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group of order 2

Contents

Contents

Definition

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

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

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

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

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

Properties

ADE-Classification

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:

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

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