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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 diagram via
McKay correspondence
Platonic solidfinite subgroups of SO(3)finite subgroups of SU(2)simple Lie group
A nA_ncyclic group
n+1\mathbb{Z}_{n+1}
cyclic group
n+1\mathbb{Z}_{n+1}
special unitary group
D n+4D_{n+4}dihedron,
hosohedron
dihedral group
D n+2D_{n+2}
binary dihedral group
2D n+22 D_{n+2}
special orthogonal group
E 6E_6tetrahedrontetrahedral group
TT
binary tetrahedral group
2T2T
E6
E 7E_7cube,
octahedron
octahedral group
OO
binary octahedral group
2O2O
E7
E 8E_8dodecahedron,
icosahedron
icosahedral group
II
binary icosahedral group
2I2I
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)

See also

Last revised on April 17, 2018 at 01:50:43. See the history of this page for a list of all contributions to it.