Editing Dihedral group (section)

===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''.