This articleneeds additional citations forverification. Please helpimprove this article byadding citations to reliable sources. Unsourced material may be challenged and removed. Find sources: "Infinite dihedral group" – news ·newspapers ·books ·scholar ·JSTOR(July 2011) (Learn how and when to remove this message)

p1m1, (*∞∞)	p2, (22∞)	p2mg, (2*∞)
In 2-dimensions threefrieze 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.

Inmathematics, theinfinite dihedral group Dih_∞ is aninfinite group with properties analogous to those of the finitedihedral groups.

Intwo-dimensional geometry, theinfinite dihedral group represents thefrieze group symmetry,p1m1, seen as an infinite set of parallel reflections along an axis.

Every dihedral group is generated by a rotationr and a reflection s; if therotation is a rational multiple of a full rotation, then there is some integern such thatrⁿ is the identity, and we have a finite dihedral group of order 2n. If the rotation isnot a rational multiple of a full rotation, then there is no suchn and the resulting group hasinfinitely many elements and is called Dih_∞. It haspresentations

${\displaystyle ⟨r,s\mid s^2=1,srs=r^{-1}⟩}$

${\displaystyle ⟨x,y\mid x^2=y^2=1⟩}$^[1]

and is isomorphic to asemidirect product ofZ andZ/2, and to thefree productZ/2 * Z/2. It is theautomorphism group of the graph consisting of a path infinite to both sides. Correspondingly, it is theisometry group ofZ (see alsosymmetry groups in one dimension), the group of permutationsα: Z → Z satisfying |i − j| = |α(i) − α(j)|, for alli, j inZ.^[2]

The infinite dihedral group can also be defined as theholomorph of theinfinite cyclic group.

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

When sampling a function at frequencyf_s (intervals1/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 frequencyf isperiodic, which gives the translation elementr =f_s. The functions and their frequencies are said to bealiases 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, namelyf ↦ −f.  For example, withf = 0.6f_s  and  N = −1, f + Nf_s = −0.4f_s reflects to  0.4f_s, resulting in the two left-most black dots in the figure.^{[note 1]}  The other two dots correspond toN = −2  and  N = 1. As the figure depicts, there are reflection symmetries, at 0.5f_s, f_s,  1.5f_s,  etc.  Formally, the quotient under aliasing is theorbifold [0, 0.5f_s], with aZ/2 action at the endpoints (the orbifold points), corresponding to reflection.

• Theorthogonal group O(2), another infinite generalization of the finite dihedral groups
• Theaffine symmetric group, a family of groups including the infinite dihedral group

• Infinite group theory

• Articles with short description
• Short description matches Wikidata
• Articles needing additional references from July 2011
• All articles needing additional references