nLab
dihedral group

Contents

Idea

For nn \in \mathbb{N}, n1n \geq 1, the dihedral group D 2nD_{2n} is the subgroup of the orthogonal group O(2)O(2) which is generated from the finite cyclic subgroup C nC_n of SO(2)SO(2) and the reflection at the xx-axis (say).

Under the further embedding O(2)SO(3)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

ADE classification

Dynkin diagramPlatonic solidfinite subgroup of SO(3)SO(3)finite subgroup of SU(2)SU(2)simple Lie group
A lA_lcyclic groupcyclic groupspecial unitary group
D lD_ldihedron/hosohedrondihedral groupbinary dihedral groupspecial orthogonal group
E 6E_6tetrahedrontetrahedral groupbinary tetrahedral groupE6
E 7E_7cube/octahedronoctahedral groupbinary octahedral groupE7
E 8E_8dodecahedron/icosahedronicosahedral groupbinary icosahedral groupE8

References

Revised on August 30, 2016 10:29:20 by Urs Schreiber (82.113.99.98)