nLab
binary cyclic group

Contents

Contents

Idea

One calls a cyclic group of order an even number 2(n+1)2(n+1) (nn \in \mathbb{N}) a binary cyclic group if one thinks of it as being the central extension of a cyclic group by the group of order 2:

/2 /2(n+1) /(n+1) \array{ \mathbb{Z}/2\mathbb{Z} &\hookrightarrow& \mathbb{Z}/2(n+1)\mathbb{Z} \\ && \downarrow \\ && \mathbb{Z}/(n+1)\mathbb{Z} }

This way binary cyclic groups are related to cyclic groups as the binary dihedral groups are related to the dihedral groups. Note that if n+1n+1 is odd, then the extension is (necessarily) the trivial extension, and if n+1n+1 is even, then this extension is the (unique) nontrivial extension.

From the point of view of the classification of finite rotation groups, the cyclic groups, dihedral groups etc. are the finite subgroups of the special orthogonal group SO(3)SO(3), while their binary versions are those subgroups of the special unitary group SU(2)SU(2) that arise from the former by taking preimages under the double cover-projection SU(2)Spin(3)SO(3)SU(2) \simeq Spin(3) \to SO(3).

ADE classification and McKay correspondence

Dynkin diagram/
Dynkin quiver
Platonic solidfinite subgroups of SO(3)finite subgroups of SU(2)simple Lie group
A n1A_{n \geq 1}cyclic group
n+1\mathbb{Z}_{n+1}
cyclic group
n+1\mathbb{Z}_{n+1}
special unitary group
SU(n+1)SU(n+1)
D4Klein four-group
D 4 2× 2D_4 \simeq \mathbb{Z}_2 \times \mathbb{Z}_2
quaternion group
2D 42 D_4 \simeq Q8
SO(8)
D n4D_{n \geq 4}dihedron,
hosohedron
dihedral group
D 2(n2)D_{2(n-2)}
binary dihedral group
2D 2(n2)2 D_{2(n-2)}
special orthogonal group
SO(2n)SO(2n)
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

See also

Last revised on April 17, 2018 at 01:47:57. See the history of this page for a list of all contributions to it.