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I want to prove following fundamental group property. \begin{align} \pi_1(G/K) = K \end{align}

Let $G$ be a universal covering group, and $K$ be either its center or a subgroup of its center. If $G$ is semi-simple, $K$ is finite group, by defining the elements $g$ and $kg$, to be equivalent (where $g\in G$ and $k\in K$), \begin{align} \pi_1(G/K) = K \end{align}

Is there any idea for proving this theorem?

I saw this theorem, from computing fundamental group of $SO(3)$ \begin{align} \pi_1(SO(3)) = \pi_1(SU(2)/Z_2) = Z_2 \end{align} Note $\pi_1(SU(2))=0$ $i.e$, simply connected.

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    How about $\pi_1(U(1)/\mathbb{Z}_n)=\pi_1(U(1)=\mathbb{Z}$ but which is not $\mathbb{Z}_n$??? – 2017-05-06

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