Let $G$ be a group, either a Lie group or a discrete group. Let a principal $G$-bundle $$ G\to E\to B,$$ then $B=E/G$, the orbit space under action of $G$.

Let $BG$ be the classifying space of $G$.

My question:

How to get the fiber sequence

$G\simeq \Omega BG\to E\to B\to BG$?