In mathematics, a dihedral group is the group of symmetries of a regular polygon, including both rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, geometry, and chemistry.
See also: Dihedral symmetry in three dimensions.
There are two competing notations for the dihedral group associated to a polygon with n sides. In geometry the group is denoted D_{n}, while in algebra the same group is denoted by D_{2n} to indicate the number of elements.
In this article, D_{n} (and sometimes Dih_{n}) refers to the symmetries of a regular polygon with n sides.
A regular polygon with n sides has 2n different symmetries: n rotational symmetries and n reflection symmetries. The associated rotations and reflections make up the dihedral group D_{n}. The following picture shows the effect of the sixteen elements of 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.
The following Cayley table shows the effect of composition in the group D_{3} (the symmetries of an equilateral triangle). R_{0} denotes the identity; R_{1} and R_{2} denote counterclockwise rotations by 120 and 240 degrees; and S_{0}, S_{1}, and S_{2} denote reflections across the three lines shown in the picture to the right.
R_{0} | R_{1} | R_{2} | S_{0} | S_{1} | S_{2} | |
---|---|---|---|---|---|---|
R_{0} | R_{0} | R_{1} | R_{2} | S_{0} | S_{1} | S_{2} |
R_{1} | R_{1} | R_{2} | R_{0} | S_{1} | S_{2} | S_{0} |
R_{2} | R_{2} | R_{0} | R_{1} | S_{2} | S_{0} | S_{1} |
S_{0} | S_{0} | S_{2} | S_{1} | R_{0} | R_{2} | R_{1} |
S_{1} | S_{1} | S_{0} | S_{2} | R_{1} | R_{0} | R_{2} |
S_{2} | S_{2} | S_{1} | S_{0} | R_{2} | R_{1} | R_{0} |
For example, S_{2}S_{1} = R_{1} because the reflection S_{1} followed by the reflection S_{2} results in a 120-degree rotation. (This is the normal backwards order for composition.) Note that the composition operation is not commutative.
In general, the group D_{n} has elements R_{0},...,R_{n−1} and S_{0},...,S_{n−1}, with composition given by the following formulae:
In all cases, addition and subtraction of subscripts should be performed using modular arithmetic with modulus n.
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 D_{4} can be represented by the following eight matrices:
In general, the matrices for elements of D_{n} have the following form:
The first matrix is a rotation matrix, expressing a counterclockwise rotation through an angle of . The second matrix is a reflection across a line that makes an angle of with the x-axis.
For n = 1 we have Dih_{1}. This notation is rarely used except in the framework of the series, because it is equal to Z_{2}. For n = 2 we have Dih_{2}, the Klein four-group. Both are exceptional within the series:
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.
Dih_{1} | Dih_{2} | Dih_{3} | Dih_{4} | Dih_{5} | Dih_{6} | Dih_{7} |
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An example of abstract group Dih_{n}, and a common way to visualize it, is the group D_{n} of Euclidean plane isometries which keep the origin fixed. These groups form one of the two series of discrete point groups in two dimensions. D_{n} consists of n 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. This 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).
Dihedral group D_{n} is generated by a rotation r of order n and a reflection f of order 2 such that
In matrix form, an anti-clockwise rotation and a reflection in the x-axis are given by
(in terms of complex numbers: multiplication by $e^\{2pi\; i\; over\; n\}$ and complex conjugation).
By setting
(Compare coordinate rotations and reflections.)
The dihedral group D_{2} is generated by the rotation r of 180 degrees, and the reflection f across the x-axis. 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 y-axis.
D_{2} 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 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 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, r^{2},...,r^{n−1}, f, r f, r^{2} f,...,r^{n−1} f. 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. However, notation D_{n} is also used for a subgroup of SO(3) which is also of abstract group type Dih_{n}: 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).
$Z\_n\; rtimes\_phi\; Z\_2$ is isomorphic to Dih_{n} if φ(0) is the identity and φ(1) is inversion.
If we consider Dih_{n} (n ≥ 3) as the symmetry group of a regular n-gon and number the polygon's vertices, we see that Dih_{n} is a subgroup of the symmetric group S_{n}.
The properties of the dihedral groups Dih_{n} with n ≥ 3 depend on whether n is even or odd. For example, the center of Dih_{n} consists only of the identity if n is odd, but if n is even the center has two elements, namely the identity and the element r^{n / 2} (with D_{n} as a subgroup of O(2), 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 Dih_{2n} is isomorphic with the direct product of Dih_{n} and Z_{2}.
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. 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.
If m divides n, then Dih_{n} has n / m subgroups of type Dih_{m}, and one subgroup Z_{m}. Therefore the total number of subgroups of Dih_{n} (n ≥ 1), 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. See List of small groups for the cases n ≤ 8.
Dih_{10} has 10 inner automorphisms. As 2D isometry group D_{10}, 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).
In general, the automorphism group of Dih_{n} is isomorphic to the affine group Aff(Z/nZ).
In addition to the finite dihedral groups, there is the infinite dihedral group Dih_{∞}. 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 of order 2n. 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 Dih_{∞}. It has presentations
Thus we get:
(Writing Z_{2} multiplicatively, we have (h_{1}, t_{1}) * (h_{2}, t_{2}) = (h_{1} + t_{1}h_{2}, t_{1}t_{2}) .)
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).
The subgroup of Dih(H) of elements (h, 0) is a normal subgroup of index 2, isomorphic to H, while the elements (h, 1) are all their own inverse.
The conjugacy classes are:
Thus for every subgroup M of H, the corresponding set of elements (m,0) is also a normal subgroup. We have:
Examples:
Dih(H) is Abelian, with the semidirect product a direct product, if and only if all elements of H are their own inverse:
etc.
For the group Dih_{∞} we can distinguish two cases:
Both topological groups are totally disconnected, but in the first case the (singleton) components are open, while in the second case they are not. Also, the first topological group is a closed subgroup of Dih(R) but the second is not a closed subgroup of O(2).