Dihedral group Totally Explained

Everything about Dihedral Group totally explained

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.

Notation

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.

Definition

Elements

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 D₈ 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.

Group structure

As with any geometric object, the composition of two symmetries of a regular polygon is again a symmetry. This operation gives the symmetries of a polygon the algebraic structure of a finite group.
The following Cayley table shows the effect of composition in the group D₃ (the symmetries of an equilateral triangle). R₀ denotes the identity; R₁ and R₂ denote counterclockwise rotations by 120 and 240 degrees; and S₀, S₁, and S₂ denote reflections across the three lines shown in the picture to the right.

	R₀	R₁	R₂	S₀	S₁	S₂
R₀	R₀	R₁	R₂	S₀	S₁	S₂
R₁	R₁	R₂	R₀	S₁	S₂	S₀
R₂	R₂	R₀	R₁	S₂	S₀	S₁
S₀	S₀	S₂	S₁	R₀	R₂	R₁
S₁	S₁	S₀	S₂	R₁	R₀	R₂
S₂	S₂	S₁	S₀	R₂	R₁	R₀

For example, S₂S₁ = R₁ because the reflection S₁ followed by the reflection S₂ results in a 120-degree rotation. (This is the normal backwards order for composition.) Note that the composition operation isn't commutative.
In general, the group D_n has elements R₀,...,R_n−1 and S₀,...,S_n−1, with composition given by the following formulae:
ť

R_i,R_j = R_, and each generates a normal subgroup of type Dih

_∞. As subgroups of the isometry group of Z they're 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 (the translations of both are the same: by even numbers). However, they're isomorphic as abstract groups.

Dih(S¹), or orthogonal group O(2,R), or O(2): 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(2,R), also written SO(2), and R/Z ; it's also the multiplicative group of complex numbers of absolute value 1. In the latter case one of the reflections (generating the others) 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(S¹).

Dih(Rⁿ ): the group of isometries of Rⁿ consisting of all translations and inversion in all points; for n = 1 this is the Euclidean group E(1); for n > 1 the group Dih(Rⁿ ) is a proper subgroup of E(n ), for example it doesn't contain all isometries.

H can be any subgroup of Rⁿ, for example a discrete subgroup; in that case, if it extends in n directions it's a lattice.

Discrete subgroups of Dih(R² ) which contain translations in one direction are of frieze group type $inftyinfty$ and 22 $infty$ .
Discrete subgroups of Dih(R² ) which contain translations in two directions are of wallpaper group type p1 and p2.
Discrete subgroups of Dih(R³ ) which contain translations in three directions are space groups of the triclinic crystal system.

Dih(H) is Abelian, with the semidirect product a direct product, if and only if all elements of H are their own inverse:

Dih(Z₁) = Dih₁ = Z₂

Dih(Z₂) = Dih₂ = Z₂ × Z₂ (Klein four-group)

Dih(Dih₂) = Dih₂ × Z₂ = Z₂ × Z₂ × Z₂ etc.

Topology

Dih(Rⁿ ) and its dihedral subgroups are disconnected topological groups. Dih(Rⁿ ) consists of two connected components: the identity component isomorphic to Rⁿ, and the component with the reflections. Similarly O(2) 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 2-dimensional 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 (singleton) components are open, while in the second case they're not. Also, the first topological group is a closed subgroup of Dih(R) but the second isn't a closed subgroup of O(2).

Further Information

Get more info on 'Dihedral Group'.

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