Infinite dihedral group

In mathematics, the infinite dihedral group Dih∞ is an infinite group with properties analogous to those of the finite dihedral groups. In two-dimensional geometry, the infinite dihedral group represents the frieze group symmetry, p1m1, seen as an infinite set of parallel reflections along an axis. Every dihedral group is generated by a rotation r and a reflection; 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 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. An example of infinite dihedral symmetry is in aliasing of real-valued signals. When sampling a function at frequency f_s (intervals 1/f_s), the following functions yield identical sets of samples: {sin(2π( f+Nf_s) t + φ), N = 0, ±1, ±2, ±3,...}. Thus, the detected value of frequency f is periodic, which gives the translation element r = f_s. The functions and their frequencies are said to be aliases of each other. Noting the trigonometric identity: we can write all the alias frequencies as positive values: . This gives the reflection (f) element, namely f ↦ −f. For example, with f = 0.6f_s and N = −1, f+Nf_s = −0.4f_s reflects to 0.4f_s, resulting in the two left-most black dots in the figure. The other two dots correspond to N = −2 and N = 1. As the figure depicts, there are reflection symmetries, at 0.5f_s, f_s, 1.5f_s, etc.