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)