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Dihedral group

In mathematics Mathematics

Mathematics is the discipline that deals with concepts such as quantity [i], structure [i], space [i] a ... 

, the dihedral group of order 2n is the abstract group of which one representation is the symmetry group Symmetry group

The symmetry [i] group of an object is the group [i] of all isometries [i] under which it is invariant [i] ... 

 in 2D of a regular polygon Regular polygon

A regular polygon is a simple polygon [i] which is [i] and equilateral [i] ... 

 with n sides. The group consists of n elements corresponding to rotations of the polygon, and n corresponding to reflections.

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In mathematics Mathematics

Mathematics is the discipline that deals with concepts such as quantity [i], structure [i], space [i] a ... 

, the dihedral group of order 2n is the abstract group of which one representation is the symmetry group Symmetry group

The symmetry [i] group of an object is the group [i] of all isometries [i] under which it is invariant [i] ... 

 in 2D of a regular polygon Regular polygon

A regular polygon is a simple polygon [i] which is [i] and equilateral [i] ... 

 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 Dihn is used for the dihedral group of order 2n as abstract group. The notations Dn and D2n are also seen.

For the isometry group in 2D of this abstract group type, the notation Dn is used. There are four series of isometry groups in 3D which are dihedral as abstract group Point groups in three dimensions

In geometry [i] a point group [i] in 3D is an isometry group [i] in three dimensions that leaves the ori ... 

. Only for one of them the notation Dn 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 Klein four-group

[i] Z2 × Z2, the [[direct product]... 

. 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 Cycle graph

In graph theory [i], a cycle graph, is a graph [i] that consists of a single cycle [i], or in ... 

 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.




















Dih1Dih2Dih3Dih4Dih5Dih6Dih7

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 Euclidean plane isometry

In geometry [i], a Euclidean plane isometry is an isometry [i] of the Euclidean plane [i], or more infor ... 

 which keep the origin fixed. These groups form one of the two series of discrete point groups in two dimensions. Dn consists of n rotation Rotation

Rotation is the movement of an object in a circular motion.... 

s 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 Symmetry group

The symmetry [i] group of an object is the group [i] of all isometries [i] under which it is invariant [i] ... 

 of a regular polygon Regular polygon

A regular polygon is a simple polygon [i] which is [i] and equilateral [i] ... 

 with n sides .

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

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

.

By setting
and defining and for we can write the product rules for as

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 , 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 Klein four-group

[i] Z2 × Z2, the [[direct product]... 

.

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 Symmetry

Symmetry is a characteristic feature of geometrical [i] shapes, system [i]s, equation [i]s, and ... 

 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 . 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, i.e. the group of rotations and reflections of the plane. However, notation Dn is also used for a subgroup of SO which is also of abstract group type Dihn: the proper symmetry group Symmetry group

The symmetry [i] group of an object is the group [i] of all isometries [i] under which it is invariant [i] ... 

 of a regular polygon embedded in three-dimensional space . Such a figure may be considered as a degenerate regular solid with its face counted twice. Therefore it is also called a dihedron , which explains the name dihedral group .

Equivalent definitions and properties

Further equivalent definitions of Dihn are:

  • The automorphism group of the graph Graph theory

    In mathematics [i] and computer science [i], graph theory is the study of graphs [i], mathema ... 

     consisting only of a cycle with n vertices .
  • The group with presentation

or
From the second presentation follows that Dihn belongs to the class of coxeter group Coxeter group

In mathematics [i], a Coxeter group, named after H.S.M. Coxeter [i], is a... 

s.

  • The semidirect product of cyclic group Cyclic group

    In group theory [i], a cyclic group or monogenous group is a group [i] that can be generated [i] ... 

    s Zn and Z2, with Z2 acting on Zn by inversion :


Zn
φ Z2 is isomorphic to Dihn if φ is the identity and φ is inversion.

If we consider Dihn 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 contains the element rn / 2 if n is even .

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 , is equal to d + σ , where d is the number of positive divisors of n and σ is the sum of the positive divisors of n. See List of small groups List of small groups

The following list in mathematics [i] contains the finite group [i]s of small order [i] up to [i] group isomorphism [i] ... 

 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 Euler's totient function

In number theory [i], the totient of a positive integer [i] n is defined to be the number of positi ... 

, 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 .

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 infinite Infinity

he word infinity comes from the Latin [i] infinitas or "unboundedness." It refers to several distinc ... 

ly many elements and is called Dih. It has presentations
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 .

Generalized dihedral group

For any abelian group H, the generalized dihedral group of H, written Dih, is the semidirect product of H and Z2, with Z2 acting on H by inverting elements. I.e., Dih = H
φ Z2 with φ the identity and φ inversion.

Thus we get:
* =
* =

for all h1, h2 in H and t2 in Z2.

Note that * = , i.e. first the inversion and then the operation in H. Also * = ; indeed inverts h, and toggles t between "normal" and "inverted" .

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

The conjugacy classes are:
  • the sets
  • the sets


Thus for every subgroup M of H, the corresponding set of elements is also a normal subgroup. We have:
Dih / M = Dih

Examples:
  • Dihn = Dih
    • For even n there are two sets , 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 . However, they are isomorphic as abstract groups.
    • For odd n there is only one set
  • Dih = Dih; there are two sets , 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 . However, they are isomorphic as abstract groups.
  • Dih, or orthogonal group O, or O: the isometry group of a circle Circle

    In Euclidean geometry [i], a circle is the set [i] of all points [i] in a plane at a fixed distance [i] ... 

    , or equivalently, the group of isometries in 2D that keep the origin fixed. The rotations form the circle group Circle group

    In mathematics [i], the circle group, denoted by T, is the multiplicative group [i] of all complex number [i] ... 

     S1, or equivalently SO, also written SO, and R / Z ; it is also the multiplicative group of complex number Complex number

    In mathematics [i], a complex number is a number [i] of the form

... 

s of absolute value Absolute value

In mathematics [i], the absolute value of a real number [i] is its numerical value without regard to it ... 

 1. In the latter case one of the reflections 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.
  • Dih: the group of isometries of Rn consisting of all translations and inversion in all points; for n = 1 this is the Euclidean group E; for n > 1 the group Dih is a proper subgroup of E, 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 which contain translations in one direction are of frieze group Frieze group

      A frieze group is a mathematical concept to classify designs on two-dimensional surfaces which are repet... 

       type and 22.
    • Discrete subgroups of Dih which contain translations in two directions are of wallpaper group Wallpaper group

      A wallpaper group is a mathematical classification of a two-dimensional repetitive pattern, based on the... 

       type p1 and p2.
    • Discrete subgroups of Dih which contain translations in three directions are space groups of the triclinic Triclinic crystal system

      In crystallography [i], the triclinic crystal system [i] is one of the 7 lattice point group [i] ... 

       crystal system Crystal system

      A crystal system is a category of space group [i]s, which characterize symmetry [i] of structures in thr ... 

      .


Dih is Abelian, with the semidirect product a direct product, iff all elements of H are their own inverse:
  • Dih = Dih1 = Z2
  • Dih = Dih2 = Z2 × Z2
  • Dih = Dih2 × Z2 = Z2 × Z2 × Z2

etc.

Topology

Dih and its dihedral subgroups are disconnected topological groups. Dih consists of two connected Connected space

In topology [i] and related branches of mathematics [i], a connected space is a topological space [i] wh ... 

 components: the identity component isomorphic to Rn, and the component with the reflections. Similarly O 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 components are open, while in the second case they are not. Also, the first topological group is a closed subgroup of Dih but the second is not a closed subgroup of O.

See also

  • quasidihedral group
  • dicyclic group
  • coordinate rotations and reflections
  • dihedral group of order 6 Dihedral group of order 6

    The smallest non-Abelian [i] group [i] has 6 elements. ... 

  • dihedral group of order 8 Examples of groups

    Some elementary examples of groups in mathematics [i] are given on Group [i].

... 


  • dihedral symmetry in three dimensions Dihedral symmetry in three dimensions

    This article deals with three infinite series of point groups in three dimensions [i] which have a symmetry group [i] ... 

  • dihedral symmetry groups in 3D Point groups in three dimensions

    In geometry [i] a point group [i] in 3D is an isometry group [i] in three dimensions that leaves the ori ...