group of order 2 in nLab

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

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