Dihedral group

In mathematics, the dihedral group D_{n} 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 D_{2n} instead of D_{n} for the dihedral group of order 2n.
Specifically the dihedral group D_{n} 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 D_{2}, which is generated by the rotation r of 180 degrees, and the reflection f across the yaxis. The elements of D_{2} can then be represented as {e, r, f, rf}, where e is the identity or null transformation and rf is the reflection across the xaxis.
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Dihedral4.png
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D_{2} is isomorphic to the Klein fourgroup.
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 D_{4}, 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 nonabelian 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, r^{2},...,r^{n−1}, f, fr, fr^{2},...,fr^{n−1}. The first n listed elements are rotations and the remaining n elements are axisreflections (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 threedimensional space. From this point of view, D_{n} is the proper symmetry group of a regular polygon embedded in threedimensional 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 D_{n} are:
 The automorphism group of the graph consisting only of a cycle with n vertices (if n ≥ 3).
 The group with presentation
 <math>\langle r, f \mid r^n = 1, f^2 = 1, frf = r^{1} \rangle<math>
 or
 <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)
 The semidirect product of cyclic groups C_{n} and C_{2}, with C_{2} acting on C_{n} by inversion (thus, D_{n} always has a normal subgroup isomorphic to C_{n})
If we consider D_{n} (n ≥ 3) as the symmetry group of a regular ngon 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. 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 ngon 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 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 r^{n} 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 C_{2}, and to the free product C_{2} * C_{2}. 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 C_{2}, with C_{2} 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 D_{n} = Dih(C_{n}) 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(S^{1}) (where S^{1} denotes the multiplicative group of complex numbers of absolute value 1).