Contents

group theory

# Contents

## 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 $SU(2) \simeq Spin(3) \to SO(3)$.

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
$SU(n+1)$
A1cyclic group of order 2
$\mathbb{Z}_2$
cyclic group of order 2
$\mathbb{Z}_2$
SU(2)
A2cyclic 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)
D4dihedron 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)
D5dihedron on
triangle
dihedral group of order 6
$D_6$
binary dihedral group of order 12
$2 D_6$
SO(10), Spin(10)
D6dihedron 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
$SO(2n)$, $Spin(2n)$
$E_6$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