nLab
group of order 2

Definition

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

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

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

$$\cdot 2:\mathbb{Z}\to \mathbb{Z}$$\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 $p=2$.

Related concepts

• Bockstein homomorphism

• Stiefel-Whitney class