Infinite dihedral group

Definition

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

An example of infinite dihedral symmetry is in aliasing of real-valued signals.

When sampling a function at frequency f (intervals 1/f), the following functions yield identical sets of samples: }. Thus, the detected value of frequency f 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 \end{cases}

we can write all the alias frequencies as positive values: |f+Nf_s|. This gives the reflection (f) element, namely f ↦ −f. For example, with and , reflects to 0.4f, resulting in the two left-most black dots in the figure.In signal processing, the symmetry about axis f/2 is known as folding, and the axis is known as the folding frequency. The other two dots correspond to and . As the figure depicts, there are reflection symmetries, at 0.5f, f, 1.5f, etc. Formally, the quotient under aliasing is the orbifold [0, 0.5f], with a Z/2 action at the endpoints (the orbifold points), corresponding to reflection.

Notes

References

References

1. (August 2004). "The surgery obstruction groups of the infinite dihedral group". Geometry & Topology.
2. Meenaxi Bhattacharjee, Dugald Macpherson, Rögnvaldur G. Möller, Peter M. Neumann. Notes on Infinite Permutation Groups, Issue 1689. Springer, 1998. [{{Google books. 978-3-540-64965-6