nLab dihedral group

Idea

Dihedral groups

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

For instance $D_6$ is the symmetry group of the equilateral triangle and is isomorphic to the symmetric group, $S_3$.

For $n \in \mathbb{N}$, $n \geq 1$, the dihedral group $D_{2n}$ is thus the subgroup of the orthogonal group $O(2)$ which is generated from the finite cyclic subgroup $C_n \,\coloneqq\, \mathbb{Z}/n$ of $SO(2)$ and the reflection at the $x$-axis (say). It is a semi-direct product of $C_n$ and a $C_2 \,\coloneqq\, \mathbb{Z}/2$ corresponding to that reflection, hence fitting into a short exact sequence as follows:

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

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: ${\vert D_{2 n}\vert} = 2 n$.

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

   Also beware that there is yet another group denoted $D_n$ mentioned at Coxeter group.

Binary dihedral/dicyclic groups

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

$$2 D_{2n} = Dic_n$$

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

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

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

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

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 multiples of the imaginary quaternion $\mathrm{j}$ with this copy:

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

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

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

2. $x \coloneqq \mathrm{j} \in Pin_-(2) \subset Spin(3)$.

It is manifest that these two generators satisfy the relations

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

and in fact these generators and relations fully determine $2 D_{2n}$, 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	dihedron, Platonic solid	finite subgroups of SO(3)	finite subgroups of SU(2)	simple Lie group
$A_{n \geq 1}$		cyclic group $\mathbb{Z}_{n+1}$	cyclic group $\mathbb{Z}_{n+1}$	special unitary group $SU(n+1)$
A1		cyclic group of order 2 $\mathbb{Z}_2$	cyclic group of order 2 $\mathbb{Z}_2$	SU(2)
A2		cyclic group of order 3 $\mathbb{Z}_3$	cyclic group of order 3 $\mathbb{Z}_3$	SU(3)
A3 = D3		cyclic group of order 4 $\mathbb{Z}_4$	cyclic group of order 4 $2 D_2 \simeq \mathbb{Z}_4$	SU(4) $\simeq$ Spin(6)
D4	dihedron on bigon	Klein four-group $D_4 \simeq \mathbb{Z}_2 \times \mathbb{Z}_2$	quaternion group $2 D_4 \simeq$ Q8	SO(8), Spin(8)
D5	dihedron on triangle	dihedral group of order 6 $D_6$	binary dihedral group of order 12 $2 D_6$	SO(10), Spin(10)
D6	dihedron on square	dihedral group of order 8 $D_8$	binary dihedral group of order 16 $2 D_{8}$	SO(12), Spin(12)
$D_{n \geq 4}$	dihedron, hosohedron	dihedral group $D_{2(n-2)}$	binary dihedral group $2 D_{2(n-2)}$	special orthogonal group, spin group $SO(2n)$, $Spin(2n)$
$E_6$	tetrahedron	tetrahedral group $T$	binary tetrahedral group $2T$	E6
$E_7$	cube, octahedron	octahedral group $O$	binary octahedral group $2O$	E7
$E_8$	dodecahedron, icosahedron	icosahedral group $I$	binary icosahedral group $2I$	E8

Group presentation

The dihedral group $D_{2n}$ has a group presentation

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

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

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

Examples

Quaternion group $Q_8$ and triality

The first binary dihedral group $2 D_4$ is isomorphic to the quaternion group of order 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 $2 D_4$

$=$ dicyclic group $Dic_2$ $=$ quaternion group $Q_8$

$\,$

group order: ${\vert 2D_4\vert} = 8$

conjugacy classes:	1	2	4A	4B	4C
their cardinality:	1	1	2	2	2

$\,$

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

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

irrep	1	2	4A	4B	4C	Schur index
$\rho_1$	1	1	1	1	1	1
$\rho_2$	1	1	-1	1	-1	1
$\rho_3$	1	1	1	-1	-1	1
$\rho_4$	1	1	-1	-1	1	1
$\rho_5$	2	-2	0	0	0	2

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

irrep	1	2	4A	4B	4C
$\rho_1$	1	1	1	1	1
$\rho_2$	1	1	-1	1	-1
$\rho_3$	1	1	1	-1	-1
$\rho_4$	1	1	-1	-1	1
$\rho_5 \oplus \rho_5$	4	-4	0	0	0

References

• GroupNames, Q8,

• Groupprops, Linear representation theory of dicyclic groups

• James Montaldi, Real representations – Binary cubic – Q8

category: character tables

Related concepts

• anti-cyclotomic extension

• dihedral homology

• group presentation

• Coxeter group

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

See also

• Wikipedia, Dihedral group

• Wikipedia, Binary dihedral group

• Wikipedia, Dicyclic group

• Groupprops, Dicyclic group

• Groupprops, Linear representation theory of dicyclic groups

• Groupprops, Group cohomology of dihedral group:D8

• GroupNames, Dicyclic groups $Dic_n$

Discussion as the equivariance group in equivariant cohomology theory:

• John Greenlees, Section 2 of: Rational $SO(3)$-Equivariant Cohomology Theories, in Homotopy methods in algebraic topology (Boulder, CO, 1999), Contemp. Math. 271, Amer. Math. Soc. (2001) 99 (web, GBooks)

and specifically in equivariant K-theory and KR-theory:

• Robert Bruner, John Greenlees, Chapter 8 of: Connective Real K-Theory of Finite Groups, Mathematical Surveys and Monographs 169 AMS 2010 (ISBN:978-0-8218-5189-0)

Discussion of equivariant ordinary cohomology (Bredon cohomology) over the point but in arbitrary RO(G)-degree, for equivariance group a dihedral group of order $2p$:

• Igor Kriz, Yunze Lu, On the $RO(G)$-graded coefficients of dihedral equivariant cohomology, Mathematical Research Letters 27 4 (2020) (arXiv:2005.01225, doi:10.4310/MRL.2020.v27.n4.a7)

see also

• Yajit Jain, Bredon Equivariant Homology of Representation Spheres, 2014 (pdf, pdf)