Contents

group theory

# Contents

## Definition

There is, up to isomorphism, a unique simple group of order 2:

it has two elements $(1,\sigma)$, where $\sigma \cdot \sigma = 1$.

This is usually denoted $\mathbb{Z}_2$ or $\mathbb{Z}/2\mathbb{Z}$, because it is the cokernel (the quotient by the image of) the homomorphism

$\cdot 2 : \mathbb{Z} \to \mathbb{Z}$

on the additive group of integers. As such $\mathbb{Z}_2$ is the special case of a cyclic group $\mathbb{Z}_p$ for $p = 2$ and hence also often denoted $C_2$.

## Properties

In the ADE-classification of finite subgroups of SU(2), the group of order 2 is the smallest non-trivial group, and the smallest in the A-series:

Dynkin diagram/
Dynkin quiver
Platonic solidfinite 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
$SU(n+1)$
D4Klein four-group
$D_4 \simeq \mathbb{Z}_2 \times \mathbb{Z}_2$
quaternion group
$2 D_4 \simeq$ Q8
SO(8)
$D_{n \geq 4}$dihedron,
hosohedron
dihedral group
$D_{2(n-2)}$
binary dihedral group
$2 D_{2(n-2)}$
special orthogonal group
$SO(2n)$
$E_6$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

Last revised on May 8, 2019 at 10:53:02. See the history of this page for a list of all contributions to it.