Infinite dihedral group explained

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

\langle r, s \mid s ^{2} = 1 , srs = r ^{-1} \rangle \langle x, y \mid x ^{2} = y ^{2} = 1 \rangle ^[1]

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 (intervals), 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:

\sin( 2 \pi (f + Nf _{s)t} + \varphi) = \begin{cases} + \sin( 2 \pi (f + Nf _{s)t} + \varphi), & f + Nf _{s} \ge 0 , \\[ 4 pt] - \sin( 2 \pi |f + Nf _{s|t} - \varphi), & f + Nf _{s} < 0 , \end{cases}

we can write all the alias frequencies as positive values: $|f+Nf\_s|$. This gives the reflection element, namely  ↦ .  For example, with   and  ,    reflects to  , resulting in the two left-most black dots in the figure.^[3]   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 [0, 0.5{{math|''f''{{sub|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
• The affine symmetric group, a family of groups including the infinite dihedral group

References

1. Connolly . Francis . Davis . James . The surgery obstruction groups of the infinite dihedral group . Geometry & Topology . 8 . August 2004 . 3 . 1043–1078 . 10.2140/gt.2004.8.1043. math/0306054 .
2. Meenaxi Bhattacharjee, Dugald Macpherson, Rögnvaldur G. Möller, Peter M. Neumann. Notes on Infinite Permutation Groups, Issue 1689. Springer, 1998. [{{Google books|plainurl=y|id=lKsEawMzEOQC|page=38|text=Infinite dihedral group}} p. 38].
3. In signal processing, the symmetry about axis is known as folding, and the axis is known as the folding frequency.