# Infinite dihedral group

In mathematics, the **infinite dihedral group** Dih_{∞} is an infinite group with properties analogous to those of the finite dihedral groups.

p1m1, (*∞∞) | p2, (22∞) | p2mg, (2*∞) |
---|---|---|

In 2-dimensions three frieze groups p1m1, p2, and p2mg are isomorphic to the Dih_{∞} group. They all have 2 generators. The first has two parallel reflection lines, the second two 2-fold gyrations, and the last has one mirror and one 2-fold gyration. |

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 *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

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: **| f+N f_{s}|.** This gives the reflection (

*f*) element, namely

*f*↦ −

*f*. For example, with

*f*= 0.6

*f*

_{s}and

*N*= −1,

*f+Nf*

_{s}= −0.4

*f*

_{s}

*reflects*to 0.4

*f*

_{s}, resulting in the two left-most black dots in the figure.[note 1] The other two dots correspond to

*N*= −2 and

*N*= 1. As the figure depicts, there are reflection symmetries, at 0.5

*f*

_{s},

*f*

_{s}, 1.5

*f*

_{s}, etc. 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.

## See also

- The orthogonal group O(2), another infinite generalization of the finite dihedral groups

## Notes

- In signal processing, the symmetry about axis
*f*_{s}/2 is known as*folding,*and the axis is known as the*folding frequency*.

## References

- Connolly, Francis; Davis, James (August 2004). "The surgery obstruction groups of the infinite dihedral group" (PDF).
*Geometry & Topology*.**8**: 1043–1078. arXiv:math/0306054. doi:10.2140/gt.2004.8.1043. Retrieved 2 May 2013. - Meenaxi Bhattacharjee, Dugald Macpherson, Rögnvaldur G. Möller, Peter M. Neumann. Notes on Infinite Permutation Groups, Issue 1689. Springer, 1998. p. 38. ISBN 978-3-540-64965-6