Dihedral group

In mathematics, the dihedral group of order 2n is the abstract group of which one representation is the symmetry group in 2D of a regular polygon with n sides. The group consists of n elements corresponding to rotations of the polygon, and n corresponding to reflections.

Notation

In this article the notation Dih_n is used for the dihedral group of order 2n as abstract group. The notations D_n and D_2n are also seen.

For the isometry group in 2D of this abstract group type, the notation D_n is used. There are four series of isometry groups in 3D which are dihedral as abstract group. Only for one of them the notation D_n is used.

Small dihedral groups

For n = 1 we have Dih₁. This notation is rarely used except in the framework of the series, because it is equal to Z₂. For n = 2 we have Dih₂, the Klein four-group. Both are exceptional within the series:

they are abelian; for all other values of n the group Dih_n is not abelian
they are not subgroups of the symmetric group S_n, corresponding to the fact that 2n > n ! for these n.

The cycle graphs of dihedral groups consist of an n-element cycle and n 2-element cycles. The dark vertex in the cycle graphs below of various dihedral groups stand for the identity element, and the other vertices are the other elements of the group. A cycle consists of successive powers of either of the elements connected to the identity element.

Dih₁	Dih₂	Dih₃	Dih₄	Dih₅	Dih₆	Dih₇

The dihedral group as symmetry group in 2D and rotation group in 3D

An example of abstract group Dih_n, and a common way to visualize it, is the group D_n of Euclidean plane isometries which keep the origin fixed. These groups form one of the two series of discrete point groups in two dimensions. D_n consists of n rotations of multiples of 360°/n about the origin, and reflections across n lines through the origin, making angles of multiples of 180°/n with each other. This is the symmetry group of a regular polygon with n sides (for n ≥3, and also for the degenerate case n = 2, where we have a line segment in the plane).

Dihedral group D_n is generated by a rotation r of order n and a reflection f of order 2 such that

frf = r^{-1}

(in geometric terms: in the mirror a rotation looks like an inverse rotation)

In matrix form, an anti-clockwise rotation and a reflection in the x-axis are given by

r = \begin{bmatrix}\cos{2\pi \over n} & -\sin{2\pi \over n} \\ \sin{2\pi \over n} & \cos{2\pi \over n}\end{bmatrix} \qquad f = \begin{bmatrix}1 & 0 \\ 0 & -1\end{bmatrix}

(in terms of complex numbers: multiplication by $e^{2\pi i \over n}$ and complex conjugation).

By setting

r_0 = \begin{bmatrix}\cos{2\pi \over n} & -\sin{2\pi \over n} \\ \sin{2\pi \over n} & \cos{2\pi \over n}\end{bmatrix} \qquad f_0 = \begin{bmatrix}1 & 0 \\ 0 & -1\end{bmatrix}

and defining

r_j = r_0^j

and

f_j = r_j \, f_0

for

j \in \{1,\ldots,n-1\}

we can write the product rules for

D_n

r_j \, r_k = r_{(j+k) \mbox{ mod n}}

r_j \, f_k = f_{(j+k) \mbox{ mod n}}

f_j \, r_k = f_{(j-k) \mbox{ mod n}}

f_j \, f_k = r_{(j-k) \mbox{ mod n}}

(Compare coordinate rotations and reflections.)

The dihedral group D₂ is generated by the rotation r of 180 degrees, and the reflection f across the x-axis. The elements of D₂ can then be represented as {e, r, f, rf}, where e is the identity or null transformation and rf is the reflection across the y-axis.

D₂ is isomorphic to the Klein four-group.

If the order of D_n is greater than 4, the operations of rotation and reflection in general do not commute and D_n is not abelian; for example, in D₄, a rotation of 90 degrees followed by a reflection yields a different result from a reflection followed by a rotation of 90 degrees:

Thus, beyond their obvious application to problems of symmetry in the plane, these groups are among the simplest examples of non-abelian groups, and as such arise frequently as easy counterexamples to theorems which are restricted to abelian groups.

The 2n elements of D_n can be written as e, r, r²,...,rⁿ⁻¹, f, r f, r² f,...,rⁿ⁻¹ f. The first n listed elements are rotations and the remaining n elements are axis-reflections (all of which have order 2). The product of two rotations or two reflections is a rotation; the product of a rotation and a reflection is a reflection.

So far, we have considered D_n to be a subgroup of O(2), i.e. the group of rotations (about the origin) and reflections (across axes through the origin) of the plane. However, notation D_n is also used for a subgroup of SO(3) which is also of abstract group type Dih_n: the proper symmetry group of a regular polygon embedded in three-dimensional space (if n ≥ 3). Such a figure may be considered as a degenerate regular solid with its face counted twice. Therefore it is also called a dihedron (Greek: solid with two faces), which explains the name dihedral group (in analogy to tetrahedral, octahedral and icosahedral group, referring to the proper symmetry groups of a regular tetrahedron, octahedron, and icosahedron respectively).

Equivalent definitions and properties

Further equivalent definitions of Dih_n are:

The automorphism group of the graph consisting only of a cycle with n vertices (if n ≥ 3).
The group with presentation

\langle r, f \mid r^n = 1, f^2 = 1, frf = r^{-1} \rangle

\langle x, y \mid x^2 = y^2 = (xy)^n = 1 \rangle

(Indeed the only finite groups that can be generated by two elements of order 2 are the dihedral groups and the cyclic groups)

From the second presentation follows that Dih_n belongs to the class of coxeter groups.

The semidirect product of cyclic groups Z_n and Z₂, with Z₂ acting on Z_n by inversion (thus, Dih_n always has a normal subgroup isomorphic to Z_n ):

_φ Z₂ is isomorphic to Dih_n if φ(0) is the identity and φ(1) is inversion.

If we consider Dih_n (n ≥ 3) as the symmetry group of a regular n-gon and number the polygon's vertices, we see that Dih_n is a subgroup of the symmetric group S_n.

The properties of the dihedral groups Dih_n with n ≥ 3 depend on whether n is even or odd. For example, the center of Dih_n consists only of the identity if n is odd, but contains the element r^{n / 2} if n is even (with D_n as a subgroup of O(2), this is inversion; since it is scalar multiplication by −1, it is clear that it commutes with any linear transformation).

For odd n, abstract group Dih_2n is isomorphic with the direct product of Dih_n and Z₂.

In the case of 2D isometries, this corresponds to adding inversion, giving rotations and mirrors in between the existing ones.

All the reflections are conjugate to each other in case n is odd, but they fall into two conjugacy classes if n is even. If we think of the isometries of a regular n-gon: for odd n there are rotations in the group between every pair of mirrors, while for even n only half of the mirrors can be reached from one by these rotations.

If m divides n, then Dih_n has n / m subgroups of type Dih_m, and one subgroup Z_m. Therefore the total number of subgroups of Dih_n (n ≥ 1), is equal to d (n) + σ (n), where d (n) is the number of positive divisors of n and σ (n) is the sum of the positive divisors of n. See List of small groups for the cases n ≤ 8.

Examples of automorphism groups

Dih₉ has 18 inner automorphisms. As 2D isometry group D₉, the group has mirrors at 20° intervals. The 18 inner automorphisms provide rotation of the mirrors by multiples of 20°, and reflections. As isometry group these are all automorphisms. As abstract group there are in addition to these, 36 outer automorphisms, e.g. multiplying angles of rotation by 2.

Dih₁₀ has 10 inner automorphisms. As 2D isometry group D₁₀, the group has mirrors at 18° intervals. The 10 inner automorphisms provide rotation of the mirrors by multiples of 36°, and reflections. As isometry group there are 10 more automorphisms; they are conjugates by isometries outside the group, rotating the mirrors 18° with respect to the inner automorphisms. As abstract group there are in addition to these 10 inner and 10 outer automorphisms, 20 more outer automorphisms, e.g. multiplying rotations by 3.

Compare the values 6 and 4 for Euler's totient function, the multiplicative group of integers modulo n for n = 9 and 10, respectively. This triples and doubles the number of automorphisms compared with the two automorphisms as isometries (keeping the order of the rotations the same or reversing the order).

Infinite dihedral group

In addition to the finite dihedral groups, there is the infinite dihedral group Dih_∞. 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, f \mid f^2 = 1, frf = r^{-1} \rangle

\langle x, y \mid x^2 = y^2 = 1 \rangle

and is isomorphic to a semidirect product of Z and Z₂, and to the free product Z₂ * Z₂. 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).

Generalized dihedral group

_φ Z₂ with φ(0) the identity and φ(1) inversion.

Thus we get:

(h₁, 0) * (h₂, t₂) = (h₁ + h₂, t₂)

(h₁, 1) * (h₂, t₂) = (h₁ - h₂, 1 + t₂)

for all h₁, h₂ in H and t₂ in Z₂.

(Writing Z₂ multiplicatively, we have (h₁, t₁) * (h₂, t₂) = (h₁ + t₁h₂, t₁t₂) .)

Note that (h, 0) * (0,1) = (h,1), i.e. first the inversion and then the operation in H. Also (0, 1) * (h, t) = (- h, 1 + t); indeed (0,1) inverts h, and toggles t between "normal" (0) and "inverted" (1) (this combined operation is its own inverse).

The subgroup of Dih(H) of elements (h, 0) is a normal subgroup of index 2, isomorphic to H, while the elements (h, 1) are all their own inverse.

The conjugacy classes are:

the sets {(h,0 ), (-h,0 )}
the sets {(h + k + k, 1) | k in H }

Thus for every subgroup M of H, the corresponding set of elements (m,0) is also a normal subgroup. We have:

Dih(H) / M = Dih ( H / M )

Examples:

Dih_n = Dih(Z_n)
- For even n there are two sets {(h + k + k, 1) | k in H }, and each generates a normal subgroup of type Dih_{n / 2}. As subgroups of the isometry group of the set of vertices of a regular n-gon they are different: the reflections in one subgroup all have two fixed points, while none in the other subgroup has (the rotations of both are the same). However, they are isomorphic as abstract groups.
- For odd n there is only one set {(h + k + k, 1) | k in H }
Dih_∞ = Dih(Z); there are two sets {(h + k + k, 1) | k in H }, and each generates a normal subgroup of type Dih_∞. As subgroups of the isometry group of Z they are different: the reflections in one subgroup all have a fixed point, the mirrors are at the integers, while none in the other subgroup has, the mirrors are in between (the translations of both are the same: by even numbers). However, they are isomorphic as abstract groups.
Dih(S¹), or orthogonal group O(2,R), or O(2): the isometry group of a circle, or equivalently, the group of isometries in 2D that keep the origin fixed. The rotations form the circle group S¹, or equivalently SO(2,R), also written SO(2), and R / Z ; it is also the multiplicative group of complex numbers of absolute value 1. In the latter case one of the reflections (generating the others) is complex conjugation. There are no proper normal subgroups with reflections. The discrete normal subgroups are cyclic groups of order n for all positive integers n. The quotient groups are isomorphic with the same group Dih(S¹).
Dih(Rⁿ ): the group of isometries of Rⁿ consisting of all translations and inversion in all points; for n = 1 this is the Euclidean group E(1); for n > 1 the group Dih(Rⁿ ) is a proper subgroup of E(n ), i.e. it does not contain all isometries.
H can be any subgroup of Rⁿ, e.g. a discrete subgroup; in that case, if it extends in n directions it is a lattice.
- Discrete subgroups of Dih(R² ) which contain translations in one direction are of frieze group type $\infty\infty$ and 22 $\infty$ .
- Discrete subgroups of Dih(R² ) which contain translations in two directions are of wallpaper group type p1 and p2.
- Discrete subgroups of Dih(R³ ) which contain translations in three directions are space groups of the triclinic crystal system.

Dih(H) is Abelian, with the semidirect product a direct product, iff all elements of H are their own inverse:

Dih(Z₁) = Dih₁ = Z₂
Dih(Z₂) = Dih₂ = Z₂ × Z₂ (Klein four-group)
Dih(Dih₂) = Dih₂ × Z₂ = Z₂ × Z₂ × Z₂

etc.

Topology

Dih(Rⁿ ) and its dihedral subgroups are disconnected topological groups. Dih(Rⁿ ) consists of two connected components: the identity component isomorphic to Rⁿ, and the component with the reflections. Similarly O(2) consists of two connected components: the identity component isomorphic to the circle group, and the component with the reflections.

For the group Dih_∞ we can distinguish two cases:

Dih_∞ as the isometry group of Z
Dih_∞ as a 2D isometry group generated by a rotation by an irrational number of turns, and a reflection

Both topological groups are totally disconnected, but in the first case the (singleton) components are open, while in the second case they are not. Also, the first topological group is a closed subgroup of Dih(R) but the second is not a closed subgroup of O(2).

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