In reading Georgi (Lie algebra in particle physics) I reaf at page 43 the following definition of Lie Gruoup: "a lie gruoup is a group whose elements depend smoothly on a set of continuous parameters." $\phi$ Wikipedia says that $O(n)$ in a Lie group. For $n\geq2$ this is intuitive (for example elements of $O(2)$ are parametrized by one parameter $\phi$, while elements of $O(3)$ are parametrized by two parameters $\theta, \phi$ and so on. My question is: in case of $O(1)$ I have that the group has only two elements (in its fundamental representation the real numbers $+1 $ and $ -1$ but I cannot see how to parametrize those elements with a continuous parameter...Hence is $O(1)$ a Lie group?
Moreover what is the difference between Z2 and O(1)? (this last question arises because I read in a book that Ising 1D model has Z2 symmetry and in another section of that book I read that Ising 1D model has O(1) symmetry [when seen as case n=1 of the O(n) model]
Any discrete space $X$ is a $0$-dimensional smooth manifold. Any $x\in X$ has a nbhd $\{x\}$ with the obvious chart from $\{x\}$ to the Euclidean space $\Bbb R^0=\{0\}$. These charts are the only ones possible in the atlas, and there is only one possible transition map $\{0\}\to\{0\}$, which is vacuously smooth.