Infinite dihedral group

Every dihedral group is generated by a rotation r and a reflection s; if the rotation is a rational multiple of a full rotation, then there is some integer n such that rn is the identity, and we have a finite dihedral group of order 2n. If the rotation is not a rational multiple of a full rotation, then there is no such n and the resulting group has infinitely many elements and is called Dih∞. It has presentations :\langle r, s \mid s^2 = 1, srs = r^{-1} \rangle \,\! :\langle x, y \mid x^2 = y^2 = 1 \rangle \,\! and is isomorphic to a semidirect product of Z and Z/2, and to the free product Z/2 * Z/2. It is the automorphism group of the graph consisting of a path infinite to both sides. Correspondingly, it is the isometry group of Z (see also symmetry groups in one dimension), the group of permutations α: Z → Z satisfying |i − j| = |α(i) − α(j)|, for all i, j in Z. The infinite dihedral group can also be defined as the holomorph of the infinite cyclic group. ==Aliasing==

), and they exhibit dihedral symmetry. The many-to-one phenomenon is known as aliasing. An example of infinite dihedral symmetry is in aliasing of real-valued signals. When sampling a function at frequency (intervals ), the following functions yield identical sets of samples: {{math|{sin(2( f + Nf) t + φ), N  0, ±1, ±2, ±3, . . . }}}. Thus, the detected value of frequency is periodic, which gives the translation element . The functions and their frequencies are said to be aliases of each other. Noting the trigonometric identity: : \sin(2\pi (f+Nf_s)t + \varphi) = \begin{cases} +\sin(2\pi (f+Nf_s)t + \varphi), & f+Nf_s \ge 0, \\[4pt] -\sin(2\pi |f+Nf_s|t - \varphi), & f+Nf_s we can write all the alias frequencies as positive values: |f+Nf_s|. This gives the reflection () element, namely  ↦ .  For example, with   and  ,    reflects to  , resulting in the two left-most black dots in the figure.  The other two dots correspond to   and  . As the figure depicts, there are reflection symmetries, at 0.5,  ,  1.5,  etc.  Formally, the quotient under aliasing is the orbifold [0, 0.5], with a Z/2 action at the endpoints (the orbifold points), corresponding to reflection. ==See also==