Encyclopedia
In
mathematics, the
dihedral group of order 2
n is the abstract group of which one representation is the
symmetry group in 2D of a
regular polygon with
n sides. The group consists of
n elements corresponding to rotations of the polygon, and
n corresponding to reflections.
Notation
In this article the notation Dih
n is used for the dihedral group of order 2
n as abstract group. The notations
Dn and
D2n are also seen.
For the isometry group in 2D of this abstract group type, the notation
Dn is used. There are four series of
isometry groups in 3D which are dihedral as abstract group. Only for one of them the notation
Dn is used.
Small dihedral groups
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:
- they are abelian; for all other values of n the group Dihn is not abelian
- they are not subgroups of the symmetric group Sn, corresponding to the fact that 2n > n ! for these n.
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.
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Dih1 |
Dih2 |
Dih3 |
Dih4 |
Dih5 |
Dih6 |
Dih7 |
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The dihedral group as symmetry group in 2D and rotation group in 3D
An example of abstract group Dih
n, and a common way to visualize it, is the group
Dn of
Euclidean plane isometries which keep the origin fixed. These groups form one of the two series of discrete point groups in two dimensions.
Dn 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 .
Dihedral group
Dn 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
.
By setting
and defining and for we can write the product rules for as
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 , 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 2
n elements of D
n can be written as
e,
r,
r2,...,
rn−1,
f,
r f,
r2 f,...,
rn−1 f. The first
n listed elements are rotations and the remaining
n elements are axis-reflections . 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, i.e. the group of rotations and reflections of the plane. However, notation
Dn is also used for a subgroup of SO which is also of abstract group type Dih
n: the
proper symmetry group of a
regular polygon embedded in three-dimensional space . Such a figure may be considered as a degenerate regular solid with its face counted twice. Therefore it is also called a
dihedron , which explains the name
dihedral group .
Equivalent definitions and properties
Further equivalent definitions of Dih
n are:
- The automorphism group of the graph consisting only of a cycle with n vertices .
- The group with presentation
- or
- From the second presentation follows that Dihn belongs to the class of coxeter groups.
- The semidirect product of cyclic groups Zn and Z2, with Z2 acting on Zn by inversion :
Z
n φ Z
2 is isomorphic to Dih
n if φ is the identity and φ is inversion.
If we consider Dih
n 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 contains the element
rn / 2 if
n is even .
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 , is equal to
d + σ , where
d is the number of positive divisors of
n and σ is the sum of the positive divisors of
n. See
List of small groups for the cases
n ≤ 8.
Examples of automorphism groups
Dih
9 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.
Dih
10 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 .
Infinite dihedral group
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
rn is the identity, and we have a finite dihedral group of order 2
n. 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
and is isomorphic to a semidirect product of Z and Z
2, and to the free product Z
2 * Z
2. It is the automorphism group of the graph consisting of a path infinite to both sides. Correspondingly, it is the isometry group of Z .
Generalized dihedral group
For any abelian group
H, the
generalized dihedral group of
H, written Dih, is the semidirect product of
H and Z
2, with Z
2 acting on
H by inverting elements. I.e., Dih =
H φ Z
2 with φ the identity and φ inversion.
Thus we get:
- * =
- * =
for all
h1,
h2 in
H and
t2 in Z
2.
Note that * = , i.e. first the inversion and then the operation in
H. Also * = ; indeed inverts
h, and toggles
t between "normal" and "inverted" .
The subgroup of Dih of elements is a normal subgroup of index 2, isomorphic to
H, while the elements are all their own inverse.
The conjugacy classes are:
Thus for every subgroup
M of
H, the corresponding set of elements is also a normal subgroup. We have:
Dih
/ M = Dih
Examples:
- Dihn = Dih
- For even n there are two sets , and each generates a normal subgroup of type Dihn / 2. As subgroups of the isometry group of the set of vertices of a regular n-gon they are different: the reflections in one subgroup all have two fixed points, while none in the other subgroup has . However, they are isomorphic as abstract groups.
- For odd n there is only one set
- Dih∞ = Dih; there are two sets , and each generates a normal subgroup of type Dih∞. As subgroups of the isometry group of Z they are different: the reflections in one subgroup all have a fixed point, the mirrors are at the integers, while none in the other subgroup has, the mirrors are in between . However, they are isomorphic as abstract groups.
- Dih, or orthogonal group O, or O: the isometry group of a circle, or equivalently, the group of isometries in 2D that keep the origin fixed. The rotations form the circle group S1, or equivalently SO, also written SO, and R / Z ; it is also the multiplicative group of complex number
...
s of
absolute value 1. In the latter case one of the reflections is complex conjugation. There are no proper normal subgroups with reflections. The discrete normal subgroups are cyclic groups of order
n for all positive integers
n. The quotient groups are isomorphic with the same group Dih.
- Dih: the group of isometries of Rn consisting of all translations and inversion in all points; for n = 1 this is the Euclidean group E; for n > 1 the group Dih is a proper subgroup of E, i.e. it does not contain all isometries.
- H can be any subgroup of Rn, e.g. a discrete subgroup; in that case, if it extends in n directions it is a lattice.
- Discrete subgroups of Dih which contain translations in one direction are of frieze group type and 22.
- Discrete subgroups of Dih which contain translations in two directions are of wallpaper group type p1 and p2.
- Discrete subgroups of Dih which contain translations in three directions are space groups of the triclinic crystal system.
Dih is Abelian, with the semidirect product a direct product, iff all elements of
H are their own inverse:
- Dih = Dih1 = Z2
- Dih = Dih2 = Z2 × Z2
- Dih = Dih2 × Z2 = Z2 × Z2 × Z2
etc.
Topology
Dih and its dihedral subgroups are disconnected topological groups. Dih consists of two
connected components: the identity component isomorphic to R
n, and the component with the reflections. Similarly O consists of two connected components: the identity component isomorphic to the circle group, and the component with the reflections.
For the group Dih
∞ we can distinguish two cases:
- Dih∞ as the isometry group of Z
- Dih∞ as a 2D isometry group generated by a rotation by an irrational number of turns, and a reflection
Both topological groups are totally disconnected, but in the first case the components are open, while in the second case they are not. Also, the first topological group is a closed subgroup of Dih but the second is not a closed subgroup of O.
See also
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