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 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

:$\langle r, s \mid s^2 = 1, srs = r^ \rangle \,\!$
:$\langle x, y \mid x^2 = y^2 = 1 \rangle \,\!$

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.Meenaxi Bhattacharjee, Dugald Macpherson, Rögnvaldur G. Möller, Peter M. Neumann. Notes on Infinite Permutation Groups, Issue 1689. Springer, 1998. p. 38

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 + \phi) = \left\.  This gives the reflection () element, namely ↦ .  For example, with   and  ,    ''reflects'' to  , resulting in the two left-most black dots in the figure.In signal processing, the symmetry about axis 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 0.5,  ,  1.5,  etc.  Formally, the quotient under aliasing is the ''orbifold'' , 0.5 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

Notes

{{reflist, group=note

References

Infinite group theory