How to calculate the fundamental group of discrete group like the cyclic group $Z_n$?
Physically, it seems that $\pi_1(Z_2)$ will be trivial, as there can be no vortex structure for $Z_2$ symmetry. But there can be some "vortex" structure in $Z_n, n\geq3$, so $\pi_1(Z_n), n\geq3$ may be non-trivial.
A discrete space is not path-connected, so you need to choose a basepoint in your discrete group. Then, any loop lands in the path component of the chosen basepoint which is contractible, hence $\pi_1(G)=0$ for any discrete group $G$.