# Infinite dihedral group

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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 4th 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 *r ^{n}* is the identity, and we have a finite dihedral group of order 2

*n*. 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**.^{[2]}

The infinite dihedral group can also be defined as the holomorph of the infinite cyclic group.

## Aliasing

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A concrete example of infinite dihedral symmetry is in aliasing of real-valued signals; this is realized as follows. If sampling a signal (signal processing term for a function) at frequency *f _{s}*, then the functions sin(

*ft*) and sin((

*f*+

*f*)

_{s}*t*) cannot be distinguished (and likewise for cos), which gives the translation (

*r*) element – translation by

*f*(the detected frequency is

_{s}*periodic*). Further, for a

*real*signal, cos(−

*ft*) = cos(

*ft*) and sin(−

*ft*) = −sin(

*ft*), so every negative frequency has a corresponding positive frequency (this is not true for complex signals), and gives the reflection (

*f*) element, namely

*f*↦ −

*f*. Together these give further reflection symmetries, at 0.5

*f*,

_{s}*f*, 1.5

_{s}*f*, etc.; this phenomenon is called

_{s}*folding,*as the graph of the detected signal "folds back" on itself, as depicted in the diagram at right.

Formally, the quotient under aliasing is the *orbifold* [0, 0.5*f _{s}*], with a

**Z**/2 action at the endpoints (the orbifold points), corresponding to reflection.