Dihedral group

From Academic Kids

In mathematics, the dihedral group Dn is a certain nonabelian group of order 2n. It is usually thought of a group of transformations of the Euclidean plane consisting of rotations (about the origin) and reflections (across lines through the origin). As such it is the symmetry group of a regular polygon with n sides (for n > 2).

Warning: Many authors use the notation D2n instead of Dn for the dihedral group of order 2n.

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

<math>frf = r^{-1}.<math>

One specific matrix representation is given by

<math>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}<math>

The simplest dihedral group is D2, which is generated by the rotation r of 180 degrees, and the reflection f across the y-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 x-axis.

Missing image

D2 is isomorphic to the Klein four-group.

If the order of Dn is greater than 4, the operations of rotation and reflection in general do not commute and Dn 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:


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 Dn 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 Dn 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 Dn 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, Dn 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).

Further equivalent definitions of Dn are:

<math>\langle r, f \mid r^n = 1, f^2 = 1, frf = r^{-1} \rangle<math>
<math>\langle x, y \mid x^2 = y^2 = (xy)^n = 1 \rangle<math>
(Indeed the only finite groups that can be generated by two elements of order 2 are the dihedral groups and the cyclic groups)

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

The properties of the dihedral groups Dn with n ≥ 3 depend on whether n is even or odd. For example, the center of Dn consists only of the identity if n is odd, but contains the element rn/2 if n is even. 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.

If m divides n, then Dm is a subgroup of Dn. The total number of subgroups of Dn (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.


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

<math>\langle r, f \mid f^2 = 1, frf = r^{-1} \rangle<math>
<math>\langle x, y \mid x^2 = y^2 = 1 \rangle<math>

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.

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. Clearly, we have Dn = Dih(Cn) and D = Dih(Z). The symmetry group of a straight line is isomorphic to Dih(R) and the symmetry group of a circle is Dih(S1) (where S1 denotes the multiplicative group of complex numbers of absolute value 1).

See also


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