Dihedral group

In mathematics, the dihedral group of order 2n is a certain group for which here the notation D_n is used, but elsewhere the notation D_2n is also used. For n=1 and n=2 it is abelian, but for all other values of n it is not.

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 stand for 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.

D1	D2	D3	D4	D5	D6	D7

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

It is usually thought of as a group of Euclidean plane isometries consisting of 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). As such it is the symmetry group of a regular polygon with n sides (for n ≥3, and also for the degenerate case n = 2, where we have a line segment in the plane).

Depending on context, either this symmetry group is meant (specific up to isometry conjugacy), or just the group as algebraic structure (i.e. specific up to isomorphism).

Specifically the dihedral group D_n is generated by a rotation r of order n and a reflection f of order 2 such that

$frf=r^{-1}.$

One specific matrix representation is given by

$r={\begin{bmatrix}\cos {2\pi \over n}&-\sin {2\pi \over n}\\\sin {2\pi \over n}&\cos {2\pi \over n}\end{bmatrix}}\qquad f={\begin{bmatrix}1&0\\0&-1\end{bmatrix}}$

(in terms of complex numbers: multiplication by $e^{2\pi \over n}$ and complex conjugation).

By setting

$r_{0}={\begin{bmatrix}\cos {2\pi \over n}&-\sin {2\pi \over n}\\\sin {2\pi \over n}&\cos {2\pi \over n}\end{bmatrix}}\qquad f_{0}={\begin{bmatrix}1&0\\0&-1\end{bmatrix}}$

and defining $r_{j}=r_{0}^{j}$ and $f_{j}=f_{0}\,r_{j}$ for $j\in \{1,\ldots ,n-1\}$ we can write the product rules for $D_{n}$ as

$r_{j}\,r_{k}=r_{(j+k){\mbox{ mod n}}}$

$r_{j}\,f_{k}=f_{(j+k){\mbox{ mod n}}}$

$f_{j}\,r_{k}=f_{(j-k){\mbox{ mod n}}}$

$f_{j}\,f_{k}=r_{(j-k){\mbox{ mod n}}}$

The dihedral group D₂ is generated by the rotation r of 180 degrees, and the reflection f across the x-axis. The elements of D₂ can then be represented as {e, r, f, rf}, where e is the identity or null transformation and rf is the reflection across the y-axis.

x-axis is vertical

D₂ is isomorphic to the Klein four-group.

If the order of D_n is greater than 4, the operations of rotation and reflection in general do not commute and D_n is not abelian; for example, in D₄, a rotation of 90 degrees followed by a reflection yields a different result from a reflection followed by a rotation of 90 degrees:

x-axis is vertical

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 2n elements of D_n can be written as e, r, r²,...,rⁿ⁻¹, f, fr, fr²,...,frⁿ⁻¹. 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. One can also think of D_n as a subgroup of SO(3), i.e. the group of rotations (about the origin) of the three-dimensional space. From this point of view, D_n is 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).

See also dihedral symmetry.

Equivalent definitions and properties

Further equivalent definitions of D_n are:

• The automorphism group of the graph consisting only of a cycle with n vertices (if n ≥ 3).
• The group with presentation

$\langle r,f\mid r^{n}=1,f^{2}=1,frf=r^{-1}\rangle$

or

$\langle x,y\mid x^{2}=y^{2}=(xy)^{n}=1\rangle$

(Indeed the only finite groups that can be generated by two elements of order 2 are the dihedral groups and the cyclic groups)

• The semidirect product of cyclic groups C_n and C₂, with C₂ acting on C_n by inversion (thus, D_n always has a normal subgroup isomorphic to C_n):

C_n X_φ C₂ is isomorphic to D_n if φ(0) is the identity and φ(1) is inversion.

If we consider D_n (n ≥ 3) as the symmetry group of a regular n-gon and number the polygon's vertices, we see that D_n is a subgroup of the symmetric group S_n.

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 contains the element r^n/2 if n is even (with D_n as a subgroup of O(2) or SO(3), this is inversion; since it is scalar multiplication by −1, it is clear that it commutes with any linear transformation).

For odd n, abstract group D_2n is isomorphic with the direct product of D_n and C₂.

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

All the reflections are conjugate to each other in case n is odd, but they fall into two conjugacy classes if n is even. This corresponds to the geometrical fact that every symmetry axis of a regular n-gon passes through a vertex and an opposite side if n is odd, but half of them pass through opposite sides and half pass through opposite vertices if n is even. All the rotations are conjugate regardless of whether n is even of odd.

If m divides n, then D_m is a subgroup of D_n. The total number of subgroups of D_n (n ≥ 3), 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.

Generalizations

In addition to the finite dihedral groups, there is the infinite dihedral group D_∞. Every dihedral group is generated by a rotation r and a reflection; if the rotation is a rational multiple of a full rotation, then there is some integer n such that rⁿ is the identity, and we have a finite dihedral group. If the rotation is not a rational multiple of a full rotation, then there is no such n and the resulting group has infinitely many elements and is called D_∞. It has presentations

$\langle r,f\mid f^{2}=1,frf=r^{-1}\rangle$

$\langle x,y\mid x^{2}=y^{2}=1\rangle$

and is isomorphic to a semidirect product of Z and C₂, and to the free product C₂ * C₂. It can also be visualized as the automorphism group of the graph consisting of a path infinite to both sides, and is isomorphic to one of the (classes of) discrete symmetry groups in one dimension: that of repetitive patterns which also have mirror image symmetry.

Finally, if H is any abelian group, we can speak of the generalized dihedral group of H (sometimes written Dih(H)). This group is a semidirect product of H and C₂, with C₂ acting on H by inverting elements. Dih(H) has a normal subgroup of index 2 isomorphic to H, and contains in addition an element f of order 2 such that, for all x in H,  x f = f x ⁻¹.

We get:

(h₁, 0) * (h₂, t₂) = (h₁ + h₂, t₂)

(h₁, 1) * (h₂, t₂) = (h₁ - h₂, 1 + t₂)

for all h₁, h₂ in H and t₂ in C₂.

Note that (h, 0) * (0,1) = (h,1), i.e. first the inversion and then the operation in H. Also (0, 1) * (h, t) = (- h, 1 + t); indeed (0,1) inverts h, and toggles t between "normal" (0) and "inverted" (1) (this combined operation is its own inverse).

Clearly, we have D_n = Dih(C_n) and D_∞ = Dih(Z). The symmetry group of a straight line is isomorphic to Dih(R).

The symmetry group of a circle is Dih(S¹) (where S¹ denotes the circle group: the multiplicative group of complex numbers of absolute value 1), also called the orthogonal group O(2,R), or O(2) for short.

Dihedral symmetry groups in 3D

D_n is the rotation group of the n-sided prism with regular base, and n-sided bipyramid with regular base, and also of a regular, n-sided antiprism and of a regular, n-sided trapezohedron. The group is also the full symmetry group of such objects after making them chiral by e.g. an identical chiral marking on every face, or some modification in the shape.

There are two more infinite series of symmetry groups algebraically of type D_n:

• C_nv of order 2n, the symmetry group of a regular, n-sided prism with one base colored differently, and also of a regular n-sided pyramid
• D_nd of order 4n, the symmetry group of a regular, n-sided antiprism

Thus we have, with bolding of the 11 crystallographic point groups, for which the crystallographic restriction applies:

• order 4: D₂, C_2v, D_1d
• order 6: D₃, C_3v
• order 8: D₄, C_4v, D_2d
• order 10: D₅, C_5v
• order 12: D₆, C_6v, D_3d
• order 14: D₇, C_7v
• order 16: D₈, C_8v, D_4d

etc.

See also

• quasidihedral group
• dicyclic group
• coordinate rotations and reflections
• dihedral group of order 6
• dihedral group of order 8