View source for Dihedral group

{{short description|Group of symmetries of a regular polygon}}
{{more inline|date=April 2015}}

{{Group theory sidebar|Finite}}
[[File:Snowflake8.png|thumb|The [[symmetry group]] of a [[snowflake]] is D<sub>6</sub>, a dihedral symmetry, the same as for a regular [[hexagon]].]]

In [[mathematics]], a '''dihedral group''' is the [[group (mathematics)|group]] of [[symmetry|symmetries]] of a [[regular polygon]],<ref>{{MathWorld|urlname=DihedralGroup|title=Dihedral Group}}</ref><ref>{{cite book | last1=Dummit | first1=David S. | last2=Foote | first2=Richard M. | title=Abstract Algebra | publisher=[[John Wiley & Sons]] | year=2004 | edition=3rd | isbn=0-471-43334-9}}</ref> which includes [[rotational symmetry|rotations]] and [[reflection symmetry|reflections]]. Dihedral groups are among the simplest examples of [[finite group]]s, and they play an important role in [[group theory]], [[geometry]], and [[chemistry]].

The notation for the dihedral group differs in [[geometry]] and [[abstract algebra]]. In [[geometry]], {{math|D{{sub|''n''}}}} or {{math|Dih{{sub|''n''}}}} refers to the symmetries of the [[n-gon]], a group of order {{math|2''n''}}. In [[abstract algebra]], {{math|D{{sub|2''n''}}}} refers to this same dihedral group.<ref name="mathimages">{{cite web|title=Dihedral Groups: Notation|url=http://mathforum.org/mathimages/index.php/Dihedral_Groups#Notation|website=Math Images Project|access-date=2016-06-11|archive-url=https://web.archive.org/web/20160320105838/http://mathforum.org/mathimages/index.php/Dihedral_Groups#Notation#Notation|archive-date=2016-03-20|url-status=dead}}</ref> The geometric convention is used in this article.

==Definition==

===Elements===
[[File:Hexagon reflections.svg|thumb|The six axes of [[reflection symmetry|reflection]] of a regular hexagon]]
A regular polygon with <math>n</math> sides has <math>2n</math> different symmetries: <math>n</math> [[rotational symmetry|rotational symmetries]] and <math>n</math> [[reflection symmetry|reflection symmetries]]. Usually, we take <math>n \ge 3</math> here. The associated [[rotation]]s and [[reflection (mathematics)|reflections]] make up the dihedral group <math>\mathrm{D}_n</math>. If <math>n</math> is odd, each axis of symmetry connects the midpoint of one side to the opposite vertex. If <math>n</math> is even, there are ''n''/2 axes of symmetry connecting the midpoints of opposite sides and <math>n/2</math> axes of symmetry connecting opposite vertices. In either case, there are <math>n</math> axes of symmetry and <math>2n</math> elements in the symmetry group.<ref>{{citation|title=Introduction to Algebra|first=Peter Jephson|last=Cameron|publisher=Oxford University Press|year=1998|isbn=9780198501954|page=95|url=https://books.google.com/books?id=syYYl-NVM5IC&pg=PA95}}</ref> Reflecting in one axis of symmetry followed by reflecting in another axis of symmetry produces a rotation through twice the angle between the axes.<ref>{{citation|title=Glimpses of Algebra and Geometry|series=Undergraduate Texts in Mathematics|first=Gabor|last=Toth|edition=2nd|publisher=Springer|year=2006|isbn=9780387224558|page=98|url=https://books.google.com/books?id=IRwBCAAAQBAJ&pg=PA98}}</ref>

The following picture shows the effect of the sixteen elements of <math>\mathrm{D}_8</math> on a [[stop sign]]:

[[File:Dihedral8.png|550px]]

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.

===Group structure===
As with any geometric object, the [[composition of functions|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 [[group (mathematics)|finite group]].<ref name=lovett>{{citation|title=Abstract Algebra: Structures and Applications|first=Stephen|last=Lovett|publisher=CRC Press|year=2015|isbn=9781482248913|page=71|url=https://books.google.com/books?id=jRUqCgAAQBAJ&pg=PA71}}</ref>

[[File:Labeled Triangle Reflections.svg|thumb]]
[[File:Two Reflection Rotation.svg|thumb|The composition of these two reflections is a rotation.]]
The following [[Cayley table]] shows the effect of composition in the group [[Dihedral group of order 6|D<sub>3</sub>]] (the symmetries of an [[equilateral triangle]]).  r<sub>0</sub> denotes the identity; r<sub>1</sub> and r<sub>2</sub> denote counterclockwise rotations by 120° and 240° respectively, and s<sub>0</sub>, s<sub>1</sub> and s<sub>2</sub> denote reflections across the three lines shown in the adjacent picture.

{| class=wikitable width=200
!||r<sub>0</sub>||r<sub>1</sub>||r<sub>2</sub>||s<sub>0</sub>||s<sub>1</sub>||s<sub>2</sub>
|-
!r<sub>0</sub>
| r<sub>0</sub> || r<sub>1</sub> || r<sub>2</sub> 
| s<sub>0</sub> || s<sub>1</sub> || s<sub>2</sub>
|-
!r<sub>1</sub>
| r<sub>1</sub> || r<sub>2</sub> || r<sub>0</sub> 
| s<sub>1</sub> || s<sub>2</sub> || s<sub>0</sub>
|-
!r<sub>2</sub>
| r<sub>2</sub> || r<sub>0</sub> || r<sub>1</sub> 
| s<sub>2</sub> || s<sub>0</sub> || s<sub>1</sub>
|-
!s<sub>0</sub>
| s<sub>0</sub> || s<sub>2</sub> || s<sub>1</sub> 
| r<sub>0</sub> || r<sub>2</sub> || r<sub>1</sub>
|-
!s<sub>1</sub>
| s<sub>1</sub> || s<sub>0</sub> || s<sub>2</sub> 
| r<sub>1</sub> || r<sub>0</sub> || r<sub>2</sub>
|-
!s<sub>2</sub>
| s<sub>2</sub> || s<sub>1</sub> || s<sub>0</sub> 
| r<sub>2</sub> || r<sub>1</sub> || r<sub>0</sub>
|}

For example, {{nowrap|1=s<sub>2</sub>s<sub>1</sub> = r<sub>1</sub>}}, because the reflection s<sub>1</sub> followed by the reflection s<sub>2</sub> results in a rotation of 120°.  The order of elements denoting the [[composition of functions|composition]] is right to left, reflecting the convention that the element acts on the expression to its right.  The composition operation is not [[commutativity|commutative]].<ref name=lovett/>

In general, the group D<sub>''n''</sub> has elements r<sub>0</sub>, ..., r<sub>''n''&minus;1</sub> and s<sub>0</sub>, ..., s<sub>''n''&minus;1</sub>, with composition given by the following formulae:

:<math>\mathrm{r}_i\,\mathrm{r}_j = \mathrm{r}_{i+j}, \quad \mathrm{r}_i\,\mathrm{s}_j = \mathrm{s}_{i+j}, \quad \mathrm{s}_i\,\mathrm{r}_j = \mathrm{s}_{i-j}, \quad \mathrm{s}_i\,\mathrm{s}_j = \mathrm{r}_{i-j}.</math>

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

===Matrix representation===
[[File:Pentagon Linear.png|thumb|The symmetries of this pentagon are [[linear transformation]]s of the plane as a vector space.]]
If we center the regular polygon at the origin, then elements of the dihedral group act as [[linear map|linear transformations]] of the [[Cartesian coordinate system|plane]].  This lets us represent elements of D<sub>''n''</sub> as [[Matrix (mathematics)|matrices]], with composition being [[matrix multiplication]].
This is an example of a (2-dimensional) [[group representation]].

For example, the elements of the group [[Dihedral group of order 8|D<sub>4</sub>]] can be represented by the following eight matrices:

:<math>\begin{matrix}
  \mathrm{r}_0 = \left(\begin{smallmatrix}  1 &  0 \\[0.2em]  0 &  1 \end{smallmatrix}\right), &
    \mathrm{r}_1 = \left(\begin{smallmatrix}  0 & -1 \\[0.2em]  1 &  0 \end{smallmatrix}\right), &
    \mathrm{r}_2 = \left(\begin{smallmatrix} -1 &  0 \\[0.2em]  0 & -1 \end{smallmatrix}\right), &
    \mathrm{r}_3 = \left(\begin{smallmatrix}  0 &  1 \\[0.2em] -1 &  0 \end{smallmatrix}\right), \\[1em]
  \mathrm{s}_0 = \left(\begin{smallmatrix}  1 &  0 \\[0.2em]  0 & -1 \end{smallmatrix}\right), &
    \mathrm{s}_1 = \left(\begin{smallmatrix}  0 &  1 \\[0.2em]  1 &  0 \end{smallmatrix}\right), &
    \mathrm{s}_2 = \left(\begin{smallmatrix} -1 &  0 \\[0.2em]  0 &  1 \end{smallmatrix}\right), &
    \mathrm{s}_3 = \left(\begin{smallmatrix}  0 & -1 \\[0.2em] -1 &  0 \end{smallmatrix}\right).
\end{matrix}</math>

In general, the matrices for elements of D<sub>''n''</sub> have the following form:

:<math>\begin{align}
  \mathrm{r}_k & = \begin{pmatrix}
    \cos \frac{2\pi k}{n} & -\sin \frac{2\pi k}{n} \\
    \sin \frac{2\pi k}{n} &  \cos \frac{2\pi k}{n}
  \end{pmatrix}\ \ \text{and} \\
  \mathrm{s}_k & =  \begin{pmatrix}
    \cos \frac{2\pi k}{n} &  \sin \frac{2\pi k}{n} \\
    \sin \frac{2\pi k}{n} & -\cos \frac{2\pi k}{n}
  \end{pmatrix}
  .
\end{align}</math>

r<sub>''k''</sub> is a [[rotation matrix]], expressing a counterclockwise rotation through an angle of  {{nowrap|2''&pi;k''/''n''}}.  s<sub>''k''</sub> is a reflection across a line that makes an angle of {{nowrap|''&pi;k''/''n''}} with the ''x''-axis.

===Other definitions===
Further equivalent definitions of {{math|D{{sub|''n''}}}} are:{{unordered list
| The [[automorphism group]] of the [[graph theory|graph]] consisting only of a cycle with ''n'' vertices (if ''n'' ≥ 3).
| The group with [[presentation of a group|presentation]]
:<math>\begin{align}
  \mathrm{D}_{n} &= \langle \mathrm{r}, \mathrm{s} \mid \operatorname{ord}(\mathrm{r}) = n, \operatorname{ord}(\mathrm{s}) = 2, \mathrm{srs} = \mathrm{r}^{-1} \rangle \\
        &= \langle \mathrm{r, s} \mid \mathrm{r}^n = \mathrm{s}^2 = (\mathrm{sr})^2 = 1 \rangle.
\end{align}</math>
From the second presentation follows that {{math|D{{sub|''n''}}}} belongs to the class of [[Coxeter group]]s.
| The [[semidirect product]] of [[cyclic group]]s {{math|Z{{sub|''n''}}}} and {{math|Z{{sub|2}}}}, with {{math|Z{{sub|2}}}} acting on {{math|Z{{sub|''n''}}}} by [[inversion (discrete mathematics)|inversion]] (thus, {{math|D{{sub|''n''}}}} always has a [[normal subgroup]] isomorphic to the group {{math|Z{{sub|''n''}}}}). {{nowrap|Z{{sub|''n''}} ⋊{{sub|''φ''}} Z{{sub|2}}}} is isomorphic to {{math|D{{sub|''n''}}}} if {{math|''φ''(0)}} is the [[identity function|identity]] and {{math|''φ''(1)}} is inversion.
}}

==Small dihedral groups==
[[File:Regular hexagon symmetries.svg|thumb|350px|Example subgroups from a hexagonal dihedral symmetry]]
{{math|D{{sub|1}}}} is [[isomorphic]] to {{math|Z{{sub|2}}}}, the [[cyclic group]] of order 2.

{{math|D{{sub|2}}}} is [[isomorphic]] to {{math|K{{sub|4}}}}, the [[Klein four-group]].

{{math|D{{sub|1}}}} and {{math|D{{sub|2}}}} are exceptional in that:
* {{math|D{{sub|1}}}} and {{math|D{{sub|2}}}} are the only [[abelian group|abelian]] dihedral groups. Otherwise, {{math|D{{sub|''n''}}}} is non-abelian.
* {{math|D{{sub|''n''}}}} is a [[subgroup]] of the [[symmetric group]] {{math|S{{sub|''n''}}}} for {{math|''n'' ≥ 3}}. Since {{math|2''n'' > ''n''!}} for  {{math|''n'' {{=}} 1}} or {{math|''n'' {{=}} 2}}, for these values, {{math|D{{sub|''n''}}}} is too large to be a subgroup.
* The inner automorphism group of {{math|D{{sub|2}}}} is trivial, whereas for other even values of {{math|''n''}}, this is {{math|D{{sub|''n''}} / Z{{sub|2}}}}.

The [[cycle graph (group)|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]].
{| class=wikitable
|+ Cycle graphs
|-
! D<sub>1</sub> = [[Cyclic group|Z<sub>2</sub>]] || D<sub>2</sub> = Z<sub>2</sub><sup>2</sup> = [[Klein four-group|K<sub>4</sub>]] || D<sub>3</sub> || D<sub>4</sub> || D<sub>5</sub>
|-
| [[File:GroupDiagramMiniC2.svg|100px]] || [[File:GroupDiagramMiniD4.svg|100px]] || [[File:GroupDiagramMiniD6.svg|100px]] || [[File:GroupDiagramMiniD8.svg|100px]] || [[File:GroupDiagramMiniD10.svg|100px]]
|-
| [[File:GroupDiagramMiniD12.svg|100px]] || [[File:GroupDiagramMiniD14.svg|100px]] || [[File:GroupDiagramMiniD16.svg|100px]] || [[File:GroupDiagramMiniD18.png|100px]] || [[File:GroupDiagramMiniD20.png|100px]]
|-
! D{{sub|6}} {{=}} D{{sub|3}} × Z{{sub|2}} || D{{sub|7}} || D<sub>8</sub> || D<sub>9</sub> || D{{sub|10}} {{=}} D{{sub|5}} × Z{{sub|2}}
|}

{| class=wikitable style="text-align:center; vertical-align:top;"
|-
! [[Dihedral group of order 6|D<sub>3</sub>]] = [[Symmetric group|S]]<sub>3</sub>
! [[Dihedral group of order 8|D<sub>4</sub>]]
|-
| [[File:Symmetric group 3; cycle graph.svg|240px]]
| [[File:Dih4 cycle graph.svg|240px]]
|}

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

An example of abstract group {{math|D{{sub|''n''}}}}, and a common way to visualize it, is the group of [[Euclidean plane isometry|Euclidean plane isometries]] which keep the origin fixed. These groups form one of the two series of discrete [[point groups in two dimensions]]. {{math|D{{sub|''n''}}}} consists of {{math|''n''}} [[rotation]]s of multiples of {{math|360°/''n''}} about the origin, and [[Reflection (mathematics)|reflection]]s across {{math|''n''}} lines through the origin, making angles of multiples of {{math|180°/''n''}} with each other. This is the [[symmetry group]] of a [[regular polygon]] with {{math|''n''}} sides (for {{math|''n'' ≥ 3}}; this extends to the cases {{math|''n'' {{=}} 1}} and {{math|''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).

{{math|D{{sub|''n''}}}} is [[generating set of a group|generated]] by a rotation {{math|r}} of [[order (group theory)|order]] {{math|''n''}} and a reflection {{math|s}} of order 2 such that

:<math>\mathrm{srs} = \mathrm{r}^{-1} \, </math>

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

In terms of [[complex numbers]]: multiplication by <math>e^{2\pi i \over n}</math> and [[complex conjugation]].

In matrix form, by setting

:<math>\mathrm{r}_1 = \begin{bmatrix}\cos{2\pi \over n} & -\sin{2\pi \over n} \\[8pt] \sin{2\pi \over n} & \cos{2\pi \over n}\end{bmatrix} \qquad \mathrm{s}_0 = \begin{bmatrix}1 & 0 \\ 0 & -1\end{bmatrix}</math>

and defining <math>\mathrm{r}_j = \mathrm{r}_1^j</math> and <math>\mathrm{s}_j = \mathrm{r}_j \, \mathrm{s}_0</math> for <math>j \in \{1,\ldots,n-1\}</math> we can write the product rules for &nbsp;D<sub>''n''</sub> as

:<math>\begin{align}
  \mathrm{r}_j \, \mathrm{r}_k &= \mathrm{r}_{(j+k) \text{ mod }n} \\
  \mathrm{r}_j \, \mathrm{s}_k &= \mathrm{s}_{(j+k) \text{ mod }n} \\
  \mathrm{s}_j \, \mathrm{r}_k &= \mathrm{s}_{(j-k) \text{ mod }n} \\
  \mathrm{s}_j \, \mathrm{s}_k &= \mathrm{r}_{(j-k) \text{ mod }n}
\end{align}</math>

(Compare [[coordinate rotations and reflections]].)

The dihedral group D<sub>2</sub> is generated by the rotation r of 180 degrees, and the reflection s across the ''x''-axis. The elements of D<sub>2</sub> can then be represented as {e,&nbsp;r,&nbsp;s,&nbsp;rs}, where e is the identity or null transformation and rs is the reflection across the ''y''-axis.

[[File:Dihedral4.png|thumb|500px|The four elements of D<sub>2</sub> (x-axis is vertical here)]]

D<sub>2</sub> is [[group isomorphism|isomorphic]] to the [[Klein four-group]].

For ''n'' > 2 the operations of rotation and reflection in general do not [[commutative|commute]] and D<sub>''n''</sub> is not [[abelian group|abelian]]; for example, in [[Dihedral group of order 8|D<sub>4</sub>]], a rotation of 90 degrees followed by a reflection yields a different result from a reflection followed by a rotation of 90 degrees.

[[File:d8isNonAbelian.png|thumb|500px|D<sub>4</sub> 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 {{math|2''n''}} elements of {{math|D{{sub|''n''}}}} can be written as {{math|e}}, {{math|r}}, {{math|r{{sup|2}}}}, ... , {{math|r{{sup|''n''−1}}}}, {{math|s}}, {{math|r s}}, {{math|r{{sup|2}}s}}, ... ,  {{math|r{{sup|''n''−1}}s}}. The first {{math|''n''}} listed elements are rotations and the remaining {{math|''n''}} elements are axis-reflections (all of which have order&nbsp;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 {{math|D{{sub|''n''}}}} to be a [[subgroup]] of [[orthogonal group|{{math|O(2)}}]], i.e. the group of rotations (about the origin) and reflections (across axes through the origin) of the plane. However, notation {{math|D{{sub|''n''}}}} is also used for a subgroup of [[SO(3)]] which is also of abstract group type {{math|D{{sub|''n''}}}}: the [[symmetry group|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).

===Examples of 2D dihedral symmetry===
<gallery widths="180px">
File:Red Star of David.svg|2D D<sub>6</sub> symmetry &ndash; [[Red Cross (symbol)#The Red Star of David (Magen David Adom)|The Red Star of David]]
File:Ashoka Chakra.svg|2D D<sub>24</sub> symmetry &ndash; [[Ashoka Chakra]], as depicted on the [[Flag of India|National flag of the Republic of India]].
</gallery>

==Properties==

The properties of the dihedral groups {{math|D{{sub|''n''}}}} with {{math|''n'' ≥ 3}} depend on whether {{math|''n''}} is even or odd. For example, the [[center of a group|center]] of {{math|D{{sub|''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<sup>''n''/2</sup> (with D<sub>''n''</sub> as a subgroup of O(2), this is [[Inversion (discrete mathematics)|inversion]]; since it is [[scalar multiplication]] by &minus;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 {{math|D{{sub|''n''}}}} is isomorphic with the [[direct product of groups|direct product]] of {{math|D{{sub|''n'' / 2}}}} and {{math|Z{{sub|2}}}}.
Generally, if ''m'' [[divisor|divides]] ''n'', then {{math|D{{sub|''n''}}}} has ''n''/''m'' [[subgroup]]s of type {{math|D{{sub|''m''}}}}, and one subgroup ℤ<sub>''m''</sub>. Therefore, the total number of subgroups of {{math|D{{sub|''n''}}}} (''n''&nbsp;≥&nbsp;1), is equal to ''d''(''n'')&nbsp;+&nbsp;σ(''n''), where ''d''(''n'') is the number of positive [[divisor]]s of ''n'' and ''σ''(''n'') is the sum of the positive divisors of&nbsp;''n''.  See [[list of small groups]] for the cases&nbsp;''n''&nbsp;≤&nbsp;8.

The dihedral group of order 8 (D<sub>4</sub>) is the smallest example of a group that is not a [[T-group (mathematics)|T-group]]. Any of its two [[Klein four-group]] subgroups (which are normal in D<sub>4</sub>) has as normal subgroup order-2 subgroups generated by a reflection (flip) in D<sub>4</sub>, but these subgroups are not normal in D<sub>4</sub>.

===Conjugacy classes of reflections===
All the reflections are [[conjugacy class|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. 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 subgroup|Sylow 2-subgroup]] ({{nowrap|2 {{=}} 2{{sup|1}}}} is the maximum power of 2 dividing {{nowrap|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).

==Automorphism group==
The [[automorphism group]] of {{math|D{{sub|''n''}}}} is isomorphic to the [[Holomorph (mathematics)|holomorph]] of ℤ/''n''ℤ, i.e., to {{nowrap|Hol(ℤ/''n''ℤ) {{=}} {''ax'' + ''b'' {{!}} (''a'', ''n'') {{=}} 1} }} and has order ''nϕ''(''n''), where ''ϕ'' is Euler's [[totient]] function, the number of ''k'' in {{nowrap|1, …, ''n'' &minus; 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 {{nowrap|''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 {{nowrap|''k'' {{=}} ±1}}.

===Examples of automorphism groups===
{{math|D{{sub|9}}}} has 18 [[inner automorphism]]s. As 2D isometry group D<sub>9</sub>, 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 automorphism]]s; e.g., multiplying angles of rotation by 2.

{{math|D{{sub|10}}}} has 10 inner automorphisms. As 2D isometry group D<sub>10</sub>, 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|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 {{math|D{{sub|3}}}} (order 6), {{math|D{{sub|4}}}} (order 8), and {{math|D{{sub|6}}}} (order 12).<ref>{{cite book|title=A Course in Group Theory|first1=John F.|last1=Humphreys|year=1996|isbn=9780198534594|publisher=Oxford University Press|page=195|url=https://books.google.com/books?id=2jBqvVb0Q-AC&pg=PA195}}</ref><ref>{{cite web|url=http://www.math.ucsd.edu/~atparris/small_groups.html|title=Groups of small order|first1=John|last1=Pedersen|publisher=Dept of Mathematics, University of South Florida}}</ref><ref>{{cite web|url=http://math.uchicago.edu/~may/REU2013/REUPapers/Sommer-Simpson.pdf|title=Automorphism groups for semidirect products of cyclic groups|first1=Jasha|last1=Sommer-Simpson|format=pdf|page=13|quote='''Corollary 7.3.''' Aut(D<sub>''n''</sub>) = D<sub>''n''</sub> if and only if ''φ''(''n'') = 2|date=2 November 2013}}</ref>

===Inner automorphism group===
The inner automorphism group of {{math|D{{sub|''n''}}}} is isomorphic to:<ref>{{cite journal | pmc = 1078492 | pmid=16588559 | volume=28 | title=Automorphisms of the Dihedral Groups | journal=Proc Natl Acad Sci U S A | pages=368–71 | last1 = Miller | first1 = GA | date=September 1942 | doi=10.1073/pnas.28.9.368}}</ref>
* {{math|D{{sub|''n''}}}} if ''n'' is odd;
* {{math|D{{sub|''n''}} / Z{{sub|2}}}} if {{math|''n''}} is even (for {{math|''n'' {{=}} 2}}, {{math|D{{sub|''2''}} / Z{{sub|2}} {{=}} ''1''}} ).

==Generalizations==
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 [[integer]]s.
* 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 group]]s includes both of the examples above, as well as many other groups.
* The [[quasidihedral group]]s are family of finite groups with similar properties to the dihedral groups.

==See also==
{{commons category|Dihedral groups}}
* [[Coordinate rotations and reflections]]
* [[Cycle index#Dihedral group Dn|Cycle index of the dihedral group]]
* [[Dicyclic group]]
* [[Dihedral group of order 6]]
* [[Dihedral group of order 8]]
* [[Point groups in three dimensions#Symmetry groups in 3D that are dihedral as abstract group|Dihedral symmetry groups in 3D]]
* [[Dihedral symmetry in three dimensions]]

==References==
{{reflist|2}}

==External links==
* [http://demonstrations.wolfram.com/DihedralGroupNOfOrder2n/ Dihedral Group n of Order 2n] by Shawn Dudzik, [[Wolfram Demonstrations Project]].
* [http://groupprops.subwiki.org/wiki/Dihedral_group Dihedral group] at Groupprops
* {{MathWorld|urlname=DihedralGroup|title=Dihedral Group}}
* {{MathWorld|urlname=DihedralGroupD3|title=Dihedral Group D3}}
* {{MathWorld|urlname=DihedralGroupD4|title=Dihedral Group D4}}
* {{MathWorld|urlname=DihedralGroupD5|title=Dihedral Group D5}}
* {{MathWorld|urlname=DihedralGroupD6|title=Dihedral Group D6|author=Davis, Declan}}
* [http://groupnames.org/#?dihedral Dihedral groups on GroupNames]

[[Category:Euclidean symmetries]]
[[Category:Finite reflection groups]]
[[Category:Properties of groups]]