Dihedral group

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This snowflake has the dihedral symmetry of a regular hexagon.

In mathematics, a dihedral group is the group of symmetries of a regular polygon, including both rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, geometry, and chemistry.

1 Notation
2 Definition
3 Small dihedral groups
4 The dihedral group as symmetry group in 2D and rotation group in 3D
5 Equivalent definitions and properties
6 Examples of automorphism groups
7 Infinite dihedral group
8 Generalized dihedral group
9 Topology
10 See also
11 External links

[edit] Notation

There are two competing notations for the dihedral group associated to a polygon with n sides. In geometry the group is denoted D_n, while in algebra the same group is denoted by D_2n to indicate the number of elements.

In this article, D_n (and sometimes Dih_n) refers to the symmetries of a regular polygon with n sides.

[edit] Definition

[edit] Elements

The six reflection symmetries of a regular hexagon.

A regular polygon with n sides has 2n different symmetries: n rotational symmetries and n reflection symmetries. The associated rotations and reflections make up the dihedral group D_n. The following picture shows the effect of the sixteen elements of D₈ on a stop sign:

The first row shows the effect of the eight rotations, and the second row shows the effect of the eight reflections.

[edit] Group structure

As with any geometric object, the composition of two symmetries of a regular polygon is again a symmetry. This operation gives the symmetries of a polygon the algebraic structure of a finite group.

The composition of these two reflections is a rotation.

The following Cayley table shows the effect of composition in the group D₃ (the symmetries of an equilateral triangle). R₀ denotes the identity; R₁ and R₂ denote counterclockwise rotations by 120 and 240 degrees; and S₀, S₁, and S₂ denote reflections across the three lines shown in the picture to the right.

	R₀	R₁	R₂	S₀	S₁	S₂
R₀	R₀	R₁	R₂	S₀	S₁	S₂
R₁	R₁	R₂	R₀	S₁	S₂	S₀
R₂	R₂	R₀	R₁	S₂	S₀	S₁
S₀	S₀	S₂	S₁	R₀	R₂	R₁
S₁	S₁	S₀	S₂	R₁	R₀	R₂
S₂	S₂	S₁	S₀	R₂	R₁	R₀

For example, S₂S₁ = R₁ because the reflection S₁ followed by the reflection S₂ results in a 120-degree rotation. (This is the normal backwards order for composition.) Note that the composition operation is not commutative.

In general, the group D_n has elements R₀,...,R_n−1 and S₀,...,S_n−1, with composition given by the following formulae:

$R_i\,R_j = R_{i+j},\;\;\;\;R_i\,S_j = S_{i+j},\;\;\;\;S_i\,R_j = S_{i-j},\;\;\;\;S_i\,S_j = R_{i-j}$

In all cases, addition and subtraction of subscripts should be performed using modular arithmetic with modulus n.

[edit] Matrix representation

The symmetries of this pentagon are linear transformations.

If we center the regular polygon at the origin, then elements of the dihedral group act as linear transformations of the plane. This lets us represent elements of D_n as matrices, with composition being matrix multiplication. This is an example of a (2-dimensional) group representation.

For example, the elements of the group D₄ can be represented by the following eight matrices:

$\begin{matrix} R_0=\bigl(\begin{smallmatrix}1&0\\[0.2em]0&1\end{smallmatrix}\bigr), & R_1=\bigl(\begin{smallmatrix}0&-1\\[0.2em]1&0\end{smallmatrix}\bigr), & R_2=\bigl(\begin{smallmatrix}-1&0\\[0.2em]0&-1\end{smallmatrix}\bigr), & R_3=\bigl(\begin{smallmatrix}0&1\\[0.2em]-1&0\end{smallmatrix}\bigr), \\[1em] S_0=\bigl(\begin{smallmatrix}1&0\\[0.2em]0&-1\end{smallmatrix}\bigr), & S_1=\bigl(\begin{smallmatrix}0&1\\[0.2em]1&0\end{smallmatrix}\bigr), & S_2=\bigl(\begin{smallmatrix}-1&0\\[0.2em]0&1\end{smallmatrix}\bigr), & S_3=\bigl(\begin{smallmatrix}0&-1\\[0.2em]-1&0\end{smallmatrix}\bigr). \end{matrix}$

In general, the matrices for elements of D_n have the following form:

$R_k \;=\; \left(\!\! \begin{array}{rr} \cos \frac{2\pi k}{n} & -\sin \frac{2\pi k}{n} \\[0.5em] \sin \frac{2\pi k}{n} & \cos \frac{2\pi k}{n} \end{array}\!\!\right)$ and $S_k \;=\; \left(\!\! \begin{array}{rr} \cos \frac{2\pi k}{n} & \sin \frac{2\pi k}{n} \\[0.5em] \sin \frac{2\pi k}{n} & -\cos \frac{2\pi k}{n} \end{array} \!\!\right).$

The first matrix is a rotation matrix, expressing a counterclockwise rotation through an angle of 2πk ⁄ n. The second matrix is a reflection across a line that makes an angle of πk ⁄ n with the x-axis.

[edit] 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

[edit] The dihedral group as symmetry group in 2D and rotation group in 3D

2D D₆ symmetry - The Red Star of David

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

f r f = 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 $D n$ as

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

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

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

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

(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.

The four elements of D2 (x-axis is vertical here)

The four elements of D₂ (x-axis is vertical here)

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:

D4 is nonabelian (x-axis is vertical here)

D₄ is nonabelian (x-axis is vertical here)

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).

[edit] 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 ):

$Z_n \rtimes_\phi Z_2$ 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 if n is even the center has two elements, namely the identity and the element r^{n / 2} (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.

[edit] 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).

In general, the automorphism group of Dih_n is isomorphic to the affine group Aff(Z/nZ).

[edit] 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).

[edit] Generalized dihedral group

For any abelian group H, the generalized dihedral group of H, written Dih(H), is the semidirect product of H and Z₂, with Z₂ acting on H by inverting elements. I.e., $\mathrm{Dih}(H) = H \rtimes_\phi Z_2$ 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, if and only if 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.

[edit] 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 2-dimensional 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).

[edit] See also

[edit] External links

Dihedral Group n of Order 2n by Shawn Dudzik, The Wolfram Demonstrations Project.

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