What is $\mathbb{Z}_2$ symmetry?

Question

I encountered the notion of $\mathbb{Z}_2$ symmetry in an article. Can someone give a definition?

Answer 1

There is an operator $P$ such that $P^2$ is the identity and commutes with the Hamiltonians.

In this case $P=\prod \sigma_x$.

This is a $\mathbb{Z}_2$ because $P$ and the identity form a group isomorphic to the integers modulo 2 (odds and evens)

Identity corresponds to evens, $P$ to odds. $P^2=1$ corresponds to odd plus odd equals even.

If there were some other operators commuting with the Hamiltonians, then would be some other kind of symmetry, not $\mathbb{Z}_2$. But in this case, it is not that complicated.