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.

Definition

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, the group of permutations α: Z → Z satisfying |i - j| = |α - α|, for all i, j in Z.
The infinite dihedral group can also be defined as the holomorph of the infinite cyclic group.

Aliasing

An example of infinite dihedral symmetry is in aliasing of real-valued signals.
When sampling a function at frequency , the following functions yield identical sets of samples:. 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:
we can write all the alias frequencies as positive values: . 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 , with a Z/2 action at the endpoints, corresponding to reflection.