nLab binary cyclic group

Idea

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

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

This way binary cyclic groups are related to cyclic groups as the binary dihedral groups are related to the dihedral groups. Note that if $n+1$ is odd, then the extension is (necessarily) the trivial extension, and if $n+1$ 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 $SO(3)$, while their binary versions are those subgroups of the special unitary group $SU(2)$ that arise from the former by taking preimages under the double cover-projection $SU(2) \simeq Spin(3) \to SO(3)$.

ADE classification and McKay correspondence

Dynkin diagram/ Dynkin quiver	dihedron, Platonic solid	finite subgroups of SO(3)	finite subgroups of SU(2)	simple Lie group
$A_{n \geq 1}$		cyclic group $\mathbb{Z}_{n+1}$	cyclic group $\mathbb{Z}_{n+1}$	special unitary group $SU(n+1)$
A1		cyclic group of order 2 $\mathbb{Z}_2$	cyclic group of order 2 $\mathbb{Z}_2$	SU(2)
A2		cyclic group of order 3 $\mathbb{Z}_3$	cyclic group of order 3 $\mathbb{Z}_3$	SU(3)
A3 = D3		cyclic group of order 4 $\mathbb{Z}_4$	cyclic group of order 4 $2 D_2 \simeq \mathbb{Z}_4$	SU(4) $\simeq$ Spin(6)
D4	dihedron on bigon	Klein four-group $D_4 \simeq \mathbb{Z}_2 \times \mathbb{Z}_2$	quaternion group $2 D_4 \simeq$ Q8	SO(8), Spin(8)
D5	dihedron on triangle	dihedral group of order 6 $D_6$	binary dihedral group of order 12 $2 D_6$	SO(10), Spin(10)
D6	dihedron on square	dihedral group of order 8 $D_8$	binary dihedral group of order 16 $2 D_{8}$	SO(12), Spin(12)
$D_{n \geq 4}$	dihedron, hosohedron	dihedral group $D_{2(n-2)}$	binary dihedral group $2 D_{2(n-2)}$	special orthogonal group, spin group $SO(2n)$, $Spin(2n)$
$E_6$	tetrahedron	tetrahedral 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

See also

• Wikipedia, Binary cyclic group