group theory

# Contents

## Idea

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)

## Properties

### Group cohomology

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

### As part of the ADE pattern

Dynkin diagramPlatonic solidfinite subgroup of $SO(3)$finite subgroup of $SU(2)$simple Lie group
$A_l$cyclic groupcyclic groupspecial unitary group
$D_l$dihedron/hosohedrondihedral groupbinary dihedral groupspecial orthogonal group
$E_6$tetrahedrontetrahedral groupbinary tetrahedral groupE6
$E_7$cube/octahedronoctahedral groupbinary octahedral groupE7
$E_8$dodecahedron/icosahedronicosahedral groupbinary icosahedral groupE8