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 diagram via
McKay correspondence
Platonic solidfinite 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
$E_6$tetrahedrontetrahedral 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

## 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)