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dihedral group

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Contents

Idea

Dihedral groups

The dihedral group, D 2nD_{2n}, is a finite group of order 2n2n. It may be defined as the symmetry group of a regular nn-gon.

For instance D 6D_6 is the symmetry group of the equilateral triangle and is isomorphic to the symmetric group, S 3S_3.

For nn \in \mathbb{N}, n1n \geq 1, the dihedral group D 2nD_{2n} is thus the subgroup of the orthogonal group O(2)O(2) which is generated from the finite cyclic subgroup C nC_n of SO(2)SO(2) and the reflection at the xx-axis (say). It is a semi-direct product of C nC_n and a C 2C_2 corresponding to that reflection.

Under the further embedding O(2)SO(3)O(2)\hookrightarrow SO(3) the (cyclic and) dihedral groups are precisely those finite subgroups of SO(3) that, among their ADE classification, are not in the exceptional series.

(see e.g. Greenless 01, section 2)

Remark

Warning on notation

There are two different conventions for numbering the dihedral groups.

  1. The above is the algebraic convention in which the suffix gives the order of the group: |D 2n|=2n{\vert D_{2 n}\vert} = 2 n.

  2. In the geometric convention one writes “D nD_n” instead of “D 2nD_{2n}”, recording rather the geometric nature of the object of which it is the symmetry group.

    Also beware that there is yet another group denoted D nD_n mentioned at Coxeter group.

Binary dihedral/dicyclic groups

Under the further lift through the spin group-double cover map SU(2)Spin(3)SO(3)SU(2) \simeq Spin(3) \to SO(3) of the special orthogonal group, the dihedral group D 2nD_{2n} is covered by the binary dihedral group, also known as the dicyclic group and denoted

2D 2n=Dic n 2 D_{2n} = Dic_n

Equivalently this is the lift of the dihedral group D 2nD_{2n} (above) through the pin group double cover of the orthogonal group O(2) to Pin(2)

2D 2n AA Pin (2) AA Spin(3) (pb) (pb) D 2n AA O(2) AA SO(3) \array{ 2 D_{2n} &\overset{\phantom{AA}}{\hookrightarrow}& Pin_-(2) &\overset{\phantom{AA}}{\hookrightarrow}& Spin(3) \\ \big\downarrow &{}^{(pb)}& \big\downarrow &{}^{(pb)}& \big\downarrow \\ D_{2n} &\overset{\phantom{AA}}{\hookrightarrow}& O(2) &\overset{\phantom{AA}}{\hookrightarrow}& SO(3) }

Explicity, let j\mathbb{H} \simeq \mathbb{C} \oplus j \mathbb{C} be the quaternions realized as the Cayley-Dickson double of the complex numbers, and identify the circle group

SO(2)S() SO(2) \simeq S\big( \mathbb{C}\big) \hookrightarrow \mathbb{H}

with the unit circle in \mathbb{C} \hookrightarrow \mathbb{H} this way, with group structure given by multiplication of quaternions. Then the Pin group Pin(2) is isomorphic to the subgroup of the group of units ×\mathbb{H}^\times of the quaternions which consists of this copy of SO(2) together with the multiplies of the imaginary quaternion jj with this copy of SO(2)SO(2):

Pin (2)S()jS()S()Spin(3). Pin_-(2) \;\simeq\; S\big( \mathbb{C}\big) \;\cup\; j \cdot S\big( \mathbb{C}\big) \;\subset\; S(\mathbb{H}) \;\simeq\; Spin(3) \,.

The binary dihedral group 2D 2n2 D_{2n} is the subgroup of that generated from

  1. aexp(πi1n)S()Spin (2)Spin(3) a \coloneqq \exp\left( \pi i \tfrac{1}{n} \right) \in S(\mathbb{C}) \subset Spin_-(2) \subset Spin(3)

  2. xjSpin (2)Spin(3)x \coloneqq j \in Spin_-(2) \subset Spin(3).

It is manifest that these two generators satisfy the relations

a 2n=1,AAx 2=a n(=1),AAx 1ax=a 1 a^{2n} = 1 \,, \phantom{AA} x^2 = a^n \; (= -1) \,, \phantom{AA} x^{-1} a x = a^{-1}

and in fact these generators and relations fully determine 2D 2n2 D_{2n}, up to isomorphism.

Properties

Group cohomology

The group cohomology of the dihedral group is discussed for instance at Groupprops.

As part of the ADE pattern

ADE classification and McKay correspondence

Dynkin diagram/
Dynkin quiver
Platonic solidfinite subgroups of SO(3)finite subgroups of SU(2)simple Lie group
A n1A_{n \geq 1}cyclic group
n+1\mathbb{Z}_{n+1}
cyclic group
n+1\mathbb{Z}_{n+1}
special unitary group
SU(n+1)SU(n+1)
D4Klein four-group
D 4 2× 2D_4 \simeq \mathbb{Z}_2 \times \mathbb{Z}_2
quaternion group
2D 42 D_4 \simeq Q8
SO(8)
D n4D_{n \geq 4}dihedron,
hosohedron
dihedral group
D 2(n2)D_{2(n-2)}
binary dihedral group
2D 2(n2)2 D_{2(n-2)}
special orthogonal group
SO(2n)SO(2n)
E 6E_6tetrahedrontetrahedral group
TT
binary tetrahedral group
2T2T
E6
E 7E_7cube,
octahedron
octahedral group
OO
binary octahedral group
2O2O
E7
E 8E_8dodecahedron,
icosahedron
icosahedral group
II
binary icosahedral group
2I2I
E8

Group presentation

The dihedral group D 2nD_{2n} has a group presentation

x,y:x n=y 2=(xy) 2=1.\langle x,y : x^n=y^2=(xy)^2=1\rangle.

From this it is easy to see that it is a semi-direct product of the C nC_n generated by xx and the C 2C_2 generated by yy. The action of yy on xx is given by yx=x 1\,^y x= x^{-1}.

It is a standard example considered in elementary combinatorial group theory.

Examples

Quaternion group Q 8Q_8 and triality

The first binary dihedral group 2D 42 D_4 is isomorphic to the quaternion group of order 8:

2D 4(=Dic 2)Q 8. 2 D_4 (= Dic_2) \simeq Q_8 \,.

In the ADE-classification this is the entry D4.

linear representation theory of binary dihedral group 2D 42 D_4

== dicyclic group Dic 2Dic_2 == quaternion group Q 8Q_8

\,

group order: |2D 4|=8{\vert 2D_4\vert} = 8

conjugacy classes:124A4B4C
their cardinality:11222

\,

splitting field(α,β)\mathbb{Q}(\alpha, \beta) with α 2+β 2=1\alpha^2 + \beta^2 = -1
field generated by characters\mathbb{Q}

character table over splitting field (α,β)\mathbb{Q}(\alpha,\beta)/complex numbers \mathbb{C}

irrep124A4B4CSchur index
ρ 1\rho_1111111
ρ 2\rho_211-11-11
ρ 3\rho_3111-1-11
ρ 4\rho_411-1-111
ρ 5\rho_52-20002

character table over rational numbers \mathbb{Q}/real numbers \mathbb{R}

irrep124A4B4C
ρ 1\rho_111111
ρ 2\rho_211-11-1
ρ 3\rho_3111-1-1
ρ 4\rho_411-1-11
ρ 5ρ 5\rho_5 \oplus \rho_54-4000

References

References

Discussion in the context of the classification of finite rotation groups goes back to

  • Felix Klein, chapter I.4 of Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom fünften Grade, 1884, translated as Lectures on the Icosahedron and the Resolution of Equations of Degree Five by George Morrice 1888, online version

Discussion in the context of equivariant cohomology theory:

  • John Greenlees, Rational SO(3)-Equivariant Cohomology Theories, in Homotopy methods in algebraic topology (Boulder, CO, 1999), Contemp. Math. 271, Amer. Math. Soc. (2001) 99 (web)

See also

Last revised on February 13, 2019 at 08:39:50. See the history of this page for a list of all contributions to it.