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In mathematics, a **dihedral group** is the group of symmetries of a regular polygon,^{ [1] }^{ [2] } which includes 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.

- Definition
- Elements
- Group structure
- Matrix representation
- Other definitions
- Small dihedral groups
- The dihedral group as symmetry group in 2D and rotation group in 3D
- Examples of 2D dihedral symmetry
- Properties
- Conjugacy classes of reflections
- Automorphism group
- Examples of automorphism groups
- Inner automorphism group
- Generalizations
- See also
- References
- External links

The notation for the dihedral group differs in geometry and abstract algebra. In geometry, D_{n} or Dih_{n} refers to the symmetries of the *n*-gon, a group of order 2*n*. In abstract algebra, D_{2n} refers to this same dihedral group.^{ [3] } This article uses the geometric convention, D_{n}.

A regular polygon with sides has different symmetries: rotational symmetries and reflection symmetries. Usually, we take here. The associated rotations and reflections make up the dihedral group . If is odd, each axis of symmetry connects the midpoint of one side to the opposite vertex. If is even, there are axes of symmetry connecting the midpoints of opposite sides and axes of symmetry connecting opposite vertices. In either case, there are axes of symmetry and elements in the symmetry group.^{ [4] } Reflecting in one axis of symmetry followed by reflecting in another axis of symmetry produces a rotation through twice the angle between the axes.^{ [5] }

The following picture shows the effect of the sixteen elements of 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, in each case acting on the stop sign with the orientation as shown at the top left.

As with any geometric object, the composition of two symmetries of a regular polygon is again a symmetry of this object. With composition of symmetries to produce another as the binary operation, this gives the symmetries of a polygon the algebraic structure of a finite group.^{ [6] }

The following Cayley table shows the effect of composition in the group D_{3} (the symmetries of an equilateral triangle). r_{0} denotes the identity; r_{1} and r_{2} denote counterclockwise rotations by 120° and 240° respectively, and s_{0}, s_{1} and s_{2} denote reflections across the three lines shown in the adjacent picture.

r_{0} | r_{1} | r_{2} | s_{0} | s_{1} | s_{2} | |
---|---|---|---|---|---|---|

r_{0} | r_{0} | r_{1} | r_{2} | s_{0} | s_{1} | s_{2} |

r_{1} | r_{1} | r_{2} | r_{0} | s_{1} | s_{2} | s_{0} |

r_{2} | r_{2} | r_{0} | r_{1} | s_{2} | s_{0} | s_{1} |

s_{0} | s_{0} | s_{2} | s_{1} | r_{0} | r_{2} | r_{1} |

s_{1} | s_{1} | s_{0} | s_{2} | r_{1} | r_{0} | r_{2} |

s_{2} | s_{2} | s_{1} | s_{0} | r_{2} | r_{1} | r_{0} |

For example, s_{2}s_{1} = r_{1}, because the reflection s_{1} followed by the reflection s_{2} results in a rotation of 120°. The order of elements denoting the composition is right to left, reflecting the convention that the element acts on the expression to its right. The composition operation is not commutative.^{ [6] }

In general, the group D_{n} has elements r_{0}, ..., r_{n−1} and s_{0}, ..., s_{n−1}, with composition given by the following formulae:

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

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 D_{4} can be represented by the following eight matrices:

In general, the matrices for elements of D_{n} have the following form:

r_{k} is a rotation matrix, expressing a counterclockwise rotation through an angle of 2*πk*/*n*. s_{k} is a reflection across a line that makes an angle of *πk*/*n* with the *x*-axis.

Further equivalent definitions of D_{n} are:

- The group with presentation
_{n}belongs to the class of Coxeter groups.

D_{1} is isomorphic to Z_{2}, the cyclic group of order 2.

D_{2} is isomorphic to K_{4}, the Klein four-group.

D_{1} and D_{2} are exceptional in that:

- D
_{1}and D_{2}are the only abelian dihedral groups. Otherwise, D_{n}is non-abelian. - D
_{n}is a subgroup of the symmetric group S_{n}for*n*≥ 3. Since 2*n*>*n*! for*n*= 1 or*n*= 2, for these values, D_{n}is too large to be a subgroup. - The inner automorphism group of D
_{2}is trivial, whereas for other even values of*n*, this is D_{n}/ Z_{2}.

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 represents 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.

D_{1} = Z_{2} | D_{2} = Z_{2}^{2} = K_{4} | D_{3} | D_{4} | D_{5} |
---|---|---|---|---|

D_{6}= D_{3} × Z_{2} | D_{7} | D_{8} | D_{9} | D_{10}= D_{5} × Z_{2} |

D_{3} = S _{3} | D_{4} |
---|---|

An example of abstract group D_{n}, and a common way to visualize it, is the group 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; this extends to the cases *n* = 1 and *n* = 2 where we have a plane with respectively a point offset from the "center" of the "1-gon" and a "2-gon" or line segment).

D_{n} is generated by a rotation r of order *n* and a reflection s of order 2 such that

In geometric terms: in the mirror a rotation looks like an inverse rotation.

In terms of complex numbers: multiplication by and complex conjugation.

In matrix form, by setting

and defining and for we can write the product rules for D_{n} as

(Compare coordinate rotations and reflections.)

The dihedral group D_{2} is generated by the rotation r of 180 degrees, and the reflection s across the *x*-axis. The elements of D_{2} can then be represented as {e, r, s, rs}, where e is the identity or null transformation and rs is the reflection across the *y*-axis.

D_{2} is isomorphic to the Klein four-group.

For *n* > 2 the operations of rotation and reflection in general do not commute and D_{n} is not abelian; for example, in D_{4}, 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 2*n* elements of D_{n} can be written as e, r, r^{2}, ... , r^{n−1}, s, r s, r^{2}s, ... , r^{n−1}s. 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 D_{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).

- 2D D
_{16}symmetry – Imperial Seal of Japan, representing eightfold chrysanthemum with sixteen petals. - 2D D
_{12}symmetry — The Naval Jack of the Republic of China (White Sun)

The properties of the dihedral groups D_{n} with *n* ≥ 3 depend on whether *n* is even or odd. For example, the center of D_{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).

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

For *n* twice an odd number, the abstract group D_{n} is isomorphic with the direct product of D_{n / 2} and Z_{2}. Generally, if *m* divides *n*, then D_{n} has *n*/*m* subgroups of type D_{m}, and one subgroup _{m}. Therefore, the total number of subgroups of D_{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.

The dihedral group of order 8 (D_{4}) is the smallest example of a group that is not a T-group. Any of its two Klein four-group subgroups (which are normal in D_{4}) has as normal subgroup order-2 subgroups generated by a reflection (flip) in D_{4}, but these subgroups are not normal in D_{4}.

All the reflections are conjugate to each other whenever *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. Geometrically, in an odd polygon every axis of symmetry passes through a vertex and a side, while in an even polygon there are two sets of axes, each corresponding to a conjugacy class: those that pass through two vertices and those that pass through two sides.

Algebraically, this is an instance of the conjugate Sylow theorem (for *n* odd): for *n* odd, each reflection, together with the identity, form a subgroup of order 2, which is a Sylow 2-subgroup (2 = 2^{1} is the maximum power of 2 dividing 2*n* = 2[2*k* + 1]), while for *n* even, these order 2 subgroups are not Sylow subgroups because 4 (a higher power of 2) divides the order of the group.

For *n* even there is instead an outer automorphism interchanging the two types of reflections (properly, a class of outer automorphisms, which are all conjugate by an inner automorphism).

The automorphism group of D_{n} is isomorphic to the holomorph of /*n*, i.e., to Hol(/*n*) = {*ax* + *b* | (*a*, *n*) = 1} and has order *nϕ*(*n*), where *ϕ* is Euler's totient function, the number of *k* in 1, ..., *n*− 1 coprime to *n*.

It can be understood in terms of the generators of a reflection and an elementary rotation (rotation by *k*(2*π*/*n*), for *k* coprime to *n*); which automorphisms are inner and outer depends on the parity of *n*.

- For
*n*odd, the dihedral group is centerless, so any element defines a non-trivial inner automorphism; for*n*even, the rotation by 180° (reflection through the origin) is the non-trivial element of the center. - Thus for
*n*odd, the inner automorphism group has order 2*n*, and for*n*even (other than*n*= 2) the inner automorphism group has order*n*. - For
*n*odd, all reflections are conjugate; for*n*even, they fall into two classes (those through two vertices and those through two faces), related by an outer automorphism, which can be represented by rotation by*π*/*n*(half the minimal rotation). - The rotations are a normal subgroup; conjugation by a reflection changes the sign (direction) of the rotation, but otherwise leaves them unchanged. Thus automorphisms that multiply angles by
*k*(coprime to*n*) are outer unless*k*= ±1.

D_{9} has 18 inner automorphisms. As 2D isometry group D_{9}, 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.

D_{10} has 10 inner automorphisms. As 2D isometry group D_{10}, 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).

The only values of *n* for which *φ*(*n*) = 2 are 3, 4, and 6, and consequently, there are only three dihedral groups that are isomorphic to their own automorphism groups, namely D_{3} (order 6), D_{4} (order 8), and D_{6} (order 12).^{ [7] }^{ [8] }^{ [9] }

The inner automorphism group of D_{n} is isomorphic to:^{ [10] }

- D
_{n}if*n*is odd; - D
_{n}/ Z_{2}if*n*is even (for*n*= 2, D_{2}/ Z_{2}=*1*).

There are several important generalizations of the dihedral groups:

- The infinite dihedral group is an infinite group with algebraic structure similar to the finite dihedral groups. It can be viewed as the group of symmetries of the integers.
- The orthogonal group O(2),
*i.e.,*the symmetry group of the circle, also has similar properties to the dihedral groups. - The family of generalized dihedral groups includes both of the examples above, as well as many other groups.
- The quasidihedral groups are family of finite groups with similar properties to the dihedral groups.

Wikimedia Commons has media related to Dihedral groups .

In geometry, an **n-gonal antiprism** or **n-antiprism** is a polyhedron composed of two parallel direct copies of an n-sided polygon, connected by an alternating band of 2*n* triangles. They are represented by the Conway notation A*n*.

In group theory, the **symmetry group** of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the ambient space which takes the object to itself, and which preserves all the relevant structure of the object. A frequent notation for the symmetry group of an object *X* is *G* = Sym(*X*).

In mathematics, the **orthogonal group** in dimension *n*, denoted O(*n*), is the group of distance-preserving transformations of a Euclidean space of dimension *n* that preserve a fixed point, where the group operation is given by composing transformations. The orthogonal group is sometimes called the **general orthogonal group**, by analogy with the general linear group. Equivalently, it is the group of *n*×*n* orthogonal matrices, where the group operation is given by matrix multiplication. The orthogonal group is an algebraic group and a Lie group. It is compact.

In geometry, a **heptadecagon**, **septadecagon** or **17-gon** is a seventeen-sided polygon.

In mathematics the **spin group** Spin(*n*) is the double cover of the special orthogonal group SO(*n*) = SO(*n*, **R**), such that there exists a short exact sequence of Lie groups

In mathematics, the **circle group**, denoted by or , is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the **unit complex numbers**

In mathematics, a **Euclidean group** is the group of (Euclidean) isometries of a Euclidean space ; that is, the transformations of that space that preserve the Euclidean distance between any two points. The group depends only on the dimension *n* of the space, and is commonly denoted E(*n*) or ISO(*n*).

The **crystallographic restriction theorem** in its basic form was based on the observation that the rotational symmetries of a crystal are usually limited to 2-fold, 3-fold, 4-fold, and 6-fold. However, quasicrystals can occur with other diffraction pattern symmetries, such as 5-fold; these were not discovered until 1982 by Dan Shechtman.

In mathematics, **D _{3}** (sometimes alternatively denoted by

In group theory and geometry, a **reflection group** is a discrete group which is generated by a set of reflections of a finite-dimensional Euclidean space. The symmetry group of a regular polytope or of a tiling of the Euclidean space by congruent copies of a regular polytope is necessarily a reflection group. Reflection groups also include Weyl groups and crystallographic Coxeter groups. While the orthogonal group is generated by reflections, it is a continuous group, not a discrete group, and is generally considered separately.

In geometry, a **point group in three dimensions** is an isometry group in three dimensions that leaves the origin fixed, or correspondingly, an isometry group of a sphere. It is a subgroup of the orthogonal group O(3), the group of all isometries that leave the origin fixed, or correspondingly, the group of orthogonal matrices. O(3) itself is a subgroup of the Euclidean group E(3) of all isometries.

In a group, the **conjugate** by *g* of *h* is *ghg*^{−1}.

In geometry, a **two-dimensional point group** or **rosette group** is a group of geometric symmetries (isometries) that keep at least one point fixed in a plane. Every such group is a subgroup of the orthogonal group O(2), including O(2) itself. Its elements are rotations and reflections, and every such group containing only rotations is a subgroup of the special orthogonal group SO(2), including SO(2) itself. That group is isomorphic to R/Z and the first unitary group, U(1), a group also known as the circle group.

In mathematics, the **infinite dihedral group** Dih_{∞} is an infinite group with properties analogous to those of the finite dihedral groups.

In projective geometry and linear algebra, the **projective orthogonal group** PO is the induced action of the orthogonal group of a quadratic space *V* = (*V*,*Q*) on the associated projective space P(*V*). Explicitly, the projective orthogonal group is the quotient group

In mathematics, the **generalized dihedral groups** are a family of groups with algebraic structures similar to that of the dihedral groups. They include the finite dihedral groups, the infinite dihedral group, and the orthogonal group *O*(2). Dihedral groups play an important role in group theory, geometry, and chemistry.

In geometry, a **point reflection** is a type of isometry of Euclidean space. An object that is invariant under a point reflection is said to possess **point symmetry**; if it is invariant under point reflection through its center, it is said to possess **central symmetry** or to be **centrally symmetric.**

In mathematics, the **Bolza surface**, alternatively, complex algebraic **Bolza curve**, is a compact Riemann surface of genus with the highest possible order of the conformal automorphism group in this genus, namely of order 48. The full automorphism group is the semi-direct product of order 96. An affine model for the Bolza surface can be obtained as the locus of the equation

In the theory of Riemann surfaces and hyperbolic geometry, the triangle group (2,3,7) is particularly important. This importance stems from its connection to Hurwitz surfaces, namely Riemann surfaces of genus *g* with the largest possible order, 84(*g* − 1), of its automorphism group.

In geometry, **Coxeter notation** is a system of classifying symmetry groups, describing the angles between fundamental reflections of a Coxeter group in a bracketed notation expressing the structure of a Coxeter-Dynkin diagram, with modifiers to indicate certain subgroups. The notation is named after H. S. M. Coxeter, and has been more comprehensively defined by Norman Johnson.

- ↑ Weisstein, Eric W. "Dihedral Group".
*MathWorld*. - ↑ Dummit, David S.; Foote, Richard M. (2004).
*Abstract Algebra*(3rd ed.). John Wiley & Sons. ISBN 0-471-43334-9. - ↑ "Dihedral Groups: Notation".
*Math Images Project*. Archived from the original on 2016-03-20. Retrieved 2016-06-11. - ↑ Cameron, Peter Jephson (1998),
*Introduction to Algebra*, Oxford University Press, p. 95, ISBN 9780198501954 - ↑ Toth, Gabor (2006),
*Glimpses of Algebra and Geometry*, Undergraduate Texts in Mathematics (2nd ed.), Springer, p. 98, ISBN 9780387224558 - 1 2 Lovett, Stephen (2015),
*Abstract Algebra: Structures and Applications*, CRC Press, p. 71, ISBN 9781482248913 - ↑ Humphreys, John F. (1996).
*A Course in Group Theory*. Oxford University Press. p. 195. ISBN 9780198534594. - ↑ Pedersen, John. "Groups of small order". Dept of Mathematics, University of South Florida.
- ↑ Sommer-Simpson, Jasha (2 November 2013). "Automorphism groups for semidirect products of cyclic groups" (PDF). p. 13.
**Corollary 7.3.**Aut(D_{n}) = D_{n}if and only if*φ*(*n*) = 2 - ↑ Miller, GA (September 1942). "Automorphisms of the Dihedral Groups".
*Proc Natl Acad Sci U S A*.**28**(9): 368–71. doi: 10.1073/pnas.28.9.368 . PMC 1078492 . PMID 16588559.

- Dihedral Group n of Order 2n by Shawn Dudzik, Wolfram Demonstrations Project.
- Dihedral group at Groupprops
- Weisstein, Eric W. "Dihedral Group".
*MathWorld*. - Weisstein, Eric W. "Dihedral Group D3".
*MathWorld*. - Weisstein, Eric W. "Dihedral Group D4".
*MathWorld*. - Weisstein, Eric W. "Dihedral Group D5".
*MathWorld*. - Davis, Declan. "Dihedral Group D6".
*MathWorld*. - Dihedral groups on GroupNames

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