nLab cyclic group of order 4

Idea

The cyclic group of order 4.

Properties

General

Besides the Klein four-group $\mathbb{Z}/2 \times \mathbb{Z}/2$, the cyclic group of order 4 is the only other group of order 4, up to isomorphism.

ADE-Classification

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

• group names, C4