(2) or SO(2); but I'm none to clear on the terminology of non-discrete groups :).

It is O(2), SO(2) only includes rotations.

JeffBobFrank 23:02, 21 Feb 2004 (UTC)

Also, the semidirect product of R and C₂ is the symmetry group of a straight line, not of a circle. AxelBoldt 22:23, 23 Dec 2004 (UTC)

Notation

In mathematics, the dihedral group of order 2n is a certain group for which here the notation D_n is used, but elsewhere the notation D_2n is also used, e.g. in the list of small groups.

It would be better to use, at least in Wikipedia, a uniform notation. Is D_n for order 2n more common?--Patrick 12:10, 5 August 2005 (UTC)

It really depends on context. Geometers usually prefer D_n, while algebraists prefer D_2n. I prefer the former, but that's only because I am more geometrical in thinking. Striving to keep the notation the same in every article is going to be difficult. -- Fropuff 17:03, 5 August 2005 (UTC)

I also prefer D_n and have the impression that that is more common in general. Also in April the notation in this article was changed into this, and that was not disputed. Therefore, when I encounter the notation D_2n I may change it.--Patrick 08:19, 6 August 2005 (UTC)

I prefer the D_2n notation. I must confess that I read little in geometry, but as far as I can tell algebraists use D_2n exclusively. Nevcamion (talk) 02:04, 15 September 2008 (UTC)

For 3d geometry symmetry, there's actually 3 forms: D_n, D_nh, D_nd, for rotation, reflectional, and mixed symmetry. List_of_spherical_symmetry_groups#Dihedral_symmetry_.5B2.2Cn.5D. Tom Ruen (talk) 02:33, 15 September 2008 (UTC)

The D_2n notation should be mentioned somewhere in the article (probably in the first paragraph) as it causes ambiguity otherwise. --82.28.107.134 (talk) 14:05, 20 December 2020 (UTC)

D₁ and D₂

Is the nth dihedral group usually defined for n=1? The two texts I have on hand (Grillet's Algebra and Fraleigh's A First Course in Abstract Algebra) do not mention D₁. In fact, Grillet defines the nth dihedral group only for n>=2. Fraleigh does not even mention D₂ for that matter.

Considering D₂ now, can someone give some more insight into the nature of this group? I originally operated under the assumption that D_n was a subgroup of the symmetric group of n elements, S_n. This is clearly true when n>=3, but apparently not for n=2, since S₂ is isomorphic to C₂, while we claim here that D₂ is isomorphic to C₂ x C₂.

Specifically I do not see how D₂ fits in with the standard notion that the nth dihedral groups are the symmetries of a regular n-gon. What are the symmetries of a 2-gon, i.e., a line with two endpoints? Seems to me one will only encounter the identity, and what is equivalent to a reflection through the middle. -- Shawn M. O'Hare 13:34, 5 November 2005 (UTC)

Systematically, n=1 and n=2 make sense, they are among the discrete point groups in two dimensions: cyclic groups with additionally reflection. As abstract group, the article mentions that Dih₁ is a rarely used notation (except in the framework of the series) because it is equal to Z₂.

I agree that n=1 and n=2 are special in that they are larger than the symmetric groups, corresponding to the fact that 2n > n! for these n. Therefore Dih₂ does not correspond to the isometry group of a 2-gon, but to the isometry group of the plane leaving the 2-gon fixed. For a "1-gon" this does not work.--Patrick 14:26, 5 November 2005 (UTC)

Your clarification is greatly appreciated. -- Shawn M. O'Hare 12:01, 6 November 2005 (UTC)

Automorphisms

The article should be more specific about the automorphism groups of dihedral groups. The examples are helpful and they suggest the general case, but the general result is not explicitly stated. The group Dih_n has n*phi(n) automorphisms. The automorphism group is isomorphic to the group of transformations x → a*x + b (mod n) where a is coprime to n. (Reference: http://www.research.att.com/~njas/sequences/A002618) David Radcliffe 10:09, 12 March 2006 (UTC)

Etymology

I was hoping the article would give me some idea why dihedral groups are called dihedral, but to no avail. Might be a good addition. —The preceding unsigned comment was added by 201.143.107.13 (talk) 04:45, 5 December 2006 (UTC).

Good point! I had the same question. I just added a paragraph on the etymology. ★NealMcB★ (talk) 03:22, 19 February 2024 (UTC)

D₂ images

The images showing the 'F' letter acted upon by D₂ are wrong. Next to the images it's said the D₂ is isomorphic to the Klein group, which is abelian. But images state that D₂ is non-abelian.

Also, it is clear from second image that if 'r' is the rotation through 90 degrees and 'f' is the reflection, then rf ≠ fr and so the group has more than 4 elements -- in contradiction with the first image.

As I'm trying to learn the subject, I don't know what's actually the correct answer. The three different notations for the dihedral group names employed in the article don't help either.. -- Guygurari 19:34, 26 May 2007 (UTC)

The second image shows that D₄ (of order 8) is non-abelian. I've corrected the caption. --Zundark 21:30, 26 May 2007 (UTC)

Context needed

I've added the {{context}} flag. I was perusing the Luhn algorithm, which led me to the Verhoeff algorithm, which led me to.. what is this? a few pages torn out of the introduction to some advanced maths textbook? It certainly isn't part of an encyclopedia, which is what I thought I was browsing. It really needs a decent introduction before diving into formulae and esoterica. Who studies this, and why? Does it have any real world applications, or is it purely theoretical? A link to a broader topic which subsumes this would be fine, but there's gotta be something...

I suspect many of the more specialized articles in specialized topics will have this same flaw, but that doesn't excuse the problem. MrRedwood 08:54, 15 September 2007 (UTC)

I added a link. See also the other links in the intro.--Patrick 11:44, 15 September 2007 (UTC)

Possible revision and expansion

Hi everyone!

I've been working on a possible revision and expansion of this article for a while. Parts of it are now in good enough shape that it might make sense to move them into the main article in some form:

• User:Jim.belk/Dihedral Group Draft

Please be aware that when I'm writing a draft, I tend to rewrite as much text as possible, on the theory that we can use whichever version comes out better. I am not proposing a wholesale replacement of this page by the draft—I just think that some parts of the draft are good enough to be moved into the main article.

In addition to the above draft, I have also written drafts of some possible supporting articles:

• User:Jim.belk/Representation Theory of the Dihedral Group Draft
• User:Jim.belk/Generalized Dihedral Group Draft

Let me know what you think. Jim 20:09, 22 September 2007 (UTC)

See my comments on your draft page's talk page. Cheers. Chas zzz brown 01:48, 23 September 2007 (UTC)

With Chas's help (see User_talk:Jim.belk/Dihedral_Group_Draft), the "Definition" seems ready to be moved into the article. In addition to moving this section, I have reworded the introduction, changed the introductory picture to a snowflake, and reworded the short section on notation. Anyone looking on should feel free to revert or alter any of these edits, or offer comments on how they could be improved.

Chas and I are continuing to work on "symmetry", "algebraic properties", and "generalizations" sections at User:Jim.belk/Generalized Dihedral Group Draft. Anyone else looking on is invited to help with this effort or offer comments on either talk page. Jim 01:14, 24 September 2007 (UTC)

I especially liked that Jim mentions solvability of the dihedral groups on his draft page. This would make a nice addition to the main article since it is a general property and since dihedral groups are as simple a nontrivial (nonabelian) illustration of solvability as we can ask for. Hardmath (talk) 20:35, 13 February 2010 (UTC)

Equivalent definitions error

Hi, I think there's an error in the first presentation of the "Equivalent definitions" section. The last relation should be: $srs^{-1} = r^{-1}$. — Preceding unsigned comment added by Suitangi (talk • contribs) 01:59, 22 December 2010 (UTC)

The given presentation is correct. Since $s^{2}=1$, the element $s$ is its own inverse. Jim.belk (talk) 02:12, 22 December 2010 (UTC)

Mention that they are examples of Coxeter / Complex reflection groups

Also in those two articles the dihedral groups are written as I_2(m) 129.215.104.159 (talk) 16:38, 27 July 2011 (UTC)

History and applications

This are two different suggestions, but I think we could try to add a "history" section and an "applications" section.

About history: I think this group is one of the most popular booktext-examples of non-abelian groups and it virtually appears in all Group Theory basic texts. However, I have never found even a brief summary of the history and origin of this group. It could be a significant improvement of the article.

Does anybody know a good reference for this? ¡Add it here! I personally don't.

About applications: this group is of great importance of quantum computation in the field of quantum algorithms. This is a topic I know quite well, but I does not make a section itself.

Does anybody know more applications of the dihedral group in science? If we had some more we could put them together.

Garrapito (talk) 18:04, 16 November 2011 (UTC)

Equivalent definitions

(third)

The semidirect product of cyclic groups Zn and Z2, with Z2 acting on Zn by inversion (thus, Dihn always has a normal subgroup isomorphic to the group Zn

Right prentesize is missing... Jumpow (talk) 11:03, 11 January 2013 (UTC) Jumpow

Snowflakes

According to the relevant Wikipedia page, only a tiny fraction of snowflakes have sixfold symmetry. (And even then it seems to be approximate, as far as I can tell.) So, while the picture at the top of this page is pretty, it's extremely misleading... I'm going to remove it. — Preceding unsigned comment added by 131.111.184.88 (talk) 22:30, 26 July 2013 (UTC)

Automorphism group

The article says "The automorphism group of Dih_n is isomorphic to the affine group Aff(Z/nZ) $=\{ax+b\mid (a,n)=1\}$", but this is a more general notion of an affine group than the definition in the article Affine group, where there is only question of vector spaces. In fact, the group $=\{ax+b\mid (a,n)=1\}$ is the holomorph of Z/nZ, thus the automorphism group of Dih_n is isomorphic to the holomorph of Z/nZ. Marvoir (talk) 14:39, 18 January 2014 (UTC)

Notation and font use

I notice some recent edits by GeoffreyT2000 away from the general notation (especially font use) for groups on WP. In particular:

• Groups names, e.g. Z₂ are in roman font - they are symbol, not variables. See:
  • WP:Manual of Style/Mathematics#Notational conventions: "The abstract cyclic group of order n, when written additively, has notation Z_n, or in contexts where there may be confusion with p-adic integers, Z/nZ; when written multiplicatively, e.g. as roots of unity, C_n is used (this does not affect the notation of isometry groups called C_n)." There is no possibility of the mentioned confusion, and use of ring names (Z in this instance) seems like a confusion of disciplines in the purely group context. Note the lack of italics on the group names. I am aware that many papers and publications do use italics, but here the MoS applies. Note too that the bold Z_n generally refers to the ring rather than to the group.
  • WP:Manual of Style/Mathematics#Variables: "To start with, we generally use italic text for variables, but never for numbers or symbols"

Further, reverting another editor's edits without an edit summary is kind of, well, saying that the reverted edit it was so blatantly bad as to be beneath explanation for a revert, such as for vandalism; i.e., it can be interpreted as insulting. Please refrain from summaryless reverts. —Quondum 16:40, 10 April 2015 (UTC)

Assessment comment

The comment(s) below were originally left at Talk:Dihedral group/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

I've upgraded this article to mid priority. Dihedral groups play a major role in group theory, geometry, and chemistry. Jim 19:54, 22 September 2007 (UTC)

Last edited at 19:54, 22 September 2007 (UTC). Substituted at 02:00, 5 May 2016 (UTC)

Sentence that does not mean anything:

In the section Examples of automorphism groups, this sentence appears as follows:

"Compare the values 6 and 4 for Euler's totient function, the multiplicative group of integers modulo n for n = 9 and 10, respectively. ".

I hope someone knowledgeable about the dihedral group will fix this, so that it means something.2600:1700:E1C0:F340:1147:130B:19C6:51FB (talk) 18:12, 4 February 2019 (UTC)

Intro sentence

The intro says: "Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, geometry, and chemistry." This was added in 2007, by an editor apparently no longer active. The chemistry claim seems dubious -- and yet I can imagine a chemist telling us something persuasive to the effect that this is real. Not being a chemist, I can only search, and find lots of pages about dihedral groups which have copied this text from WP, and lots of pages about chemistry which talk about dihedral angles and alkyl or other groups, but these are not mathematical groups, let alone dihedral ones. ¶ Can anyone see a reason to have "chemistry" here? I think perhaps the whole sentence could be deleted, since the only real content is "geometry"; after all, dihedral groups are important in group theory like molecules are important in chemisty. Imaginatorium (talk) 04:17, 21 April 2020 (UTC)

Dihn

How to pronounce this word 2001:4455:2D8:F500:68:F270:CBA:747C (talk) 08:09, 15 November 2022 (UTC)

There's no "word" Dihn on the page, only the abbreviation "Dih" (for dihedral) with an associated number ("n" as a generalization if there's no specific number). Probably "Dihedral group of order n" or "Dihedral group of the nth order... AnonMoos (talk) 14:44, 16 November 2022 (UTC)

Is this correct?

The section Other definitions contains this passage:

"D_n can also be defined as the group with presentation

${\begin{aligned}\mathrm {D} _{n}&=\left\langle r,s\mid \operatorname {ord} (r)=n,\operatorname {ord} (s)=2,srs^{-1}=r^{-1}\right\rangle \\&=\left\langle r,s\mid r^{n}=s^{2}=(sr)^{2}=1\right\rangle .\end{aligned}}$

But in the first presentation, src = r^-1 is not equivalent to (sr)² = 1 (without the additional information that s² = 1, or in other words that s = s^-1).

But srs = r^-1 is always equivalent to (sr)² = 1.

I hope someone knowledgeable about this subject will fix this.