For $n \in \mathbb{N}$, $n \geq 1$, the dihedral group $D_{2n}$ is 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).
Under the further embedding $O(2)\hookrightarrow SO(3)$ the (cyclic and) dihedral groups are precisely those finite groups in the ADE classification which are not in the exceptional series.
(see e.g. Greenless 01, section 2)
The group cohomology of the dihedral group is discussed for instance at Groupprops.
Dynkin diagram via McKay correspondence | Platonic solid | finite subgroups of SO(3) | finite subgroups of SU(2) | simple Lie group |
---|---|---|---|---|
$A_n$ | cyclic group $\mathbb{Z}_{n+1}$ | cyclic group $\mathbb{Z}_{n+1}$ | special unitary group | |
$D_{n+4}$ | dihedron, hosohedron | dihedral group $D_{n+2}$ | binary dihedral group $2 D_{n+2}$ | special orthogonal group |
$E_6$ | 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 |
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
Groupprops, Group cohomology of dihedral group:D8
Last revised on April 17, 2018 at 01:50:43. See the history of this page for a list of all contributions to it.