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 $C_n$ of $SO(2)$ and the reflection at the $x$-axis (say). It is a semi-direct product of $C_n$ and a $C_2$ corresponding to that reflection.
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.
(see e.g. Greenless 01, section 2)
Warning on notation
There are two different conventions for numbering the dihedral groups.
The above is the algebraic convention in which the suffix gives the order of the group: ${\vert D_{2 n}\vert} = 2 n$.
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.
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
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)
Explicity, let $\mathbb{H} \simeq \mathbb{C} \oplus j \mathbb{C}$ be the quaternions realized as the Cayley-Dickson double of the complex numbers, and identify the circle group
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 $j$ with this copy of $SO(2)$:
The binary dihedral group $2 D_{2n}$ is the subgroup of that generated from
$a \coloneqq \exp\left( \pi i \tfrac{1}{n} \right) \in S(\mathbb{C}) \subset Spin_-(2) \subset Spin(3)$
$x \coloneqq j \in Spin_-(2) \subset Spin(3)$.
It is manifest that these two generators satisfy the relations
and in fact these generators and relations fully determine $2 D_{2n}$, up to isomorphism.
The group cohomology of the dihedral group is discussed for instance at Groupprops.
ADE classification and McKay correspondence
Dynkin diagram/ Dynkin quiver | 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 $SU(n+1)$ | |
D4 | Klein four-group $D_4 \simeq \mathbb{Z}_2 \times \mathbb{Z}_2$ | quaternion group $2 D_4 \simeq$ Q8 | SO(8) | |
$D_{n \geq 4}$ | dihedron, hosohedron | dihedral group $D_{2(n-2)}$ | binary dihedral group $2 D_{2(n-2)}$ | special orthogonal group $SO(2n)$ |
$E_6$ | 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 |
The dihedral group $D_{2n}$ has a group presentation
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.
The first binary dihedral group $2 D_4$ is isomorphic to the quaternion group of order 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
Discussion in the context of the classification of finite rotation groups goes back to
Discussion in the context of equivariant cohomology theory:
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 $Dic_n$
Last revised on February 13, 2019 at 08:39:50. See the history of this page for a list of all contributions to it.