Suppose a lie group $G$ acts smoothly, freely and transitively on a manifold $M$.
Show that $\pi : M \rightarrow M / G$ is a principal $G$-bundle.
I am having trouble with the definition of a principal $G$-bundle. All I know is that it is a fiber bundle with a local trivializing condition (which I am having trouble understanding). I am unable to find an explanation that I understand.
I know that by the quotient manifold theorem, $M/G$ is a smooth manifold. I believe that $\pi$ is a fiber bundle with fiber $G$ but I don't know how to proceed. Any hints/suggestions/answers would be appreciated.
$G$ acts transitively on $M$ i.e for every $x,y\in M$, there exists $g\in G$ such that $y=g.x$ it implies that the quotient $M/G$ is a point in fact $M=G$ and $M\rightarrow M/G$ is the trivial $G$-bundle over a point.