Dihedral group

Dihedral group

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.

See also: Dihedral symmetry in three dimensions.

Notation

There are two competing notations for the dihedral group associated to a polygon with n sides. In geometry the group is denoted Dn, while in algebra the same group is denoted by D2n to indicate the number of elements.

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

Definition

Elements

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 Dn. 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.

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 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 Dn has elements R0,...,Rn−1 and S0,...,Sn−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.

Matrix representation

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

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

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

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

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 Dihn is not abelian
  • they are not subgroups of the symmetric group Sn, 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 Dihn, and a common way to visualize it, is the group Dn of Euclidean plane isometries which keep the origin fixed. These groups form one of the two series of discrete point groups in two dimensions. Dn 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 Dn 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{2pi over n} & -sin{2pi over n} sin{2pi over n} & cos{2pi over n}end{bmatrix} qquad f = begin{bmatrix}1 & 0 0 & -1end{bmatrix}

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

By setting

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

D2 is isomorphic to the Klein four-group.

If the order of Dn is greater than 4, the operations of rotation and reflection in general do not commute and Dn 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 Dn 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 Dn 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 Dn is also used for a subgroup of SO(3) which is also of abstract group type Dihn: 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 Dihn are:

langle r, f mid r^n = 1, f^2 = 1, frf = r^{-1} rangle
or
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 Dihn belongs to the class of coxeter groups.

Z_n rtimes_phi Z_2 is isomorphic to Dihn if φ(0) is the identity and φ(1) is inversion.

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

The properties of the dihedral groups Dihn with n ≥ 3 depend on whether n is even or odd. For example, the center of Dihn 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 rn / 2 (with Dn 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 Dih2n is isomorphic with the direct product of Dihn 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 Dihn has n / m subgroups of type Dihm, and one subgroup Zm. Therefore the total number of subgroups of Dihn (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).

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

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

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:

  • Dihn = Dih(Zn)
    • For even n there are two sets {(h + k + k, 1) | k in H }, and each generates a normal subgroup of type Dihn / 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 inftyinfty and 22infty.
    • 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.

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).

See also

External links

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