Editing Dihedral group (section)

==Definition==
The word "dihedral" comes from "di-" and "-hedron".
The latter comes from the  Greek word hédra, which means "face of a geometrical solid". Overall it thus refers to the two faces of a polygon.

===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 <math>n/2</math> 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
 | last = Cameron | first = Peter Jephson
 | title = Introduction to Algebra
 | 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
 | last = Toth | first = Gabor
 | title = Glimpses of Algebra and Geometry
 | series = Undergraduate Texts in Mathematics
 | edition = 2nd
 | publisher = Springer
 | year = 2006
 | isbn = 9780387224558
 | page = 98
 | url = https://books.google.com/books?id=IRwBCAAAQBAJ&pg=PA98
}}</ref>

[[File:Dihedral8.png|550px|thumb|center|This picture shows the effect of the sixteen elements of <math>\mathrm{D}_8</math> on a [[stop sign]]. Here, 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 [[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|The lines of reflection labelled S<sub>0</sub>, S<sub>1</sub>, and S<sub>2</sub> remain fixed in space (on the page) and do not themselves move when a symmetry operation (rotation or reflection) is done on the triangle (this matters when doing compositions of symmetries).]]
[[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} \\[5pt]
  \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===
{{math|D{{sub|''n''}}}} is the [[semidirect product]] of <math>\mathrm C_2 = \{1, s\}</math> acting on <math>\mathrm C_n</math> via the automorphism <math>\varphi_s(r) = r^{-1}</math>.<ref name="mac-lane">{{cite book |last1=Mac Lane |first1=Saunders |author-link1=Saunders Mac Lane |last2=Birkhoff |first2=Garrett |author-link2=Garrett Birkhoff |title=Algebra |edition=3rd |year=1999 |publisher=American Mathematical Society |isbn=0-8218-1646-2 |pages=414–415}}</ref>

It hence has [[presentation of a group|presentation]]<ref>{{cite book |last1=Johnson |first1=DL |title=Presentations of groups |date=1990 |publisher=Cambridge University Press |location=Cambridge, U.K. ; New York, NY, USA |isbn=9780521585422 |page=140 |url=https://archive.org/details/presentationsofg0000john_z8f6/page/140}}</ref>
: <math>\begin{align}
  \mathrm{D}_n &= \left\langle r, s \mid \operatorname{ord}(r) = n, \operatorname{ord}(s) = 2, srs^{-1} = r^{-1} \right\rangle \\
               &= \left\langle r, s \mid \operatorname{ord}(r) = n, \operatorname{ord}(s) = 2, srs = r^{-1} \right\rangle \\
               &= \left\langle r,s \mid r^n = s^2 = (sr)^2 = 1 \right\rangle.
\end{align}</math>

Using the relation <math>s^2 = 1</math>, we obtain the relation <math>r= s \cdot sr</math>. 
It follows that <math>\mathrm D_n</math> is generated by <math>s</math> and <math>t:=sr</math>. This substitution also shows that <math>\mathrm D_n</math> has the presentation
:<math>
  \mathrm{D}_n = \left\langle s,t \mid  s^2=1, t^2 = 1, (st)^n=1\right\rangle
.
</math>
In particular, {{math|D{{sub|''n''}}}} belongs to the class of [[Coxeter group]]s.