group of order 2 in nLab

Contents

Contents

Definition

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

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

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

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

on the additive group of integers. As such ℤ 2\mathbb{Z}_2 is the special case of a cyclic group ℤ 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:

• prime field

• Bockstein homomorphism

• Stiefel-Whitney class

• Steenrod square