Elements

The six axes of reflection of a regular hexagon

A regular polygon with ${\displaystyle n}$ sides has ${\displaystyle 2n}$ different symmetries: ${\displaystyle n}$ rotational symmetries and ${\displaystyle n}$ reflection symmetries. Usually, we take ${\displaystyle n\geq 3}$ here. The associated rotations and reflections make up the dihedral group ${\displaystyle \mathrm {D} _{n}}$. If ${\displaystyle n}$ is odd, each axis of symmetry connects the midpoint of one side to the opposite vertex. If ${\displaystyle n}$ is even, there are ${\displaystyle n/2}$ axes of symmetry connecting the midpoints of opposite sides and ${\displaystyle n/2}$ axes of symmetry connecting opposite vertices. In either case, there are ${\displaystyle n}$ axes of symmetry and ${\displaystyle 2n}$ 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 ${\displaystyle \mathrm {D} _{8}}$ 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.

Group structure

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 lines of reflection labelled S₀, S₁, and S₂ 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).

The composition of these two reflections is a rotation.

The following Cayley table shows the effect of composition in the group D₃ (the symmetries of an equilateral triangle). r₀ denotes the identity; r₁ and r₂ denote counterclockwise rotations by 120° and 240° respectively, and s₀, s₁ and s₂ denote reflections across the three lines shown in the adjacent picture.

For example, s₂s₁ = r₁, because the reflection s₁ followed by the reflection s₂ 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 r₀, ..., r_n−1 and s₀, ..., s_n−1, with composition given by the following formulae:

${\displaystyle \mathrm {r} _{i}\,\mathrm {r} _{j}=\mathrm {r} _{i+j},\quad \mathrm {r} _{i}\,\mathrm {s} _{j}=\mathrm {s} _{i+j},\quad \mathrm {s} _{i}\,\mathrm {r} _{j}=\mathrm {s} _{i-j},\quad \mathrm {s} _{i}\,\mathrm {s} _{j}=\mathrm {r} _{i-j}.}$

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

Matrix representation

The symmetries of this pentagon are linear transformations 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 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 D₄ can be represented by the following eight matrices:

${\displaystyle {\begin{matrix}\mathrm {r} _{0}=\left({\begin{smallmatrix}1&0\\[0.2em]0&1\end{smallmatrix}}\right),&\mathrm {r} _{1}=\left({\begin{smallmatrix}0&-1\\[0.2em]1&0\end{smallmatrix}}\right),&\mathrm {r} _{2}=\left({\begin{smallmatrix}-1&0\\[0.2em]0&-1\end{smallmatrix}}\right),&\mathrm {r} _{3}=\left({\begin{smallmatrix}0&1\\[0.2em]-1&0\end{smallmatrix}}\right),\\[1em]\mathrm {s} _{0}=\left({\begin{smallmatrix}1&0\\[0.2em]0&-1\end{smallmatrix}}\right),&\mathrm {s} _{1}=\left({\begin{smallmatrix}0&1\\[0.2em]1&0\end{smallmatrix}}\right),&\mathrm {s} _{2}=\left({\begin{smallmatrix}-1&0\\[0.2em]0&1\end{smallmatrix}}\right),&\mathrm {s} _{3}=\left({\begin{smallmatrix}0&-1\\[0.2em]-1&0\end{smallmatrix}}\right).\end{matrix}}}$

In general, the matrices for elements of D_n have the following form:

${\displaystyle {\begin{aligned}\mathrm {r} _{k}&={\begin{pmatrix}\cos {\frac {2\pi k}{n}}&-\sin {\frac {2\pi k}{n}}\\\sin {\frac {2\pi k}{n}}&\cos {\frac {2\pi k}{n}}\end{pmatrix}}\ \ {\text{and}}\\[5pt]\mathrm {s} _{k}&={\begin{pmatrix}\cos {\frac {2\pi k}{n}}&\sin {\frac {2\pi k}{n}}\\\sin {\frac {2\pi k}{n}}&-\cos {\frac {2\pi k}{n}}\end{pmatrix}}.\end{aligned}}}$

r_k is a rotation matrix, expressing a counterclockwise rotation through an angle of 2πk/n. s_k is a reflection across a line that makes an angle of πk/n with the x-axis.

Other definitions

D_n can also be defined as the group with presentation

${\displaystyle {\begin{aligned}\mathrm {D} _{n}&=\left\langle r,s\mid \operatorname {ord} (r)=n,\operatorname {ord} (s)=2,srs^{-1}=r^{-1}\right\rangle \\&=\left\langle r,s\mid r^{n}=s^{2}=(sr)^{2}=1\right\rangle .\end{aligned}}}$

Using the relation ${\displaystyle s^{2}=1}$, we obtain the relation ${\displaystyle r=s\cdot sr}$. It follows that ${\displaystyle \mathrm {D} _{n}}$ is generated by ${\displaystyle s}$ and ${\displaystyle t:=sr}$. This substitution also shows that ${\displaystyle \mathrm {D} _{n}}$ has the presentation

${\displaystyle \mathrm {D} _{n}=\left\langle s,t\mid s^{2}=1,t^{2}=1,(st)^{n}=1\right\rangle .}$

In particular, D_n belongs to the class of Coxeter groups.

Examples of 2D dihedral symmetry

• 
  2D D₁₆ symmetry – Imperial Seal of Japan, representing eightfold chrysanthemum with sixteen petals.
• 
  2D D₆ symmetry – The Red Star of David
• 
  2D D₁₂ symmetry — The Naval Jack of the Republic of China (White Sun)
• 
  2D D₂₄ symmetry – Ashoka Chakra, as depicted on the National flag of the Republic of India.

Conjugacy classes of reflections

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 = 2¹ is the maximum power of 2 dividing 2n = 2[2k + 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).

Examples of automorphism groups

D₉ 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.

D₁₀ 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, 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 D₃ (order 6), D₄ (order 8), and D₆ (order 12).^[7][8][9]

Inner automorphism group

The inner automorphism group of D_n is isomorphic to:^[10]

• D_n if n is odd;
• D_n / Z₂ if n is even (for n = 2, D₂ / Z₂ = 1 ).