Talk:Dihedral group

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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)

Contents 1 Notation 2 D1 and D2 3 Automorphisms 4 Etymology

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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).