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 $C_2$.