Talk:Dihedral group

message: 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)
thread_title: Untitled

message: :In mathematics, the dihedral group of order 2n is a certain group for which here the notation Dn is used, but elsewhere the notation D2n 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  Dn for order 2n more common?--Patrick 12:10, 5 August 2005 (UTC)
replies: text: It really depends on context. Geometers usually prefer Dn, while algebraists prefer D2n. 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)
replies: text: I also prefer Dn 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 D2n I may change it.--Patrick 08:19, 6 August 2005 (UTC)
replies: text: I prefer the D2n notation. I must confess that I read little in geometry, but as far as I can tell algebraists use D2n exclusively. Nevcamion (talk) 02:04, 15 September 2008 (UTC)
thread_title: Notation

message: 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 Dihn 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)
thread_title:  Automorphisms 

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

message: 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)
replies: text: The second image shows that D4 (of order 8) is non-abelian. I've corrected the caption. --Zundark 21:30, 26 May 2007 (UTC)
thread_title:  D<sub>2</sub> images 

message: I've added the  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)
replies: text: I added a link. See also the other links in the intro.--Patrick 11:44, 15 September 2007 (UTC)
thread_title:  Context needed 

message: 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)
replies: text: See my comments on your draft page's talk page. Cheers. Chas zzz brown 01:48, 23 September 2007 (UTC)
replies: text: 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)
thread_title:  Possible revision and expansion 

message: 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)
replies: text: 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)
thread_title:  Equivalent definitions error 

message: 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)
thread_title:  Mention that they are examples of Coxeter / Complex reflection groups 

message: 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)
thread_title:  History and applications 

message: (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)
thread_title:  Equivalent definitions 

message: 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)
thread_title:  Snowflakes 

message: The article says "The automorphism group of Dihn 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 Dihn is isomorphic to the holomorph of Z/nZ. Marvoir (talk) 14:39, 18 January 2014 (UTC)
thread_title:  Automorphism group 

message: I notice some recent edits by  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 Zn, or in contexts where there may be  confusion with p-adic integers, Z/nZ; when written multiplicatively, e.g. as roots of unity, Cn is used (this does not affect the notation of isometry groups called Cn)." 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 Zn 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)
thread_title:  Notation and font use 

message: Substituted at 02:00, 5 May 2016 (UTC)
thread_title: Assessment comment

message: 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)
thread_title:  Sentence that does not mean anything: 

message: 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)
thread_title:  Intro sentence 

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